Exclude coupled star callers at every length and port rank

research/longer-coupled-port-caller/CERTIFICATE-SUMMARY.json

@@ -0,0 +1,11 @@
+{
+  "orders": 75600,
+  "degree_counts": {
+    "2": 72020,
+    "3": 3580
+  },
+  "largest_witness_terms": 77,
+  "uncompressed_bytes": 6472926,
+  "compressed_bytes": 108057,
+  "sha256": "2e9279611423b0fc0488780e10cc8ac4ce3397f9c3f1809dd0ab294f4987ed1a"
+}

research/longer-coupled-port-caller/DISCOVERY.md

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+# Longer coupled-port search
+
+Ali Romman with substantial OpenAI Codex assistance. Apache-2.0.
+
+This file records discovery steps. The final result is in PROOF.md.
+No new kappa is established.
+Base: 6cf7a683f447b9f1e2333791907fd92be5009a63.
+Claim: https://github.com/CrocSwap/integer-mult-bounds/discussions/5593#discussioncomment-19186397
+
+## A useful necessary condition
+
+Work in characteristic two. Let R_i be square-zero matrices and let their
+sum be a diagonal projector E. Let P_i be diagonal coordinate projectors
+with P_i R_i = R_i P_i = R_i. A coordinate on which E is one must occur in
+at least three P_i.
+
+Proof: zero occurrences fail the diagonal equation. With one occurrence,
+the row and column of that R_i are those of E, so its square has diagonal
+entry one. With two occurrences, call their row vectors a and e_j+a,
+and their column vectors b and e_j+b. Square zero gives a*b=0 and
+(e_j+a)*(e_j+b)=0. But a_j=b_j, so the latter expression is 1+a*b=1.
+Both cases are contradictions.
+
+For full rank-q queries in rank-2q coordinate supports, 3*rank(E) is at
+most 2*q*L. The proof does not assume separate ports or any chronology.
+If every full call has a private stage axis, the target basis makes its
+projector diagonal. Thus this condition applies at every coupled rank.
+
+A second consequence applies when each stage occurs twice and has no
+rank loss. Its two rank-2q projectors sum to a rank-4q idempotent. The
+projectors are orthogonal. The sum of the two compatible queries is
+square zero and is supported on the diagonal target stage projector.
+The same three-occurrence argument applies to these grouped queries.
+It rules out full stage capacity whenever 3*N > 2*q*L, at all lengths
+and all coupled ranks. No chronological assumption is needed for this
+necessary condition. Rank losses and noncommuting local projectors need
+further work.
+
+## Query frame complement
+
+For sum R_i = I_N, equal rank q, write R_i=U_i V_i with V_i U_i=0.
+Set U=[U_i], V=[V_i]^T, K=VU. Then UV=I and C=I-K is an idempotent
+of rank h=q*L-N, with q by q identity diagonal blocks.
+Factor C=A B, BA=I_h. Each B_i A_i is a rank-q idempotent and their sum
+is I_h. When L is even, the complementary idempotents I_h-B_i A_i
+have rank h-q and also sum to I_h. Therefore L*(h-q)>=h. In particular,
+N<=q*L-q-ceil(q/(L-1)). The trace condition also requires N even over F2.
+These are necessary frame conditions, not a caller construction.
+
+## Numerical discovery screens
+
+The initial screen used the pinned PR5599 local invoice. It assumed all
+2*q*L stage incidences could be used and distributed them as evenly as
+possible. For q=104, eight calls on 720 fields appeared to have a raw
+crossing near .00348661. That frame is impossible: h=112, while the
+complement inequality requires 8*(112-104)>=112. Other paired-stage
+screens also lose capacity before an operator is built. These floating
+numbers are rejected discovery hypotheses, not kappa claims.
+
+## Next experiments
+
+The remaining construction must use noncommuting local stage projectors
+and pay for lost stage capacity, or change another stated hypothesis.
+The current search keeps the local P104 supplier and independent helpers.
+Use PR5631's mixed chronology condition before a full word replay.
+
+## Completed continuation
+
+The initial finite search checked 24^3 coupled splits in each of 144
+two-sweep orders: 1,990,656 assignments, with no chronology survivor.
+The rank-two coupled search used Z3 5.3.2.0 on 144 orders and found all
+unsatisfiable; this result alone was not used as a proof. Exact free-algebra
+witnesses then covered all 75,600 order types, with no matrix-size bound.
+The final theorem and its reduction are in PROOF.md.
+
+DISCOVERY-RESULTS.json.gz preserves all five JSON discovery reports. Its
+SHA256 is ec6fbe62a74cb05750db45c24c2a14ca18cb6b4f6bfb88a8151b76df40120ef8. The retained scripts
+can regenerate these reports. The registered checker uses only the exact
+order witnesses and standard-library matrix checks.
+
+Commands, from the repository root:
+
+```sh
+python3 -B research/longer-coupled-port-caller/search_star.py
+python3 -B research/longer-coupled-port-caller/search_orders.py --degree 2
+python3 -B research/longer-coupled-port-caller/nc_star.py --degree 3 --limit 144
+python3 -B research/longer-coupled-port-caller/build_certificates.py
+```
+
+The optional search_coupled.py command requires z3-solver==5.3.2.0. It is
+not a verification dependency. Its finite negative output is superseded
+by the exact algebra certificates. The numeric invoice screens do not
+supply an operator, compiler or kappa result.

research/longer-coupled-port-caller/PROOF.md

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+# All-order exclusion of the coupled star query frame
+
+Ali Romman with substantial OpenAI Codex assistance. Apache-2.0.
+
+## 1. Result and scope
+
+The constant-loss frame in PR5631 passes the earlier stage-rank and
+intersection cuts at arbitrarily large sizes. This package permits arbitrary
+matrix coupling inside every good stage pair. It excludes every call order,
+every coupled channel dimension, and every odd leaf count g>=3 for this
+frame. It also supplies a small exact proof format for other matrix searches.
+
+This is a theorem in the binary normalized model. It is not a reduction of
+all complex or dyadic native callers to that model. It gives no new kappa.
+Other query frames and connector contracts remain open.
+
+Let F be any field of characteristic two. Let W have positive dimension d.
+The main space is W_0 plus W_1 plus ... plus W_g, with W_0 the core. Let
+E_j select W_0 and W_j. On these two blocks put
+
+    T_j = [[I,I],[I,I]].
+
+All other blocks of T_j are zero. Split each good stage into two calls.
+Their projectors P_j and P'_j have sum E_j, are orthogonal, and their
+queries R_j and R'_j have sum T_j. Require P_j R_j = R_j P_j = R_j and
+P'_j R'_j = R'_j P'_j = R'_j. These conditions cover every compatible
+split at full stage rank. No separate-port restriction is imposed.
+
+The four remaining queries are F_C, G_C, F_(I+C), G_(I+C), where C is any
+idempotent on W. F_H has block H from each leaf input to the core output.
+G_H has block H from the core input to each leaf output. All other blocks
+are zero. These are the connector queries of the source star frame, after
+an arbitrary common channel basis change. Their stage projectors are not
+used in the exclusion. They can be thin calls. There can be arbitrary
+constant basis changes already absorbed into the displayed matrices.
+
+For odd g, the sum of all queries is I: the sum of the T_j has identity
+diagonal and identity core/leaf cross blocks; the four connectors cancel
+the cross blocks. Thus the frame passes the query-sum condition.
+
+For the native equal-rank instance, d=2q, rank A_j=rank C=q. Each good
+projector has rank 2q, each query has rank q, N=2q(g+1), and L=2g+4.
+The theorem is stronger than that instance: its algebra does not use the
+ranks of A_j or C.
+
+**Theorem.** No order of these calls has an endpoint with the central
+involution Q on every main field and the declared independent good-stage
+projectors E_j, if g>=3 is odd. This holds for every d>0 and every F of
+characteristic two. Failure already occurs in a mixed first derivative.
+
+## 2. All coupled splits have a matrix parameterization
+
+Since the two projectors are orthogonal and sum to E_j, both act inside
+W_0 plus W_j. Also P_j T_j = T_j P_j = R_j. The centralizer of T_j has
+block form
+
+    P_j  = [[A_j+B_j, B_j], [B_j, A_j+B_j]],
+    R_j  = [[A_j, A_j],     [A_j, A_j]],
+    P'_j = E_j + P_j,        R'_j = T_j + R_j.
+
+To see completeness, write a general four-block matrix and equate its
+products with T_j on the two sides. Its two diagonal blocks are equal,
+and its two off-diagonal blocks are equal. Idempotence is equivalent to
+
+    A_j^2 = A_j,
+    A_j B_j + B_j A_j = B_j.
+
+For example, in a basis with T_j=[[0,I],[0,0]], the projector is
+[[A_j,B_j],[0,A_j]]. This also gives rank P_j=2 rank A_j. These are full
+matrix parameters. Their entries can mix every channel. They are not
+scalar choices or copies of a one-port calculation.
+
+## 3. Necessary chronology equations
+
+Put Q=1+e and z_j=1+d_j, with e^2=d_j^2=0. After normalization a good
+call is (I+d_j P)(I+e R). Calls act in listed order, with later factors
+on the left. The query sum is I. The coefficient of e*d_j is
+
+    E_j + sum_(i in stage j) [P_i, sum_(k later than i) R_k].
+
+The desired endpoint has coefficient E_j. Therefore its defect must be
+zero. This is PR5631's mixed chronology condition. The bracket is AB+BA
+in characteristic two. An exact local matrix word supplies a separate
+check of this formula in the verifier.
+
+For each good stage j, retain only defect blocks whose row and column
+are not both in {0,j}. A nonzero retained block involves stage j and
+one other leaf k. A query at a third leaf contributes zero to this block:
+its only nonzero rows and columns are at the core and that third leaf.
+A connector contributes H or zero, independent of the number of leaves.
+Thus these retained equations depend only on the relative order of the
+two good stages and the four connectors.
+
+Choose any three good stages. Delete every other good call from the order.
+The retained equations on those three stages do not change. Rename the
+three stages in order of first occurrence. If a stage's second labelled
+call occurs first, replace A_j by I+A_j and leave B_j unchanged. This
+preserves all local equations and exchanges the two calls. Consequently
+there are exactly
+
+    10! / (2^3 * 3!) = 75600
+
+order types to check. This finite reduction is independent of g and d.
+It is the step that turns a finite order certificate into an unbounded
+length and matrix-size theorem.
+
+## 4. Exact certificates in a free algebra
+
+Use the free associative algebra F2<A0,B0,A1,B1,A2,B2,C>. A word records
+the order of its factors. In particular A0*B1 and B1*A0 are different
+words. The equations are the six local relations in Section 2, C^2+C,
+and the retained chronology blocks from Section 3.
+
+For each of the 75600 order types, ORDER-CERTIFICATES.json.gz gives
+explicit finite words u_l,v_l and equation indices i_l such that
+
+    sum_l u_l * equation_(i_l) * v_l = 1.
+
+The separate checker expands this sum and cancels equal words modulo two.
+There are 72020 witnesses of degree at most two and 3580 of degree at most
+three. The largest has 77 terms. No SAT or numerical solver is needed to
+check them. The search code uses finite linear algebra on words to find
+these identities; the checker does not import that code.
+
+If a tuple of matrices solved the equations, evaluation of each witness
+would give 0=I_d. This is impossible for d>0. Evaluation preserves word
+order, so no commutativity of the matrices is assumed. Since F contains
+F2, the same identities work over every characteristic-two field.
+This proves the theorem.
+
+For a short example, put all first good calls before all connectors, and
+put second good calls in reverse order. Four retained equations are
+
+    I + A0 + A1 + B1 = 0,
+    B1 = 0,
+    I + A0 + A2 + B2 = 0,
+    I + A1 + A2 + B2 = 0.
+
+Their sum is I=0. The other order types need the certificates above;
+this example is not used as a substitute for their complete coverage.
+
+## 5. What the search removes
+
+The source relaxed profile has 2q jobs of rank 1+8g and 2gq jobs of rank
+9, against 2g+4 full calls and their complete independent helper bill.
+For q=104 and g=1023, the PR5599 bill gives a positive relaxed moment at
+saving .003125. The exact integer-power enclosure is in RESULT.json.
+This exceeds the selected .00310045 but is not an attainable caller.
+The theorem excludes every coupled split and every order of this frame,
+including large cases that were not individually instantiated.
+
+The preliminary eight-call, 720-field price screen also failed a frame
+condition. If rank-q square-zero queries sum to I_N, their Gram complement
+has rank h=qL-N and identity q by q diagonal blocks. Factoring it gives
+rank-q idempotents E_i on an h-dimensional space with sum I_h. For even L,
+the complements I_h-E_i also sum to I_h. Hence L(h-q)>=h. The proposed
+q=104,L=8,N=720 has h=112 and 8(h-q)=64, so it is impossible.
+
+A separate useful test is the three-occurrence lemma. If square-zero
+matrices R_i sum to a diagonal projector E and are supported on diagonal
+projectors P_i, each coordinate where E=1 must occur in at least three
+P_i. With two occurrences, their rows are a and e_j+a and their columns
+are b and e_j+b. Square zero gives a*b=0, but
+(e_j+a)*(e_j+b)=1+a*b because a_j=b_j. This is a contradiction. One
+occurrence also contradicts square zero. Thus 3 rank E <= sum rank P_i.
+This covers arbitrary coupled queries whose local projectors commute.
+For paired stages at full rank, group the two orthogonally supported
+queries. Their sum is square zero and supported on the target stage
+projector. The same test applies. Rank loss requires another argument;
+the star theorem above supplies one for the declared frame.
+
+## 6. Verification boundary and next routes
+
+The order reduction and free-algebra evaluation argument are written
+proofs. The executable checker verifies the complete finite order cover
+and every exact identity. A different matrix implementation checks the
+local identities, query frame and mixed coefficient formula on coupled
+noncommuting examples. The one-leaf case is a genuine full polynomial
+word and is retained as a positive control. Corrupted certificates and
+changed query data must fail. No proof assistant or independent human
+acceptance is claimed.
+
+This theorem does not exclude a different query Gram frame, a different
+connector map, more connector queries interleaved with good calls, nonfull
+stage ranks, different main outputs, noninvolutive stages, helper feedback,
+or arbitrary complex/dyadic callers. It does not price or certify a new
+native multiplication algorithm. A future search can change one of these
+premises, or use the same free-algebra certificate format on another exact
+frame before it allocates large matrices and a complete native bill.
+
+Credit Cinna Davis / 1337isnot1337 for the controller, simplex and private
+projector lineage; PR5617 for stage/intersection cuts; PR5631 for the
+constant-loss star and mixed chronology equation; and the P104, helper,
+native, compiler and outer-framework authors retained by those sources.
+Free associative algebras and linear combinations of polynomial equations
+are standard tools. The all-order coupled-frame reduction and its exact
+witnesses are the contribution of this package.

research/longer-coupled-port-caller/README.md

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+# An all-order obstruction for coupled star callers
+
+The constant-loss star frame cannot be repaired by port coupling or by a
+longer order. In the binary normalized model, this package excludes every
+odd leaf count g>=3, every coupled channel dimension and every order of
+its calls. It keeps the star query frame and four complementary connector
+queries fixed. **No new kappa is claimed.**
+
+[PROOF.md](PROOF.md) gives the complete contract and the three-stage
+reduction. There are only 75,600 relative order types after that reduction.
+For each type, an exact identity in a free noncommutative algebra proves
+0=I. The identities apply to matrices of every size. This is stronger than
+a bounded search over matrix entries or port counts.
+
+[RESULT.json](RESULT.json) records the independent replay. The compressed
+[certificates](ORDER-CERTIFICATES.json.gz) contain every witness. The
+[checker](check_certificates.py) expands them without importing the search
+code or using a solver. There are 72,020 witnesses of degree at most two
+and 3,580 of degree at most three. None has more than 77 terms.
+
+The package also checks actual coupled matrices, the mixed coefficient
+formula, a complete one-leaf positive word, and corrupted data. The pinned
+PR5599 bill gives a positive relaxed moment at saving 1/320 for g=1023;
+the theorem shows why that price screen cannot become a caller in this
+frame. The selected record advanced upstream to .00310045 during this
+work. It was not changed by this contribution.
+
+## Reproduce
+
+Use Python 3.12 and the standard library:
+
+```sh
+python3 scripts/research_package.py --package research/longer-coupled-port-caller --check-only
+python3.12 scripts/research_package.py --package research/longer-coupled-port-caller
+python3.12 -I -O -B research/longer-coupled-port-caller/verify.py
+```
+
+The registered check uses the saved exact witnesses. To rebuild witnesses:
+
+```sh
+python3.12 -B research/longer-coupled-port-caller/build_certificates.py
+```
+
+The optional discovery searches and their complete receipts are described
+in [DISCOVERY.md](DISCOVERY.md). Their Z3 results are not proof inputs.
+Exact source commits, Git blobs and SHA256 hashes are in
+[SOURCES.json](SOURCES.json). The package began at canonical main
+6cf7a683f447b9f1e2333791907fd92be5009a63; the refresh at
+df0137efd35e11b4445734bf612bdd4c7ebc330c changed the selected record but
+not the priority or working instructions.
+
+## Open next questions
+
+- Change the query Gram frame. Test its chronological matrix equations
+  before any full native construction or favorable price claim.
+- Change the connector maps or add interleaved paid queries. Recompute
+  every equation and keep the complete bill.
+- Establish a valid characteristic-zero statement for a concrete caller.
+  General dyadic coefficients cannot be reduced modulo two.
+
+Other frames, connector contracts, nonfull stage ranks, noninvolutive
+stages, changed targets and productive helper feedback remain outside the
+theorem. No native replay, proof-assistant proof, independent human review
+or global obstruction to improving kappa is claimed.
+
+## Credit
+
+Ali Romman with substantial OpenAI Codex assistance; no subagents.
+Apache-2.0. Credit Cinna Davis / 1337isnot1337 for the controller and simplex
+lineage, PR5617 for the stage/intersection cuts, PR5631 for the star frame
+and mixed chronology condition, and all P104/helper/native/compiler/outer
+source authors retained by those packages. Free-algebra identities and
+finite linear algebra are standard tools. The new work is the coupled
+parameterization, the all-order reduction and its exact certificates.

research/longer-coupled-port-caller/REFRESH.json

@@ -0,0 +1,17 @@
+{
+  "checked_at_utc": "2026-10-26T01:59:00Z",
+  "main": "df0137efd35e11b4445734bf612bdd4c7ebc330c",
+  "selected": "0.00310045",
+  "selected_record_sha256": "2a0220e866417774606e0f991e41b9c1d26b1f43b97857238cf478dbc7d330b5",
+  "priorities_or_working_instructions_changed": false,
+  "active_claims_checked": [
+    5600,
+    5636,
+    5638,
+    5642,
+    5125,
+    1166
+  ],
+  "own_scope": "Global caller with fixed supplier; no overlap with local contact/history/G20 construction",
+  "coverage": "Live latest Discussion updates plus the initial full open-PR and Discussion overlap cache; sparse local notes remain a limitation"
+}