From c951dc916aadc7d8af5b126360558f7664b08a4f Mon Sep 17 00:00:00 2001
From: Ali Romman <226877337+aliromman@users.noreply.github.com>
Date: Mon, 26 Oct 2026 10:14:10 -0500
Subject: [PATCH 1/2] Prove chronological connector rank bounds and full star
 endpoint controls

---
 research/coupled31-chronology/COMMUTING.md    |  90 ++++
 research/coupled31-chronology/COST.md         | 118 +++++
 research/coupled31-chronology/DISCOVERY.md    |  64 +++
 research/coupled31-chronology/PROOF.md        | 205 ++++++++
 research/coupled31-chronology/README.md       |  92 ++++
 research/coupled31-chronology/RESULT.json     | 153 ++++++
 research/coupled31-chronology/SOURCES.json    |  53 ++
 research/coupled31-chronology/TASK.json       | 102 ++++
 research/coupled31-chronology/VERIFY.json     |  36 ++
 research/coupled31-chronology/WORD31.json     |   1 +
 research/coupled31-chronology/check.py        | 397 ++++++++++++++
 .../discovery/coupled_seed.py                 | 150 ++++++
 .../discovery/one-row-three-stage.json        | 250 +++++++++
 .../discovery/pair-search.json                |  14 +
 .../discovery/row_signature_census.py         |  54 ++
 .../discovery/triangle-search.json            |  14 +
 research/coupled31-chronology/pricing.py      | 112 ++++
 .../sources/escape-word.json                  | 351 +++++++++++++
 .../sources/old-word-replay.py                |  49 ++
 .../sources/p104-result.json                  | 487 ++++++++++++++++++
 .../sources/pr5673-proof.md                   | 231 +++++++++
 .../sources/pr5681-proof.md                   | 169 ++++++
 research/coupled31-chronology/test_verify.py  |  57 ++
 research/coupled31-chronology/verify.py       |  65 +++
 24 files changed, 3314 insertions(+)
 create mode 100644 research/coupled31-chronology/COMMUTING.md
 create mode 100644 research/coupled31-chronology/COST.md
 create mode 100644 research/coupled31-chronology/DISCOVERY.md
 create mode 100644 research/coupled31-chronology/PROOF.md
 create mode 100644 research/coupled31-chronology/README.md
 create mode 100644 research/coupled31-chronology/RESULT.json
 create mode 100644 research/coupled31-chronology/SOURCES.json
 create mode 100644 research/coupled31-chronology/TASK.json
 create mode 100644 research/coupled31-chronology/VERIFY.json
 create mode 100644 research/coupled31-chronology/WORD31.json
 create mode 100644 research/coupled31-chronology/check.py
 create mode 100644 research/coupled31-chronology/discovery/coupled_seed.py
 create mode 100644 research/coupled31-chronology/discovery/one-row-three-stage.json
 create mode 100644 research/coupled31-chronology/discovery/pair-search.json
 create mode 100644 research/coupled31-chronology/discovery/row_signature_census.py
 create mode 100644 research/coupled31-chronology/discovery/triangle-search.json
 create mode 100644 research/coupled31-chronology/pricing.py
 create mode 100644 research/coupled31-chronology/sources/escape-word.json
 create mode 100644 research/coupled31-chronology/sources/old-word-replay.py
 create mode 100644 research/coupled31-chronology/sources/p104-result.json
 create mode 100644 research/coupled31-chronology/sources/pr5673-proof.md
 create mode 100644 research/coupled31-chronology/sources/pr5681-proof.md
 create mode 100644 research/coupled31-chronology/test_verify.py
 create mode 100644 research/coupled31-chronology/verify.py

diff --git a/research/coupled31-chronology/COMMUTING.md b/research/coupled31-chronology/COMMUTING.md
new file mode 100644
index 0000000000..6b97f3f02e
--- /dev/null
+++ b/research/coupled31-chronology/COMMUTING.md
@@ -0,0 +1,90 @@
+# Commuting full calls need at least half as many thin maps as leaves
+
+Retain the paired-star contract in PROOF.md. Suppose all full-call projectors
+commute, including projectors from different stages. Thin maps can still be
+arbitrary and can have any rank.
+
+**Theorem.** A successful word needs at least `ceil(g/2)` thin maps.
+This count is sharp when their ranks are unrestricted.
+
+## Proof
+
+For g>=2, each full projector commutes with every other full-stage support E_k.
+For a different star edge k, this forces its off-diagonal block B to zero.
+The remaining A projectors commute. Commuting idempotents have a common
+eigenbasis over the base field, since each minimal polynomial divides
+`x(x-1)`. Use this same basis on the core and all leaves.
+
+Fix one coordinate in that basis. Exactly one half of each stage acts on
+this coordinate. Read these g active stage events in chronological order.
+The pure stage prefix is diagonal. Its entries are monomials in the
+independent D_j. The e coefficient of the whole word, after removal of
+its pure stage product, is exactly
+
+```text
+sum_i prefix_i^(-1) R_i prefix_i.
+```
+
+This calculation is in the Laurent polynomial ring. The original polynomial
+ring embeds in it, so an exact equality here proves the original endpoint.
+
+For the entry from leaf j to the core, the full query in stage j contributes
+one monomial. Its exponent records which other stage events precede j.
+The D_j exponent cancels because both endpoints receive D_j together.
+A thin map can cancel this term only if it occurs immediately before or
+after event j in the induced active-event sequence. Every other gap has a
+different monomial in the independent stage variables. There is no other
+full query in this matrix entry.
+
+One thin map lies in one gap of this sequence. It can therefore cancel the
+required entry for at most the two neighboring events. The target has zero
+in every such off-diagonal entry. All g events need a cancellation, which
+gives `2m>=g`. Cross-coordinate entries in thin maps cannot alter this
+argument: diagonal conjugation preserves each matrix entry separately.
+
+For g=1, no thin map would leave the full query sum T_1, which is not I.
+Thus at least one thin map is required in that case too.
+
+This excludes five thin maps at both 13 and 31 leaves, regardless of their
+ranks or the number of coupled ports. Those targets need noncommuting full
+projectors, a changed stage frame, or a different endpoint contract.
+
+## A complete construction at the count bound
+
+Choose the same coordinate half projection A in every stage and B=0.
+Run both halves of stage 1, then both of stage 2, and continue in order.
+After each odd-numbered stage j, insert one thin map. If j<g, it contains
+both directions between the core and leaves j and j+1, with identity blocks
+and no other entries. If j=g, use only leaf j. For even g, also put identity
+on the core diagonal of the first thin map.
+
+The map between stages j and j+1 cancels both full-query off-diagonal terms
+in the normalized Laurent sum. The last singleton cancels its own terms.
+The diagonal full-query sum is I when g is odd. When g is even, it lacks
+the core identity, which the first thin map supplies. The resulting sum is
+exactly I. The pure stage product is the required diagonal product. Thus
+the complete endpoint is correct for every g>=1 and every q>=1.
+
+Every thin map has rank 2d=4q, including the even-g correction and singleton.
+There are `ceil(g/2)` maps. They are not asserted to have square zero.
+An arbitrary rank decomposition gives the stated rank-one query tariff;
+native transport and frame costs are separate. The complete recursive price
+loses for g>=3. This is a sharp map-count construction, not a profitable caller.
+
+## Exact replay
+
+WORD31.json contains all matrices and their order for g=31,q=2. The checker
+first checks projectors, ranks, resolutions and each prescribed query.
+It then collects every Laurent monomial in every matrix entry. Exactly the
+128 diagonal identity terms remain. There is no truncation of stage degree.
+
+A separate literal interpreter evaluates the dual-number word on a Cartesian
+grid of three distinct F4 values per stage. Each stage occurs twice, so its
+degree is at most two. Polynomial interpolation makes this a complete check
+for the small fixtures, including the retained noncommuting eight-field word.
+The control `D^2+D` shows why the two binary values alone would be insufficient.
+Deleted and moved thin queries fail the long endpoint check.
+
+The count argument and sharp construction are part of the present research.
+Credit the retained star contract and source lineage in PROOF.md. Ali Romman
+with substantial OpenAI Codex assistance; no subagents. Apache-2.0.
diff --git a/research/coupled31-chronology/COST.md b/research/coupled31-chronology/COST.md
new file mode 100644
index 0000000000..68ad4afe0f
--- /dev/null
+++ b/research/coupled31-chronology/COST.md
@@ -0,0 +1,118 @@
+# Complete tariff consequences and unpriced obligations
+
+The source is PR5599 at `09dd0965711ca7eeb169057a420abab1c9052137`.
+Its exact local paid and independent-helper histograms are retained in
+`sources/p104-result.json`. Their net rank mass is 1720 per full call.
+This section includes every term of the declared recursive star tariff.
+It does not assign zero cost to a missing physical implementation.
+
+## 1. The entire recursive profile
+
+Let g be the number of leaves, q the number of coupled ports per binary
+block, and R the total thin matrix rank per block. Scale to 104 native ports.
+The signed target-minus-paid moment at exponent t is
+
+```text
+M(g,R/q;t) = 208 H_(8g+1) + 208g H_9
+            + 2g helpers - 2g paid - 104(R/q) H_1,
+H_r = r^t.
+```
+
+The first term is the core target. The second term is every leaf target.
+Every full call retains the complete paid profile and independent helpers.
+Every thin rank unit costs one rank-one query. The proof permits a favorable
+fractional block count when it derives a lower bound; actual constructions
+must use integer blocks or provide the missing mixed block contract.
+
+The linear margin is `96g+208-104(R/q)`. This formula is a complete recursive
+tariff for this declared algebraic comparison. Native frame realization and
+nonrecursive obligations remain distinct.
+
+## 2. A uniform ceiling for one or two coupled ports
+
+The chronological theorem gives
+
+```text
+R/q >= 4 + (2/q) ceil((g-2)/3).
+```
+
+For q<=2, this is at least `4+(g-2)/3`. Define
+
+```text
+C(t) = 208*9^t + 2 sum helpers_r*r^t - 2 sum paid_r*r^t.
+```
+
+For all g>=3, the moment therefore satisfies
+
+```text
+M <= 208(8g+1)^t + g(C(t)-104/3) - 1040/3.                     (1)
+```
+
+Take saving `alpha=8/3125=.00256`, so `t=3117/3125`. The function
+`y -> y^t` is concave on positive y. Its tangent at `8*32+1=257` bounds
+the right side of (1) by `s*g+c`, where
+
+```text
+s = 1664*t*257^(t-1) + C(t) - 104/3,
+c = 208*257^(t-1)*(1+256*alpha) - 1040/3.
+```
+
+Exact integer-power intervals give `s<0` and `c<-7`. The interval endpoints
+are in RESULT.json. Thus **every integer leaf count g>=3 loses at .00256**
+for q<=2, even in the favorable fractional relaxation.
+
+After equal ranks are cancelled, positive terms have rank at least eight
+and negative terms have rank at most seven. If the margin is negative at
+t0, its margin at any `0<=t<=t0` is at most `8^(t-t0)` times that negative
+margin. Consequently every saving from .00256 through 1 is rejected.
+This is a family-wide tariff ceiling, not a root search on a finite range.
+In particular this family cannot reach selected kappa .00310045.
+
+For the requested g=31,q=2 case, R>=28. Its 104-port bill therefore needs
+at least 1456 rank-one thin queries, instead of the proposed 520. For the
+old g=127,q=1 case, R>=88 instead of ten. Both complete optimistic profiles
+are recorded. Additional nonnegative charges cannot repair their loss.
+
+## 3. A complete binary endpoint as a control
+
+WORD31.json gives a real 31-leaf binary word with 62 full calls and 16 thin
+maps of rank eight on each two-port block. Its full normalized endpoint is
+proved in COMMUTING.md and checked without expansion into 2^31 terms.
+At 104 ports it uses 52 blocks and 6656 rank-one thin queries. All source
+full-call helpers and paid terms remain in its profile. Its linear margin
+is -3472, and the same separated-rank argument rejects every concave moment.
+It is a positive operator control with a negative complete recursive price.
+
+This is not the requested five-rank-two word. The theorem proves that such
+a word cannot exist under the paired-star contract.
+
+## 4. Larger coupling and the remaining obligations
+
+At 31 leaves, five rank-q maps require q>=20. Using q=26 divides the 104
+ports. Its optimistic moment is positive at saving .0035. The smaller
+13-leaf, q=8, five-rank-eight profile is positive at .0034. Exact intervals
+are saved. Neither profile has a supplied word. COMMUTING.md proves that
+neither can use commuting full-call projectors. A larger scalar field with
+all maps linear over that field is not a substitute for this missing word.
+
+| Obligation | Status in this package |
+| --- | --- |
+| Chronology of the requested 31-leaf/five-rank-two word | Excluded by PROOF.md |
+| Every coefficient of the 31-leaf control word | Proved and exactly replayed |
+| Source full-call paid and helper profiles | Pinned complete source input |
+| Thin query tariff | Charged by actual rank, or by a proved lower bound |
+| Native realization of new coupled frames | Unknown; no zero-cost assumption |
+| Owned and borrowed state, input reads and dirty compensation | Unknown for new frames |
+| COPY, CLEAR, retained values and complete restoration | Unknown for new frames |
+| Physical routes, tapes, precision, stock and uniform compiler | Unknown for new frames |
+| Outer transfer and a new complete kappa | Not supplied |
+
+PR5701 gives separate characteristic-zero trace qualifications for direct
+square-zero lifts of the static PR5681 construction. This package does not
+repeat or bypass them. A positive binary word must still receive its actual
+native contract and a complete physical bill before it can improve kappa.
+
+Credit the PR5599 tariff, PR5673 profile, and all retained P104/helper/native/
+compiler contributors. The new cost conclusion uses the chronological bound
+in this package. Ali Romman with substantial OpenAI Codex assistance.
+Apache-2.0.
diff --git a/research/coupled31-chronology/DISCOVERY.md b/research/coupled31-chronology/DISCOVERY.md
new file mode 100644
index 0000000000..a21c6dae35
--- /dev/null
+++ b/research/coupled31-chronology/DISCOVERY.md
@@ -0,0 +1,64 @@
+# Search record and limits
+
+The assigned target was 31 leaves, two coupled ports, and five rank-two thin
+connectors. Main and active claims were refreshed first. PR5681 had supplied
+static maps for this target; no new static construction was needed. PR5701
+had separately qualified two characteristic-zero lift contracts.
+
+The first new calculation used the thin row-space basis in the mixed
+chronology relation. Left multiplication by one stage projection isolates
+the prefix of every other stage. Untouched stages force the affine row map
+to lose a dimension whenever a prefix determinant becomes too large.
+Iteration gives the proof in PROOF.md. Thus the requested endpoint is
+impossible, before any larger synthesis or native compilation.
+
+The complete source tariff with the new rank lower bound then rejects every
+one- or two-port paired-star caller at saving .00256. The proof uses an exact
+tangent, so it does not leave an unsearched large-leaf range.
+
+Two bounded seed searches tested larger coupling as a possible next route.
+Both used g=3,d=4, arbitrary compatible rank-two full query halves, and all
+eleven call positions as solver variables. The fixed five rank-two thin maps
+use the PR5681 F/G/E formula. Their leaf signatures are `(0,1,1)` in one run
+and `(1,2,3)` in the other. The sum and square-zero identities hold in both.
+Each run checks the full first mixed coefficient and the pairwise pure stage
+coefficient. Any SAT model is then passed to the retained complete polynomial
+replay, which keeps all stage exponents. Both runs reached their 60-second
+solver limit without a model. They are **unknown**, not obstructions.
+
+```sh
+python3.12 -m venv /tmp/coupled31-discovery
+/tmp/coupled31-discovery/bin/python -m pip install z3-solver==5.3.2.0
+/tmp/coupled31-discovery/bin/python -B research/coupled31-chronology/discovery/coupled_seed.py --pattern pair --timeout 60000 --models 8 --output /tmp/coupled31-pair.json
+/tmp/coupled31-discovery/bin/python -B research/coupled31-chronology/discovery/coupled_seed.py --pattern triangle --timeout 60000 --models 8 --output /tmp/coupled31-triangle.json
+```
+
+The actual environment was Python 3.12.13 on macOS arm64, with Z3 5.3.2.
+The two saved JSON files retain the precise version, pattern, timeout and
+unknown reason. Timings and solver progress can vary. The default verifier
+does not import the solver or treat a timeout as mathematical evidence.
+
+A smaller projected-chronology census shows why a dimension bound alone is
+not a sufficient endpoint test. Of 19,440 ordered three-stage rank-one flags
+over F2, 9864 admit a one-extra-row representation of the extracted S_jk
+differences. They are necessary-condition candidates only. The census covers
+six flags per stage and all 90 words with two occurrences of each label.
+It does not specify thin matrices, their call order, or a full endpoint.
+
+```sh
+python3.12 -B research/coupled31-chronology/discovery/row_signature_census.py
+```
+
+Its saved JSON contains four examples. Compare the counts and examples;
+elapsed seconds are not a correctness criterion. No claim of a new caller
+comes from this census.
+
+The next investigation of a cheap five-map target must use larger port
+coupling and noncommuting full projectors. The commuting-projector count
+proof in COMMUTING.md excludes five maps at both 13 and 31 leaves. It also
+supplies a complete word at the larger map-count bound. WORD31.json and its
+negative complete recursive price are retained as a control.
+
+No changed native supplier, practical runtime gain, or kappa improvement
+was found. All failed or incomplete searches keep their stated scopes.
+Ali Romman with substantial OpenAI Codex assistance; Apache-2.0.
diff --git a/research/coupled31-chronology/PROOF.md b/research/coupled31-chronology/PROOF.md
new file mode 100644
index 0000000000..641598c096
--- /dev/null
+++ b/research/coupled31-chronology/PROOF.md
@@ -0,0 +1,205 @@
+# A chronological lower bound for star connectors
+
+This is a finite-matrix theorem in characteristic two. It rules out the
+31-leaf, five-rank-two target from PR5673 and the static maps in PR5681.
+It does not rule out a different stage frame or a general native caller.
+No new kappa is established.
+
+## 1. Contract and statement
+
+Use the full star contract of PR5673. The core and each of the g leaves
+have dimension d. A stage j has orthogonal projectors P_jh which resolve
+its core-plus-leaf support E_j. They commute with
+
+```text
+T_j = [[I,I],[I,I]].
+```
+
+The full call is `(I+x_j P_jh)(I+e R_jh)`, where `R_jh=P_jh T_j`,
+`x_j=D_j+1`, and `e^2=0`. Thin calls are `I+e Z`. All x_j are independent.
+The target is `(I+e I) product_j (I+x_j E_j)`. Thin maps can be arbitrary;
+the proof needs no bound on their number and no square-zero condition.
+
+The local form is
+
+```text
+P_jh = [[A_jh+B_jh,B_jh],[B_jh,A_jh+B_jh]],
+R_jh = [[A_jh,A_jh],[A_jh,A_jh]].
+```
+
+Within a stage, the A_jh are orthogonal idempotents which resolve I.
+Suppose every nonzero A_jh has rank at least r, and put
+`L=floor(d/r)+1`. Let V be the span of all thin row spaces, and U the
+span of all thin image spaces. For g>=3, write
+
+```text
+dim V = 2d+b,    dim U = 2d+a.
+```
+
+**Theorem.** Every successful word satisfies
+
+```text
+g <= L b+2,       g <= L a+2,
+sum_thin rank Z >= 2d + 2 ceil((g-2)/L).                         (1)
+```
+
+This holds over every field of characteristic two. It does not use a finite
+field count. For two equal query halves per stage, d=2q and r=q, so L=3.
+Thus the extra row and image dimensions must each grow at least as
+`ceil((g-2)/3)`. A large number of distinct static signatures is not enough.
+
+## 2. The exact chronology relation
+
+The coefficient of e gives `sum_full R + sum_thin Z = I`. The coefficient
+of e*x_j then gives the necessary relation
+
+```text
+sum_(i in stage j) [P_i, sum_(l after i) R_l] = 0.               (2)
+```
+
+Here the sum includes full and thin queries. This is a necessary coefficient
+of the literal ordered word, not an endpoint converse. The retained source
+and PR5681 give separate derivations of (2).
+
+For j!=k, let
+
+```text
+S_jk = sum_(jh before kl) A_jh A_kl.
+```
+
+In (2), add block rows core and j and read column block k. A later full
+call in stage k contributes `A_jh A_kl`. Other full stages contribute zero.
+A later thin map contributes `A_jh (Z_0k+Z_jk)`.
+
+The thin sum is `K=I+sum T_j`. Its rank is 2d. Its row space is the core
+plus the leaf-sum space; its image is the core plus the diagonal leaf space.
+Choose b further row vectors which extend the row space of K to V. Let
+M_k be their b-by-d restriction to leaf k. Every thin block column can be
+written as a term constant in k plus a matrix times M_k. Therefore (2) gives
+matrices L_j and C_j such that
+
+```text
+S_jk = L_j + C_j M_k,                  k!=j.                    (3)
+```
+
+For a particular full call A=A_jh, let P_k be the sum of the A projections
+in stage k before that call. Orthogonality within stage j gives
+`A S_jk=A(I+P_k)`. Multiply (3) by A and absorb its constant term. For each
+call there are matrices H and C with
+
+```text
+A P_k = H+C M_k for every k!=j,        A P_j=0.                 (4)
+```
+
+This statement remains valid after selection of any subset of leaves.
+Use the induced call order and omit the other stages. A removed stage has
+no contribution to S_jk for retained j,k. This restriction property is
+essential to the proof below.
+
+## 3. A dimension drop forced by the first calls
+
+For a nonempty leaf set J, define h(J) as the dimension of the span of
+all columns of `M_k-M_l`, for k,l in J. This span is a subspace of F^b.
+It is not the dimension of the vectorized matrix differences. Choose one
+leaf as a reference and a basis for that column span. After absorption of
+the reference matrix into H, equation (4) has signatures with h(J) rows.
+
+Read the full calls on J in their induced order. Let `S=sum_(k in J) P_k`
+be the total prefix. If `A S=0` at each of the first L calls, then
+
+```text
+(I+xA)(I+xS) = I+x(S+A)
+```
+
+at each step, for an independent central variable x. The product of the
+first L factors is `I+xS_L`. Its determinant has degree at most d.
+Each A is idempotent, so `det(I+xA)=(1+x)^rank(A)`. The product determinant
+has degree at least L*r>d. This is impossible. Consequently one of these
+first L calls has `A S!=0`.
+
+Let U be the leaves not yet called at that point, after exclusion of the
+current stage. At most L leaves have been removed from J. If U is nonempty,
+choose u in U. For every k in U, `P_k=0`, so (4) gives
+
+```text
+H=-C M_u,          C(M_k-M_u)=0.
+```
+
+The map C cannot vanish on the column-difference span of J. If it did,
+(4) would give `A P_k=0` for every k!=j; the own-stage product is also zero.
+That would give `A S=0`. Thus C is nonzero on that span, but it kills the
+column-difference span of U. We have proved
+
+```text
+|J minus U| <= L,             h(U) <= h(J)-1.                  (5)
+```
+
+Only the stage ranks and equation (4) were used. No connector placement,
+matrix-size search, or pure higher stage coefficient was assumed.
+
+## 4. Induction and the paid-rank bound
+
+If h(J)=0, all signatures on J are equal. The one-sided theorem from
+PR5673 gives |J|<=2. Its short proof also follows from (4): for any three
+stages, `A P_k=A P_l` at every update, so `A S=0`. At the end S=I in
+characteristic two. The determinant would be both `(1+x)^(3d)` and
+`(1+x)^d`, which is impossible.
+
+We now prove `|J|<=L h(J)+2` by induction on h(J). If |J|<=L+2 and h(J)>0,
+the bound holds. Otherwise (5) supplies a nonempty U. Equation (4) is
+preserved on U, and the induction hypothesis gives
+
+```text
+|J| <= L+|U| <= L+L(h(J)-1)+2 = L h(J)+2.
+```
+
+For the full leaf set, h(J)<=b. This proves the row-space bound in (1).
+Transpose every matrix and reverse the complete word for the image-space
+bound. The target is unchanged, since its full stage supports commute.
+
+Factor each thin map at its rank and concatenate the factors. If R is the
+sum of those ranks, Sylvester's inequality gives
+
+```text
+rank K >= dim U+dim V-R,
+R >= 2d+a+b.
+```
+
+Combining the two dimension bounds proves (1).
+
+## 5. The two previously open targets
+
+For g=31 and q=2, d=4 and L=3. Equation (1) gives a>=10, b>=10 and
+**R>=28**. Five rank-two thin connectors have R=10. Thus no choice of their
+matrices, call order, or compatible paired stage projections can give the
+required endpoint. The PR5681 static maps remain valid static maps; they
+cannot be completed to this chronological word.
+
+For g=127 and q=1, the bound is a,b>=42 and **R>=88**, instead of the ten
+rank-one connectors in the earlier target. It is excluded for the same reason.
+
+For g=31 with five rank-q connectors, the necessary inequality is
+`5q>=4q+20`, or q>=20. A paired-port block with q=26 divides the 104 native
+ports and passes this new necessary test. It still has no supplied word.
+The q=8, g=13, five-rank-eight target also passes. Passing (1) is not a
+construction and does not supply a native lift.
+
+## 6. Scope and attribution
+
+The result concerns independent stage variables, the fixed star support,
+full orthogonal stage resolutions, the stated query map, and the exact
+normalized target. Nonfull stages, paid idle stage maps, different graph
+frames, and general complex or dyadic callers are outside this proof.
+Arbitrary thin maps are included. Square-zero thin maps are a special case.
+
+The argument uses elementary determinant degree, subspace dimension and
+Sylvester rank. The new step is the recursive restriction in (5). It changes
+the old logarithmic static signature bound into a linear chronological bound.
+The exact tariff consequence is in COST.md. Finite checks support the written
+proof; they do not constitute human review or proof-assistant acceptance.
+
+Credit PR5673 for the one-sided proof and rank framework, PR5681 for the
+static witnesses and separate coefficient audit, Cinna Davis for the
+controller/simplex source, and the retained P104/helper/native/compiler line.
+Prepared by Ali Romman with substantial OpenAI Codex assistance; no subagents.
+Apache-2.0.
diff --git a/research/coupled31-chronology/README.md b/research/coupled31-chronology/README.md
new file mode 100644
index 0000000000..357d9898ac
--- /dev/null
+++ b/research/coupled31-chronology/README.md
@@ -0,0 +1,92 @@
+# Chronology excludes the 31-leaf five-connector target
+
+**No new kappa.** The requested 31-leaf caller with five rank-two connectors
+cannot have the full endpoint under the paired-star contract. Its thin rank
+must be at least 28, rather than ten. The new proof also excludes the earlier
+127-leaf, ten-rank-one target, whose thin rank must be at least 88.
+
+The main result is a linear information bound for every finite dimension and
+every characteristic-two field. If b is the extra row dimension in the thin
+connectors, the paired-stage contract requires `g<=3b+2`. The same bound holds
+for extra image dimension. A prefix determinant argument forces a dimension
+drop after at most three full calls; restriction to untouched leaves repeats
+that drop. [PROOF.md](PROOF.md) gives the complete induction and the extension
+to arbitrary full-stage splits with a specified minimum query rank.
+
+This goes beyond the static cut in [PR5673](https://github.com/CrocSwap/integer-mult-bounds/pull/5673)
+and its sharp witnesses in [PR5681](https://github.com/CrocSwap/integer-mult-bounds/pull/5681).
+Those static witnesses remain valid. Chronology prevents their completion.
+
+[COST.md](COST.md) charges the entire pinned P104 recursive profile. An exact
+concave tangent proves that every paired-star caller with one or two coupled
+ports loses at saving **.00256**, for every integer leaf count g>=3. This is
+a uniform ceiling below selected .00310045, not a bounded parameter search.
+The invoice marks all unconstructed physical obligations as unknown.
+
+A larger coupling is still possible under this test. At 31 leaves, five
+rank-q connectors require q>=20. The q=26 profile fits the 104-port population
+and has a positive optimistic .0035 price. It has no supplied operator.
+[COMMUTING.md](COMMUTING.md) supplies a second restriction: commuting full
+projectors require at least `ceil(g/2)` thin maps, regardless of port count
+or thin rank. Thus the five-map large-coupling targets need noncommuting full
+projectors or another contract.
+
+[WORD31.json](WORD31.json) supplies a complete 31-leaf binary word at that
+map-count bound: 62 full calls and 16 thin maps of rank eight on each pair
+of ports. Its exact Laurent endpoint passes; its complete recursive bill
+loses every concave moment. It is a control, not a native supplier.
+
+## Reproduce
+
+Python 3.12 and the standard library suffice. A cold registered metadata
+check also needs the two open provenance commits:
+
+```sh
+git fetch https://github.com/CrocSwap/integer-mult-bounds.git c10f75f09e41803ae8efe478331a7509b7379b2f
+git fetch https://github.com/CrocSwap/integer-mult-bounds.git 67f07448eec5ad8d84bdf96b46981ac0b893592f
+python3.12 -B scripts/research_package.py --package research/coupled31-chronology
+python3.12 -I -O -B research/coupled31-chronology/verify.py
+```
+
+Expected: `PASS_CHRONOLOGICAL_CONNECTOR_RANK_BOUND` and the registered package
+pass. The checker imports no solver or discovery script. It checks formal
+polynomial determinants, local dimension-drop systems, the static source
+fixture, the full long endpoint, a separate F4 interpolation endpoint proof,
+and exact integer-power price bounds. Damaged sources, endpoints and receipts
+are rejected. [RESULT.json](RESULT.json) records the exact evidence.
+
+Finite checks support the written universal proofs. They are not independent
+human or proof-assistant acceptance. The binary word does not supply a native
+lift, legal physical reads, restoration, adapters, tapes, compiler or outer
+transfer. The selected frontier is unchanged.
+
+## Open next questions
+
+- Audit the restriction and dimension-drop steps in PROOF.md. Check that the
+  signature dimension is the column span of matrix differences, and that
+  removed stages do not enter the retained mixed blocks.
+- Construct a genuinely noncommuting coupled word. The q=26, g=31, five-map
+  profile and the q=8, g=13, five-map profile pass the new rank bound and an
+  optimistic price. Both remain unconstructed. First test every independent
+  stage coefficient, then provide the native lift and complete physical bill.
+- Resolve the saved small coupled seed searches. Two fixed static connector
+  choices with all call orders free timed out. They remain unknown. Do not
+  infer impossibility or a large-word extension from those runs.
+
+Structured questions and the changed card scope are in TASK.json. Discovery
+commands and failed or incomplete routes are in DISCOVERY.md. The canonical
+investigation is [Discussion5593](https://github.com/CrocSwap/integer-mult-bounds/discussions/5593).
+Work starts from canonical main `df0137efd35e11b4445734bf612bdd4c7ebc330c`,
+priority seven. Initial remote overlap covered 401 open PRs and 5060 Discussion
+records; sparse local coverage was incomplete.
+
+Credit Cinna Davis's controller/simplex work, PR5617/5631/5649/5673/5681,
+the separate PR5701 lift qualification, and the complete retained P104/helper/
+native/compiler lineage. Ali Romman with substantial OpenAI Codex assistance;
+no subagents. Apache-2.0.
+
+The pinned P104 tariff retains Pierre-Loic Doulcet / hexapode, Joel Pulikkan /
+GamingPuzzled, DreamingOfClouds, thu-nmrc, Gareth Davidson / bitplane, David
+Arias, Jacob Alan Sussman, zxv and Cao Yifan, with the earlier geometry, BIT
+and analytic authors in its source notices. Chris R / Clear1nsight supplies
+the separate PR5701 characteristic-zero qualification.
diff --git a/research/coupled31-chronology/RESULT.json b/research/coupled31-chronology/RESULT.json
new file mode 100644
index 0000000000..5f09e0d7b4
--- /dev/null
+++ b/research/coupled31-chronology/RESULT.json
@@ -0,0 +1,153 @@
+{
+  "authenticated_inputs": 5,
+  "evidence": {
+    "dimension_drop": {
+      "consistent": 702,
+      "inconsistent": 1998,
+      "local_systems": 2700,
+      "strict_drop_instances": 104
+    },
+    "endpoints": {
+      "binary_alias_rejected": true,
+      "damaged_words_rejected": 2,
+      "full_grid_assignments": 66,
+      "long_word": {
+        "fields": 128,
+        "final_coefficients": 128,
+        "full_calls": 62,
+        "query_entry_contributions": 744,
+        "thin_maps": 16,
+        "thin_rank": 128
+      }
+    },
+    "prefix": {
+      "formal_prefix_updates": 216,
+      "noncommuting_updates": 131
+    },
+    "static_target": {
+      "chronology_bound": 5,
+      "chronology_rejected": true,
+      "d": 4,
+      "extra_image_dimension": 1,
+      "extra_row_dimension": 1,
+      "leaves": 31,
+      "static_checks_pass": true,
+      "thin_maps": 5,
+      "thin_rank": 10
+    }
+  },
+  "limits": [
+    "No kappa increase",
+    "No native lift or complete physical caller",
+    "Written universal proofs with finite controls; no proof-assistant acceptance",
+    "Two finite seed solver timeouts remain unknown"
+  ],
+  "pricing": {
+    "complete_binary_word31": {
+      "H1_thin_calls_at_104_ports": 6656,
+      "all_concave_moments_rejected": true,
+      "binary_blocks": 52,
+      "full_calls": 62,
+      "linear_margin": -3472,
+      "rank_per_thin_map": 8,
+      "signed_profile": {
+        "1": -143118,
+        "2": -28148,
+        "3": -30566,
+        "4": -186,
+        "5": -9734,
+        "6": -1116,
+        "7": -558,
+        "8": 992,
+        "9": 31992,
+        "249": 208
+      },
+      "thin_maps": 16
+    },
+    "net_rank_mass_per_full_call": 1720,
+    "old_127_leaf": {
+      "minimum_thin_rank": 88,
+      "optimistic_signed_profile": {
+        "1": -568206,
+        "2": -115316,
+        "3": -125222,
+        "4": -762,
+        "5": -39878,
+        "6": -4572,
+        "7": -2286,
+        "8": 4064,
+        "9": 131064,
+        "1017": 208
+      },
+      "ports": 1,
+      "requested_thin_rank": 10
+    },
+    "requested_31_leaf": {
+      "minimum_H1_calls_at_104_ports": 1456,
+      "minimum_thin_rank": 28,
+      "optimistic_signed_profile": {
+        "1": -137918,
+        "2": -28148,
+        "3": -30566,
+        "4": -186,
+        "5": -9734,
+        "6": -1116,
+        "7": -558,
+        "8": 992,
+        "9": 31992,
+        "249": 208
+      },
+      "ports": 2,
+      "requested_thin_rank": 10
+    },
+    "unconstructed_targets": [
+      {
+        "coupled_ports": 8,
+        "leaves": 13,
+        "margin": [
+          "23457899974847/2199023255552",
+          "23457899996609/2199023255552"
+        ],
+        "minimum_thin_rank": 40,
+        "rank_per_thin_map": 8,
+        "saving": "17/5000",
+        "status": "UNCONSTRUCTED; COMMUTING FULL PROJECTORS EXCLUDED",
+        "thin_maps": 5
+      },
+      {
+        "coupled_ports": 26,
+        "leaves": 31,
+        "margin": [
+          "268079792823357/1099511627776",
+          "16754987053077/68719476736"
+        ],
+        "minimum_thin_rank": 124,
+        "rank_per_thin_map": 26,
+        "saving": "7/2000",
+        "status": "UNCONSTRUCTED; COMMUTING FULL PROJECTORS EXCLUDED",
+        "thin_maps": 5
+      }
+    ],
+    "uniform_one_or_two_port_ceiling": {
+      "all_integer_leaves_at_least_three": true,
+      "all_larger_savings_rejected": true,
+      "intercept_interval": [
+        "-2387010674050001077/331141978521600000",
+        "-4774021348099800407/662283957043200000"
+      ],
+      "saving_rejected": "8/3125",
+      "slope_interval": [
+        "-593132702156541649/5298271656345600000",
+        "-593132698134742867/5298271656345600000"
+      ],
+      "tangent_leaves": 32
+    }
+  },
+  "proof_sha256": {
+    "COMMUTING.md": "d4329ca042cd80527a086fbd6b4644f67660145adfc72196c0a8744bf4d40017",
+    "COST.md": "076e0f322209c64ab91870e282f251d07ecaae282b536f0f993ba36576fae457",
+    "PROOF.md": "f909adc46fb9df2955430da2ce1f29caf78b4cb668e8eda0361fa5d6dd800095"
+  },
+  "scope": "Paired full star stages over characteristic two; arbitrary thin maps",
+  "status": "PASS_CHRONOLOGICAL_CONNECTOR_RANK_BOUND"
+}
diff --git a/research/coupled31-chronology/SOURCES.json b/research/coupled31-chronology/SOURCES.json
new file mode 100644
index 0000000000..37c7f67d64
--- /dev/null
+++ b/research/coupled31-chronology/SOURCES.json
@@ -0,0 +1,53 @@
+{
+  "schema_version": 1,
+  "base_commit": "df0137efd35e11b4445734bf612bdd4c7ebc330c",
+  "files": [
+    {
+      "path": "certificates/selected-result.json",
+      "usage": "provenance",
+      "git_blob": "581f8d87008354eb5b869f7ed9de1317ad837ce8",
+      "sha256": "2a0220e866417774606e0f991e41b9c1d26b1f43b97857238cf478dbc7d330b5"
+    },
+    {
+      "path": "docs/research/agent-task-catalog.json",
+      "usage": "provenance",
+      "git_blob": "f126dd2e2dad2b669e75d1d0d0bd05c600725602",
+      "sha256": "f7a48bfa2659b88378282a8eee7d9fe52cdf505594124c4d1686b58510fffd72"
+    },
+    {
+      "path": "research/coupled31-chronology/sources/pr5673-proof.md",
+      "usage": "input",
+      "pin": "head:c10f75f09e41803ae8efe478331a7509b7379b2f:research/crossed-connector-caller/PROOF.md",
+      "git_blob": "e3800fbcf1c5f5633e14d9856ca1400985292a0d",
+      "sha256": "e086b3376173f6b790e05ef09f23b4d5970c8ac5145ab2970e084743f782190b"
+    },
+    {
+      "path": "research/coupled31-chronology/sources/escape-word.json",
+      "usage": "input",
+      "pin": "head:c10f75f09e41803ae8efe478331a7509b7379b2f:research/crossed-connector-caller/ESCAPE-WORD.json",
+      "git_blob": "aae71c6ad294d6bc9ca5e2216a7785b9e6611afd",
+      "sha256": "20285745b7a6c4bfefcccc365caacefaf982d30adb60fea29ffa14f0ea6b639c"
+    },
+    {
+      "path": "research/coupled31-chronology/sources/old-word-replay.py",
+      "usage": "input",
+      "pin": "head:c10f75f09e41803ae8efe478331a7509b7379b2f:research/crossed-connector-caller/word_replay.py",
+      "git_blob": "037ed3f765822341156e059c306218504ac37e1b",
+      "sha256": "d4574ed109672c9351af8ace5fbcd31671aed6936877dfa4ff09b56b9007da0a"
+    },
+    {
+      "path": "research/coupled31-chronology/sources/pr5681-proof.md",
+      "usage": "input",
+      "pin": "head:67f07448eec5ad8d84bdf96b46981ac0b893592f:research/one-sided-connector-audit/PROOF.md",
+      "git_blob": "e3be1bb272f3a73a8c6cd248fec69dec498e401d",
+      "sha256": "2e52ac6d32923a1ce6f9693415df553b2d4c38c1a103b9e1968c4e114044d800"
+    },
+    {
+      "path": "research/coupled31-chronology/sources/p104-result.json",
+      "usage": "input",
+      "pin": "head:09dd0965711ca7eeb169057a420abab1c9052137:research/p104-helper-subspace-splits/RESULT.json",
+      "git_blob": "35f75c8555193c6e845cd52cc562ca23fd5924cb",
+      "sha256": "1d92e587915052110dfb3aca8890df5afdaf496164efa5a4a52e69ceeb3c70f3"
+    }
+  ]
+}
diff --git a/research/coupled31-chronology/TASK.json b/research/coupled31-chronology/TASK.json
new file mode 100644
index 0000000000..001d33d03d
--- /dev/null
+++ b/research/coupled31-chronology/TASK.json
@@ -0,0 +1,102 @@
+{
+  "schema_version": 1,
+  "task_id": "joint-query-controller",
+  "priority_revision": 7,
+  "base_commit": "df0137efd35e11b4445734bf612bdd4c7ebc330c",
+  "original_card": {
+    "id": "joint-query-controller",
+    "question": "Can the installed QQ16 caller win a complete matched protocol bill, or can a changed demand eliminate actual paid work?",
+    "starting_points": [
+      "docs/research/research-family-status.md",
+      "certificates/selected-result.json",
+      "research/qq16-caller-install/README.md"
+    ],
+    "methods": [
+      "Derive the matched whole-caller physical bill for installed and baseline protocols.",
+      "Change the actual demanded observations and prove a cheaper legal producer/consumer contract."
+    ],
+    "acceptance_gate": "Name the changed premise of the fixed F0/B1 observation obstruction. Show exactly which query is recovered, legal native reads and restoration, and a full favorable budget including cover maps, helpers and endpoints. Local inverse identities alone do not close the global controller. A scoped local result may be retained without a complete supplier. Expanding it into further construction rounds requires the family-level budget and obligation gate in research-family-status.md; an optimistic passing relaxation alone is not acceptance.",
+    "first_step": {
+      "command": [
+        "python3",
+        "-B",
+        "scripts/research_baseline.py"
+      ],
+      "expected": "PASS_RESEARCH_BASELINE_BINDINGS_ONLY",
+      "inspect": "Read #4166 COST.md and ADMISSION.json. An actual caller connection is now proposed. Its native-component profile loses; parent park/restore versus external-call costs remain unresolved. Retain the producer and recovery lineage rather than rebuild it.",
+      "scope": "Source-binding orientation only; no scientific replay, theorem acceptance or supplier admission. Budget any heavy replay separately."
+    },
+    "first_experiment": "At the actual installed cut, compare matched complete parent park/restore and external input/return protocols at the same bank/packet geometry, including every later root call and service cost. Alternatively pin both bridge observations and derive a genuinely smaller source-compatible demand. Stop if the complete necessary inequality fails or an essential cost is unknown; no larger decoder scan from a local tape-action saving.",
+    "bottleneck_link": "The native local words exist; retained query information and its physical access are the missing link to a composed recursive interface.",
+    "allowed_variations": [
+      "One paid query, one changed first pair, or one precisely specified affine helper state.",
+      "A counterexample or lower bound for that exact proposed repair.",
+      "One unresolved obligation of a newer local decoder, with exact source heads and a distinct claim/domain."
+    ],
+    "scope_change_triggers": [
+      "Retaining the fixed first pair and ordinary schedule while allowing only homogeneous selected-only control, already excluded by the witness.",
+      "Treating cover-map counts as zero-cost physical address operations.",
+      "Recreating the pending complete local decoder or early-record/output-bank coalescing already under investigation without a distinct audit or changed premise.",
+      "Treating a completed-output defect inequality as a producer-interior, dirty-range or physical-cost certificate.",
+      "Recreating completed-output recovery or caller wiring already supplied by #4166 without new independent coverage.",
+      "Dropping retained later root calls or claiming native-profile dominance from the isolated publication saving."
+    ],
+    "overlap_terms": [
+      "QQ16",
+      "joint-query",
+      "changing-chart controller",
+      "selected observation",
+      "1969",
+      "2540",
+      "early record",
+      "output-bank retention",
+      "complete native decoder",
+      "2965",
+      "producer defect",
+      "dirty-helper range",
+      "F17 consumer",
+      "4166",
+      "qq16-caller-install"
+    ],
+    "existing_checks": [
+      {
+        "scope": "PR #4166: author-checked installed producer/service/return/bridge connection; native-component comparison loses.",
+        "evidence": "research/qq16-caller-install/README.md",
+        "remaining": "Complete matched physical protocol bill, or changed source-compatible demand/circuit; no affordable supplier yet.",
+        "status": "author-checked",
+        "source_commit": "0b082d22a0e9a442d264ab0633e35ef26c790630",
+        "selected_record_sha256": "bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129"
+      }
+    ]
+  },
+  "scope_status": "changed",
+  "actual_scope": "Chronological dimension bound for all full star splits; uniform paired-star tariff ceiling; sharp commuting map-count proof and complete long binary endpoint",
+  "scope_change_reason": "The user explicitly assigned the 31-leaf five-rank-two target. PR5681 already supplied static maps. A new chronology argument rules out that target, then tests larger coupling and commuting full projectors.",
+  "bottleneck_link": "A different connector can retain useful full-call stages while using cheap thin queries; success requires a complete endpoint and paid bill.",
+  "budget_minutes": 120,
+  "stop_rule": "Prove or construct the requested full endpoint and price the complete profile. If excluded, give a useful general bound and test a changed coupling contract. Preserve unknown searches as unknown.",
+  "outcome": "completed",
+  "evidence": "PROOF.md, COMMUTING.md, COST.md, WORD31.json, RESULT.json",
+  "author": "Ali Romman with substantial OpenAI Codex assistance",
+  "ai_assistance": "Substantial OpenAI Codex assistance; no subagents.",
+  "open_questions": [
+    {
+      "question": "Does the recursive untouched-leaf restriction prove the stated linear dimension bound?",
+      "why_it_matters": "It excludes both previously open target budgets in all finite dimensions and gives a uniform price ceiling.",
+      "first_step": "Audit equations (3)-(5) in PROOF.md and the determinant-degree step; replay the local controls.",
+      "blocked_by": "No independent human or proof-assistant acceptance is claimed."
+    },
+    {
+      "question": "Can five thin maps support 31 leaves with 26 coupled ports, or 13 leaves with eight coupled ports?",
+      "why_it_matters": "These pass the new necessary rank bound and a favorable optimistic price.",
+      "first_step": "Use noncommuting full projectors; check every independent stage coefficient before any native lift.",
+      "blocked_by": "No such word is supplied; commuting full projectors are excluded."
+    },
+    {
+      "question": "Do either saved small coupled seed searches have a full word?",
+      "why_it_matters": "A small noncommuting seed may expose a compositional construction.",
+      "first_step": "Resolve the saved pair or triangle static-connector domain, then replay the complete polynomial endpoint.",
+      "blocked_by": "Both all-order SMT runs timed out; unknown is not infeasible."
+    }
+  ]
+}
diff --git a/research/coupled31-chronology/VERIFY.json b/research/coupled31-chronology/VERIFY.json
new file mode 100644
index 0000000000..c960064c3c
--- /dev/null
+++ b/research/coupled31-chronology/VERIFY.json
@@ -0,0 +1,36 @@
+{
+  "schema_version": 1,
+  "status": "ready",
+  "python": "3.12",
+  "requirements": [],
+  "inputs": [
+    "research/coupled31-chronology/sources/pr5673-proof.md",
+    "research/coupled31-chronology/sources/escape-word.json",
+    "research/coupled31-chronology/sources/old-word-replay.py",
+    "research/coupled31-chronology/sources/pr5681-proof.md",
+    "research/coupled31-chronology/sources/p104-result.json"
+  ],
+  "commands": [
+    {
+      "argv": [
+        "{python}",
+        "-I",
+        "-B",
+        "{package}/verify.py"
+      ],
+      "timeout_seconds": 60,
+      "checks": "Authenticate source copies; prefix determinants and dimension drop; static target; full Laurent and interpolation endpoints; exact complete tariff bounds."
+    },
+    {
+      "argv": [
+        "{python}",
+        "-I",
+        "-O",
+        "-B",
+        "{package}/test_verify.py"
+      ],
+      "timeout_seconds": 60,
+      "checks": "Reject changed input, endpoint, order, written proof or result receipt."
+    }
+  ]
+}
diff --git a/research/coupled31-chronology/WORD31.json b/research/coupled31-chronology/WORD31.json
new file mode 100644
index 0000000000..a57608244a
--- /dev/null
+++ b/research/coupled31-chronology/WORD31.json
@@ -0,0 +1 @@
+{"g":31,"q":2,"order":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77],"calls":[{"stage":0,"p":[1,2,0,0,16,32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],"r":[17,34,0,0,17,34,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]},{"stage":0,"p":[0,0,4,8,0,0,64,128,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],"r":[0,0,68,136,0,0,68,136,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]},{"stage":31,"p":[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],"r":[272,544,1088,2176,1,2,4,8,1,2,4,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]},{"stage":1,"p":[1,2,0,0,0,0,0,0,256,512,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],"r":[257,514,0,0,0,0,0,0,257,514,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]},{"stage":1,"p":[0,0,4,8,0,0,0,0,0,0,1024,2048,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],"r":[0,0,1028,2056,0,0,0,0,0,0,1028,2056,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]},{"stage":2,"p":[1,2,0,0,0,0,0,0,0,0,0,0,4096,8192,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],"r":[4097,8194,0,0,0,0,0,0,0,0,0,0,4097,8194,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]},{"stage":2,"p":[0,0,4,8,0,0,0,0,0,0,0,0,0,0,16384,32768,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],"r":[0,0,16388,32776,0,0,0,0,0,0,0,0,0,0,16388,32776,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]},{"stage":31,"p":[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],"r":[69632,139264,278528,557056,0,0,0,0,0,0,0,0,1,2,4,8,1,2,4,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]},{"stage":3,"p":[1,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,65536,131072,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],"r":[65537,131074,0,0,0,0,0,0,0,0,0,0,0,0,0,0,65537,131074,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]},{"stage":3,"p":[0,0,4,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,262144,524288,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],"r":[0,0,262148,524296,0,0,0,0,0,0,0,0,0,0,0,0,0,0,262148,524296,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]},{"stage":4,"p":[1,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1048576,2097152,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],"r":[1048577,2097154,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1048577,2097154,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]},{"stage":4,"p":[0,0,4,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4194304,8388608,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],"r":[0,0,4194308,8388616,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4194308,8388616,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]},{"stage":31,"p":[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],"r":[17825792,35651584,71303168,142606336,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,4,8,1,2,4,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]},{"stage":5,"p":[1,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,16777216,33554432,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],"r":[16777217,33554434,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,16777217,33554434,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]},{"stage":5,"p":[0,0,4,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,67108864,134217728,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],"r":[0,0,67108868,134217736,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,67108868,134217736,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]},{"stage":6,"p":[1,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,268435456,536870912,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],"r":[268435457,536870914,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,268435457,536870914,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]},{"stage":6,"p":[0,0,4,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1073741824,2147483648,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],"r":[0,0,1073741828,2147483656,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1073741828,2147483656,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]},{"stage":31,"p":[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],"r":[4563402752,9126805504,18253611008,36507222016,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,4,8,1,2,4,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]},{"stage":7,"p":[1,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4294967296,8589934592,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],"r":[4294967297,8589934594,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4294967297,8589934594,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]},{"stage":7,"p":[0,0,4,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,17179869184,34359738368,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],"r":[0,0,17179869188,34359738376,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,17179869188,34359738376,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]},{"stage":8,"p":[1,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,68719476736,137438953472,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],"r":[68719476737,137438953474,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,68719476737,137438953474,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]},{"stage":8,"p":[0,0,4,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,274877906944,549755813888,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],"r":[0,0,274877906948,549755813896,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,274877906948,549755813896,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]},{"stage":31,"p":[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],"r":[1168231104512,2336462209024,4672924418048,9345848836096,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,4,8,1,2,4,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]},{"stage":9,"p":[1,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1099511627776,2199023255552,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],"r":[1099511627777,2199023255554,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1099511627777,2199023255554,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]},{"stage":9,"p":[0,0,4,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4398046511104,8796093022208,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],"r":[0,0,4398046511108,8796093022216,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4398046511108,8796093022216,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]},{"stage":10,"p":[1,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,17592186044416,35184372088832,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],"r":[17592186044417,35184372088834,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,17592186044417,35184372088834,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]},{"stage":10,"p":[0,0,4,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,70368744177664,140737488355328,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],"r":[0,0,70368744177668,140737488355336,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,70368744177668,140737488355336,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]},{"stage":31,"p":[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],"r":[299067162755072,598134325510144,1196268651020288,2392537302040576,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,4,8,1,2,4,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]},{"stage":11,"p":[1,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,281474976710656,562949953421312,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],"r":[281474976710657,562949953421314,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,281474976710657,562949953421314,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]},{"stage":11,"p":[0,0,4,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1125899906842624,2251799813685248,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],"r":[0,0,1125899906842628,2251799813685256,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1125899906842628,2251799813685256,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]},{"stage":12,"p":[1,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4503599627370496,9007199254740992,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],"r":[4503599627370497,9007199254740994,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4503599627370497,9007199254740994,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]},{"stage":12,"p":[0,0,4,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,18014398509481984,36028797018963968,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],"r":[0,0,18014398509481988,36028797018963976,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,18014398509481988,36028797018963976,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]},{"stage":31,"p":[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],"r":[76561193665298432,153122387330596864,306244774661193728,612489549322387456,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,4,8,1,2,4,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]},{"stage":13,"p":[1,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,72057594037927936,144115188075855872,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],"r":[72057594037927937,144115188075855874,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,72057594037927937,144115188075855874,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]},{"stage":13,"p":[0,0,4,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,288230376151711744,576460752303423488,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],"r":[0,0,288230376151711748,576460752303423496,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,288230376151711748,576460752303423496,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]},{"stage":14,"p":[1,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1152921504606846976,2305843009213693952,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],"r":[1152921504606846977,2305843009213693954,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1152921504606846977,2305843009213693954,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]},{"stage":14,"p":[0,0,4,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4611686018427387904,9223372036854775808,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],"r":[0,0,4611686018427387908,9223372036854775816,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4611686018427387908,9223372036854775816,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]},{"stage":31,"p":[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],"r":[19599665578316398592,39199331156632797184,78398662313265594368,156797324626531188736,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,4,8,1,2,4,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]},{"stage":15,"p":[1,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,18446744073709551616,36893488147419103232,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],"r":[18446744073709551617,36893488147419103234,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,18446744073709551617,36893488147419103234,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]},{"stage":15,"p":[0,0,4,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,73786976294838206464,147573952589676412928,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],"r":[0,0,73786976294838206468,147573952589676412936,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,73786976294838206468,147573952589676412936,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]},{"stage":16,"p":[1,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,295147905179352825856,590295810358705651712,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],"r":[295147905179352825857,590295810358705651714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,295147905179352825857,590295810358705651714,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]},{"stage":16,"p":[0,0,4,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1180591620717411303424,2361183241434822606848,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],"r":[0,0,1180591620717411303428,2361183241434822606856,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1180591620717411303428,2361183241434822606856,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]},{"stage":31,"p":[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],"r":[5017514388048998039552,10035028776097996079104,20070057552195992158208,40140115104391984316416,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,4,8,1,2,4,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]},{"stage":17,"p":[1,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4722366482869645213696,9444732965739290427392,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],"r":[4722366482869645213697,9444732965739290427394,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4722366482869645213697,9444732965739290427394,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]},{"stage":17,"p":[0,0,4,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,18889465931478580854784,37778931862957161709568,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],"r":[0,0,18889465931478580854788,37778931862957161709576,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,18889465931478580854788,37778931862957161709576,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]},{"stage":18,"p":[1,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,75557863725914323419136,151115727451828646838272,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],"r":[75557863725914323419137,151115727451828646838274,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,75557863725914323419137,151115727451828646838274,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]},{"stage":18,"p":[0,0,4,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,302231454903657293676544,604462909807314587353088,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],"r":[0,0,302231454903657293676548,604462909807314587353096,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,302231454903657293676548,604462909807314587353096,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]},{"stage":31,"p":[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],"r":[1284483683340543498125312,2568967366681086996250624,5137934733362173992501248,10275869466724347985002496,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,4,8,1,2,4,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]},{"stage":19,"p":[1,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1208925819614629174706176,2417851639229258349412352,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],"r":[1208925819614629174706177,2417851639229258349412354,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1208925819614629174706177,2417851639229258349412354,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]},{"stage":19,"p":[0,0,4,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4835703278458516698824704,9671406556917033397649408,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],"r":[0,0,4835703278458516698824708,9671406556917033397649416,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4835703278458516698824708,9671406556917033397649416,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]},{"stage":20,"p":[1,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,19342813113834066795298816,38685626227668133590597632,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],"r":[19342813113834066795298817,38685626227668133590597634,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,19342813113834066795298817,38685626227668133590597634,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]},{"stage":20,"p":[0,0,4,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,77371252455336267181195264,154742504910672534362390528,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],"r":[0,0,77371252455336267181195268,154742504910672534362390536,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,77371252455336267181195268,154742504910672534362390536,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]},{"stage":31,"p":[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],"r":[328827822935179135520079872,657655645870358271040159744,1315311291740716542080319488,2630622583481433084160638976,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,4,8,1,2,4,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]},{"stage":21,"p":[1,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,309485009821345068724781056,618970019642690137449562112,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],"r":[309485009821345068724781057,618970019642690137449562114,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,309485009821345068724781057,618970019642690137449562114,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]},{"stage":21,"p":[0,0,4,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1237940039285380274899124224,2475880078570760549798248448,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],"r":[0,0,1237940039285380274899124228,2475880078570760549798248456,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1237940039285380274899124228,2475880078570760549798248456,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]},{"stage":22,"p":[1,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4951760157141521099596496896,9903520314283042199192993792,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],"r":[4951760157141521099596496897,9903520314283042199192993794,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4951760157141521099596496897,9903520314283042199192993794,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]},{"stage":22,"p":[0,0,4,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,19807040628566084398385987584,39614081257132168796771975168,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],"r":[0,0,19807040628566084398385987588,39614081257132168796771975176,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,19807040628566084398385987588,39614081257132168796771975176,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]},{"stage":31,"p":[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],"r":[84179922671405858693140447232,168359845342811717386280894464,336719690685623434772561788928,673439381371246869545123577856,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,4,8,1,2,4,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]},{"stage":23,"p":[1,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,79228162514264337593543950336,158456325028528675187087900672,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],"r":[79228162514264337593543950337,158456325028528675187087900674,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,79228162514264337593543950337,158456325028528675187087900674,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]},{"stage":23,"p":[0,0,4,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,316912650057057350374175801344,633825300114114700748351602688,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],"r":[0,0,316912650057057350374175801348,633825300114114700748351602696,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,316912650057057350374175801348,633825300114114700748351602696,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]},{"stage":24,"p":[1,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1267650600228229401496703205376,2535301200456458802993406410752,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],"r":[1267650600228229401496703205377,2535301200456458802993406410754,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1267650600228229401496703205377,2535301200456458802993406410754,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]},{"stage":24,"p":[0,0,4,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5070602400912917605986812821504,10141204801825835211973625643008,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],"r":[0,0,5070602400912917605986812821508,10141204801825835211973625643016,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5070602400912917605986812821508,10141204801825835211973625643016,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]},{"stage":31,"p":[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],"r":[21550060203879899825443954491392,43100120407759799650887908982784,86200240815519599301775817965568,172400481631039198603551635931136,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,4,8,1,2,4,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]},{"stage":25,"p":[1,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,20282409603651670423947251286016,40564819207303340847894502572032,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],"r":[20282409603651670423947251286017,40564819207303340847894502572034,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,20282409603651670423947251286017,40564819207303340847894502572034,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]},{"stage":25,"p":[0,0,4,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,81129638414606681695789005144064,162259276829213363391578010288128,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],"r":[0,0,81129638414606681695789005144068,162259276829213363391578010288136,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,81129638414606681695789005144068,162259276829213363391578010288136,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]},{"stage":26,"p":[1,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,324518553658426726783156020576256,649037107316853453566312041152512,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],"r":[324518553658426726783156020576257,649037107316853453566312041152514,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,324518553658426726783156020576257,649037107316853453566312041152514,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]},{"stage":26,"p":[0,0,4,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1298074214633706907132624082305024,2596148429267413814265248164610048,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],"r":[0,0,1298074214633706907132624082305028,2596148429267413814265248164610056,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1298074214633706907132624082305028,2596148429267413814265248164610056,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]},{"stage":31,"p":[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],"r":[5516815412193254355313652349796352,11033630824386508710627304699592704,22067261648773017421254609399185408,44134523297546034842509218798370816,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,4,8,1,2,4,8,0,0,0,0,0,0,0,0,0,0,0,0]},{"stage":27,"p":[1,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5192296858534827628530496329220096,10384593717069655257060992658440192,0,0,0,0,0,0,0,0,0,0,0,0,0,0],"r":[5192296858534827628530496329220097,10384593717069655257060992658440194,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5192296858534827628530496329220097,10384593717069655257060992658440194,0,0,0,0,0,0,0,0,0,0,0,0,0,0]},{"stage":27,"p":[0,0,4,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,20769187434139310514121985316880384,41538374868278621028243970633760768,0,0,0,0,0,0,0,0,0,0,0,0],"r":[0,0,20769187434139310514121985316880388,41538374868278621028243970633760776,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,20769187434139310514121985316880388,41538374868278621028243970633760776,0,0,0,0,0,0,0,0,0,0,0,0]},{"stage":28,"p":[1,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,83076749736557242056487941267521536,166153499473114484112975882535043072,0,0,0,0,0,0,0,0,0,0],"r":[83076749736557242056487941267521537,166153499473114484112975882535043074,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,83076749736557242056487941267521537,166153499473114484112975882535043074,0,0,0,0,0,0,0,0,0,0]},{"stage":28,"p":[0,0,4,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,332306998946228968225951765070086144,664613997892457936451903530140172288,0,0,0,0,0,0,0,0],"r":[0,0,332306998946228968225951765070086148,664613997892457936451903530140172296,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,332306998946228968225951765070086148,664613997892457936451903530140172296,0,0,0,0,0,0,0,0]},{"stage":31,"p":[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],"r":[1412304745521473114960295001547866112,2824609491042946229920590003095732224,5649218982085892459841180006191464448,11298437964171784919682360012382928896,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,4,8,1,2,4,8,0,0,0,0]},{"stage":29,"p":[1,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1329227995784915872903807060280344576,2658455991569831745807614120560689152,0,0,0,0,0,0],"r":[1329227995784915872903807060280344577,2658455991569831745807614120560689154,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1329227995784915872903807060280344577,2658455991569831745807614120560689154,0,0,0,0,0,0]},{"stage":29,"p":[0,0,4,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5316911983139663491615228241121378304,10633823966279326983230456482242756608,0,0,0,0],"r":[0,0,5316911983139663491615228241121378308,10633823966279326983230456482242756616,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5316911983139663491615228241121378308,10633823966279326983230456482242756616,0,0,0,0]},{"stage":30,"p":[1,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,21267647932558653966460912964485513216,42535295865117307932921825928971026432,0,0],"r":[21267647932558653966460912964485513217,42535295865117307932921825928971026434,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,21267647932558653966460912964485513217,42535295865117307932921825928971026434,0,0]},{"stage":30,"p":[0,0,4,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,85070591730234615865843651857942052864,170141183460469231731687303715884105728],"r":[0,0,85070591730234615865843651857942052868,170141183460469231731687303715884105736,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,85070591730234615865843651857942052868,170141183460469231731687303715884105736]},{"stage":31,"p":[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],"r":[21267647932558653966460912964485513216,42535295865117307932921825928971026432,85070591730234615865843651857942052864,170141183460469231731687303715884105728,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,4,8]}]}
diff --git a/research/coupled31-chronology/check.py b/research/coupled31-chronology/check.py
new file mode 100644
index 0000000000..5d1bee4ffc
--- /dev/null
+++ b/research/coupled31-chronology/check.py
@@ -0,0 +1,397 @@
+"""Check the prefix lemmas and complete endpoint controls.
+
+Ali Romman with substantial OpenAI Codex assistance. Apache-2.0.
+"""
+from itertools import product
+import json
+from pathlib import Path
+import random
+
+ROOT = Path(__file__).resolve().parent
+
+
+def require(ok, message):
+    if not ok:
+        raise ValueError(message)
+
+
+def eye(n):
+    return tuple(1 << i for i in range(n))
+
+
+def add(a, b):
+    return tuple(x ^ y for x, y in zip(a, b))
+
+
+def mul(a, b):
+    out = []
+    for row in a:
+        value = 0
+        while row:
+            bit = row & -row
+            value ^= b[bit.bit_length()-1]
+            row ^= bit
+        out.append(value)
+    return tuple(out)
+
+
+def basis(rows):
+    out = {}
+    for value in rows:
+        while value:
+            pivot = value.bit_length()-1
+            if pivot not in out:
+                out[pivot] = value
+                break
+            value ^= out[pivot]
+    return out
+
+
+def rank(rows):
+    return len(basis(rows))
+
+
+def trans(a):
+    return tuple(sum(((a[j] >> i) & 1) << j for j in range(len(a)))
+                 for i in range(len(a)))
+
+
+def polynomial_mul(a, b):
+    out = 0
+    while b:
+        if b & 1:
+            out ^= a
+        a <<= 1
+        b >>= 1
+    return out
+
+
+def determinant_x(s):
+    """Compute det(I+xS) over F2[x] by subsets of columns."""
+    n = len(s)
+    dp = {0: 1}
+    for i in range(n):
+        new = {}
+        for mask, value in dp.items():
+            for j in range(n):
+                if not (mask >> j) & 1:
+                    entry = int(i == j) ^ (2*((s[i] >> j) & 1))
+                    target = mask | (1 << j)
+                    new[target] = new.get(target, 0) ^ polynomial_mul(value, entry)
+        dp = new
+    return dp[(1 << n)-1]
+
+
+def prefix_controls():
+    rng = random.Random(310526)
+    cases = noncommuting = 0
+    for d in range(2, 8):
+        for repeat in range(8):
+            stages = []
+            for i in range(d):
+                a = [0]*d
+                a[i] = (1 << i) | (rng.randrange(1 << (d-i-1)) << (i+1))
+                stages.append(tuple(a))
+            s = (0,)*d
+            expected = 1
+            for a in stages:
+                require(mul(a, a) == a and rank(a) == 1, 'prefix idempotent')
+                require(mul(a, s) == (0,)*d, 'prefix annihilation')
+                noncommuting += mul(s, a) != (0,)*d
+                s = add(s, a)
+                expected = polynomial_mul(expected, 3)
+                require(determinant_x(s) == expected, 'formal determinant identity')
+                cases += 1
+            require(rank(s) == d, 'full prefix is invertible')
+            require(all(mul(a, s) != (0,)*d for a in stages), 'next call must fail')
+    require(noncommuting > 0, 'noncommuting prefix controls')
+    return {'formal_prefix_updates': cases, 'noncommuting_updates': noncommuting}
+
+
+def solve_binary(rows):
+    pivots = {}
+    for coefficients, value in rows:
+        while coefficients:
+            p = coefficients.bit_length()-1
+            if p not in pivots:
+                pivots[p] = (coefficients, value)
+                break
+            c, v = pivots[p]
+            coefficients ^= c
+            value ^= v
+        if not coefficients and value:
+            return None
+    answer = 0
+    for p in sorted(pivots):
+        c, v = pivots[p]
+        if v ^ ((c & answer).bit_count() & 1):
+            answer |= 1 << p
+    return answer
+
+
+def signature_dimension(ms, d):
+    if not ms:
+        return 0
+    h = len(ms[0])
+    columns = []
+    for m in ms[1:]:
+        for c in range(d):
+            columns.append(sum((((m[t] ^ ms[0][t]) >> c) & 1) << t
+                               for t in range(h)))
+    return rank(columns)
+
+
+def drop_controls():
+    rng = random.Random(73131)
+    d = 2
+    projections = [tuple(a) for a in product(range(4), repeat=2)
+                   if mul(a, a) == tuple(a)]
+    proper = [a for a in projections if rank(a) == 1]
+    checked = solved = drops = rejected = 0
+    for h in (1, 2, 3):
+        for g in (3, 5, 7):
+            for repeat in range(300):
+                a = rng.choice(proper)
+                ps = [rng.choice(((0, 0), add(eye(d), a)))]
+                ps += [rng.choice(projections) for _ in range(g-1)]
+                ms = [tuple(rng.randrange(4) for _ in range(h)) for _ in range(g)]
+                c_rows = []
+                for row in range(d):
+                    equations = []
+                    for k in range(1, g):
+                        ap = mul(a, ps[k])
+                        for col in range(d):
+                            lhs = (1 << col) | sum(((ms[k][t] >> col) & 1) << (d+t)
+                                                  for t in range(h))
+                            equations.append((lhs, (ap[row] >> col) & 1))
+                    solution = solve_binary(equations)
+                    if solution is None:
+                        break
+                    c_rows.append(solution >> d)
+                checked += 1
+                if len(c_rows) != d:
+                    rejected += 1
+                    continue
+                solved += 1
+                s = (0,)*d
+                for p in ps:
+                    s = add(s, p)
+                untouched = [k for k in range(1, g) if ps[k] == (0,)*d]
+                if untouched and mul(a, s) != (0,)*d:
+                    require(signature_dimension([ms[k] for k in untouched], d)
+                            < signature_dimension(ms, d), 'strict signature dimension drop')
+                    drops += 1
+    require(drops > 20 and rejected > 0, 'nonvacuous dimension controls')
+    return {'local_systems': checked, 'consistent': solved,
+            'inconsistent': rejected, 'strict_drop_instances': drops}
+
+
+def static_witness():
+    """Reproduce the binary PR5681 fixture from its written formula."""
+    d, g, n = 4, 31, 128
+    labels = [0] + [v for v in range(1, 16) for _ in range(2)]
+    u = sum(value << ((j+1)*d) for j, value in enumerate(labels))
+    e = tuple(u if (u >> i) & 1 else 0 for i in range(n))
+    fs, gs = [], []
+    for axis in range(d):
+        f, z = [0]*n, [0]*n
+        for j in range(1, g+1):
+            f[axis] ^= 1 << (j*d+axis)
+            z[j*d+axis] ^= 1 << axis
+        fs.append(tuple(f))
+        gs.append(tuple(z))
+    queries = [add(fs[0], e), add(gs[1], e)]
+    queries += [add(fs[i], gs[(i+1) % d]) for i in range(1, d)]
+    k = (0,)*n
+    for z in queries:
+        require(rank(z) == 2 and mul(z, z) == (0,)*n, 'static rank and square')
+        k = add(k, z)
+    a = rank([row for z in queries for row in trans(z)])-2*d
+    b = rank([row for z in queries for row in z])-2*d
+    require(rank(k) == 2*d and a == b == 1, 'static excess dimensions')
+    require(g > 3*b+2, 'new chronological rejection')
+    return {'leaves': g, 'd': d, 'thin_maps': len(queries), 'thin_rank': 10,
+            'extra_image_dimension': a, 'extra_row_dimension': b,
+            'static_checks_pass': True, 'chronology_bound': 3*b+2,
+            'chronology_rejected': True}
+
+
+def commuting_word(g, q):
+    d, n = 2*q, 2*q*(g+1)
+    calls = []
+    for j in range(g):
+        for half in range(2):
+            p, r = [0]*n, [0]*n
+            for c in range(half*q, (half+1)*q):
+                i, k = c, (j+1)*d+c
+                p[i], p[k] = 1 << i, 1 << k
+                r[i] = r[k] = (1 << i) | (1 << k)
+            calls.append({'stage': j, 'p': p, 'r': r})
+        if j % 2 == 0:
+            leaves = [j+1] + ([j+2] if j+1 < g else [])
+            z = [0]*n
+            for leaf in leaves:
+                for c in range(d):
+                    z[c] ^= 1 << (leaf*d+c)
+                    z[leaf*d+c] ^= 1 << c
+            if g % 2 == 0 and j == 0:
+                for c in range(d):
+                    z[c] ^= 1 << c
+            calls.append({'stage': g, 'p': [0]*n, 'r': z})
+    return {'g': g, 'q': q, 'order': list(range(len(calls))), 'calls': calls}
+
+
+def check_contract(word):
+    g, q = word['g'], word['q']
+    d, n = 2*q, 2*q*(g+1)
+    calls = word['calls']
+    require(sorted(word['order']) == list(range(len(calls))), 'call order permutation')
+    projections = [[] for _ in range(g)]
+    for call in calls:
+        j = call['stage']
+        p, r = tuple(call['p']), tuple(call['r'])
+        require(len(p) == len(r) == n and all(0 <= x < 1 << n for x in p+r), 'matrix shape')
+        if j == g:
+            require(not any(p), 'thin stage-free contract')
+            continue
+        require(0 <= j < g, 'stage range')
+        projections[j].append(p)
+        require(mul(p, p) == p and rank(p) == d, 'full projection rank')
+        require(mul(p, r) == mul(r, p) == r and rank(r) == q, 'full query compatibility')
+        t = [0]*n
+        for c in range(d):
+            t[c] = t[(j+1)*d+c] = (1 << c) | (1 << ((j+1)*d+c))
+        require(mul(p, t) == r, 'prescribed stage query')
+    for j, pair in enumerate(projections):
+        require(len(pair) == 2, 'two full stage halves')
+        target = tuple((1 << i) if i < d or (j+1)*d <= i < (j+2)*d else 0
+                       for i in range(n))
+        require(add(*pair) == target and not any(mul(*pair)), 'orthogonal resolution')
+
+
+def laurent_endpoint(word):
+    """Check every Laurent coefficient after exact diagonal normalization."""
+    check_contract(word)
+    g = word['g']
+    n = len(word['calls'][0]['p'])
+    degrees = [[0]*g for _ in range(n)]
+    terms = set()
+    contributions = 0
+    for index in word['order']:
+        call = word['calls'][index]
+        j, p, r = call['stage'], call['p'], call['r']
+        for i, row in enumerate(r):
+            for k in range(n):
+                if (row >> k) & 1:
+                    exponent = tuple(y-x for x, y in zip(degrees[i], degrees[k]))
+                    key = (i, k, exponent)
+                    if key in terms:
+                        terms.remove(key)
+                    else:
+                        terms.add(key)
+                    contributions += 1
+        if j < g:
+            for i, row in enumerate(p):
+                require(row in (0, 1 << i), 'diagonal full-call entry')
+                degrees[i][j] += int(bool(row))
+    d = 2*word['q']
+    for i, degree in enumerate(degrees):
+        require(degree == [int(i < d or i//d == j+1) for j in range(g)], 'pure stage endpoint')
+    require(terms == {(i, i, (0,)*g) for i in range(n)}, 'complete Laurent endpoint')
+    return {'fields': n, 'full_calls': 2*g, 'thin_maps': len(word['calls'])-2*g,
+            'thin_rank': sum(rank(c['r']) for c in word['calls'] if c['stage'] == g),
+            'query_entry_contributions': contributions, 'final_coefficients': len(terms)}
+
+
+def gf4_mul(a, b):
+    value = 0
+    for i in range(2):
+        if (b >> i) & 1:
+            value ^= a << i
+    return value ^ (7 if value & 4 else 0)
+
+
+def apply_binary(a, b):
+    n = len(a)
+    out = []
+    for bits in a:
+        row = [0]*n
+        while bits:
+            bit = bits & -bits
+            source = b[bit.bit_length()-1]
+            row = [x ^ y for x, y in zip(row, source)]
+            bits ^= bit
+        out.append(row)
+    return out
+
+
+def grid_endpoint(word):
+    """Three distinct F4 points decide degree-at-most-two stage polynomials."""
+    check_contract(word)
+    g = word['g']
+    n = len(word['calls'][0]['p'])
+    d = 2*word['q']
+    count = 0
+    for values in product((0, 1, 2), repeat=g):
+        pure = [[int(i == j) for j in range(n)] for i in range(n)]
+        derivative = [[0]*n for _ in range(n)]
+        for index in word['order']:
+            call = word['calls'][index]
+            j, p, r = call['stage'], call['p'], call['r']
+            rb = apply_binary(r, pure)
+            if j == g:
+                derivative = [[a ^ b for a, b in zip(x, y)] for x, y in zip(derivative, rb)]
+            else:
+                pe, pb = apply_binary(p, derivative), apply_binary(p, pure)
+                derivative = [[a ^ gf4_mul(values[j] ^ 1, b) ^ gf4_mul(values[j], c)
+                               for a, b, c in zip(x, y, z)] for x, y, z in zip(derivative, pe, rb)]
+                pure = [[a ^ gf4_mul(values[j] ^ 1, b) for a, b in zip(x, y)]
+                        for x, y in zip(pure, pb)]
+        for i in range(n):
+            value = 1
+            for j in range(g):
+                if i < d or i//d == j+1:
+                    value = gf4_mul(value, values[j])
+            expected = [value if i == k else 0 for k in range(n)]
+            require(pure[i] == derivative[i] == expected, 'full F4 grid endpoint')
+        count += 1
+    return count
+
+
+def endpoint_controls():
+    grid_cases = 0
+    for g in range(1, 4):
+        word = commuting_word(g, 1)
+        laurent_endpoint(word)
+        grid_cases += grid_endpoint(word)
+    old = json.loads((ROOT/'sources/escape-word.json').read_text())
+    old.update(g=3, q=1)
+    grid_cases += grid_endpoint(old)
+    # D^2+D vanishes at both binary points, but not at the third grid point.
+    require(all(gf4_mul(x, x) ^ x == 0 for x in (0, 1)), 'binary alias control')
+    require(gf4_mul(2, 2) ^ 2 != 0, 'extension point rejects alias')
+    word = json.loads((ROOT/'WORD31.json').read_text())
+    receipt = laurent_endpoint(word)
+    rejected = 0
+    for kind in ('deleted', 'moved'):
+        broken = json.loads(json.dumps(word))
+        thin = next(i for i, c in enumerate(broken['calls']) if c['stage'] == word['g'])
+        if kind == 'deleted':
+            broken['calls'][thin]['r'] = [0]*len(broken['calls'][thin]['r'])
+        else:
+            broken['order'].remove(thin)
+            broken['order'].append(thin)
+        try:
+            laurent_endpoint(broken)
+        except ValueError:
+            rejected += 1
+        else:
+            raise ValueError('damaged endpoint accepted')
+    require(rejected == 2, 'endpoint controls')
+    return {'full_grid_assignments': grid_cases, 'binary_alias_rejected': True,
+            'damaged_words_rejected': rejected, 'long_word': receipt}
+
+
+def run():
+    return {'prefix': prefix_controls(), 'dimension_drop': drop_controls(),
+            'static_target': static_witness(), 'endpoints': endpoint_controls()}
diff --git a/research/coupled31-chronology/discovery/coupled_seed.py b/research/coupled31-chronology/discovery/coupled_seed.py
new file mode 100644
index 0000000000..fca9f75e33
--- /dev/null
+++ b/research/coupled31-chronology/discovery/coupled_seed.py
@@ -0,0 +1,150 @@
+"""Search a coupled three-leaf seed with all call orders free.
+
+This search uses necessary coefficients. Each candidate needs full replay.
+Ali Romman with substantial OpenAI Codex assistance. Apache-2.0.
+"""
+import argparse
+import importlib.util
+import json
+from pathlib import Path
+import time
+import z3
+
+ROOT = Path(__file__).resolve().parents[1]
+
+
+def build(pattern, timeout, seed):
+    d, g, n = 4, 3, 16
+    solver = z3.Solver()
+    solver.set(timeout=timeout, random_seed=seed)
+    def zero(size):
+        return [z3.BitVecVal(0, size) for _ in range(size)]
+    def eye(size):
+        return [z3.BitVecVal(1 << i, size) for i in range(size)]
+    def add(a, b):
+        return [x ^ y for x, y in zip(a, b)]
+    def mul(a, b):
+        size = len(a)
+        out = []
+        for row in a:
+            value = z3.BitVecVal(0, size)
+            for k in range(size):
+                value ^= z3.If(z3.Extract(k, k, row) == 1, b[k], 0)
+            out.append(z3.simplify(value))
+        return out
+    def embed(a, b, stage):
+        p, r = zero(n), zero(n)
+        for i in range(d):
+            aa = z3.ZeroExt(n-d, a[i])
+            bb = z3.ZeroExt(n-d, b[i])
+            p[i] = (aa ^ bb) | (bb << (d*(stage+1)))
+            p[d*(stage+1)+i] = bb | ((aa ^ bb) << (d*(stage+1)))
+            r[i] = aa | (aa << (d*(stage+1)))
+            r[d*(stage+1)+i] = r[i]
+        return p, r
+    calls = []
+    for j in range(g):
+        a = [z3.BitVec(f'a{j}_{i}', d) for i in range(d)]
+        b = [z3.BitVec(f'b{j}_{i}', d) for i in range(d)]
+        for x, y in zip(mul(a, a), a):
+            solver.add(x == y)
+        for x, y in zip(add(mul(a, b), mul(b, a)), b):
+            solver.add(x == y)
+        trace = z3.BitVecVal(0, 1)
+        for i in range(d):
+            trace ^= z3.Extract(i, i, a[i])
+        solver.add(trace == 0, z3.Or(*[x != 0 for x in a]),
+                   z3.Or(*[x != (1 << i) for i, x in enumerate(a)]))
+        for half in range(2):
+            p, r = embed(a if not half else add(a, eye(d)), b, j)
+            calls.append((j, p, r))
+    signatures = (0, 1, 1) if pattern == 'pair' else (1, 2, 3)
+    u = sum(v << (d*(j+1)) for j, v in enumerate(signatures))
+    e = [u if (u >> i) & 1 else 0 for i in range(n)]
+    fs, gs = [], []
+    for i in range(d):
+        f, h = [0]*n, [0]*n
+        for j in range(g):
+            f[i] |= 1 << (d*(j+1)+i)
+            h[d*(j+1)+i] = 1 << i
+        fs.append(f)
+        gs.append(h)
+    qs = [[x ^ y for x, y in zip(fs[0], e)],
+          [x ^ y for x, y in zip(gs[1], e)]]
+    qs += [[x ^ y for x, y in zip(fs[i], gs[(i+1) % d])]
+           for i in range(1, d)]
+    for query in qs:
+        calls.append((g, zero(n), [z3.BitVecVal(x, n) for x in query]))
+    positions = [z3.Int(f'pos{i}') for i in range(len(calls))]
+    solver.add(z3.Distinct(positions), *[z3.And(p >= 0, p < len(calls))
+                                       for p in positions])
+    for j in range(g):
+        solver.add(positions[2*j] < positions[2*j+1])
+    for j in range(g):
+        equation = zero(n)
+        for i in (2*j, 2*j+1):
+            tail = zero(n)
+            for k, (_, _, r) in enumerate(calls):
+                if k != i:
+                    tail = add(tail, [z3.If(positions[k] > positions[i], x, 0)
+                                     for x in r])
+            p = calls[i][1]
+            equation = add(equation, add(mul(p, tail), mul(tail, p)))
+        solver.add(*[x == 0 for x in equation])
+    # Every pair of pure stage coefficients must agree.
+    for j in range(g):
+        for k in range(j):
+            eq = zero(n)
+            for i in (2*j, 2*j+1):
+                for h in (2*k, 2*k+1):
+                    before = mul(calls[h][1], calls[i][1])
+                    after = mul(calls[i][1], calls[h][1])
+                    eq = add(eq, [z3.If(positions[i] < positions[h], x, y)
+                                  for x, y in zip(before, after)])
+            ej = add(calls[2*j][1], calls[2*j+1][1])
+            ek = add(calls[2*k][1], calls[2*k+1][1])
+            solver.add(*[x == y for x, y in zip(eq, mul(ej, ek))])
+    return solver, calls, positions
+
+
+def main():
+    parser = argparse.ArgumentParser()
+    parser.add_argument('--pattern', choices=('pair', 'triangle'), required=True)
+    parser.add_argument('--timeout', type=int, default=60000)
+    parser.add_argument('--seed', type=int, default=20261026)
+    parser.add_argument('--models', type=int, default=8)
+    parser.add_argument('--output', type=Path, required=True)
+    args = parser.parse_args()
+    start = time.monotonic()
+    solver, calls, positions = build(args.pattern, args.timeout, args.seed)
+    spec = importlib.util.spec_from_file_location('old_word', ROOT/'sources/old-word-replay.py')
+    replay = importlib.util.module_from_spec(spec)
+    spec.loader.exec_module(replay)
+    records = []
+    for iteration in range(args.models):
+        status = solver.check()
+        row = {'iteration': iteration, 'status': str(status),
+               'seconds': round(time.monotonic()-start, 3)}
+        if status == z3.sat:
+            model = solver.model()
+            order = sorted(range(len(calls)), key=lambda i: model.eval(positions[i], model_completion=True).as_long())
+            numeric = [{'stage': j, 'p': [model.eval(x, model_completion=True).as_long() for x in p],
+                        'r': [model.eval(x, model_completion=True).as_long() for x in r]}
+                       for j, p, r in calls]
+            row.update(order=order, calls=numeric, endpoint=replay.replay(numeric, order))
+            # Do not repeat a necessary-condition false positive.
+            expressions = [x for _, p, r in calls for x in p+r] + positions
+            solver.add(z3.Or(*[x != model.eval(x, model_completion=True) for x in expressions]))
+        elif status == z3.unknown:
+            row['reason'] = solver.reason_unknown()
+        records.append(row)
+        result = {'pattern': args.pattern, 'timeout_ms': args.timeout,
+                  'seed': args.seed, 'z3': z3.get_version_string(), 'records': records}
+        args.output.write_text(json.dumps(result, indent=2)+'\n')
+        print(json.dumps({k: v for k, v in row.items() if k != 'calls'}), flush=True)
+        if status != z3.sat or row['endpoint']['passed']:
+            break
+
+
+if __name__ == '__main__':
+    main()
diff --git a/research/coupled31-chronology/discovery/one-row-three-stage.json b/research/coupled31-chronology/discovery/one-row-three-stage.json
new file mode 100644
index 0000000000..dfd4f26187
--- /dev/null
+++ b/research/coupled31-chronology/discovery/one-row-three-stage.json
@@ -0,0 +1,250 @@
+{
+  "counts": {
+    "total": 19440,
+    "has_one_extra_row": 9864
+  },
+  "examples": [
+    {
+      "flags": [
+        [
+          [
+            0,
+            2
+          ],
+          [
+            1,
+            0
+          ]
+        ],
+        [
+          [
+            0,
+            2
+          ],
+          [
+            1,
+            0
+          ]
+        ],
+        [
+          [
+            0,
+            2
+          ],
+          [
+            1,
+            0
+          ]
+        ]
+      ],
+      "order": [
+        0,
+        0,
+        1,
+        2,
+        2,
+        1
+      ],
+      "signatures": [
+        1,
+        2
+      ],
+      "differences": [
+        [
+          0,
+          0
+        ],
+        [
+          0,
+          2
+        ],
+        [
+          1,
+          0
+        ]
+      ]
+    },
+    {
+      "flags": [
+        [
+          [
+            0,
+            2
+          ],
+          [
+            1,
+            0
+          ]
+        ],
+        [
+          [
+            0,
+            2
+          ],
+          [
+            1,
+            0
+          ]
+        ],
+        [
+          [
+            0,
+            2
+          ],
+          [
+            1,
+            0
+          ]
+        ]
+      ],
+      "order": [
+        0,
+        0,
+        2,
+        1,
+        1,
+        2
+      ],
+      "signatures": [
+        2,
+        1
+      ],
+      "differences": [
+        [
+          0,
+          0
+        ],
+        [
+          1,
+          0
+        ],
+        [
+          0,
+          2
+        ]
+      ]
+    },
+    {
+      "flags": [
+        [
+          [
+            0,
+            2
+          ],
+          [
+            1,
+            0
+          ]
+        ],
+        [
+          [
+            0,
+            2
+          ],
+          [
+            1,
+            0
+          ]
+        ],
+        [
+          [
+            0,
+            2
+          ],
+          [
+            1,
+            0
+          ]
+        ]
+      ],
+      "order": [
+        0,
+        1,
+        0,
+        2,
+        2,
+        1
+      ],
+      "signatures": [
+        1,
+        2
+      ],
+      "differences": [
+        [
+          0,
+          0
+        ],
+        [
+          0,
+          2
+        ],
+        [
+          1,
+          0
+        ]
+      ]
+    },
+    {
+      "flags": [
+        [
+          [
+            0,
+            2
+          ],
+          [
+            1,
+            0
+          ]
+        ],
+        [
+          [
+            0,
+            2
+          ],
+          [
+            1,
+            0
+          ]
+        ],
+        [
+          [
+            0,
+            2
+          ],
+          [
+            1,
+            0
+          ]
+        ]
+      ],
+      "order": [
+        0,
+        1,
+        1,
+        0,
+        2,
+        2
+      ],
+      "signatures": [
+        3,
+        2
+      ],
+      "differences": [
+        [
+          1,
+          0
+        ],
+        [
+          0,
+          2
+        ],
+        [
+          0,
+          0
+        ]
+      ]
+    }
+  ],
+  "seconds": 0.516,
+  "scope": "Necessary projected chronology only; not a full word"
+}
diff --git a/research/coupled31-chronology/discovery/pair-search.json b/research/coupled31-chronology/discovery/pair-search.json
new file mode 100644
index 0000000000..bb6040788c
--- /dev/null
+++ b/research/coupled31-chronology/discovery/pair-search.json
@@ -0,0 +1,14 @@
+{
+  "pattern": "pair",
+  "timeout_ms": 60000,
+  "seed": 20261026,
+  "z3": "5.3.2",
+  "records": [
+    {
+      "iteration": 0,
+      "status": "unknown",
+      "seconds": 60.373,
+      "reason": "timeout"
+    }
+  ]
+}
diff --git a/research/coupled31-chronology/discovery/row_signature_census.py b/research/coupled31-chronology/discovery/row_signature_census.py
new file mode 100644
index 0000000000..9d8752b684
--- /dev/null
+++ b/research/coupled31-chronology/discovery/row_signature_census.py
@@ -0,0 +1,54 @@
+"""Test one extra row at the projected three-stage chronology gate.
+
+This is a necessary-condition census, not a full endpoint search.
+Ali Romman with substantial OpenAI Codex assistance. Apache-2.0.
+"""
+import importlib.util
+import itertools
+import json
+from pathlib import Path
+import time
+
+ROOT = Path(__file__).resolve().parents[1]
+
+
+def run():
+    spec = importlib.util.spec_from_file_location('checker', ROOT/'check.py')
+    c = importlib.util.module_from_spec(spec)
+    spec.loader.exec_module(c)
+    flags = [(a, c.add(a, c.eye(2))) for a in itertools.product(range(4), repeat=2)
+             if c.mul(a, a) == a and c.rank(a) == 1]
+    orders = sorted(set(itertools.permutations((0, 0, 1, 1, 2, 2))))
+    hist = {'total': 0, 'has_one_extra_row': 0}
+    examples = []
+    start = time.monotonic()
+    for fs in itertools.product(flags, repeat=3):
+        for order in orders:
+            seen, events = [0]*3, []
+            for j in order:
+                events.append((j, fs[j][seen[j]]))
+                seen[j] += 1
+            sums = {(j, k): (0, 0) for j in range(3) for k in range(3) if j != k}
+            for i, (j, a) in enumerate(events):
+                for k, b in events[i+1:]:
+                    if j != k:
+                        sums[j, k] = c.add(sums[j, k], c.mul(a, b))
+            differences = [c.add(*[sums[j, k] for k in range(3) if k != j])
+                           for j in range(3)]
+            compatible = []
+            for u, v in itertools.product(range(4), repeat=2):
+                if all(all(row in (0, m) for row in mat)
+                       for mat, m in zip(differences, (u ^ v, v, u))):
+                    compatible.append((u, v))
+            hist['total'] += 1
+            if compatible:
+                hist['has_one_extra_row'] += 1
+                if len(examples) < 4:
+                    examples.append({'flags': fs, 'order': order,
+                                     'signatures': compatible[0], 'differences': differences})
+    return {'counts': hist, 'examples': examples, 'seconds': round(time.monotonic()-start, 3),
+            'scope': 'Necessary projected chronology only; not a full word'}
+
+
+if __name__ == '__main__':
+    print(json.dumps(run(), indent=2))
diff --git a/research/coupled31-chronology/discovery/triangle-search.json b/research/coupled31-chronology/discovery/triangle-search.json
new file mode 100644
index 0000000000..56c85a4e8a
--- /dev/null
+++ b/research/coupled31-chronology/discovery/triangle-search.json
@@ -0,0 +1,14 @@
+{
+  "pattern": "triangle",
+  "timeout_ms": 60000,
+  "seed": 20261026,
+  "z3": "5.3.2",
+  "records": [
+    {
+      "iteration": 0,
+      "status": "unknown",
+      "seconds": 60.279,
+      "reason": "timeout"
+    }
+  ]
+}
diff --git a/research/coupled31-chronology/pricing.py b/research/coupled31-chronology/pricing.py
new file mode 100644
index 0000000000..29688c77c2
--- /dev/null
+++ b/research/coupled31-chronology/pricing.py
@@ -0,0 +1,112 @@
+"""Check exact tariff bounds from the pinned P104 inventory.
+
+Ali Romman with substantial OpenAI Codex assistance. Apache-2.0.
+"""
+from fractions import Fraction
+from functools import lru_cache
+import json
+from pathlib import Path
+
+ROOT = Path(__file__).resolve().parent
+
+
+@lru_cache(None)
+def power_interval(rank, numerator, denominator, bits=42):
+    scale = 1 << bits
+    target = pow(rank, numerator) << (bits*denominator)
+    low, high = 0, rank*scale+1
+    while high-low > 1:
+        mid = (low+high)//2
+        if pow(mid, denominator) <= target:
+            low = mid
+        else:
+            high = mid
+    upper = low if pow(low, denominator) == target else low+1
+    return Fraction(low, scale), Fraction(upper, scale)
+
+
+def interval_sum(profile, saving):
+    power = 1-saving
+    low = high = Fraction(0)
+    for rank, count in profile.items():
+        a, b = power_interval(rank, power.numerator, power.denominator)
+        low += count*(a if count > 0 else b)
+        high += count*(b if count > 0 else a)
+    return low, high
+
+
+def profile(source, leaves, thin_per_port):
+    counts = {int(r): -2*leaves*v for r, v in source['paid'].items()}
+    for r, v in source['helpers'].items():
+        counts[int(r)] = counts.get(int(r), 0)+2*leaves*v
+    counts[8*leaves+1] = counts.get(8*leaves+1, 0)+208
+    counts[9] = counts.get(9, 0)+208*leaves
+    counts[1] -= 104*thin_per_port
+    return {r: c for r, c in sorted(counts.items()) if c}
+
+
+def run():
+    source = json.loads((ROOT/'sources/p104-result.json').read_text())['local']
+    require(sum(int(r)*n for r, n in source['paid'].items())
+            - sum(int(r)*n for r, n in source['helpers'].items()) == 1720,
+            'retained net rank mass')
+    saving = Fraction(8, 3125)
+    t = 1-saving
+    c_profile = {int(r): -2*n for r, n in source['paid'].items()}
+    for r, n in source['helpers'].items():
+        c_profile[int(r)] = c_profile.get(int(r), 0)+2*n
+    c_profile[9] = c_profile.get(9, 0)+208
+    c_low, c_high = interval_sum(c_profile, saving)
+    z_low, z_high = power_interval(257, t.numerator, t.denominator)
+    z_low, z_high = z_low/257, z_high/257
+    slope = (1664*t*z_low+c_low-Fraction(104, 3),
+             1664*t*z_high+c_high-Fraction(104, 3))
+    intercept = (208*z_low*(1+256*saving)-Fraction(1040, 3),
+                 208*z_high*(1+256*saving)-Fraction(1040, 3))
+    require(slope[1] < 0 and intercept[1] < -7, 'uniform negative tangent')
+    bound31 = profile(source, 31, 14)
+    bound127 = profile(source, 127, 88)
+    complete31 = profile(source, 31, 64)
+    require(sum(r*c for r, c in complete31.items()) == -3472, 'complete word linear loss')
+    require(min(r for r, c in complete31.items() if c > 0) >= 8, 'positive rank floor')
+    require(max(r for r, c in complete31.items() if c < 0) <= 7, 'negative rank ceiling')
+    # These signs hold per g. They transfer rejection to larger savings.
+    require(all((r >= 8 if c > 0 else r <= 7) for r, c in c_profile.items() if c),
+            'uniform separated ranks')
+    candidates = []
+    for g, q, thin, alpha in ((13, 8, 5, Fraction(17, 5000)),
+                               (31, 26, 5, Fraction(7, 2000))):
+        lower_rank = 4*q+2*((g-2+2)//3)
+        require(thin*q >= lower_rank, 'candidate rank budget')
+        low, high = interval_sum(profile(source, g, thin), alpha)
+        require(low > 0, 'candidate optimistic margin')
+        candidates.append({'leaves': g, 'coupled_ports': q, 'thin_maps': thin,
+                           'rank_per_thin_map': q, 'minimum_thin_rank': lower_rank,
+                           'saving': str(alpha), 'margin': [str(low), str(high)],
+                           'status': 'UNCONSTRUCTED; COMMUTING FULL PROJECTORS EXCLUDED'})
+    return {'net_rank_mass_per_full_call': 1720,
+            'requested_31_leaf': {'ports': 2, 'requested_thin_rank': 10,
+                                  'minimum_thin_rank': 28,
+                                  'minimum_H1_calls_at_104_ports': 1456,
+                                  'optimistic_signed_profile': bound31},
+            'old_127_leaf': {'ports': 1, 'requested_thin_rank': 10,
+                             'minimum_thin_rank': 88,
+                             'optimistic_signed_profile': bound127},
+            'uniform_one_or_two_port_ceiling': {
+                'saving_rejected': str(saving), 'tangent_leaves': 32,
+                'slope_interval': [str(x) for x in slope],
+                'intercept_interval': [str(x) for x in intercept],
+                'all_integer_leaves_at_least_three': True,
+                'all_larger_savings_rejected': True},
+            'complete_binary_word31': {'full_calls': 62, 'thin_maps': 16,
+                                       'rank_per_thin_map': 8, 'binary_blocks': 52,
+                                       'H1_thin_calls_at_104_ports': 6656,
+                                       'signed_profile': complete31,
+                                       'linear_margin': -3472,
+                                       'all_concave_moments_rejected': True},
+            'unconstructed_targets': candidates}
+
+
+def require(ok, message):
+    if not ok:
+        raise ValueError(message)
diff --git a/research/coupled31-chronology/sources/escape-word.json b/research/coupled31-chronology/sources/escape-word.json
new file mode 100644
index 0000000000..aae71c6ad2
--- /dev/null
+++ b/research/coupled31-chronology/sources/escape-word.json
@@ -0,0 +1,351 @@
+{
+  "source": "Finite search case11, seed3; all matrices over F2",
+  "scope": "Binary normalized operator only; six full calls and eight rank-one thin calls",
+  "case": 11,
+  "order": [
+    1,
+    3,
+    8,
+    11,
+    7,
+    6,
+    12,
+    13,
+    2,
+    9,
+    4,
+    0,
+    10,
+    5
+  ],
+  "status": "sat",
+  "calls": [
+    {
+      "stage": 0,
+      "p": [
+        1,
+        1,
+        4,
+        4,
+        0,
+        0,
+        0,
+        0
+      ],
+      "r": [
+        5,
+        5,
+        5,
+        5,
+        0,
+        0,
+        0,
+        0
+      ]
+    },
+    {
+      "stage": 0,
+      "p": [
+        0,
+        3,
+        0,
+        12,
+        0,
+        0,
+        0,
+        0
+      ],
+      "r": [
+        0,
+        15,
+        0,
+        15,
+        0,
+        0,
+        0,
+        0
+      ]
+    },
+    {
+      "stage": 1,
+      "p": [
+        0,
+        3,
+        0,
+        0,
+        0,
+        48,
+        0,
+        0
+      ],
+      "r": [
+        0,
+        51,
+        0,
+        0,
+        0,
+        51,
+        0,
+        0
+      ]
+    },
+    {
+      "stage": 1,
+      "p": [
+        1,
+        1,
+        0,
+        0,
+        16,
+        16,
+        0,
+        0
+      ],
+      "r": [
+        17,
+        17,
+        0,
+        0,
+        17,
+        17,
+        0,
+        0
+      ]
+    },
+    {
+      "stage": 2,
+      "p": [
+        1,
+        1,
+        0,
+        0,
+        0,
+        0,
+        64,
+        64
+      ],
+      "r": [
+        65,
+        65,
+        0,
+        0,
+        0,
+        0,
+        65,
+        65
+      ]
+    },
+    {
+      "stage": 2,
+      "p": [
+        0,
+        3,
+        0,
+        0,
+        0,
+        0,
+        0,
+        192
+      ],
+      "r": [
+        0,
+        195,
+        0,
+        0,
+        0,
+        0,
+        0,
+        195
+      ]
+    },
+    {
+      "stage": 3,
+      "p": [
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0
+      ],
+      "r": [
+        0,
+        104,
+        0,
+        104,
+        104,
+        0,
+        104,
+        0
+      ]
+    },
+    {
+      "stage": 3,
+      "p": [
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0
+      ],
+      "r": [
+        111,
+        111,
+        111,
+        111,
+        0,
+        0,
+        0,
+        111
+      ]
+    },
+    {
+      "stage": 3,
+      "p": [
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0
+      ],
+      "r": [
+        2,
+        0,
+        2,
+        0,
+        0,
+        2,
+        0,
+        2
+      ]
+    },
+    {
+      "stage": 3,
+      "p": [
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0
+      ],
+      "r": [
+        0,
+        195,
+        0,
+        0,
+        0,
+        0,
+        0,
+        195
+      ]
+    },
+    {
+      "stage": 3,
+      "p": [
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0
+      ],
+      "r": [
+        5,
+        5,
+        5,
+        5,
+        0,
+        0,
+        0,
+        0
+      ]
+    },
+    {
+      "stage": 3,
+      "p": [
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0
+      ],
+      "r": [
+        0,
+        172,
+        172,
+        0,
+        172,
+        0,
+        172,
+        172
+      ]
+    },
+    {
+      "stage": 3,
+      "p": [
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0
+      ],
+      "r": [
+        60,
+        60,
+        60,
+        0,
+        60,
+        0,
+        60,
+        0
+      ]
+    },
+    {
+      "stage": 3,
+      "p": [
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0
+      ],
+      "r": [
+        0,
+        249,
+        249,
+        0,
+        249,
+        0,
+        249,
+        0
+      ]
+    }
+  ],
+  "choices": [
+    5,
+    1,
+    5
+  ]
+}
diff --git a/research/coupled31-chronology/sources/old-word-replay.py b/research/coupled31-chronology/sources/old-word-replay.py
new file mode 100644
index 0000000000..037ed3f765
--- /dev/null
+++ b/research/coupled31-chronology/sources/old-word-replay.py
@@ -0,0 +1,49 @@
+"""Replay the complete binary word with independent stage variables.
+
+Ali Romman with substantial OpenAI Codex assistance. Apache-2.0.
+"""
+import json
+from pathlib import Path
+
+def eye(n):return tuple(1<<i for i in range(n))
+def add(a,b):return tuple(x^y for x,y in zip(a,b))
+def mul(a,b):
+    result=[]
+    for row in a:
+        v=0
+        while row:
+            bit=row&-row;v^=b[bit.bit_length()-1];row^=bit
+        result.append(v)
+    return tuple(result)
+
+def product(a,b):
+    n=len(next(iter(a.values())));zero=(0,)*n;out={}
+    for x,ax in a.items():
+        for y,by in b.items():
+            if x[0]+y[0]>1:continue
+            key=tuple(a+b for a,b in zip(x,y));out[key]=add(out.get(key,zero),mul(ax,by))
+    return {k:v for k,v in out.items() if any(v)}
+
+def replay(calls,order):
+    n=len(calls[0]['p']);g=max(c['stage'] for c in calls);base=(0,)*(g+1);ebit=(1,)+(0,)*g
+    word={base:eye(n)};expected={base:eye(n),ebit:eye(n)};targets=[(0,)*n for _ in range(g)]
+    for c in calls:
+        if c['stage']<g:targets[c['stage']]=add(targets[c['stage']],tuple(c['p']))
+    for s,e in enumerate(targets):
+        d=list(base);d[s+1]=1;expected=product({base:eye(n),tuple(d):e},expected)
+    for i in order:
+        c=calls[i];p=tuple(c['p']);r=tuple(c['r']);gate={base:eye(n),ebit:r}
+        if c['stage']<g:
+            d=list(base);d[c['stage']+1]=1;ed=d.copy();ed[0]=1
+            gate[tuple(d)]=p;gate[tuple(ed)]=r
+        word=product(gate,word)
+    defects={k:add(word.get(k,(0,)*n),expected.get(k,(0,)*n)) for k in word.keys()|expected.keys()}
+    defects={k:v for k,v in defects.items() if any(v)}
+    return {'passed':not defects,'coefficients':len(word),'expected_coefficients':len(expected),'defect_coefficients':len(defects),'first_defect':next(iter(defects.items()),None)}
+
+if __name__=='__main__':
+    import argparse
+    parser=argparse.ArgumentParser();parser.add_argument('--file',type=Path,default=Path(__file__).with_name('ESCAPE-WORD.json'));args=parser.parse_args()
+    data=json.loads(args.file.read_text());records=data if isinstance(data,list) else [data]
+    for record in records:
+        if 'calls' in record:print(json.dumps({'case':record.get('case'),**replay(record['calls'],record['order'])}),flush=True)
diff --git a/research/coupled31-chronology/sources/p104-result.json b/research/coupled31-chronology/sources/p104-result.json
new file mode 100644
index 0000000000..35f75c8555
--- /dev/null
+++ b/research/coupled31-chronology/sources/p104-result.json
@@ -0,0 +1,487 @@
+{
+  "authenticated_inputs": 13,
+  "complete_cost": {
+    "compiler": {
+      "H": 707182000000000,
+      "N": "2^707182000000000",
+      "eta": {
+        "lower": "0.000000136964415492079857925247",
+        "upper": "0.000000136964415492079857925248"
+      },
+      "gap": "1/14000000000",
+      "integer_test_exponent": 50513,
+      "logarithm_power": 1024,
+      "theta": "63/64",
+      "uniform_saving": "21703153/7000000000",
+      "word_materialized": false
+    },
+    "full_forward_calls": 18,
+    "gain": "61/4000000",
+    "kappa": "62009/20000000",
+    "kappa_decimal": ".00310045",
+    "margin_at_previous": {
+      "lower": "3.715057037515374246068780385152",
+      "upper": "3.715057037515374246068780385153"
+    },
+    "negative_controls": {
+      "old_BIT": "Rejected by outer inequalities",
+      "old_cutoff": "Rejected by completion-tail ratio greater than one"
+    },
+    "nonrecursive": {
+      "fixed_scalar_growth_bound": "At most 2^(16*26000) before native factors; a declared constant, not an attained norm.",
+      "local_paid_rank_mass": 6014,
+      "local_routes": 5183000,
+      "local_scalar": 220232,
+      "peak_owned_fields": 1,
+      "raw_budget": 1000000000,
+      "raw_scalar_upper": 184647552,
+      "tensor_native_growth_bound": "Each rank-r Gaussian has row l1 norm at most 2^(rf); use local_paid_rank_mass for the whole word."
+    },
+    "ordinary_BIT": {
+      "admitted": "17/5000",
+      "after_cover": "69/20000",
+      "atom": "1/250",
+      "before_cover": "3459/1000000",
+      "ordinary_saving": "8590884599/2500000000000",
+      "prime_rule": "odd q > 2^26*N^5 avoiding all fixed denominators and rank witnesses"
+    },
+    "outer": {
+      "functional_checks": {
+        "actual_new_BIT_stock_sublinear": "3485999999253/3500000000000",
+        "fixed_halo_explicit_metadata_branch": "269500000001/3500000000",
+        "new_BIT_row_stock_sublinear_in_active_group": "13957999997009/14000000000000",
+        "new_semantic_group_guard": "3/14000000000"
+      },
+      "margins": {
+        "boundary_algebraic_matrix": "607688255956593697/196000000000000000000",
+        "fused_gaussian": "607688255869781091/196000000000000000000",
+        "global_coordinate_router": "1356447/437500000",
+        "group_bit_router": "18990258433430659/6125000000000000000",
+        "group_scans": "6999999999/7000000000",
+        "main_transform": "607688241956593697/196000000000000000000",
+        "microbox_face_corrections": "269500000001/3500000000",
+        "polynomial_products": "13999999999/14000000000",
+        "sparse_union_movement": "80232688255956593697/196000000000000000000"
+      },
+      "parameters": {
+        "a": "1356447/437500000",
+        "b": "8681261/2800000000",
+        "beta": "13999999997/14000000000",
+        "bit_saving": "17/5000",
+        "cap_xi": "178232688241956593697/196000000000000000000",
+        "complex_uniform_saving": "21703153/7000000000",
+        "d": "1/14000000000",
+        "epsilon": "13999999999/14000000000",
+        "kappa": "62009/20000000",
+        "lambda_": "3987598199/4000000000",
+        "lambda_prime": "13956593697/14000000000",
+        "layout": "1/14000000000",
+        "q": "43406303/14000000000",
+        "r0": "1/140000000000",
+        "stop": "1/14000000000",
+        "tested_saving": "6200901/2000000000"
+      },
+      "strict_checks": {
+        "actual_bit_above_effective": "131053/437500000",
+        "actual_logarithmic_gamma_guard": "1/14000000000",
+        "actual_logarithmic_group_guard": "17767311779043406303/98000000000000000000",
+        "alpha_logarithm_absorption": "1/140000000000",
+        "boundary_algebraic_matrix_above_kappa": "55956593697/196000000000000000000",
+        "cap_above_band": "80232688241956593697/196000000000000000000",
+        "cap_below_cell": "17767311744043406303/196000000000000000000",
+        "cap_below_phase_cell_power": "213767311758043406303/196000000000000000000",
+        "complex_above_effective": "1/14000000000",
+        "complex_below_one_over_32": "78818739/2800000000",
+        "complex_endpoint": "1/7000000000",
+        "epsilon_below_one": "1/14000000000",
+        "epsilon_positive": "13999999999/14000000000",
+        "fine_prime_halo_umbrella": "7/2",
+        "fused_gaussian_above_kappa": "55869781091/196000000000000000000",
+        "gaussian_above_main": "6956593697/98000000000000000000",
+        "global_coordinate_router_above_kappa": "1/3500000000",
+        "global_gamma_power_slack": "9/140000000000",
+        "group_below_dimension": "1/7000000000",
+        "group_bit_router_above_kappa": "2183430659/6125000000000000000",
+        "group_gauge_power_slack": "17767311778343406303/98000000000000000000",
+        "group_key_fits_record": "1/7000000000",
+        "group_positive": "13999999997/14000000000",
+        "group_scans_above_kappa": "6978296849/7000000000",
+        "halo_covers_generic_inverse_radius": "5",
+        "improvement_over_user_baseline": "684739790400418148631/500000000000000000000000",
+        "lambda_above_internal": "1/28000000000",
+        "lambda_above_sigma": "3/28000000000",
+        "lambda_above_tau": "1/28000000000",
+        "lambda_prime_above_lambda": "1/28000000000",
+        "lambda_prime_above_leaf": "5591318739/39200000000000000000",
+        "leaf_saving_above_effective": "2791318739/39200000000000000000",
+        "main_transform_above_kappa": "41956593697/196000000000000000000",
+        "membership_arithmetic_fits_record": "1/14000000000",
+        "microbox_face_corrections_above_kappa": "134744574213/1750000000",
+        "microbox_halo_volume": "1106000000003/14000000000",
+        "new_complex_uniform_saving_above_b": "1/14000000000",
+        "new_semantic_guard_with_C1_one": "1/14000000000",
+        "polynomial_products_above_kappa": "13956593699/14000000000",
+        "positive_layout_slack": "1/14000000000",
+        "q_below_effective": "1/14000000000",
+        "q_below_internal": "1/14000000000",
+        "q_below_leaf": "5591318739/39200000000000000000",
+        "q_positive": "43406303/14000000000",
+        "short_record_fallback": "2791318739/2800000000",
+        "source_axis_superpolynomial": "1/14000000000",
+        "sparse_union_movement_above_kappa": "79625000055956593697/196000000000000000000",
+        "stop_below_one": "13999999999/14000000000",
+        "stop_positive": "1/14000000000",
+        "strassen_integer_power_gap": "1951441519231",
+        "support_router_span": "3/14000000000",
+        "theta_positive_polynomial_power": "2",
+        "union_header_fits_record": "3/14000000000",
+        "union_tag_work_per_visit_fits_record": "3/14000000000"
+      }
+    },
+    "paid": {
+      "1": 41214,
+      "16": 416,
+      "2": 8262,
+      "24": 416,
+      "3": 8874,
+      "32": 416,
+      "4": 1656,
+      "40": 416,
+      "48": 416,
+      "5": 3024,
+      "56": 416,
+      "6": 324,
+      "64": 416,
+      "7": 252,
+      "72": 208
+    },
+    "paid_calls": 66726,
+    "previous_open_pr5565": "7713/2500000",
+    "productive_fields": 13048,
+    "query_shears": 15,
+    "rank_mass_surplus": 760,
+    "raw_ceiling": {
+      "derivative_upper": "-741665511346505619707405027113237790341393747177503845065075007/3138550867693340381917894711603833208051177722232017256448",
+      "interval": [
+        "0",
+        "17/5000"
+      ]
+    },
+    "raw_headroom": [
+      "157681/25000000000000",
+      "25229/4000000000000"
+    ],
+    "raw_root_interval": [
+      "77511407681/25000000000000",
+      "12401825229/4000000000000"
+    ],
+    "relative_gain_percent": "7625/15426",
+    "strictification": {
+      "alpha": {
+        "lower": "0.999999994094860604597376680636",
+        "upper": "0.999999994094860604597376680637"
+      },
+      "cutoff": 17,
+      "delta": {
+        "lower": "0.972208404163139976829701213167",
+        "upper": "0.972208404163139976829701213168"
+      },
+      "floor": 18,
+      "gamma": {
+        "lower": "0.967537821927968182942254182934",
+        "upper": "0.967537821927968182942254182935"
+      },
+      "normalized_paid_upper": "5242880004882813/5242880536870912",
+      "power": 1024,
+      "reserve": "1/10000000000",
+      "tail_block": 128,
+      "tail_decay": "39/40",
+      "tail_offset": "1/5",
+      "tested": "6200901/2000000000"
+    },
+    "targets": {
+      "1": 36,
+      "145": 208,
+      "17": 416,
+      "2": 90,
+      "25": 416,
+      "33": 416,
+      "4": 1602,
+      "41": 416,
+      "49": 416,
+      "5": 198,
+      "57": 416,
+      "65": 416,
+      "7": 90,
+      "73": 208,
+      "8": 288,
+      "9": 7416
+    },
+    "whole_idle_jobs": 3120
+  },
+  "limits": [
+    "The five selected framework assumptions and helper-endpoint contract remain.",
+    "No complete rounded multiplier or enormous uniform compiler word is materialized.",
+    "No independent human or proof-kernel acceptance or global optimality is claimed."
+  ],
+  "local": {
+    "endpoint_coefficients": 571536,
+    "helpers": {
+      "1": 2,
+      "2": 5,
+      "4": 89,
+      "5": 11,
+      "7": 5,
+      "8": 16,
+      "9": 412
+    },
+    "owned_lifetimes": 8,
+    "paid": {
+      "1": 2203,
+      "2": 459,
+      "3": 493,
+      "4": 92,
+      "5": 168,
+      "6": 18,
+      "7": 14
+    },
+    "paid_rank_mass": 6014,
+    "peak_owned_fields": 1,
+    "public_inputs": 748,
+    "real_scalar_allowance": 220232
+  },
+  "local_helper_delta": {
+    "1": 2,
+    "2": 5,
+    "4": 6,
+    "5": 6,
+    "7": 5,
+    "8": 2,
+    "9": -13
+  },
+  "native": {
+    "distinct_maps": 2923,
+    "helper_normalizers": 1080,
+    "helpers": {
+      "1": 2,
+      "2": 5,
+      "4": 89,
+      "5": 11,
+      "7": 5,
+      "8": 16,
+      "9": 412
+    },
+    "main_adapters": 208,
+    "proof": "Every signed Pauli image and exact global phase; frame induction at every positive tensor width",
+    "recipe_payload_sha256": "b2e6b538f6152ccf0024ca57381e46b2fe6b75b89504d935009dade234cc3f18",
+    "wrong_phase_rejected": true
+  },
+  "negative_controls": {
+    "contaminated_prefix": "Exact prefix functional; no other input",
+    "contaminated_suffix": "Exact suffix direction; no other output",
+    "extra_helper_credit": "Every productive helper has a native target",
+    "missing_central_translation": "Rejected by the complete Laurent endpoint",
+    "missing_input_basis": "All public and owned rows",
+    "missing_output_basis": "All public and owned rows",
+    "unearned_whole_target": "All public and owned exits",
+    "unpaid_gaussian": "Complete field and call counts"
+  },
+  "physical_caller": {
+    "central_square_columns": 53248,
+    "full_calls": 18,
+    "main_targets": {
+      "145": 208,
+      "17": 416,
+      "25": 416,
+      "33": 416,
+      "41": 416,
+      "49": 416,
+      "57": 416,
+      "65": 416,
+      "73": 208
+    },
+    "new_paid_jobs": 0,
+    "physical_contacts": 7488,
+    "ports": 104,
+    "route_payload_sha256": "7f4f0ca768ffbc4f93dcff575592c4c67411f46b4289e45b73f715a2ad47b142",
+    "tensor_rule": "Factorwise native maps; scalar signs stay outside the tensor power.",
+    "whole_idle_jobs": {
+      "16": 416,
+      "24": 416,
+      "32": 416,
+      "40": 416,
+      "48": 416,
+      "56": 416,
+      "64": 416,
+      "72": 208
+    },
+    "wrapper_sites": [
+      1,
+      3,
+      5,
+      7,
+      9,
+      11,
+      13,
+      15,
+      17,
+      19,
+      21,
+      23,
+      25,
+      27,
+      29,
+      31
+    ]
+  },
+  "published_source_delta": {
+    "helpers": {
+      "8": -1,
+      "9": 1
+    },
+    "paid": {
+      "1": -8,
+      "3": -1,
+      "4": -2,
+      "5": 4
+    }
+  },
+  "separate_columns": {
+    "arithmetic": "Dense integer rows with binary denominators; reverse chronology",
+    "coefficients": 571536,
+    "columns": 756
+  },
+  "splits": [
+    {
+      "cut_after": 13775,
+      "events": 25155,
+      "pivot": 648,
+      "public": 736,
+      "ranks": [
+        5,
+        4
+      ]
+    },
+    {
+      "cut_after": 13765,
+      "events": 25177,
+      "pivot": 592,
+      "public": 737,
+      "ranks": [
+        5,
+        4
+      ]
+    },
+    {
+      "cut_after": 13706,
+      "events": 25195,
+      "pivot": 664,
+      "public": 738,
+      "ranks": [
+        5,
+        4
+      ]
+    },
+    {
+      "cut_after": 13673,
+      "events": 25217,
+      "pivot": 680,
+      "public": 739,
+      "ranks": [
+        5,
+        4
+      ]
+    },
+    {
+      "cut_after": 12526,
+      "events": 25253,
+      "pivot": 259,
+      "public": 740,
+      "ranks": [
+        4,
+        5
+      ]
+    },
+    {
+      "cut_after": 12448,
+      "events": 25285,
+      "pivot": 432,
+      "public": 741,
+      "ranks": [
+        4,
+        5
+      ]
+    },
+    {
+      "cut_after": 13961,
+      "events": 25299,
+      "pivot": 722,
+      "public": 742,
+      "ranks": [
+        2,
+        7
+      ]
+    },
+    {
+      "cut_after": 13647,
+      "events": 25317,
+      "pivot": 14,
+      "public": 743,
+      "ranks": [
+        2,
+        7
+      ]
+    },
+    {
+      "cut_after": 13648,
+      "events": 25335,
+      "pivot": 76,
+      "public": 744,
+      "ranks": [
+        2,
+        7
+      ]
+    },
+    {
+      "cut_after": 13648,
+      "events": 25369,
+      "pivot": 345,
+      "public": 745,
+      "ranks": [
+        2,
+        7
+      ]
+    },
+    {
+      "cut_after": 13637,
+      "events": 25419,
+      "pivot": 573,
+      "public": 746,
+      "ranks": [
+        2,
+        7
+      ]
+    },
+    {
+      "cut_after": 12078,
+      "events": 25445,
+      "pivot": 405,
+      "public": 747,
+      "ranks": [
+        8,
+        1
+      ]
+    },
+    {
+      "cut_after": 12610,
+      "events": 25475,
+      "pivot": 225,
+      "public": 748,
+      "ranks": [
+        8,
+        1
+      ]
+    }
+  ],
+  "status": "PASS_P104_HELPER_SUBSPACE_KAPPA_00310045"
+}
diff --git a/research/coupled31-chronology/sources/pr5673-proof.md b/research/coupled31-chronology/sources/pr5673-proof.md
new file mode 100644
index 0000000000..e3800fbcf1
--- /dev/null
+++ b/research/coupled31-chronology/sources/pr5673-proof.md
@@ -0,0 +1,231 @@
+# A connector must distinguish leaves in both directions
+
+This is a theorem about finite matrices in characteristic two. It is not a
+native multiplication theorem. No new kappa is established.
+
+## 1. Contract
+
+Let the main space be `W_0 + W_1 + ... + W_g`, with each block of dimension
+`d > 0`. Block zero is the core. Stage j acts on the core and leaf j. On these
+two blocks put
+
+```text
+E_j = diag(I,I),                 T_j = [[I,I],[I,I]].
+```
+
+Thus `T_j^2=0`. Stage j can have any finite number of full calls. Their
+projectors `P_jh` form an orthogonal resolution of `E_j`, and commute with
+`T_j`. Their query maps are `R_jh=P_jh T_j`. There is no bound on the number
+of stages, the number of splits, the block dimension, or their call order.
+
+In the normalized ring `F[e,D_1,...,D_g]/(e^2)`, a full call is
+
+```text
+(I+(D_j+1)P_jh)(I+e R_jh).
+```
+
+A thin call is `I+e Z`. The proof permits arbitrary matrices Z. In particular,
+it does not impose a rank, a square-zero condition, or a count on thin calls.
+The required word is `Q product_j (I+(D_j+1)E_j)`, where `Q=1+e`.
+The main fields are independent. All D variables remain independent.
+
+This is the star stage contract from PR5631/5649, with arbitrary stage splits
+and a much larger connector class. It excludes no general complex or dyadic
+caller. It also excludes no caller with a different grouped stage map, a
+nonfull stage, a different output, or paid idle multipliers.
+
+## 2. The one-sided theorem
+
+**Theorem.** A successful word cannot have three distinct leaves whose block
+columns are identical in every thin matrix Z. It also cannot have three
+leaves whose block rows are identical in every thin matrix Z.
+
+Only one of these two equalities is needed for exclusion. Thus changing the
+forward split, adding more queries, increasing their rank, or adding more
+stage splits cannot repair connectors which only see the sum of the leaves.
+
+### Local form and chronology
+
+Commutation with T and the local support give
+
+```text
+P_jh = [[A_jh+B_jh, B_jh], [B_jh, A_jh+B_jh]],
+R_jh = [[A_jh, A_jh], [A_jh, A_jh]].
+```
+
+The A matrices form an orthogonal resolution of I. In particular,
+`A_jh A_jk=0` for `h!=k`, `A_jh^2=A_jh`, and `sum_h A_jh=I`.
+These statements do not require the A matrices of different stages to commute.
+The orthogonal-resolution hypothesis also follows from full rank use: if the
+projectors sum to E and their ranks sum to rank E, their images form a direct
+sum. Each projector is then the corresponding coordinate projection on that
+sum. Stages with a rank deficit are a different contract.
+
+Set all D variables to one. The coefficient of e requires `sum R=I`, including
+thin queries. The mixed coefficient of `e(D_j+1)` then requires
+
+```text
+sum_(full call i in stage j) [P_i, sum_(calls l later than i) R_l] = 0.   (1)
+```
+
+This necessary condition follows by expanding the literal ordered word.
+No full-word converse is used.
+
+For distinct selected stages j,k define
+
+```text
+S_jk = sum_(call (j,h) before call (k,l)) A_jh A_kl.                    (2)
+```
+
+In (1), add block rows zero and j, and read block column k. The contribution
+of good calls is S_jk. The contribution of a later thin map Z is
+`A_jh (Z_0k+Z_jk)`. If the three selected leaf columns of every Z are equal,
+this thin contribution is the same for the two choices of k other than j.
+Consequently
+
+```text
+S_jk = S_jl    for each permutation (j,k,l) of the three stages.        (3)
+```
+
+Other good stages do not contribute to these blocks. Thus the argument
+restricts to any three selected leaves in a larger star.
+
+### The prefix determinant argument
+
+Read just the calls of the three selected stages in their original order.
+Before a call with projection A from stage j, let P_k be the sum of the A
+projections already read from stage k. Left multiplication of (2) by A and
+orthogonality within stage j give
+
+```text
+A S_jk = A(I+P_k).
+```
+
+Equation (3) gives `A P_k=A P_l`. Also `A P_j=0`. Therefore, for the total
+prefix `S=P_0+P_1+P_2`, every update satisfies
+
+```text
+A S=0.                                                               (4)
+```
+
+Introduce an independent central variable x. Equation (4) implies
+
+```text
+(I+xA)(I+xS) = I+x(S+A).
+```
+
+Starting at S=0, the product of all these factors is `I+xS_final`.
+At the end, `S_final=I+I+I=I`. On the other hand, an idempotent of rank r
+has determinant `det(I+xA)=(1+x)^r`. Each stage is a resolution of I,
+so its ranks sum to d. Taking determinants now gives
+
+```text
+(1+x)^(3d) = det(product(I+xA)) = det((1+x)I) = (1+x)^d.
+```
+
+This is impossible for positive d. This proves the column statement for
+all finite dimensions and all finite split counts. It uses no list of orders.
+Transpose all matrices and reverse the order to obtain the row statement.
+
+## 3. A paid-rank and finite-field cut
+
+Assume g is at least three. The sum of the good queries is
+`sum T_j`. Hence the thin sum must be
+
+```text
+K = I+sum T_j = epsilon E_core + F+G,
+epsilon = (g+1) mod 2,
+F: each leaf -> core by I,       G: core -> each leaf by I.
+```
+
+The rank of K is `2d`. Its column space is the core plus the diagonal leaf
+space, and its row space is the core plus the leaf-sum space.
+
+Let U be the span of all thin images, and V the span of all thin row spaces.
+Write `dim U=2d+a`, `dim V=2d+b`, and let R be the sum of thin ranks.
+Factor each thin map at its actual rank, then concatenate those factors.
+Sylvester's rank inequality gives
+
+```text
+R >= dim U + dim V - rank K = 2d+a+b.                                 (5)
+```
+
+Both a and b must be positive. Otherwise one side of the one-sided theorem
+applies to all leaves. In particular **R >= 2d+2**, over any field of
+characteristic two.
+
+For a finite field of size s, choose b additional row basis vectors beyond
+the row space of K. Their restriction to one leaf is a `b by d` matrix.
+There are at most `s^(bd)` such signatures. Equal signatures imply equal
+leaf columns for every thin map. Each signature can occur at most twice.
+The column-space argument is the same. Therefore
+
+```text
+g <= 2 s^(bd),     g <= 2 s^(ad),
+R >= 2d + 2 h,     h = min {z >= 0 : g <= 2 s^(dz)}.                   (6)
+```
+
+This is a necessary condition, not a construction or a sufficient test.
+It concerns a common field and a fixed block basis. For one-port binary
+copies, d=2. It gives R>=6 for g=3..8, R>=8 for g=9..32, and R>=10 for
+g=33..128. For coupled ports, use the actual larger d. Do not apply the
+one-port bound per port after ports have been coupled.
+
+At g=31, five rank-one connectors fail this cut. Five rank-two connectors
+with d=4 meet it exactly: `2d+2=10`. Thus coupled pairs of ports are a precise
+remaining target. The cut does not prove that such a word exists.
+
+## 4. Constructive escape and its price
+
+`ESCAPE-WORD.json` contains a complete eight-field binary word. It has three
+paired stages, six full calls with rank-one queries, and eight rank-one thin calls.
+Every local projector, query rank and compatibility relation is checked.
+The verifier retains every exponent of each independent D variable and
+checks every coefficient of the full word. It also checks that the thin
+maps distinguish all three leaves in both directions.
+
+Thus the changed connector contract has a real positive operator. It does
+not give a profitable supplier. At the pinned PR5599 tariff, 104 copies have
+complete optimistic moment coefficients
+
+```text
+208 H25 + 624 H9 + 6 helpers - 6 paid - 832 H1.
+```
+
+Its linear margin is -336. After common ranks are cancelled, all positive
+terms have rank at least eight and all negative terms have rank at most
+seven. For every `0<=t<=1`, the margin is at most `8^(t-1)*(-336)<0`.
+Additional physical adapters cannot repair that price.
+
+The new cut still leaves a useful target. At g=127, ten thin queries are
+allowed by (6) for one-port binary copies. The optimistic PR5599 profile
+has a positive exact margin at saving `7/2000=.0035`, and its numerical
+crossing is about .00364156. **No 127-leaf word is supplied.** This screen
+also does not establish a characteristic-zero lift, native frame adapters,
+compiler admission, an outer transfer, or a new kappa.
+
+A smaller coupled target is g=31 with five rank-two connectors per two-port
+block. Its optimistic bill, scaled to 104 ports, also has positive exact
+margin at .0035. It needs only 62 full calls. No such word is supplied either.
+Coupling physical ports can require new frame maps; their costs must be
+charged before this screen can become a supplier claim.
+
+## 5. Evidence and limits
+
+The main result is a written finite-dimensional matrix proof. The checker
+also verifies fifteen exact free-algebra certificates for the paired-stage
+case, without a solver or the discovery implementation. Matrix and formal
+polynomial controls test the chronology extraction, restriction to three
+leaves, prefix factorization and determinant formula. They support the
+proof; finite tests alone do not prove its unbounded statements.
+
+The discovery archive retains changed C,D splits, arbitrary aggregate maps,
+extra thin queries, a different cycle frame, and three-/four-way splits.
+Timeouts remain unknown. Passing first mixed coefficients is not passing the
+full word. No proof assistant or independent human acceptance is claimed.
+
+Credit Cinna Davis's controller and simplex work; PR5617/5631/5649 for the
+query-frame and fixed-star line; and the complete P104/native/helper/compiler
+lineage for the tariff. This proof, cut, search and escape word were prepared
+for Ali Romman with substantial OpenAI Codex assistance. No subagents.
+New work is Apache-2.0.
diff --git a/research/coupled31-chronology/sources/pr5681-proof.md b/research/coupled31-chronology/sources/pr5681-proof.md
new file mode 100644
index 0000000000..e3be1bb272
--- /dev/null
+++ b/research/coupled31-chronology/sources/pr5681-proof.md
@@ -0,0 +1,169 @@
+# The static connector rank cut is sharp
+
+This package checks the proof in the retained PR5673 source. It also gives
+new static witnesses at its rank lower bound. A static witness gives the
+thin query maps, their sum, ranks and leaf signatures. It does not give a
+chronological word. No new kappa follows.
+
+## 1. Audit of the source implication
+
+Use the source contract, over a field of characteristic two. Each stage j
+has full projectors P_jh supported on the core and leaf j. They give an
+orthogonal resolution of that support. They commute with the local query
+T_j, whose four blocks are I. Their queries are R_jh=P_jh T_j.
+A thin map Z has no stage variable. All stage variables are independent.
+The target is the stated product of full stages, times the scalar 1+e,
+where e squared is zero. These hypotheses are required.
+
+Put x_j=D_j+1. Work first in the quotient which also sets x_j squared to
+zero and every other x to zero. This quotient supplies necessary conditions
+only. Read the literal calls in application order. If R_< and R_> are the
+sums before and after a full call (P,R), the coefficient of e x_j contributed
+by that call is
+
+```text
+P R_< + R_> P + P R.
+```
+
+The coefficient of e first gives sum R=I. Since P R=R, the expression becomes
+`P + P R_> + R_> P`. Summing the stage projectors gives E_j. The target has
+e x_j coefficient E_j. Thus its defect is exactly the sum of commutators
+`P R_> + R_> P`. No pure higher coefficient or endpoint converse is assumed.
+
+Commutation with T gives P=[[A+B,B],[B,A+B]]. The idempotent and orthogonality
+relations imply that the A matrices in each stage are orthogonal idempotents
+with sum I. Add block rows core and j, then take leaf column k, where k!=j.
+The right product R_> P has no such column. A later stage k contributes
+A_jh A_kl. Other full stages contribute zero. A thin map contributes
+`A_jh (Z_0k+Z_jk)`. This proves the source restriction even when there are
+other leaves, noncommuting stage flags and arbitrary blocks outside the
+chosen three columns.
+
+If three leaf columns of every thin map agree, the thin contribution does
+not depend on which of the other two leaves is k. Therefore S_jk=S_jl,
+where S_jk is the sum of A_jh A_kl for jh before kl. For a given call A_jh,
+orthogonality within j gives `A_jh S_jk=A_jh(I+P_k)`, with P_k the earlier
+prefix of stage k. Thus A_jh P_k=A_jh P_l. Its own earlier prefix is
+annihilated. The total prefix S obeys A_jh S=0.
+
+For a formal central variable x, this gives
+`(I+xA)(I+xS)=I+x(A+S)` at every step. The product of factors is therefore
+`(1+x)I`, since the three final stage prefixes each equal I. Each idempotent
+of rank r has determinant (1+x)^r: its image and kernel give a direct sum
+basis. Each stage contributes total rank d. The same determinant would have
+to be both (1+x)^(3d) and (1+x)^d. These polynomials have different degrees
+for d>0. This is a contradiction over every field of characteristic two.
+Transpose and reverse the whole word for the row version. The target is
+preserved by this operation because its full stage supports commute.
+
+No mathematical gap was found in this implication under these hypotheses.
+The source does not exclude a different stage support, an incomplete stage,
+a different target or a characteristic-zero caller. This audit does not
+supply human review or a proof-assistant theorem.
+
+## 2. Rank cut and its exact static realization
+
+Let F have q elements, where q is a power of two. Let d,h be positive
+integers. Put
+
+```text
+g = 2 q^(d h) - 1,        W = F^d + (F^d)^g.
+```
+
+Index the leaves by all h-by-d matrices M over F: use the zero matrix once
+and every nonzero matrix twice. On W, define F_i to send coordinate i of
+every leaf to core coordinate i. Define G_i to send core coordinate i to
+coordinate i of every leaf. Both have rank one and square zero. Put
+
+```text
+K = sum_i (F_i+G_i).
+```
+
+This is the required thin sum for odd g. It has rank 2d. Its image consists
+of the core and the diagonal leaf space. Its row space consists of the core
+and the leaf-sum space.
+
+For 1<=a<=h, let u_a have zero core and leaf block equal to row a of M,
+viewed as a column. Let v_a have zero core and the same leaf entries as a
+row. Put E_a=u_a v_a. Each E_a has rank one. Every nonzero signature occurs
+twice, so
+
+```text
+sum_leaf u_a = 0,      sum_leaf v_a = 0,      v_a u_b = 0
+```
+
+for all a,b. Consequently E_a E_b=0; F_i E_a=E_a F_i=0; and
+G_i E_a=E_a G_i=0. In particular all these maps have square zero.
+Take the following thin maps, with both copies of each E_a charged:
+
+```text
+F_0,...,F_(d-1), G_0,...,G_(d-1), E_1,E_1,...,E_h,E_h.
+```
+
+Their sum is K and their total rank is exactly 2d+2h. Their image span and
+row span each have dimension 2d+h. To see independence of the extra h
+vectors modulo the diagonal space, first inspect the zero-signature leaf.
+It forces any diagonal component to vanish. Then inspect signatures with
+one nonzero matrix entry; these force each coefficient of u_a to vanish.
+The row proof is the same.
+
+Two leaves have the same block columns in all thin maps exactly when their
+matrices M agree. Indeed the F/G columns are constant across leaves, and
+E_a recovers row a of M because u_a is nonzero. The row statement uses v_a.
+Thus there are exactly q^(dh) signatures; each occurs once or twice.
+No forbidden triple occurs.
+
+The source Sylvester bound is R>=2d+a+b, with extra image and row dimensions
+a,b. Both equal h here, so equality holds. The source signature bound asks
+for the least z such that g<=2q^(dz). For this g, that least z is h.
+Thus **both bounds are attained together by square-zero maps**, for every
+q,d,h above. This proves sharpness of these static bounds at an unbounded
+family of sizes. It does not prove sharpness for complete chronological
+words.
+
+## 3. Five rank-two maps for the 31-leaf target
+
+When h=1 and d>=2, write E=E_1 and group the charged rank-one maps as
+
+```text
+Z_0 = F_0 + E,
+Z_1 = G_1 + E,
+Z_(i+1) = F_i + G_((i+1) mod d),    1<=i<d.
+```
+
+Each has rank two. The two images and the two row directions are independent.
+All squares are zero. For the last group, the two indices differ, so both
+cross products vanish. The sum is still K. The union of images and row
+spaces still has dimension 2d+1: E is separate from the core/diagonal spaces,
+and each rank-two image and row space contains both rank-one directions.
+The first map detects the entire leaf column signature, and the second
+detects the entire row signature. Therefore all static bounds still pass.
+
+For F2,d=4,h=1 this gives **31 leaves and five rank-two maps**, with total
+thin rank ten. For F2,d=2,h=3, the ungrouped construction gives **127 leaves
+and ten rank-one maps**. These are exactly the two source search targets.
+Their static query maps now have an explicit formula. We do not replay or
+admit their source optimistic tariff, and we do not supply either word.
+A rejection of these targets must use an additional condition, such as the
+chronology equations or the complete polynomial endpoint.
+
+## 4. Verification boundary
+
+The separate checker imports no source code or certificates. It multiplies
+literal factors in the truncated polynomial ring and compares the extracted
+blocks in 72 words, with 55 noncommuting stage flags and 48 words that have
+external stages. It checks all 19,440 ordered three-stage rank-one flag words
+in dimension two over F2. Every word fails the prefix condition. A formal
+polynomial determinant computation is separate from that census.
+
+Six full static matrix fixtures include both target sizes and an F4 case.
+The F4 matrices use the exact regular representation from x^2+x+1; binary
+ranks are twice the stated F4 ranks. Each fixture checks every matrix entry,
+square, rank, sum and signature. Deleted queries and added identity maps are
+rejected. These finite checks support the written general proof; they do
+not replace it. No native lift, frame adapters, ownership/tape bill, complete
+caller or kappa improvement is established.
+
+Credit the retained PR5673 proof and its source lineage for the obstruction
+and rank bounds. The sharpness construction and separate audit code are new
+here. Ali Romman with substantial OpenAI Codex assistance. Apache-2.0.
diff --git a/research/coupled31-chronology/test_verify.py b/research/coupled31-chronology/test_verify.py
new file mode 100644
index 0000000000..a55359596b
--- /dev/null
+++ b/research/coupled31-chronology/test_verify.py
@@ -0,0 +1,57 @@
+#!/usr/bin/env python3
+"""Reject damaged source, word, proof and receipt copies.
+
+Ali Romman with substantial OpenAI Codex assistance. Apache-2.0.
+"""
+import json
+from pathlib import Path
+import shutil
+import subprocess
+import sys
+import tempfile
+
+PACKAGE = Path(__file__).resolve().parent
+
+
+def main():
+    controls = []
+    for kind in ('source', 'query', 'order', 'proof', 'receipt'):
+        with tempfile.TemporaryDirectory(prefix='coupled31-control-') as directory:
+            target = Path(directory)/'package'
+            shutil.copytree(PACKAGE, target, ignore=shutil.ignore_patterns('__pycache__'))
+            if kind == 'source':
+                path = target/'sources/p104-result.json'
+                path.write_text(path.read_text()+' ')
+            elif kind in ('query', 'order'):
+                path = target/'WORD31.json'
+                word = json.loads(path.read_text())
+                thin = next(i for i, c in enumerate(word['calls']) if c['stage'] == word['g'])
+                if kind == 'query':
+                    word['calls'][thin]['r'] = [0]*len(word['calls'][thin]['r'])
+                else:
+                    word['order'].remove(thin)
+                    word['order'].append(thin)
+                path.write_text(json.dumps(word))
+            elif kind == 'proof':
+                path = target/'PROOF.md'
+                path.write_text(path.read_text().replace('g <= L b+2', 'g <= L b+3'))
+            else:
+                path = target/'RESULT.json'
+                value = json.loads(path.read_text())
+                value['pricing']['requested_31_leaf']['minimum_thin_rank'] = 10
+                path.write_text(json.dumps(value))
+            process = subprocess.run([sys.executable, '-I', '-O', '-B', str(target/'verify.py')],
+                                     cwd='/', capture_output=True, text=True, timeout=30)
+            if process.returncode == 0:
+                raise ValueError('damaged package accepted: '+kind)
+            expected = {'source': 'source hash', 'query': 'complete Laurent endpoint',
+                        'order': 'complete Laurent endpoint', 'proof': 'saved result differs',
+                        'receipt': 'saved result differs'}[kind]
+            if expected not in process.stderr:
+                raise ValueError('wrong failure for '+kind+': '+process.stderr)
+            controls.append(kind)
+    print('PASS_COUPLED31_CORRUPTION_CONTROLS '+json.dumps(controls))
+
+
+if __name__ == '__main__':
+    main()
diff --git a/research/coupled31-chronology/verify.py b/research/coupled31-chronology/verify.py
new file mode 100644
index 0000000000..7b69c35c50
--- /dev/null
+++ b/research/coupled31-chronology/verify.py
@@ -0,0 +1,65 @@
+#!/usr/bin/env python3
+"""Replay the chronological connector theorem package without a solver.
+
+Ali Romman with substantial OpenAI Codex assistance. Apache-2.0.
+"""
+import hashlib
+import importlib.util
+import json
+from pathlib import Path
+import sys
+
+PACKAGE = Path(__file__).resolve().parent
+PREFIX = 'research/coupled31-chronology/'
+
+
+def require(ok, message):
+    if not ok:
+        raise ValueError(message)
+
+
+def load(name):
+    spec = importlib.util.spec_from_file_location('coupled31_'+name, PACKAGE/(name+'.py'))
+    module = importlib.util.module_from_spec(spec)
+    spec.loader.exec_module(module)
+    return module
+
+
+def run():
+    sources = json.loads((PACKAGE/'SOURCES.json').read_text())
+    authenticated = 0
+    for row in sources['files']:
+        if row['usage'] == 'input':
+            require(row['path'].startswith(PREFIX), 'package-local source contract')
+            path = PACKAGE/row['path'][len(PREFIX):]
+            require(hashlib.sha256(path.read_bytes()).hexdigest() == row['sha256'],
+                    'source hash: '+row['path'])
+            authenticated += 1
+    checker, prices = load('check'), load('pricing')
+    evidence = checker.run()
+    price = prices.run()
+    endpoint = evidence['endpoints']['long_word']
+    invoice = price['complete_binary_word31']
+    require(endpoint['thin_rank']*invoice['binary_blocks']
+            == invoice['H1_thin_calls_at_104_ports'], 'endpoint/invoice thin rank')
+    require(endpoint['full_calls'] == invoice['full_calls']
+            and endpoint['thin_maps'] == invoice['thin_maps'], 'endpoint/invoice call counts')
+    return {'status': 'PASS_CHRONOLOGICAL_CONNECTOR_RANK_BOUND',
+            'scope': 'Paired full star stages over characteristic two; arbitrary thin maps',
+            'authenticated_inputs': authenticated,
+            'proof_sha256': {name: hashlib.sha256((PACKAGE/name).read_bytes()).hexdigest()
+                             for name in ('PROOF.md', 'COST.md', 'COMMUTING.md')},
+            'evidence': evidence, 'pricing': price,
+            'limits': ['No kappa increase', 'No native lift or complete physical caller',
+                       'Written universal proofs with finite controls; no proof-assistant acceptance',
+                       'Two finite seed solver timeouts remain unknown']}
+
+
+if __name__ == '__main__':
+    result = run()
+    encoded = json.dumps(result, indent=2, sort_keys=True)+'\n'
+    if '--write-result' in sys.argv:
+        (PACKAGE/'RESULT.json').write_text(encoded)
+    else:
+        require((PACKAGE/'RESULT.json').read_text() == encoded, 'saved result differs')
+    print(encoded, end='')

From f9a3bc99b186b2e08a855e260f2f904ef33821c9 Mon Sep 17 00:00:00 2001
From: Ali Romman <226877337+aliromman@users.noreply.github.com>
Date: Mon, 26 Oct 2026 10:20:51 -0500
Subject: [PATCH 2/2] Bind retained source copies and record exact replay
 evidence

---
 research/coupled31-chronology/COST.md         |  5 ++
 research/coupled31-chronology/PROOF.md        |  5 ++
 research/coupled31-chronology/README.md       |  2 +
 research/coupled31-chronology/RESULT.json     | 10 +++-
 research/coupled31-chronology/SOURCES.json    | 52 ++++++++++++++++---
 research/coupled31-chronology/VALIDATION.json | 20 +++++++
 research/coupled31-chronology/pricing.py      |  5 ++
 research/coupled31-chronology/verify.py       |  4 ++
 8 files changed, 95 insertions(+), 8 deletions(-)
 create mode 100644 research/coupled31-chronology/VALIDATION.json

diff --git a/research/coupled31-chronology/COST.md b/research/coupled31-chronology/COST.md
index 68ad4afe0f..d8fe10ffed 100644
--- a/research/coupled31-chronology/COST.md
+++ b/research/coupled31-chronology/COST.md
@@ -68,6 +68,11 @@ margin. Consequently every saving from .00256 through 1 is rejected.
 This is a family-wide tariff ceiling, not a root search on a finite range.
 In particular this family cannot reach selected kappa .00310045.
 
+The two smaller positive leaf counts also lose. At g=1 or g=2, the static
+rank K=2d already requires R/q>=4. Their linear margins are -112 and -16.
+Separated ranks then reject every concave moment. Thus the stated ceiling
+covers every positive leaf count, not only the long callers.
+
 For the requested g=31,q=2 case, R>=28. Its 104-port bill therefore needs
 at least 1456 rank-one thin queries, instead of the proposed 520. For the
 old g=127,q=1 case, R>=88 instead of ten. Both complete optimistic profiles
diff --git a/research/coupled31-chronology/PROOF.md b/research/coupled31-chronology/PROOF.md
index 641598c096..4233ec2237 100644
--- a/research/coupled31-chronology/PROOF.md
+++ b/research/coupled31-chronology/PROOF.md
@@ -175,6 +175,11 @@ matrices, call order, or compatible paired stage projections can give the
 required endpoint. The PR5681 static maps remain valid static maps; they
 cannot be completed to this chronological word.
 
+Allowing arbitrarily many nonzero stage splits does not repair this target.
+At d=4, every nonzero query has rank at least one, so L<=5. Even this weaker
+contract requires `R>=8+2 ceil(29/5)=20`, still more than ten. The bound 28
+uses the requested paired rank-two full queries; the bound 20 does not.
+
 For g=127 and q=1, the bound is a,b>=42 and **R>=88**, instead of the ten
 rank-one connectors in the earlier target. It is excluded for the same reason.
 
diff --git a/research/coupled31-chronology/README.md b/research/coupled31-chronology/README.md
index 357d9898ac..6ee114965a 100644
--- a/research/coupled31-chronology/README.md
+++ b/research/coupled31-chronology/README.md
@@ -4,6 +4,7 @@
 cannot have the full endpoint under the paired-star contract. Its thin rank
 must be at least 28, rather than ten. The new proof also excludes the earlier
 127-leaf, ten-rank-one target, whose thin rank must be at least 88.
+Even arbitrarily fine nonzero stage splits at d=4 need thin rank at least 20.
 
 The main result is a linear information bound for every finite dimension and
 every characteristic-two field. If b is the extra row dimension in the thin
@@ -21,6 +22,7 @@ Those static witnesses remain valid. Chronology prevents their completion.
 concave tangent proves that every paired-star caller with one or two coupled
 ports loses at saving **.00256**, for every integer leaf count g>=3. This is
 a uniform ceiling below selected .00310045, not a bounded parameter search.
+The g=1 and g=2 profiles already lose every concave moment.
 The invoice marks all unconstructed physical obligations as unknown.
 
 A larger coupling is still possible under this test. At 31 leaves, five
diff --git a/research/coupled31-chronology/RESULT.json b/research/coupled31-chronology/RESULT.json
index 5f09e0d7b4..0c866aa06a 100644
--- a/research/coupled31-chronology/RESULT.json
+++ b/research/coupled31-chronology/RESULT.json
@@ -85,6 +85,7 @@
     "requested_31_leaf": {
       "minimum_H1_calls_at_104_ports": 1456,
       "minimum_thin_rank": 28,
+      "minimum_with_arbitrary_nonzero_stage_splits": 20,
       "optimistic_signed_profile": {
         "1": -137918,
         "2": -28148,
@@ -131,11 +132,16 @@
     "uniform_one_or_two_port_ceiling": {
       "all_integer_leaves_at_least_three": true,
       "all_larger_savings_rejected": true,
+      "all_positive_leaf_counts_rejected": true,
       "intercept_interval": [
         "-2387010674050001077/331141978521600000",
         "-4774021348099800407/662283957043200000"
       ],
       "saving_rejected": "8/3125",
+      "short_caller_linear_margins": {
+        "1": -112,
+        "2": -16
+      },
       "slope_interval": [
         "-593132702156541649/5298271656345600000",
         "-593132698134742867/5298271656345600000"
@@ -145,8 +151,8 @@
   },
   "proof_sha256": {
     "COMMUTING.md": "d4329ca042cd80527a086fbd6b4644f67660145adfc72196c0a8744bf4d40017",
-    "COST.md": "076e0f322209c64ab91870e282f251d07ecaae282b536f0f993ba36576fae457",
-    "PROOF.md": "f909adc46fb9df2955430da2ce1f29caf78b4cb668e8eda0361fa5d6dd800095"
+    "COST.md": "2ed1d2e4f0db0e3c694e6fce24e614a0dd0fd222279eefc66a8535b3479642a8",
+    "PROOF.md": "35a9b9dd0418a502e6dcf758920108a5c75f08b7fcb38de36a53f9bbc7df9642"
   },
   "scope": "Paired full star stages over characteristic two; arbitrary thin maps",
   "status": "PASS_CHRONOLOGICAL_CONNECTOR_RANK_BOUND"
diff --git a/research/coupled31-chronology/SOURCES.json b/research/coupled31-chronology/SOURCES.json
index 37c7f67d64..dabab9c90e 100644
--- a/research/coupled31-chronology/SOURCES.json
+++ b/research/coupled31-chronology/SOURCES.json
@@ -15,39 +15,79 @@
       "sha256": "f7a48bfa2659b88378282a8eee7d9fe52cdf505594124c4d1686b58510fffd72"
     },
     {
-      "path": "research/coupled31-chronology/sources/pr5673-proof.md",
-      "usage": "input",
+      "path": "research/crossed-connector-caller/PROOF.md",
+      "usage": "provenance",
       "pin": "head:c10f75f09e41803ae8efe478331a7509b7379b2f:research/crossed-connector-caller/PROOF.md",
       "git_blob": "e3800fbcf1c5f5633e14d9856ca1400985292a0d",
       "sha256": "e086b3376173f6b790e05ef09f23b4d5970c8ac5145ab2970e084743f782190b"
     },
     {
-      "path": "research/coupled31-chronology/sources/escape-word.json",
+      "path": "research/coupled31-chronology/sources/pr5673-proof.md",
       "usage": "input",
+      "pin": "head:c951dc916aadc7d8af5b126360558f7664b08a4f:research/coupled31-chronology/sources/pr5673-proof.md",
+      "git_blob": "e3800fbcf1c5f5633e14d9856ca1400985292a0d",
+      "sha256": "e086b3376173f6b790e05ef09f23b4d5970c8ac5145ab2970e084743f782190b",
+      "copied_from": "head:c10f75f09e41803ae8efe478331a7509b7379b2f:research/crossed-connector-caller/PROOF.md"
+    },
+    {
+      "path": "research/crossed-connector-caller/ESCAPE-WORD.json",
+      "usage": "provenance",
       "pin": "head:c10f75f09e41803ae8efe478331a7509b7379b2f:research/crossed-connector-caller/ESCAPE-WORD.json",
       "git_blob": "aae71c6ad294d6bc9ca5e2216a7785b9e6611afd",
       "sha256": "20285745b7a6c4bfefcccc365caacefaf982d30adb60fea29ffa14f0ea6b639c"
     },
     {
-      "path": "research/coupled31-chronology/sources/old-word-replay.py",
+      "path": "research/coupled31-chronology/sources/escape-word.json",
       "usage": "input",
+      "pin": "head:c951dc916aadc7d8af5b126360558f7664b08a4f:research/coupled31-chronology/sources/escape-word.json",
+      "git_blob": "aae71c6ad294d6bc9ca5e2216a7785b9e6611afd",
+      "sha256": "20285745b7a6c4bfefcccc365caacefaf982d30adb60fea29ffa14f0ea6b639c",
+      "copied_from": "head:c10f75f09e41803ae8efe478331a7509b7379b2f:research/crossed-connector-caller/ESCAPE-WORD.json"
+    },
+    {
+      "path": "research/crossed-connector-caller/word_replay.py",
+      "usage": "provenance",
       "pin": "head:c10f75f09e41803ae8efe478331a7509b7379b2f:research/crossed-connector-caller/word_replay.py",
       "git_blob": "037ed3f765822341156e059c306218504ac37e1b",
       "sha256": "d4574ed109672c9351af8ace5fbcd31671aed6936877dfa4ff09b56b9007da0a"
     },
     {
-      "path": "research/coupled31-chronology/sources/pr5681-proof.md",
+      "path": "research/coupled31-chronology/sources/old-word-replay.py",
       "usage": "input",
+      "pin": "head:c951dc916aadc7d8af5b126360558f7664b08a4f:research/coupled31-chronology/sources/old-word-replay.py",
+      "git_blob": "037ed3f765822341156e059c306218504ac37e1b",
+      "sha256": "d4574ed109672c9351af8ace5fbcd31671aed6936877dfa4ff09b56b9007da0a",
+      "copied_from": "head:c10f75f09e41803ae8efe478331a7509b7379b2f:research/crossed-connector-caller/word_replay.py"
+    },
+    {
+      "path": "research/one-sided-connector-audit/PROOF.md",
+      "usage": "provenance",
       "pin": "head:67f07448eec5ad8d84bdf96b46981ac0b893592f:research/one-sided-connector-audit/PROOF.md",
       "git_blob": "e3be1bb272f3a73a8c6cd248fec69dec498e401d",
       "sha256": "2e52ac6d32923a1ce6f9693415df553b2d4c38c1a103b9e1968c4e114044d800"
     },
     {
-      "path": "research/coupled31-chronology/sources/p104-result.json",
+      "path": "research/coupled31-chronology/sources/pr5681-proof.md",
       "usage": "input",
+      "pin": "head:c951dc916aadc7d8af5b126360558f7664b08a4f:research/coupled31-chronology/sources/pr5681-proof.md",
+      "git_blob": "e3be1bb272f3a73a8c6cd248fec69dec498e401d",
+      "sha256": "2e52ac6d32923a1ce6f9693415df553b2d4c38c1a103b9e1968c4e114044d800",
+      "copied_from": "head:67f07448eec5ad8d84bdf96b46981ac0b893592f:research/one-sided-connector-audit/PROOF.md"
+    },
+    {
+      "path": "research/p104-helper-subspace-splits/RESULT.json",
+      "usage": "provenance",
       "pin": "head:09dd0965711ca7eeb169057a420abab1c9052137:research/p104-helper-subspace-splits/RESULT.json",
       "git_blob": "35f75c8555193c6e845cd52cc562ca23fd5924cb",
       "sha256": "1d92e587915052110dfb3aca8890df5afdaf496164efa5a4a52e69ceeb3c70f3"
+    },
+    {
+      "path": "research/coupled31-chronology/sources/p104-result.json",
+      "usage": "input",
+      "pin": "head:c951dc916aadc7d8af5b126360558f7664b08a4f:research/coupled31-chronology/sources/p104-result.json",
+      "git_blob": "35f75c8555193c6e845cd52cc562ca23fd5924cb",
+      "sha256": "1d92e587915052110dfb3aca8890df5afdaf496164efa5a4a52e69ceeb3c70f3",
+      "copied_from": "head:09dd0965711ca7eeb169057a420abab1c9052137:research/p104-helper-subspace-splits/RESULT.json"
     }
   ]
 }
diff --git a/research/coupled31-chronology/VALIDATION.json b/research/coupled31-chronology/VALIDATION.json
new file mode 100644
index 0000000000..574968ee87
--- /dev/null
+++ b/research/coupled31-chronology/VALIDATION.json
@@ -0,0 +1,20 @@
+{
+  "python": "3.12.13 (main, May 10 2026, 19:20:41) [Clang 22.1.3 ]",
+  "platform": "macOS-26.6.1-arm64-arm-64bit",
+  "base": "df0137efd35e11b4445734bf612bdd4c7ebc330c",
+  "registered": "PASS_REGISTERED_RESEARCH_PACKAGE outcome=completed",
+  "optimized": "PASS_CHRONOLOGICAL_CONNECTOR_RANK_BOUND",
+  "corrupted_package_controls": 5,
+  "source_copy_binding": "Original Git provenance and retained input commit are separate exact pins; matching SHA256 is required. An initial alias pin was rejected and corrected before publication.",
+  "git_free_read_only": {
+    "cwd": "/",
+    "PATH": "/nonexistent",
+    "files": 24,
+    "all_bytes_unchanged": true,
+    "status": "PASS_CHRONOLOGICAL_CONNECTOR_RANK_BOUND"
+  },
+  "source_history": "Not run: the two open predecessor commits are explicitly pinned provenance, not ancestors of this fresh-main branch.",
+  "hosted_ci": "Not yet submitted",
+  "independent_human_review": "Not claimed",
+  "proof_assistant": "Not run"
+}
diff --git a/research/coupled31-chronology/pricing.py b/research/coupled31-chronology/pricing.py
index 29688c77c2..e988294d07 100644
--- a/research/coupled31-chronology/pricing.py
+++ b/research/coupled31-chronology/pricing.py
@@ -67,6 +67,8 @@ def run():
     bound31 = profile(source, 31, 14)
     bound127 = profile(source, 127, 88)
     complete31 = profile(source, 31, 64)
+    short_margins = {g: sum(r*c for r, c in profile(source, g, 4).items()) for g in (1, 2)}
+    require(short_margins == {1: -112, 2: -16}, 'short caller linear losses')
     require(sum(r*c for r, c in complete31.items()) == -3472, 'complete word linear loss')
     require(min(r for r, c in complete31.items() if c > 0) >= 8, 'positive rank floor')
     require(max(r for r, c in complete31.items() if c < 0) <= 7, 'negative rank ceiling')
@@ -87,6 +89,7 @@ def run():
     return {'net_rank_mass_per_full_call': 1720,
             'requested_31_leaf': {'ports': 2, 'requested_thin_rank': 10,
                                   'minimum_thin_rank': 28,
+                                  'minimum_with_arbitrary_nonzero_stage_splits': 20,
                                   'minimum_H1_calls_at_104_ports': 1456,
                                   'optimistic_signed_profile': bound31},
             'old_127_leaf': {'ports': 1, 'requested_thin_rank': 10,
@@ -97,6 +100,8 @@ def run():
                 'slope_interval': [str(x) for x in slope],
                 'intercept_interval': [str(x) for x in intercept],
                 'all_integer_leaves_at_least_three': True,
+                'short_caller_linear_margins': short_margins,
+                'all_positive_leaf_counts_rejected': True,
                 'all_larger_savings_rejected': True},
             'complete_binary_word31': {'full_calls': 62, 'thin_maps': 16,
                                        'rank_per_thin_map': 8, 'binary_blocks': 52,
diff --git a/research/coupled31-chronology/verify.py b/research/coupled31-chronology/verify.py
index 7b69c35c50..2f7dc23f25 100644
--- a/research/coupled31-chronology/verify.py
+++ b/research/coupled31-chronology/verify.py
@@ -27,6 +27,8 @@ def load(name):
 
 def run():
     sources = json.loads((PACKAGE/'SOURCES.json').read_text())
+    provenance = {row.get('pin'): row['sha256'] for row in sources['files']
+                  if row['usage'] == 'provenance'}
     authenticated = 0
     for row in sources['files']:
         if row['usage'] == 'input':
@@ -34,6 +36,8 @@ def run():
             path = PACKAGE/row['path'][len(PREFIX):]
             require(hashlib.sha256(path.read_bytes()).hexdigest() == row['sha256'],
                     'source hash: '+row['path'])
+            require(provenance.get(row['copied_from']) == row['sha256'],
+                    'retained copy differs from original provenance')
             authenticated += 1
     checker, prices = load('check'), load('pricing')
     evidence = checker.run()