From 7cbea87dabcbe89223c3c0ca1dfa3fb0fe9b6b8c Mon Sep 17 00:00:00 2001
From: flyingviper <44173173+flyingviper@users.noreply.github.com>
Date: Sun, 25 Oct 2026 22:10:02 +0100
Subject: [PATCH 1/2] Retain attributed PR5580 word for owned-coding audit

---
 .../inputs/README.md                          |  20 ++++++++++++++++++
 .../inputs/pr5580-WORD.json.gz                | Bin 0 -> 105115 bytes
 2 files changed, 20 insertions(+)
 create mode 100644 research/p104-owned-coding-boundaries/inputs/README.md
 create mode 100644 research/p104-owned-coding-boundaries/inputs/pr5580-WORD.json.gz

diff --git a/research/p104-owned-coding-boundaries/inputs/README.md b/research/p104-owned-coding-boundaries/inputs/README.md
new file mode 100644
index 0000000000..84d6228229
--- /dev/null
+++ b/research/p104-owned-coding-boundaries/inputs/README.md
@@ -0,0 +1,20 @@
+# Retained source
+
+`pr5580-WORD.json.gz` is a byte-identical copy of
+`research/p104-event-contact-step/WORD.json.gz` from PR #5580, head
+`6efa248cf456d6101d757e9d09ba29711d18a13a`:
+https://github.com/CrocSwap/integer-mult-bounds/pull/5580
+
+The source construction and event-contact change are by Ali Romman with
+substantial OpenAI Codex assistance. This package does not claim those as new.
+Retain Joel Pulikkan (@GamingPuzzled)'s PR5569 priced helper handoff and PR5472
+successor mechanism; @DreamingOfClouds's PR5561 free-endpoint word and event
+toolkit; @thu-nmrc's PR5551 coupled contacts; Ali Romman's PR5544/5545 complete
+caller and source closure; Gareth Davidson, David Arias and Cinna Davis for
+the P104 word, coalescing and private reuse; Cinna Davis for the controller;
+Jacob Alan Sussman for productive recursion; zxv and Cao Yifan for interval,
+compiler and native framework work. Earlier geometry, BIT and analytic source
+notices remain in the originating source closure.
+
+Apache-2.0. The retained word is evidence, not new work. Registered replay
+reads this local file and does not download or execute foreign code.
diff --git a/research/p104-owned-coding-boundaries/inputs/pr5580-WORD.json.gz b/research/p104-owned-coding-boundaries/inputs/pr5580-WORD.json.gz
new file mode 100644
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HcmV?d00001


From 9e690c049acc2e34002d2179f93c524b0b7762ff Mon Sep 17 00:00:00 2001
From: flyingviper <44173173+flyingviper@users.noreply.github.com>
Date: Sun, 25 Oct 2026 22:12:03 +0100
Subject: [PATCH 2/2] Prove rational recoding and joint owned-boundary limits
 for P104

---
 .../JOINT-BOUNDARY.md                         | 124 +++
 .../p104-owned-coding-boundaries/PROOF.md     | 107 +++
 .../p104-owned-coding-boundaries/README.md    | 105 ++
 .../p104-owned-coding-boundaries/RESULT.json  | 903 ++++++++++++++++++
 .../p104-owned-coding-boundaries/SOURCES.json |  31 +
 .../p104-owned-coding-boundaries/TASK.json    | 113 +++
 .../p104-owned-coding-boundaries/VERIFY.json  |  21 +
 .../classification.py                         | 112 +++
 .../p104-owned-coding-boundaries/delivery.py  | 171 ++++
 .../joint_boundary.py                         | 199 ++++
 .../p104-owned-coding-boundaries/verify.py    |  82 ++
 11 files changed, 1968 insertions(+)
 create mode 100644 research/p104-owned-coding-boundaries/JOINT-BOUNDARY.md
 create mode 100644 research/p104-owned-coding-boundaries/PROOF.md
 create mode 100644 research/p104-owned-coding-boundaries/README.md
 create mode 100644 research/p104-owned-coding-boundaries/RESULT.json
 create mode 100644 research/p104-owned-coding-boundaries/SOURCES.json
 create mode 100644 research/p104-owned-coding-boundaries/TASK.json
 create mode 100644 research/p104-owned-coding-boundaries/VERIFY.json
 create mode 100644 research/p104-owned-coding-boundaries/classification.py
 create mode 100644 research/p104-owned-coding-boundaries/delivery.py
 create mode 100644 research/p104-owned-coding-boundaries/joint_boundary.py
 create mode 100644 research/p104-owned-coding-boundaries/verify.py

diff --git a/research/p104-owned-coding-boundaries/JOINT-BOUNDARY.md b/research/p104-owned-coding-boundaries/JOINT-BOUNDARY.md
new file mode 100644
index 0000000000..7132967f59
--- /dev/null
+++ b/research/p104-owned-coding-boundaries/JOINT-BOUNDARY.md
@@ -0,0 +1,124 @@
+# A joint moment floor for the complete eight-owned-block boundary
+
+Source: PR5580 head `6efa248cf456d6101d757e9d09ba29711d18a13a`, compressed word
+SHA256 `b2a98c4bd0dbe86c09e9d15ee060dcf1aab09dbef90aa1e2127676f4039f406b`.
+No new kappa or construction is claimed.
+
+## The deliberately enlarged boundary
+
+The replacement interval is `[13378,13485)` in the source program, from the
+first owned INIT to just after the last owned CLEAR. Its public boundary has
+41 roles: eight distinct copied producers and 33 distinct recipients. No
+other public role is read or modified within the literal interval.
+
+Treat these 41 public entry values as independent coordinates. The verifier
+checks that their actual prefix rows have rank41 on original arbitrary public
+inputs. It also checks that the fixed suffix observes all41 public coordinates
+with column rank41. Therefore an exact whole-word replacement using only these
+roles must implement the same local public map: from `S(R-T)P=0`, right and left
+inverses imply `R=T`. It may not rely on correlated values in outside registers.
+
+Every producer enters and exits at dimension8. Every recipient keeps its own
+arbitrary input and gains a nonzero multiple of exactly one producer. For each
+producer, at least one recipient exits at dimension1. Projection of these eight
+observations onto the eight independent producer coordinates has rank8.
+
+An allowed replacement can change every internal scalar dependency, introduce
+arbitrarily many zero-initialized owned fields, share private work across all
+eight original lifetimes, and use arbitrary rational scalar operations. It
+must preserve all public entry/exit values and frames, clear private outputs,
+use common-frame scalar operations, ascending public transports and paid
+descending owned transports. It introduces no public helper or new productive
+credit. This is stronger than seven separate fixed-star theorems, but weaker
+than a global lower bound allowing earlier preparation or outside information.
+
+## Seven information cuts
+
+Fix a dimension threshold `k` in `{2,3,4,5,6,7,8}`. Project every current scalar
+value onto its eight producer coordinates. Let `V_k` be the span of every such
+projected row ever observed in a register with frame dimension less than k.
+The span retains past observations after storage is erased or moved away.
+
+Initially `V_k=0`: the only boundary coordinates containing producer information
+are the eight producer registers themselves, all at dimension8. Other input
+coordinates are independent recipient values, whose producer projections vanish.
+
+Scalar combinations at a common frame below k do not enlarge `V_k`. Operations
+above k stay above k. INIT is zero, CLEAR cannot add information, and a public
+ascent cannot carry information from above the threshold to below it. Only an
+owned descent from dimension `a>=k` to `b<k` can enlarge the span. That event
+transports one scalar row, so increases the span's rank by at most one.
+
+The final eight dimension1 observations have producer-projection rank8.
+Consequently every allowed replacement has at least eight owned descents
+crossing each of these seven thresholds. This argument permits sequential
+outputs, arbitrary private erasure, multiple private fields and shared work.
+
+## Convert crossing counts to the recursive moment
+
+Let `H_r(t)=r^t` for `0<t<=1`. An owned transport of rank `r=a-b` crosses
+`c=#{k in {2,...,8}: b<k<=a}` of the seven thresholds, where
+`c<=min(r,7)`. For every positive integer r,
+
+```
+H_r(t) >= c * 7^(t-1).
+```
+
+For `r<=7`, this follows from `r^(t-1)>=7^(t-1)` and `c<=r`. For `r>=7`,
+it follows from `r^t>=7^t` and `c<=7`. Summing over all owned transports and
+then summing the seven rank-eight cut requirements gives
+
+```
+total owned paid moment >= 56 * 7^(t-1) = 8 H7(t).
+```
+
+The source's public boundary has total ascending dimension change31, spread
+over31 fields each changing by one dimension. Every physical public field must
+retain its own endpoint. Its required rank-one ascent costs H1, independently
+of the owned transports counted above. Thus every replacement pays at least
+
+```
+31 H1(t) + 8 H7(t).
+```
+
+The literal word instead pays
+
+```
+33 H1(t) + H2(t) + H3(t) + 7 H7(t).
+```
+
+Therefore the maximum possible raw recursive saving inside this full shared
+boundary is at most
+
+```
+2 H1(t) + H2(t) + H3(t) - H7(t).
+```
+
+Both have the same rank mass87 at t=1. The cap is the exceptional role738's
+fragmentation penalty. The argument does not prove this cap attainable; extra
+middle-level delivery constraints may make the actual optimum higher.
+
+## Crucial distinction from original-input dimensions
+
+The original word's relevant inputs mostly enter on dimension0 or1 frames.
+Those original variables do NOT justify dimension cuts requiring eight
+high-dimensional sources. The theorem is valid here only because it uses the
+later 41-role boundary, treats its coordinates independently (with verified
+prefix and suffix ranks), and disallows reading outside public side information.
+Moving the boundary earlier can invalidate the source-dimension hypothesis.
+
+This is not a sum of the nonadditive original-support loss-space bounds. It is
+a new joint information-flow argument under a stronger boundary contract.
+It gives no unrestricted whole-word tariff optimum, new productive credit,
+ordinary-pass improvement, native admission or new multiplication bound.
+
+## Certificate
+
+`joint_boundary.py` reconstructs all source scalar rows and frame/lifetime
+legality; finds the interval and its41roles; proves both boundary ranks; checks
+the exact recipient map, eight one-dimensional independent observations,
+public31H1 tariff and each of the seven source crossing counts. A scope control
+duplicates one observation and correctly lowers the observation rank to7.
+
+Prepared with substantial OpenAI Codex assistance. Apache-2.0. Retain the PR5580,
+PR5569, PR5561 and complete P104 predecessor attribution in the source package.
diff --git a/research/p104-owned-coding-boundaries/PROOF.md b/research/p104-owned-coding-boundaries/PROOF.md
new file mode 100644
index 0000000000..c73dbf8167
--- /dev/null
+++ b/research/p104-owned-coding-boundaries/PROOF.md
@@ -0,0 +1,107 @@
+# Rational recoding and shared-boundary limits for P104 owned values
+
+All assertions below are about the literal PR5580 word at head
+`6efa248cf456d6101d757e9d09ba29711d18a13a`. They do not assert a new
+construction, global optimum, native implementation, or multiplication bound.
+
+## 1. A complete classification of the unchanged producer span
+
+Let `D_0,...,D_7` be the eight public scalar rows copied into owned fields
+735 through 742. Rows are rational covectors on **all 735 arbitrary public
+inputs**, including the original zero-frame helper inputs. The verifier
+reconstructs the prefix from the identity, rather than assuming those helper
+inputs vanish. It confirms that these producer values do not change between
+the first owned INIT and their respective COPY events.
+
+For a covector `x`, define its original-entry support to be the binary span
+of the original entry frames of inputs with nonzero coefficient in `x`.
+This is a property of its covector, not an assertion that `x` can be built
+at that frame with no paid work.
+
+Every proper subspace of the ambient `F_2^9` lies in a hyperplane `ker(h)`
+for some nonzero binary normal `h`. For each of the 511 normals, project
+the eight rational covectors onto original input coordinates whose entry
+frames are not contained in that hyperplane. A combination `sum a_i D_i`
+has support inside the hyperplane exactly when its coefficient vector `a`
+is in this projected matrix's rational kernel.
+
+Exact Fraction elimination gives 495 matrices of rank eight and 16 of rank
+seven. The latter have one-dimensional kernels. Every listed kernel mode
+has support dimension exactly eight, checked by binary elimination. Hence:
+
+* No nonzero rational combination of the unchanged producers has
+  original-entry support dimension at most seven.
+* Exactly 16 projective combinations have proper support: the eight
+  original producers and eight mixed modes listed in RESULT.json.
+* Every other nonzero combination has support dimension nine.
+
+This is an exhaustive rational statement, not a bounded coefficient search.
+The independence minor below also excludes a nonzero coefficient vector
+representing the zero covector.
+
+## 2. Direct nested delivery does not remove an original mode
+
+Let `L_i` be the fixed one-dimensional final delivery frame for `D_i`.
+A mode with support hyperplane `ker(h)` can descend directly from that
+hyperplane to `L_i` only if `h` annihilates `L_i`. The verifier tests every
+one of the sixteen classified modes at every final delivery line.
+
+After removing `D_i`, exactly three mixed modes remain eligible at `L_i`.
+Their coefficient vectors have rational rank three; adjoining the unit
+vector for `D_i` raises that rank to four. RESULT.json records an exact
+separating functional for each destination, annihilating all three
+alternates and taking value one on `D_i`.
+
+This obstruction survives arbitrary corrections from original inputs
+whose entry frame is zero. The eight producer covectors restricted to
+nonzero-entry inputs have rank eight, witnessed by the explicitly checked
+minor on columns `[0,1,2,4,5,6,9,12]`. Therefore a producer combination is
+a zero-entry-input correction only when all eight coefficients vanish.
+
+This is a direct nested 8-to-1 delivery obstruction in the unchanged producer
+span, starting at each mode's original-entry support hyperplane and using
+fixed final delivery lines.
+It does not rule out earlier owned preparation, intermediate payloads
+outside that span, moving through a different recipient schedule, or
+changed public dependencies. Nor does an entry-support hyperplane by
+itself establish legal preparation of its mixed mode there.
+
+## 3. Joint transport floor across all eight blocks
+
+See [JOINT-BOUNDARY.md](JOINT-BOUNDARY.md) for the independent information-cut
+argument and full contract. The interval `[13378,13485)` has 41 public
+boundary fields: eight producers and 33 distinct recipients. Their entry
+rows have rank 41; the unchanged suffix observes their exits with rank 41.
+These verified facts force a replacement using only these public fields
+to preserve the exact local map.
+
+The map carries eight independent producer coordinates from dimension-eight
+inputs to dimension-one observations. At every dimension threshold 2 through
+8, at least eight scalar values must cross downward in owned fields.
+Common-frame scalar operations and public ascents cannot create a downward
+crossing. The argument allows arbitrary rational dependencies, any number
+of zero-initialized owned fields, erasures, and sharing across the blocks.
+
+For `H_r(t)=r^t`, `0<t<=1`, the cut count implies an owned cost of at least
+`8 H_7(t)`. The fixed public endpoint changes separately require `31 H_1(t)`.
+Thus the shared-boundary floor is `31 H_1+8 H_7`, compared with the source
+cost `33 H_1+H_2+H_3+7 H_7`. Their difference is an upper bound on possible
+local savings, not an attainable replacement.
+
+The fixed-boundary theorem does not assume that original input symbols
+start at dimension eight. They mostly start at dimension zero or one.
+Earlier preparation and outside public side information invalidate the
+boundary hypothesis and remain meaningful research directions.
+
+## Verification scope
+
+The joint-boundary checker also replays all scalar rows and frame/lifetime
+legality of the retained word. It verifies the final map on arbitrary input
+values and clear owned outputs. These are finite algebraic checks, not a
+proof-kernel verification or a replay of signed native maps, the physical
+caller, interval pricing, compiler, BIT stock, or outer analytic framework.
+Those unchanged obligations belong to the pinned source and are not newly
+claimed here. No selected-frontier file is modified.
+
+Prepared for flyingviper with substantial OpenAI Codex assistance and
+independent agent derivations. Apache-2.0.
diff --git a/research/p104-owned-coding-boundaries/README.md b/research/p104-owned-coding-boundaries/README.md
new file mode 100644
index 0000000000..a4f0bc76bf
--- /dev/null
+++ b/research/p104-owned-coding-boundaries/README.md
@@ -0,0 +1,105 @@
+# P104: rational recoding and joint owned-block boundaries
+
+**Scoped negative results; no new construction or multiplication bound.**
+
+This package tests whether the eight owned transfers in
+[PR #5580](https://github.com/CrocSwap/integer-mult-bounds/pull/5580) can become
+cheaper by rational recoding or by sharing work across their full joint
+boundary. It provides three exact results:
+
+1. Every nonzero rational combination of the eight unchanged producer values
+   has original-entry support dimension at least eight. Exactly 16 projective
+   combinations have dimension eight: eight original and eight mixed modes.
+   All others have dimension nine. The verifier classifies all 511 binary
+   hyperplanes with exact rational elimination; this is not a coefficient grid.
+2. At each final delivery line, the three eligible mixed modes cannot reconstruct
+   its original producer, even modulo arbitrary corrections from original
+   zero-entry-frame inputs. Exact separators and an independence minor are
+   retained in [RESULT.json](RESULT.json).
+3. With the complete 41-role public boundary fixed, arbitrary sharing across
+   all eight owned blocks costs at least `31 H1(t) + 8 H7(t)` for `0<t<=1`.
+   The source costs `33 H1(t) + H2(t) + H3(t) + 7 H7(t)`. Their difference
+   bounds possible local savings; attainability remains open.
+
+The first two statements concern the unchanged producer span and direct nested 8-to-1
+delivery starting at each mode's original-entry support hyperplane. The third allows arbitrary internal rational dependencies and private
+fields, but excludes earlier preparation and outside public side information.
+Those distinctions matter: the certificates do **not** rule out changed public
+dependencies, partial residuals outside that span, or a larger boundary.
+
+See [PROOF.md](PROOF.md) and [JOINT-BOUNDARY.md](JOINT-BOUNDARY.md).
+
+## Reproduce
+
+Python 3.12; standard library only. From this package directory:
+
+```sh
+python3.12 -I -B verify.py
+python3.12 -I -O -B verify.py
+```
+
+Both print `PASS_P104_OWNED_CODING_BOUNDARIES`. Normal replay writes no files.
+The source word is retained locally and authenticated before use. The verifier
+also checks all scalar rows on arbitrary public inputs, chronological frame and
+ownership legality, exact boundary ranks and four semantic corruption controls.
+It compares the fresh receipt to the committed result; `--write` explicitly
+regenerates the latter.
+
+The registered metadata/provenance launcher, from repository root, is:
+
+```sh
+python3.12 scripts/research_package.py --package research/p104-owned-coding-boundaries
+```
+
+This launcher needs Git history for provenance; the direct verifier works in a
+Git-free package export. No solver, network, native checker or external package
+is required. The unchanged native maps, physical caller, pricing, compiler and
+outer analytic obligations are not part of this scoped replay. The full source
+replay was run separately during discovery; it is not a new proof claim here.
+
+## Relation to previous work
+
+[PR #5584](https://github.com/CrocSwap/integer-mult-bounds/pull/5584) established
+single-block optima, delivery obstructions and a rank witness, while explicitly
+leaving cross-block sharing open. This package supplies a joint information-cut
+argument under a stated 41-role boundary, plus the complete rational recoding
+classification. It does not add individual nonadditive loss-space bounds.
+
+Ali Romman's [PR #5579](https://github.com/CrocSwap/integer-mult-bounds/pull/5579)
+already covered fixed public windows; that obstruction is not claimed as new.
+The public compensated-source work in
+[Discussion #5575](https://github.com/CrocSwap/integer-mult-bounds/discussions/5575)
+and the event-retiming work in
+[Discussion #5574](https://github.com/CrocSwap/integer-mult-bounds/discussions/5574)
+are adjacent active approaches. The later
+[PR #5590](https://github.com/CrocSwap/integer-mult-bounds/pull/5590) addresses
+all frame paths with scalar contacts fixed; our joint-boundary theorem permits
+changed internal scalar dependencies but fixes the larger public boundary.
+These are different contracts. Open-PR and discussion checks were bounded
+coverage checks, not a proof of novelty. The canonical base and frozen sources
+are recorded in TASK.json and SOURCES.json.
+
+## Open next questions
+
+* Can an earlier residual assembly or changed outside side information lower
+  the complete paid-minus-productive moment? First identify actual value
+  versions, a legal transfer and all preparation/restoration costs.
+* Can the exceptional owned block attain the joint lower bound while meeting
+  every middle-level delivery? The information cuts alone do not resolve this.
+
+The user-authorized broader investigation continues separately; this package
+is an independently reviewable checkpoint, not a claim that the overall
+research goal is complete.
+
+## Credit
+
+Prepared for flyingviper with substantial OpenAI Codex assistance. Separate
+agents derived and checked the joint-boundary argument and delivery gate;
+the all-hyperplane classification was independently reconstructed by the
+primary agent and reviewed by another agent. These are computational research
+checks, not human review or proof-kernel acceptance. New work is Apache-2.0.
+
+The retained word is Ali Romman's PR5580 source, with Joel Pulikkan,
+DreamingOfClouds, thu-nmrc and the full P104 predecessor credit retained in
+[inputs/README.md](inputs/README.md). No predecessor construction is attributed
+to this package.
diff --git a/research/p104-owned-coding-boundaries/RESULT.json b/research/p104-owned-coding-boundaries/RESULT.json
new file mode 100644
index 0000000000..1ed4fe8833
--- /dev/null
+++ b/research/p104-owned-coding-boundaries/RESULT.json
@@ -0,0 +1,903 @@
+{
+  "classification": {
+    "first_owned_init": 13378,
+    "modes": [
+      {
+        "coefficients": [
+          "1",
+          "0",
+          "0",
+          "0",
+          "1/4",
+          "1/4",
+          "1/4",
+          "0"
+        ],
+        "entry_support_basis": [
+          347,
+          168,
+          94,
+          45,
+          16,
+          8,
+          5,
+          3
+        ],
+        "entry_support_dimension": 8,
+        "mixed": true,
+        "normal": 7,
+        "original_symbols": 107
+      },
+      {
+        "coefficients": [
+          "0",
+          "1",
+          "0",
+          "0",
+          "-1/4",
+          "1/4",
+          "0",
+          "-1/4"
+        ],
+        "entry_support_basis": [
+          499,
+          128,
+          73,
+          45,
+          21,
+          9,
+          6,
+          2
+        ],
+        "entry_support_dimension": 8,
+        "mixed": true,
+        "normal": 25,
+        "original_symbols": 111
+      },
+      {
+        "coefficients": [
+          "0",
+          "0",
+          "0",
+          "0",
+          "0",
+          "1",
+          "0",
+          "0"
+        ],
+        "entry_support_basis": [
+          347,
+          168,
+          94,
+          45,
+          16,
+          10,
+          5,
+          1
+        ],
+        "entry_support_dimension": 8,
+        "mixed": false,
+        "normal": 42,
+        "original_symbols": 111
+      },
+      {
+        "coefficients": [
+          "0",
+          "0",
+          "1",
+          "0",
+          "0",
+          "1/4",
+          "-1/4",
+          "1/4"
+        ],
+        "entry_support_basis": [
+          499,
+          131,
+          73,
+          45,
+          21,
+          8,
+          2,
+          1
+        ],
+        "entry_support_dimension": 8,
+        "mixed": true,
+        "normal": 52,
+        "original_symbols": 108
+      },
+      {
+        "coefficients": [
+          "1",
+          "0",
+          "-1",
+          "1",
+          "0",
+          "0",
+          "1",
+          "0"
+        ],
+        "entry_support_basis": [
+          374,
+          168,
+          94,
+          40,
+          16,
+          8,
+          7,
+          3
+        ],
+        "entry_support_dimension": 8,
+        "mixed": true,
+        "normal": 67,
+        "original_symbols": 123
+      },
+      {
+        "coefficients": [
+          "0",
+          "0",
+          "0",
+          "0",
+          "1",
+          "0",
+          "0",
+          "0"
+        ],
+        "entry_support_basis": [
+          475,
+          130,
+          75,
+          45,
+          18,
+          12,
+          2,
+          1
+        ],
+        "entry_support_dimension": 8,
+        "mixed": false,
+        "normal": 76,
+        "original_symbols": 106
+      },
+      {
+        "coefficients": [
+          "0",
+          "0",
+          "0",
+          "1",
+          "-1/4",
+          "0",
+          "1/4",
+          "1/4"
+        ],
+        "entry_support_basis": [
+          261,
+          168,
+          75,
+          45,
+          23,
+          8,
+          4,
+          1
+        ],
+        "entry_support_dimension": 8,
+        "mixed": true,
+        "normal": 82,
+        "original_symbols": 105
+      },
+      {
+        "coefficients": [
+          "0",
+          "0",
+          "0",
+          "0",
+          "0",
+          "0",
+          "1",
+          "0"
+        ],
+        "entry_support_basis": [
+          475,
+          130,
+          75,
+          45,
+          18,
+          8,
+          4,
+          2
+        ],
+        "entry_support_dimension": 8,
+        "mixed": false,
+        "normal": 97,
+        "original_symbols": 103
+      },
+      {
+        "coefficients": [
+          "1",
+          "0",
+          "0",
+          "0",
+          "0",
+          "0",
+          "0",
+          "0"
+        ],
+        "entry_support_basis": [
+          347,
+          168,
+          94,
+          45,
+          16,
+          5,
+          2,
+          1
+        ],
+        "entry_support_dimension": 8,
+        "mixed": false,
+        "normal": 104,
+        "original_symbols": 118
+      },
+      {
+        "coefficients": [
+          "0",
+          "0",
+          "0",
+          "0",
+          "0",
+          "0",
+          "0",
+          "1"
+        ],
+        "entry_support_basis": [
+          261,
+          168,
+          75,
+          45,
+          23,
+          9,
+          6,
+          3
+        ],
+        "entry_support_dimension": 8,
+        "mixed": false,
+        "normal": 127,
+        "original_symbols": 105
+      },
+      {
+        "coefficients": [
+          "0",
+          "1",
+          "-1",
+          "-1",
+          "0",
+          "0",
+          "0",
+          "-1"
+        ],
+        "entry_support_basis": [
+          347,
+          168,
+          73,
+          45,
+          16,
+          9,
+          4,
+          2
+        ],
+        "entry_support_dimension": 8,
+        "mixed": true,
+        "normal": 137,
+        "original_symbols": 124
+      },
+      {
+        "coefficients": [
+          "0",
+          "1",
+          "0",
+          "0",
+          "0",
+          "0",
+          "0",
+          "0"
+        ],
+        "entry_support_basis": [
+          374,
+          168,
+          115,
+          42,
+          16,
+          8,
+          5,
+          1
+        ],
+        "entry_support_dimension": 8,
+        "mixed": false,
+        "normal": 162,
+        "original_symbols": 113
+      },
+      {
+        "coefficients": [
+          "0",
+          "0",
+          "1",
+          "0",
+          "0",
+          "0",
+          "0",
+          "0"
+        ],
+        "entry_support_basis": [
+          374,
+          133,
+          94,
+          42,
+          16,
+          10,
+          5,
+          1
+        ],
+        "entry_support_dimension": 8,
+        "mixed": false,
+        "normal": 266,
+        "original_symbols": 120
+      },
+      {
+        "coefficients": [
+          "0",
+          "0",
+          "0",
+          "1",
+          "0",
+          "0",
+          "0",
+          "0"
+        ],
+        "entry_support_basis": [
+          296,
+          222,
+          76,
+          45,
+          20,
+          8,
+          4,
+          2
+        ],
+        "entry_support_dimension": 8,
+        "mixed": false,
+        "normal": 289,
+        "original_symbols": 110
+      },
+      {
+        "coefficients": [
+          "1",
+          "1",
+          "1",
+          "0",
+          "0",
+          "1",
+          "0",
+          "0"
+        ],
+        "entry_support_basis": [
+          389,
+          200,
+          56,
+          21,
+          8,
+          7,
+          2,
+          1
+        ],
+        "entry_support_dimension": 8,
+        "mixed": true,
+        "normal": 448,
+        "original_symbols": 120
+      },
+      {
+        "coefficients": [
+          "1",
+          "-1",
+          "0",
+          "-1",
+          "1",
+          "0",
+          "0",
+          "0"
+        ],
+        "entry_support_basis": [
+          374,
+          133,
+          115,
+          40,
+          16,
+          9,
+          7,
+          3
+        ],
+        "entry_support_dimension": 8,
+        "mixed": true,
+        "normal": 491,
+        "original_symbols": 115
+      }
+    ],
+    "rank_counts": {
+      "7": 16,
+      "8": 495
+    },
+    "scope": "Every rational combination of the unchanged eight producer values; original-entry support geometry only.",
+    "source_roles": [
+      173,
+      151,
+      146,
+      133,
+      256,
+      92,
+      9,
+      240
+    ]
+  },
+  "direct_delivery": {
+    "delivery_tests": [
+      {
+        "admissible_alternates": [
+          "E67",
+          "E448",
+          "E491"
+        ],
+        "delivery_line": 7,
+        "rank": 3,
+        "separator": [
+          "1",
+          "-2",
+          "1",
+          "0",
+          "-3",
+          "0",
+          "0",
+          "0"
+        ],
+        "target": 0,
+        "target_in_span": false
+      },
+      {
+        "admissible_alternates": [
+          "E137",
+          "E448",
+          "E491"
+        ],
+        "delivery_line": 25,
+        "rank": 3,
+        "separator": [
+          "-2",
+          "1",
+          "1",
+          "0",
+          "3",
+          "0",
+          "0",
+          "0"
+        ],
+        "target": 1,
+        "target_in_span": false
+      },
+      {
+        "admissible_alternates": [
+          "E67",
+          "E137",
+          "E448"
+        ],
+        "delivery_line": 52,
+        "rank": 3,
+        "separator": [
+          "1",
+          "1",
+          "1",
+          "0",
+          "0",
+          "-3",
+          "0",
+          "0"
+        ],
+        "target": 2,
+        "target_in_span": false
+      },
+      {
+        "admissible_alternates": [
+          "E67",
+          "E137",
+          "E491"
+        ],
+        "delivery_line": 82,
+        "rank": 3,
+        "separator": [
+          "-1",
+          "1",
+          "0",
+          "1",
+          "3",
+          "0",
+          "0",
+          "0"
+        ],
+        "target": 3,
+        "target_in_span": false
+      },
+      {
+        "admissible_alternates": [
+          "E7",
+          "E25",
+          "E82"
+        ],
+        "delivery_line": 491,
+        "rank": 3,
+        "separator": [
+          "-1/4",
+          "1/4",
+          "0",
+          "1/4",
+          "1",
+          "0",
+          "0",
+          "0"
+        ],
+        "target": 4,
+        "target_in_span": false
+      },
+      {
+        "admissible_alternates": [
+          "E7",
+          "E25",
+          "E52"
+        ],
+        "delivery_line": 448,
+        "rank": 3,
+        "separator": [
+          "-1/4",
+          "-1/4",
+          "-1/4",
+          "0",
+          "0",
+          "1",
+          "0",
+          "0"
+        ],
+        "target": 5,
+        "target_in_span": false
+      },
+      {
+        "admissible_alternates": [
+          "E7",
+          "E52",
+          "E82"
+        ],
+        "delivery_line": 67,
+        "rank": 3,
+        "separator": [
+          "-1/4",
+          "0",
+          "1/4",
+          "-1/4",
+          "0",
+          "0",
+          "1",
+          "0"
+        ],
+        "target": 6,
+        "target_in_span": false
+      },
+      {
+        "admissible_alternates": [
+          "E25",
+          "E52",
+          "E82"
+        ],
+        "delivery_line": 137,
+        "rank": 3,
+        "separator": [
+          "0",
+          "1/4",
+          "-1/4",
+          "-1/4",
+          "0",
+          "0",
+          "0",
+          "1"
+        ],
+        "target": 7,
+        "target_in_span": false
+      }
+    ],
+    "mixed_normals": [
+      7,
+      25,
+      52,
+      67,
+      82,
+      137,
+      448,
+      491
+    ],
+    "quotient_minor": [
+      [
+        "0",
+        "1/2",
+        "0",
+        "1/2",
+        "1/2",
+        "0",
+        "0",
+        "-1/2"
+      ],
+      [
+        "0",
+        "0",
+        "-1/2",
+        "-1/2",
+        "0",
+        "1/2",
+        "-1/2",
+        "0"
+      ],
+      [
+        "-1/2",
+        "0",
+        "0",
+        "0",
+        "-1/2",
+        "0",
+        "-1/2",
+        "0"
+      ],
+      [
+        "0",
+        "0",
+        "-1/2",
+        "0",
+        "-1/2",
+        "0",
+        "0",
+        "-1/2"
+      ],
+      [
+        "0",
+        "-1",
+        "-1",
+        "-1",
+        "-1",
+        "1",
+        "-1",
+        "0"
+      ],
+      [
+        "1",
+        "0",
+        "1",
+        "0",
+        "0",
+        "0",
+        "1",
+        "1"
+      ],
+      [
+        "0",
+        "0",
+        "1",
+        "0",
+        "0",
+        "0",
+        "-1",
+        "1"
+      ],
+      [
+        "0",
+        "0",
+        "0",
+        "-1",
+        "1",
+        "0",
+        "0",
+        "0"
+      ]
+    ],
+    "quotient_pivot_inputs": [
+      0,
+      1,
+      2,
+      4,
+      5,
+      6,
+      9,
+      12
+    ],
+    "quotient_rank": 8,
+    "scope": "direct nested 8-to-1 sixteen-mode delivery, modulo zero-entry input corrections",
+    "source_normals": [
+      104,
+      162,
+      266,
+      289,
+      76,
+      42,
+      97,
+      127
+    ]
+  },
+  "joint_boundary": {
+    "duplicate_observation_scope_control_rank": 7,
+    "events_half_open": [
+      13378,
+      13485
+    ],
+    "maximum_possible_local_saving": "2 H1(t) + H2(t) + H3(t) - H7(t)",
+    "new_bound": false,
+    "observed_roles": [
+      704,
+      707,
+      709,
+      710,
+      712,
+      717,
+      722,
+      727
+    ],
+    "owned_paid_histogram": {
+      "1": 2,
+      "2": 1,
+      "3": 1,
+      "7": 7
+    },
+    "prefix_rank": 41,
+    "producer_projection_at_observations": [
+      {
+        "0": "1/2"
+      },
+      {
+        "1": "1/2"
+      },
+      {
+        "2": "-1/2"
+      },
+      {
+        "3": "-1/2"
+      },
+      {
+        "4": "1/4"
+      },
+      {
+        "5": "1/4"
+      },
+      {
+        "6": "1/4"
+      },
+      {
+        "7": "-1/4"
+      }
+    ],
+    "public_paid_histogram": {
+      "1": 31
+    },
+    "public_roles": [
+      173,
+      151,
+      146,
+      133,
+      256,
+      92,
+      9,
+      240,
+      704,
+      705,
+      706,
+      707,
+      708,
+      733,
+      709,
+      732,
+      159,
+      203,
+      148,
+      710,
+      734,
+      712,
+      713,
+      714,
+      715,
+      716,
+      717,
+      718,
+      719,
+      720,
+      721,
+      722,
+      723,
+      724,
+      725,
+      726,
+      727,
+      728,
+      729,
+      730,
+      731
+    ],
+    "recipients": [
+      704,
+      705,
+      706,
+      707,
+      708,
+      733,
+      709,
+      732,
+      159,
+      203,
+      148,
+      710,
+      734,
+      712,
+      713,
+      714,
+      715,
+      716,
+      717,
+      718,
+      719,
+      720,
+      721,
+      722,
+      723,
+      724,
+      725,
+      726,
+      727,
+      728,
+      729,
+      730,
+      731
+    ],
+    "scope": "Fixed 41 independent public entry/exit rows and frames; arbitrary private auxiliaries and cross-block sharing; no outside public side information, changed helper credits or earlier public preparation.",
+    "source_entry_dimensions": [
+      8,
+      8,
+      8,
+      8,
+      8,
+      8,
+      8,
+      8
+    ],
+    "sources": [
+      173,
+      151,
+      146,
+      133,
+      256,
+      92,
+      9,
+      240
+    ],
+    "status": "PASS_JOINT_41_ROLE_OWNED_MOMENT_FLOOR",
+    "suffix_rank": 41,
+    "theorem_floor": "31 H1(t) + 8 H7(t), 0<t<=1",
+    "threshold_crossing_count": {
+      "2": 8,
+      "3": 8,
+      "4": 8,
+      "5": 8,
+      "6": 8,
+      "7": 8,
+      "8": 8
+    }
+  },
+  "negative_controls": {
+    "missing_mode": "sixteen supplied projective modes",
+    "owned_descent_declared_public": "MOVE frame and ownership",
+    "wrong_copied_payload": "full arbitrary-input scalar exit",
+    "wrong_mode_coefficient": "every tested payload has exact eight-dimensional entry support"
+  },
+  "new_bound": false,
+  "source": {
+    "head": "6efa248cf456d6101d757e9d09ba29711d18a13a",
+    "pr": 5580,
+    "sha256": "b2a98c4bd0dbe86c09e9d15ee060dcf1aab09dbef90aa1e2127676f4039f406b"
+  },
+  "status": "PASS_P104_OWNED_CODING_BOUNDARIES"
+}
diff --git a/research/p104-owned-coding-boundaries/SOURCES.json b/research/p104-owned-coding-boundaries/SOURCES.json
new file mode 100644
index 0000000000..0b5dd477c1
--- /dev/null
+++ b/research/p104-owned-coding-boundaries/SOURCES.json
@@ -0,0 +1,31 @@
+{
+  "schema_version": 1,
+  "base_commit": "5e0052ca0aa3036a3afa8b0438927377c8d97d53",
+  "files": [
+    {
+      "path": "certificates/selected-result.json",
+      "usage": "provenance",
+      "git_blob": "fc3d4ba105e1f097d0d7d051a308281e86ab1446",
+      "sha256": "66974efdced4083c00025f97d95d83909a226cdf801edb9301c195c6641f6903"
+    },
+    {
+      "path": "docs/research/agent-task-catalog.json",
+      "usage": "provenance",
+      "git_blob": "f126dd2e2dad2b669e75d1d0d0bd05c600725602",
+      "sha256": "f7a48bfa2659b88378282a8eee7d9fe52cdf505594124c4d1686b58510fffd72"
+    },
+    {
+      "path": "research/p104-owned-coding-boundaries/inputs/pr5580-WORD.json.gz",
+      "usage": "input",
+      "git_blob": "9434855e7dad0d0236d23210b3e71a2b83a9ae64",
+      "sha256": "b2a98c4bd0dbe86c09e9d15ee060dcf1aab09dbef90aa1e2127676f4039f406b",
+      "pin": "head:7cbea87dabcbe89223c3c0ca1dfa3fb0fe9b6b8c:research/p104-owned-coding-boundaries/inputs/pr5580-WORD.json.gz"
+    }
+  ],
+  "retained_origin": {
+    "pr_url": "https://github.com/CrocSwap/integer-mult-bounds/pull/5580",
+    "head": "6efa248cf456d6101d757e9d09ba29711d18a13a",
+    "original_path": "research/p104-event-contact-step/WORD.json.gz",
+    "scope": "Byte-identical source with attribution retained in this branch history; no foreign checkout required for replay."
+  }
+}
diff --git a/research/p104-owned-coding-boundaries/TASK.json b/research/p104-owned-coding-boundaries/TASK.json
new file mode 100644
index 0000000000..cffa28d16c
--- /dev/null
+++ b/research/p104-owned-coding-boundaries/TASK.json
@@ -0,0 +1,113 @@
+{
+  "schema_version": 1,
+  "task_id": "scalar-to-moment",
+  "priority_revision": 7,
+  "base_commit": "5e0052ca0aa3036a3afa8b0438927377c8d97d53",
+  "original_card": {
+    "id": "scalar-to-moment",
+    "question": "Can a changed frame itinerary remove a paid transition beyond the already covered scalar A2 contractions?",
+    "starting_points": [
+      "docs/research/research-family-status.md",
+      "certificates/selected-result.json",
+      "research/staged544-a2-graph-rewrite/README.md",
+      "research/staged544-separated-a2/README.md",
+      "research/staged544-a2-composition-audit/README.md",
+      "research/staged544-a1858-contraction-audit/README.md"
+    ],
+    "methods": [
+      "Derive the changed legal frame itinerary and exact child histogram.",
+      "Prove a scoped obstruction showing why that scalar gain cannot change the paid contract."
+    ],
+    "acceptance_gate": "An unchanged histogram is not a kappa gain; record any separate runtime or proof benefit explicitly.",
+    "first_step": {
+      "command": [
+        "python3",
+        "-B",
+        "scripts/research_baseline.py"
+      ],
+      "expected": "PASS_RESEARCH_BASELINE_BINDINGS_ONLY",
+      "inspect": "Read the A2 and separated-A2 constructions and their composition/contraction audits. Scalar contractions can preserve the complete recursive histogram; fewer ADDs alone do not justify further scans.",
+      "scope": "Source-binding orientation only; no scientific replay, theorem acceptance or supplier admission. Budget any heavy replay separately."
+    },
+    "first_experiment": "Identify an actual paid transition to remove and construct a legal replacement frame itinerary, charging restoration, source windows and changed helpers. Compare complete before/after histograms. Stop if only scalar count changes; report that benefit honestly, without expanding the same scalar-only census.",
+    "bottleneck_link": "A legal change to the histogram, not addition count alone, is the bridge to an exponent gain.",
+    "allowed_variations": [
+      "One existing scalar simplification and one proposed frame-path change."
+    ],
+    "scope_change_triggers": [
+      "Submitting only another scalar count as completion of a paid-profile task."
+    ],
+    "overlap_terms": [
+      "scalar simplification",
+      "paid profile",
+      "R439",
+      "3998",
+      "4019",
+      "4112",
+      "4152",
+      "staged544-a2-graph-rewrite",
+      "staged544-separated-a2",
+      "staged544-a2-composition-audit",
+      "staged544-a1858-contraction-audit"
+    ],
+    "existing_checks": [
+      {
+        "scope": "PR #3998: consecutive A2 scalar contraction",
+        "evidence": "research/staged544-a2-graph-rewrite/README.md",
+        "remaining": "A genuinely changed paid itinerary or a distinct independent audit; no default scalar-only expansion.",
+        "status": "author-checked",
+        "source_commit": "389268a6b5e0373a3125d5c6fcd6995f4ea324f5",
+        "selected_record_sha256": "bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129"
+      },
+      {
+        "scope": "PR #4019: separated A2 sites with unchanged recursive histogram",
+        "evidence": "research/staged544-separated-a2/README.md",
+        "remaining": "A genuinely changed paid itinerary or a distinct independent audit; no default scalar-only expansion.",
+        "status": "author-checked",
+        "source_commit": "f408e09df300b575235f7f77f81948071397e16d",
+        "selected_record_sha256": "bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129"
+      },
+      {
+        "scope": "PR #4112: composition audit",
+        "evidence": "research/staged544-a2-composition-audit/README.md",
+        "remaining": "A genuinely changed paid itinerary or a distinct independent audit; no default scalar-only expansion.",
+        "status": "author-checked",
+        "source_commit": "205b7380656ba4b6316a8a493b1a3c679b8f7a0e",
+        "selected_record_sha256": "bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129"
+      },
+      {
+        "scope": "PR #4152: A1858 contraction audit",
+        "evidence": "research/staged544-a1858-contraction-audit/README.md",
+        "remaining": "A genuinely changed paid itinerary or a distinct independent audit; no default scalar-only expansion.",
+        "status": "author-checked",
+        "source_commit": "caaa14cde5962adc372fdca71f63d0f29e73c8b2",
+        "selected_record_sha256": "bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129"
+      }
+    ]
+  },
+  "scope_status": "refined",
+  "actual_scope": "Exact rational recoding classification and joint transport floor for all eight owned blocks of the pinned P104 source.",
+  "scope_change_reason": "User explicitly requested changed public dependencies and sharing across owned blocks. This checkpoint closes fixed-producer and fixed-boundary templates before pursuing earlier preparation.",
+  "bottleneck_link": "A cheaper scalar circuit is useful for kappa only if its paid frame itinerary changes. These exact certificates distinguish recodings that cannot lower owned transport from still-open changes in dependencies.",
+  "budget_minutes": 180,
+  "stop_rule": "Stop expanding the fixed-producer coefficient family after exact all-rational classification. Continue the separately authorized broader research until the user usage cutoff; this package is one verified checkpoint.",
+  "outcome": "negative-result",
+  "evidence": "PROOF.md, JOINT-BOUNDARY.md, RESULT.json; exact replay in verify.py with four semantic corruption controls.",
+  "author": "flyingviper with substantial OpenAI Codex assistance and independent agents",
+  "ai_assistance": "Substantial OpenAI Codex assistance for derivation, implementation, literature work and independent agent checks; no human or proof-kernel acceptance claimed.",
+  "open_questions": [
+    {
+      "question": "Can earlier public preparation or changed outside side information beat the fixed 41-role boundary floor?",
+      "why_it_matters": "The rank-cut hypotheses no longer apply when producer information is already available below the cuts.",
+      "first_step": "Specify an exact coupled residual transfer at actual value versions and legal frames, then charge preparation, storage and restoration.",
+      "blocked_by": "No complete legal cheaper replacement has been found."
+    },
+    {
+      "question": "Can the exceptional owned block attain the joint 31H1+8H7 floor?",
+      "why_it_matters": "The source has a fragmentation penalty 2H1+H2+H3-H7 that this lower bound does not prove unavoidable.",
+      "first_step": "Construct all middle-level deliveries along with the final line observations, without changing public endpoints or adding unpaid information.",
+      "blocked_by": "The cut certificate does not establish attainability."
+    }
+  ],
+  "overlap_note": "PR5584 predecessor; PR5579 fixed public windows; Discussions5574/5575 event retiming and compensated source substitution; PR5590 fixed-contact path floor. These were checked against the newest100 openPRs and relevant discussion pages; bounded coverage, not novelty proof."
+}
diff --git a/research/p104-owned-coding-boundaries/VERIFY.json b/research/p104-owned-coding-boundaries/VERIFY.json
new file mode 100644
index 0000000000..67e8fd7f0e
--- /dev/null
+++ b/research/p104-owned-coding-boundaries/VERIFY.json
@@ -0,0 +1,21 @@
+{
+  "schema_version": 1,
+  "status": "ready",
+  "python": "3.12",
+  "requirements": [],
+  "inputs": [
+    "research/p104-owned-coding-boundaries/inputs/pr5580-WORD.json.gz"
+  ],
+  "commands": [
+    {
+      "argv": [
+        "{python}",
+        "-I",
+        "-B",
+        "{package}/verify.py"
+      ],
+      "timeout_seconds": 120,
+      "checks": "All-511 exact rational hyperplane census; eight delivery separators modulo zero-entry input corrections; full arbitrary-input scalar/frame replay; independent 41-role prefix and suffix ranks; seven rank-eight information cuts; four semantic rejection controls and committed result comparison."
+    }
+  ]
+}
diff --git a/research/p104-owned-coding-boundaries/classification.py b/research/p104-owned-coding-boundaries/classification.py
new file mode 100644
index 0000000000..5070e80520
--- /dev/null
+++ b/research/p104-owned-coding-boundaries/classification.py
@@ -0,0 +1,112 @@
+"""All-rational hyperplane classification of the eight frozen producer values.
+
+Forward reconstruction and Fraction elimination are independent of the
+backward producer extraction used during discovery. Apache-2.0.
+"""
+from collections import Counter
+from fractions import Fraction as Q
+
+
+def need(ok, message):
+    if not ok:
+        raise ValueError(message)
+
+
+def rref(matrix, n=8):
+    a = [list(row) for row in matrix]
+    pivots = []
+    for c in range(n):
+        r = len(pivots)
+        pivot = next((i for i in range(r, len(a)) if a[i][c]), None)
+        if pivot is None:
+            continue
+        a[r], a[pivot] = a[pivot], a[r]
+        q = a[r][c]
+        a[r] = [v/q for v in a[r]]
+        for i in range(len(a)):
+            if i != r and a[i][c]:
+                q = a[i][c]
+                a[i] = [v-q*w for v, w in zip(a[i], a[r])]
+        pivots.append(c)
+        if len(pivots) == n:
+            break
+    kernels = []
+    for free in sorted(set(range(n))-set(pivots)):
+        v = [Q(0)]*n
+        v[free] = Q(1)
+        for row, c in enumerate(pivots):
+            v[c] = -a[row][free]
+        q = next(x for x in v if x)
+        kernels.append([x/q for x in v])
+    return len(pivots), kernels
+
+
+def binary_basis(vectors):
+    out = []
+    for v in vectors:
+        for w in out:
+            v = min(v, v ^ w)
+        if v:
+            out.append(v)
+            out.sort(reverse=True)
+    return out
+
+
+def classify(word):
+    frames, events = word['frames'], word['program']
+    need(all(0 < v < 512 for f in frames for v in f), 'ambient binary dimension nine')
+    rows = [{i: Q(1)} for i in range(word['public'])]
+    first = next(i for i, e in enumerate(events) if e[0] == 'INIT')
+    for e in events[:first]:
+        if e[0] == 'MOVE':
+            continue
+        d = e[1]
+        if e[0] == 'SCALE':
+            rows[d] = {i: Q(e[-2])*v for i, v in rows[d].items()}
+        elif e[0] == 'ADD':
+            for i, v in rows[e[2]].items():
+                q = rows[d].get(i, Q(0))+Q(e[-2])*v
+                if q:
+                    rows[d][i] = q
+                else:
+                    rows[d].pop(i, None)
+        else:
+            raise ValueError('noninvertible public prefix')
+    copies = [(i, e) for i, e in enumerate(events) if e[0] == 'COPY']
+    need(len(copies) == 8, 'eight frozen producers')
+    producers = []
+    values = []
+    for i, e in copies:
+        need(not any(x[0] != 'MOVE' and x[1] == e[2] for x in events[first:i]),
+             'source value unchanged between joint boundary and COPY')
+        clear = next(x for x in events[i+1:] if x[0] == 'CLEAR' and x[1] == e[1])
+        values.append(rows[e[2]])
+        producers.append({'owned': e[1], 'source_role': e[2], 'copy_event': i,
+                          'copy_frame': e[-1], 'clear_frame': clear[-1],
+                          'initial_covector': {str(k): str(v) for k, v in sorted(rows[e[2]].items())}})
+    counts, modes = Counter(), []
+    for normal in range(1, 512):
+        outside = [i for i, f in enumerate(word['initial'])
+                   if any((normal & v).bit_count() % 2 for v in frames[f])]
+        matrix = [[row.get(i, Q(0)) for row in values] for i in outside]
+        rank, kernels = rref(matrix)
+        counts[rank] += 1
+        for coefficients in kernels:
+            combination = {}
+            for c, row in zip(coefficients, values):
+                for i, v in row.items():
+                    combination[i] = combination.get(i, Q(0))+c*v
+            combination = {i: v for i, v in combination.items() if v}
+            support = binary_basis(v for i in combination for v in frames[word['initial'][i]])
+            need(not set(combination).intersection(outside), 'exact nullspace lies inside hyperplane')
+            need(len(support) == 8, 'every nonzero proper-support mode has dimension eight')
+            modes.append({'normal': normal, 'coefficients': [str(v) for v in coefficients],
+                          'entry_support_dimension': len(support), 'entry_support_basis': support,
+                          'original_symbols': len(combination),
+                          'mixed': sum(bool(v) for v in coefficients) > 1})
+    need(counts == {8: 495, 7: 16}, 'complete all-511 rational rank census')
+    need(len(modes) == 16 and sum(m['mixed'] for m in modes) == 8, 'eight original and eight mixed lines')
+    receipt = {'first_owned_init': first, 'source_roles': [e[2] for _, e in copies],
+               'rank_counts': {str(k): v for k, v in sorted(counts.items())}, 'modes': modes,
+               'scope': 'Every rational combination of the unchanged eight producer values; original-entry support geometry only.'}
+    return receipt, modes, producers
diff --git a/research/p104-owned-coding-boundaries/delivery.py b/research/p104-owned-coding-boundaries/delivery.py
new file mode 100644
index 0000000000..b316373d40
--- /dev/null
+++ b/research/p104-owned-coding-boundaries/delivery.py
@@ -0,0 +1,171 @@
+"""Exact direct-delivery gate for the supplied P104 sixteen-mode family.
+
+This does not establish that the family is complete: the all-hyperplane
+classification is a separate obligation. Producer covectors must be extracted
+independently from the word. This routine verifies their support geometry, the
+supplied mixed-mode identities, and all eight quotient-space delivery tests.
+"""
+from fractions import Fraction as Q
+
+
+def need(ok, why):
+    if not ok:
+        raise ValueError(why)
+
+
+def rref(rows, width):
+    a = [[Q(x) for x in row] for row in rows]
+    need(all(len(row) == width for row in a), "matrix width")
+    pivots = []
+    for col in range(width):
+        hit = next((j for j in range(len(pivots), len(a)) if a[j][col]), None)
+        if hit is None:
+            continue
+        k = len(pivots)
+        a[k], a[hit] = a[hit], a[k]
+        v = a[k][col]
+        a[k] = [x / v for x in a[k]]
+        for j in range(len(a)):
+            if j != k and a[j][col]:
+                v = a[j][col]
+                a[j] = [x - v * y for x, y in zip(a[j], a[k])]
+        pivots.append(col)
+        if len(pivots) == len(a):
+            break
+    return a, pivots
+
+
+def rank(rows, width):
+    return len(rref(rows, width)[1])
+
+
+def dot(a, b):
+    return sum((x * y for x, y in zip(a, b)), Q(0))
+
+
+def parity_dot(a, b):
+    return (a & b).bit_count() % 2
+
+
+def binary_basis(vectors):
+    basis = {}
+    for x in vectors:
+        x = int(x)
+        while x:
+            p = x.bit_length() - 1
+            if p in basis:
+                x ^= basis[p]
+            else:
+                basis[p] = x
+                break
+    return [basis[p] for p in sorted(basis, reverse=True)]
+
+
+def sparse(row):
+    return {int(i): Q(v) for i, v in row.items() if Q(v)}
+
+
+def mix(coefficients, producers):
+    out = {}
+    for a, row in zip(coefficients, producers):
+        for j, v in row.items():
+            out[j] = out.get(j, Q(0)) + a * v
+    return {j: v for j, v in out.items() if v}
+
+
+def separator(rows, target, n):
+    unit = [Q(int(j == target)) for j in range(n)]
+    a, pivots = rref([list(row) + [Q(0)] for row in rows] + [unit + [Q(1)]], n + 1)
+    need(n not in pivots, "target is in the alternate delivery span")
+    answer = [Q(0)] * n
+    for row, p in zip(a, pivots):
+        answer[p] = row[-1]
+    need(answer[target] == Q(1), "normalized separator")
+    need(all(dot(row, answer) == Q(0) for row in rows), "separator annihilates alternates")
+    return answer
+
+
+def verify_delivery(word, exact_modes, producers):
+    """Return an exact receipt or raise ValueError; no asserts or float arithmetic.
+
+    Scope: unchanged final payloads in span(D), direct nested 8-to-1 deliveries
+    starting at each payload's original-entry support, fixed final lines, and
+    arbitrary free corrections from zero-entry-frame original inputs. Earlier
+    owned preparation and intermediate components outside span(D) remain open.
+    """
+    n = len(producers)
+    need(n == 8, "eight supplied source producers")
+    need(len(exact_modes) == 16, "sixteen supplied projective modes")
+    frames = word["frames"]
+    initial = [frames[f] for f in word["initial"]]
+    ninputs = word["public"]
+    need(len(initial) == ninputs, "original public input frames")
+    D = [sparse(p["initial_covector"]) for p in producers]
+    need(all(0 <= j < ninputs for row in D for j in row), "producer input indices")
+    nonzero_inputs = [j for j, frame in enumerate(initial) if frame]
+    projected = [[row.get(j, Q(0)) for j in nonzero_inputs] for row in D]
+    _, pivot_columns = rref(projected, len(nonzero_inputs))
+    need(len(pivot_columns) == n, "producer independence modulo zero-entry inputs")
+    pivot_inputs = [nonzero_inputs[k] for k in pivot_columns]
+    minor = [[row.get(j, Q(0)) for j in pivot_inputs] for row in D]
+    need(rank(minor, n) == n, "explicit quotient independence minor")
+
+    def support(row):
+        basis = binary_basis(v for j in row for v in initial[j])
+        need(len(basis) == 8, "every tested payload has exact eight-dimensional entry support")
+        normals = [h for h in range(1, 512) if all(parity_dot(h, v) == 0 for v in basis)]
+        need(len(normals) == 1, "unique ambient-nine hyperplane normal")
+        return basis, normals[0]
+
+    family = []
+    lines = []
+    for i, (p, row) in enumerate(zip(producers, D)):
+        event = word["program"][p["copy_event"]]
+        need(event == ["COPY", p["owned"], p["source_role"], "1", p["copy_frame"]], "producer COPY metadata")
+        final_basis = binary_basis(frames[p["clear_frame"]])
+        need(len(final_basis) == 1, "fixed one-dimensional delivery frame")
+        lines.append(final_basis[0])
+        basis, normal = support(row)
+        coeff = [Q(int(i == j)) for j in range(n)]
+        family.append({"name": f"D{i}", "coefficients": coeff, "normal": normal, "basis": basis})
+    seen_original = set()
+    for m in exact_modes:
+        coeff = [Q(a) for a in m["coefficients"]]
+        need(len(coeff) == n and any(coeff), "mixed mode coefficient vector")
+        row = mix(coeff, D)
+        basis, normal = support(row)
+        need(normal == m["normal"], "mixed mode's recomputed entry support normal")
+        if "covector" in m:
+            need(row == sparse(m["covector"]), "mixed mode's exact rational covector identity")
+        original = next((i for i, f in enumerate(family[:n]) if f["coefficients"] == coeff), None)
+        if original is not None:
+            need(original not in seen_original, "unique original mode")
+            need(normal == family[original]["normal"], "original mode support identity")
+            seen_original.add(original)
+        else:
+            family.append({"name": f"E{normal}", "coefficients": coeff, "normal": normal, "basis": basis})
+    need(len(seen_original) == n and len(family) == 16, "eight original and eight mixed modes")
+    need(len({m["normal"] for m in family}) == 16, "sixteen distinct support hyperplanes")
+    receipt = []
+    for i, line in enumerate(lines):
+        own = family[i]
+        need(parity_dot(own["normal"], line) == 0, "original producer admits its direct nested delivery")
+        alternate = [m for m in family if m["name"] != own["name"] and parity_dot(m["normal"], line) == 0]
+        need(len(alternate) == 3 and all(m["name"].startswith("E") for m in alternate), "exactly three mixed admissible alternates")
+        rows = [m["coefficients"] for m in alternate]
+        need(rank(rows, n) == 3, "alternate coefficient rank three")
+        y = separator(rows, i, n)
+        unit = [Q(int(j == i)) for j in range(n)]
+        need(rank(rows + [unit], n) == 4, "target raises admissible rank")
+        receipt.append({"target": i, "delivery_line": line,
+                        "admissible_alternates": [m["name"] for m in alternate],
+                        "rank": 3, "target_in_span": False,
+                        "separator": [str(v) for v in y]})
+    return {"scope": "direct nested 8-to-1 sixteen-mode delivery, modulo zero-entry input corrections",
+            "quotient_rank": n, "quotient_pivot_inputs": pivot_inputs,
+            "quotient_minor": [[str(v) for v in row] for row in minor],
+            "source_normals": [m["normal"] for m in family[:n]],
+            "mixed_normals": [m["normal"] for m in family[n:]],
+            "delivery_tests": receipt}
+
+
diff --git a/research/p104-owned-coding-boundaries/joint_boundary.py b/research/p104-owned-coding-boundaries/joint_boundary.py
new file mode 100644
index 0000000000..d72ad43535
--- /dev/null
+++ b/research/p104-owned-coding-boundaries/joint_boundary.py
@@ -0,0 +1,199 @@
+"""Exact coupled-owned-block boundary certificate for the frozen PR5580 word.
+
+The theorem is a fixed 41-public-coordinate boundary theorem. It does not
+charge original input frames as though they were eight-dimensional registers.
+"""
+from fractions import Fraction as Q
+from collections import Counter
+import copy
+
+
+def need(ok, message):
+    if not ok:
+        raise ValueError(message)
+
+
+def add(row, source, coefficient):
+    result = dict(row)
+    for key, value in source.items():
+        out = result.get(key, Q(0)) + coefficient * value
+        if out:
+            result[key] = out
+        else:
+            result.pop(key, None)
+    return result
+
+
+def apply(rows, event):
+    op, dst = event[:2]
+    if op == 'MOVE':
+        return
+    if op in ('INIT', 'CLEAR'):
+        rows[dst] = {}
+    elif op == 'ADD':
+        rows[dst] = add(rows[dst], rows[event[2]], Q(event[3]))
+    elif op == 'SCALE':
+        rows[dst] = {key: Q(event[2])*value for key, value in rows[dst].items()}
+    elif op == 'COPY':
+        rows[dst] = {key: Q(event[3])*value for key, value in rows[event[2]].items()}
+    else:
+        raise ValueError('unknown scalar opcode')
+
+
+def rank(rows):
+    pivots = {}
+    for row in rows:
+        row = dict(row)
+        while row:
+            col = min(row)
+            if col not in pivots:
+                a = row[col]
+                pivots[col] = {key: value/a for key, value in row.items()}
+                break
+            row = add(row, pivots[col], -row[col])
+    return len(pivots)
+
+
+def span(basis):
+    result = {0}
+    for vector in basis:
+        result |= {x ^ vector for x in tuple(result)}
+    need(len(result) == 1 << len(basis), 'binary basis independent')
+    return result
+
+
+def observations_rank(observations):
+    return rank([{int(k):Q(v) for k,v in row.items()} for row in observations])
+
+
+def verify(word):
+    public, events, frames = word['public'], word['program'], word['frames']
+    need(public == 735, 'frozen public inventory')
+    n = public+8
+    copies = [(i,e) for i,e in enumerate(events) if e[0] == 'COPY']
+    need([e[1] for _,e in copies] == list(range(735,743)), 'eight owned producers')
+    sources = [e[2] for _,e in copies]
+    need(sources == [173,151,146,133,256,92,9,240], 'frozen source order')
+    start = next(i for i,e in enumerate(events) if e[0] == 'INIT')
+    stop = 1+max(i for i,e in enumerate(events) if e[0] == 'CLEAR')
+    block = events[start:stop]
+    deliveries = [(start+i,e) for i,e in enumerate(block) if e[0] == 'ADD']
+    recipients = [e[1] for _,e in deliveries]
+    need(len(recipients) == len(set(recipients)) == 33, '33 distinct recipients')
+    roles = sources+recipients
+    need(len(set(roles)) == 41, '41 distinct public boundary coordinates')
+    need(all(e[1] in roles or e[1] >= public for e in block), 'no outside destination')
+    need(all(e[2] in roles or e[2] >= public for e in block if e[0] in ('ADD','COPY')),
+         'no outside public read')
+
+    rows = [{i:Q(1)} for i in range(public)]+[{} for _ in range(8)]
+    current = list(word['initial'])+[None]*8
+    spans = [span(f) for f in frames]
+    active = set()
+    entry, exit_frames, prefix = None, None, None
+    for i,e in enumerate(events):
+        if i == start:
+            entry = list(current)
+            prefix = [dict(rows[r]) for r in roles]
+        op,r = e[:2]
+        if op == 'MOVE':
+            need(current[r] == e[2] and e[4] == (r in active), 'MOVE frame and ownership')
+            need(spans[e[3]] < spans[e[2]] if e[4] else spans[e[2]] < spans[e[3]], 'nested MOVE direction')
+            current[r] = e[3]
+        elif op == 'INIT':
+            need(r >= public and r not in active, 'fresh owned field')
+            current[r] = e[2]
+            active.add(r)
+        elif op == 'CLEAR':
+            need(r in active and current[r] == e[2], 'owned CLEAR frame')
+            current[r] = None
+            active.remove(r)
+        else:
+            need(Q(e[-2]) != 0 and current[r] == e[-1], 'nonzero scalar at current chart')
+            if op in ('ADD','COPY'):
+                need(e[2] != r and current[e[2]] == current[r], 'common scalar chart')
+            if op == 'COPY':
+                need(r in active and not rows[r], 'COPY into zero owned field')
+        apply(rows,e)
+        if i+1 == stop:
+            exit_frames = list(current)
+    need(not active and current[:public] == word['final'], 'full frame exit')
+    for r,row in enumerate(rows):
+        expected = {r:Q(1)} if r < public else {}
+        if r-112 in word['live']:
+            expected[r-112] = Q(1)
+        need(row == expected, 'full arbitrary-input scalar exit')
+    need(rank(prefix) == 41, 'independent actual local entry coordinates')
+
+    local = [{} for _ in range(n)]
+    for j,r in enumerate(roles):
+        local[r] = {j:Q(1)}
+    for e in block:
+        apply(local,e)
+    need(all(local[r] == {j:Q(1)} for j,r in enumerate(sources)), 'all eight producers unchanged')
+    need(all(local[r] == {} for r in range(public,n)), 'all private exits zero')
+    need(all(len(frames[entry[r]]) == len(frames[exit_frames[r]]) == 8 for r in sources),
+         'producer entry and exit dimensions eight')
+    target_sources = {}
+    for j,r in enumerate(recipients,8):
+        need(local[r].get(j) == 1 and len(local[r]) == 2, 'recipient keeps arbitrary input plus one producer')
+        source = next(k for k in local[r] if k != j)
+        need(source < 8 and local[r][source] != 0, 'nonzero original producer transfer')
+        target_sources[r] = source
+    observed_roles = []
+    observed = []
+    for s in range(8):
+        role = next((r for r in recipients if target_sources[r] == s and len(frames[exit_frames[r]]) == 1), None)
+        need(role is not None, 'one one-dimensional observation per producer')
+        observed_roles.append(role)
+        observed.append({str(k):str(v) for k,v in local[role].items() if k < 8})
+    need(observations_rank(observed) == 8, 'eight independent low-dimensional producer observations')
+    for k in range(2,9):
+        need(all(len(frames[entry[r]]) >= k for r in sources), 'no producer coordinate initially below threshold')
+        need(all(len(frames[exit_frames[r]]) < k for r in observed_roles), 'all observations below threshold')
+
+    suffix = [{} for _ in range(n)]
+    for j,r in enumerate(roles):
+        suffix[r] = {j:Q(1)}
+    for e in events[stop:]:
+        apply(suffix,e)
+    need(rank(suffix[:public]) == 41, 'fixed suffix observes every local public coordinate')
+
+    public_hist, owned_hist = Counter(), Counter()
+    threshold_crossings = {k:0 for k in range(2,9)}
+    for e in block:
+        if e[0] != 'MOVE':
+            continue
+        a,b = len(frames[e[2]]),len(frames[e[3]])
+        (owned_hist if e[4] else public_hist)[abs(a-b)] += 1
+        if e[4]:
+            for k in threshold_crossings:
+                threshold_crossings[k] += b < k <= a
+    need(public_hist == {1:31}, 'source public histogram 31H1')
+    need(owned_hist == {7:7,1:2,2:1,3:1}, 'source owned histogram 7H7+2H1+H2+H3')
+    need(all(count == 8 for count in threshold_crossings.values()), 'source attains each threshold crossing rank')
+    public_rank = sum(len(frames[exit_frames[r]])-len(frames[entry[r]]) for r in roles)
+    need(public_rank == 31, 'public entry-exit rank change 31')
+    need(all(len(frames[exit_frames[r]])-len(frames[entry[r]]) in (0,1) for r in roles),
+         'each physical public field requires at most one rank of ascent')
+    corrupted = copy.deepcopy(observed)
+    corrupted[-1] = dict(corrupted[0])
+    need(observations_rank(corrupted) == 7, 'duplicate observation reduces the rank bound')
+    return {
+        'status':'PASS_JOINT_41_ROLE_OWNED_MOMENT_FLOOR',
+        'events_half_open':[start,stop], 'public_roles':roles,
+        'sources':sources,'recipients':recipients,
+        'prefix_rank':41,'suffix_rank':41,
+        'source_entry_dimensions':[len(frames[entry[r]]) for r in sources],
+        'observed_roles':observed_roles,'producer_projection_at_observations':observed,
+        'threshold_crossing_count':{str(k):v for k,v in threshold_crossings.items()},
+        'public_paid_histogram':{str(k):v for k,v in sorted(public_hist.items())},
+        'owned_paid_histogram':{str(k):v for k,v in sorted(owned_hist.items())},
+        'theorem_floor':'31 H1(t) + 8 H7(t), 0<t<=1',
+        'maximum_possible_local_saving':'2 H1(t) + H2(t) + H3(t) - H7(t)',
+        'duplicate_observation_scope_control_rank':7,
+        'scope':'Fixed 41 independent public entry/exit rows and frames; arbitrary private auxiliaries and cross-block sharing; no outside public side information, changed helper credits or earlier public preparation.',
+        'new_bound':False,
+    }
+
+
diff --git a/research/p104-owned-coding-boundaries/verify.py b/research/p104-owned-coding-boundaries/verify.py
new file mode 100644
index 0000000000..0f297cc2fb
--- /dev/null
+++ b/research/p104-owned-coding-boundaries/verify.py
@@ -0,0 +1,82 @@
+#!/usr/bin/env python3
+"""Verify the scoped owned-coding boundaries; no new construction or bound."""
+from pathlib import Path
+from hashlib import sha256
+import argparse
+import copy
+import gzip
+import importlib.util
+import json
+
+HERE = Path(__file__).resolve().parent
+SOURCE_SHA = 'b2a98c4bd0dbe86c09e9d15ee060dcf1aab09dbef90aa1e2127676f4039f406b'
+
+
+def load(name):
+    path = HERE/(name+'.py')
+    spec = importlib.util.spec_from_file_location('p104_coding_'+name, path)
+    module = importlib.util.module_from_spec(spec)
+    spec.loader.exec_module(module)
+    return module
+
+
+def need(ok, message):
+    if not ok:
+        raise ValueError(message)
+
+
+def verify():
+    raw = (HERE/'inputs/pr5580-WORD.json.gz').read_bytes()
+    need(sha256(raw).hexdigest() == SOURCE_SHA, 'retained PR5580 source bytes')
+    word = json.loads(gzip.decompress(raw))
+    classification = load('classification')
+    delivery = load('delivery')
+    boundary = load('joint_boundary')
+    census, modes, producers = classification.classify(word)
+    direct = delivery.verify_delivery(word, modes, producers)
+    joint = boundary.verify(word)
+    controls = {}
+
+    def rejects(label, fn):
+        try:
+            fn()
+        except ValueError as error:
+            controls[label] = str(error)
+        else:
+            raise ValueError('accepted semantic corruption: '+label)
+
+    rejects('missing_mode', lambda: delivery.verify_delivery(word, modes[:-1], producers))
+    wrong = copy.deepcopy(modes)
+    wrong[0]['coefficients'][0] = '1'
+    if wrong == modes:
+        wrong[0]['coefficients'][0] = '2'
+    rejects('wrong_mode_coefficient', lambda: delivery.verify_delivery(word, wrong, producers))
+    wrong_word = copy.deepcopy(word)
+    first_copy = producers[0]['copy_event']
+    wrong_word['program'][first_copy][3] = '2'
+    rejects('wrong_copied_payload', lambda: boundary.verify(wrong_word))
+    wrong_word2 = copy.deepcopy(word)
+    move = next(e for e in wrong_word2['program'] if e[0] == 'MOVE' and e[4])
+    move[4] = False
+    rejects('owned_descent_declared_public', lambda: boundary.verify(wrong_word2))
+    return {'status': 'PASS_P104_OWNED_CODING_BOUNDARIES', 'new_bound': False,
+            'source': {'pr': 5580, 'head': '6efa248cf456d6101d757e9d09ba29711d18a13a', 'sha256': SOURCE_SHA},
+            'classification': census, 'direct_delivery': direct,
+            'joint_boundary': joint, 'negative_controls': controls}
+
+
+def main():
+    parser = argparse.ArgumentParser(description=__doc__)
+    parser.add_argument('--write', action='store_true', help='explicitly regenerate RESULT.json')
+    args = parser.parse_args()
+    result = verify()
+    encoded = json.dumps(result, indent=2, sort_keys=True)+'\n'
+    if args.write:
+        (HERE/'RESULT.json').write_text(encoded, encoding='utf-8')
+    else:
+        need(json.loads((HERE/'RESULT.json').read_bytes()) == result, 'committed result equals fresh exact replay')
+    print(result['status'])
+
+
+if __name__ == '__main__':
+    main()