Exclude arbitrary linear recoding across the helper-572 seam (no κ)

research/p104-unrestricted-572-seam/PROOF.md

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+# Arbitrary linear recoding cannot repair the frozen helper-572 seam
+
+This scoped proof was produced by Nick Petalas. All scalars are rational.
+The finite certificate establishes the boundary facts below from the pinned
+selected PR6030 word. It does not replay a predecessor's exclusion theorem.
+
+## 1. Contract
+
+Keep the original word, its public inputs, frames, owned lifetimes and event
+order. Event indices are zero based; a window is immediately after its event.
+After event 13637, `COPY 778 <- 572` in chart 595, overwrite field 572 with a
+new independent scalar input z. Every original public input remains arbitrary.
+
+Allow any finite number of homogeneous linear operations, with arbitrary
+rational coefficients, within these banks:
+
+- Chart 595 after events 13637 through 13651, on fields
+  `(127,572,581,582,731,778)`.
+- Chart 603 after event 13652 or 13654, on `(572,740)`, and after event 13655,
+  on `(572,709,740)`.
+
+Operations may change original-input rows temporarily. They need not preserve
+an original-zero carrier. No extra field, new MOVE or changed prefix/suffix is
+permitted. We grant arbitrary linear bank maps, including singular ones, so the
+exclusion also covers ADD/SCALE programs of unbounded finite length in these
+regions. COPY or erasure within the granted banks cannot evade it.
+
+Put V equal to field 572's original final chart, 127. The necessary condition is
+that every public output whose original final chart differs from V equal its
+original unsplit value, on all original inputs and z. There are 186 such outputs.
+The desired additional target would require nonzero public dependence on z.
+
+**Theorem.** Under this contract, satisfying the protected-output condition
+forces every final public output to have zero z coefficient. Thus no nonzero
+independent target can be supplied by these corrections, regardless of their
+number or coefficients.
+
+## 2. Collapse the chronology to two matrices
+
+The first bank remains exactly the six listed fields through event 13651.
+All intervening operations in other charts are unaffected. Absorb the retained
+`582 += 2*731` and every allowed first-bank operation into one arbitrary 6 by 6
+map A at the first bank's exit.
+
+At event 13652, only field 572 moves from chart 595 to 603. Field 740 is already
+there. The retained `740 -= 572` at event 13654 and every allowed operation on
+these two fields are linear. Field 709 enters from chart 597 at event 13655;
+its value has not been changed by any allowed operation. Extend the earlier
+2 by 2 maps by the identity on that entering field and absorb them with the
+last operations into one arbitrary 3 by 3 map B.
+
+Consequently only one scalar form r from the first bank reaches the second.
+The second bank's three values are linear combinations of `(r,h,k)`, where h
+and k are the original cut values of fields 740 and 709. Five first-bank fields
+remain in chart 595. CLEAR 778 immediately follows, at event 13656.
+
+This is a relaxation: arbitrary A and B need not be realizable by reversible
+scalar gates. Proving failure in this larger class suffices.
+
+## 3. The exact source and suffix certificates
+
+Let u be the eight-component vector of cut forms in this order:
+
+```text
+u = (old127, z, old581, old582, old731, ell, h, k),
+ell = old572 = old778 after the retained COPY.
+```
+
+These eight forms are linearly independent in the original 770 inputs and z.
+The exact minor on input columns
+`(0,2,34,572,581,582,740,770)` has determinant `1/32`; column 770 denotes z.
+The verifier derives the forms from the prefix and checks this determinant
+by the Leibniz formula, separately from its pivot selection.
+
+Use the following order for state differences immediately after event 13655:
+
+```text
+(127,581,582,731,778,572,740,709).
+```
+
+Propagating their eight unit perturbations through the unchanged suffix shows:
+
+1. The perturbation on 778 reaches no final public output.
+2. The perturbation on 740 reaches only final public output 740, with coefficient 1.
+3. Output 740 is V-final. The six outputs below are protected.
+
+After deleting columns 778 and 740, the suffix rows on protected outputs
+`(116,120,127,134,138,176)` are exactly
+
+```text
+          127 581 582 731 572 709
+116        0   1   0   2   0   0
+120        0   0   0   0  -1   2
+127        1   0   0   0   0   0
+134        0   0   0  -2   0   0
+138        0   0   0   0  -1  -2
+176        0   0  -1   0   0   0
+```
+
+Its determinant is 8. Therefore preserving these six outputs alone forces the
+six corresponding state forms `(127,581,582,731,572,709)` to equal their original
+unsplit forms. This deduction is an identity in all original input columns and
+z, not just a statement about the new input's leakage.
+
+For completeness, the original eight post-seam state forms are, in the same
+order as the eight state differences,
+
+```text
+(old127, old581, old582+2*old731, old731, ell, ell, h-ell, k).
+```
+
+The verifier checks every coefficient of these identities against the prefix.
+It also checks the two exceptional suffix columns at every public output.
+Together the two nonzero minors and these identities are a finite certificate
+of all source-specific premises used in the argument.
+
+## 4. The single-channel obstruction
+
+Write the transmitted form as `r = a*u_first`, where `u_first` consists of the
+first six entries of u. Since second-bank values are B applied to `(r,h,k)`,
+there are rational coefficients d,e,f with
+
+```text
+post572 = d*r + e*h + f*k.
+```
+
+Section 3 forces `post572 = ell`. Independence of u therefore gives
+
+```text
+d*a = (0,0,0,0,0,1),  e=f=0.
+```
+
+The right side is nonzero, so d is nonzero. Thus a is a scalar multiple of the
+ell coordinate vector. In particular r has zero z coefficient. Neither h nor
+k contains z, so no second-bank field can contain z, including field 740.
+
+The other four non-erased first-bank forms are forced to their original values
+by the same minor and hence contain no z. The remaining first-bank field, 778,
+has identically zero public suffix response. Every other state field is
+unchanged by the allowed corrections. Therefore every final public z
+coefficient vanishes, proving the theorem.
+
+Equivalently, the second stage's first-six-input coefficient block has rank at most one.
+It must contain the ell row; a nonzero z coordinate in its 740 row would raise
+its rank to at least two. A second independent transported scalar could change
+that conclusion. No price or physical realization for such a transport is
+asserted here.
+
+## 5. What is new and what remains
+
+JoyboyBrian's PR6088 excludes two added ADDs in these regions by inconsistent
+endpoint equations. The present result removes the scalar-operation-count
+bound and permits arbitrary linear bank recoding. It uses a protected-output
+minor and a one-channel factorization rather than the predecessor's pair
+census or bilinear relaxation. The predecessor's counts are not rerun.
+
+The carrier-only theorems of apoursh PR6067 and Jonathan Capalbo PR6069 require
+original-zero correction sources. This proof permits original-input changes.
+JosieElliston PR6108 distinguishes the remaining cross-time problem and supplies
+a different scalar witness on PR5907. Nick Petalas PR6040 is the earlier
+unit-injection census, not an arbitrary-input recoding theorem.
+
+The exclusion fixes the overwrite, these two chart regions and all later
+original events. Later correction windows, a pre-cut save, a second transported
+form, changed routes or storage, nonlinear encodings, and another word are
+outside scope. There is no physical/native supplier, productive normalization,
+complete signed invoice, compiler/precision/caller/outer proof, new kappa or
+complete multiplication theorem. The five selected framework assumptions
+remain. Independent human and formal proof-kernel review are unclaimed.
+
+Credit JoyboyBrian, apoursh, Jonathan Capalbo, JosieElliston, Matarener,
+flyingviper, Ali Romman, DreamingOfClouds and the retained P104 lineage.
+Apache-2.0.

research/p104-unrestricted-572-seam/README.md

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+# Unlimited linear corrections cannot repair the frozen helper-572 seam
+
+This contribution was produced by Nick Petalas. It proves a scoped obstruction
+on the selected PR6030 word: after overwriting helper 572 at cut 13637, arbitrary
+finite linear recoding inside the declared chart-595 and chart-603 regions
+cannot preserve the protected original outputs and retain any nonzero public
+response to the new independent input.
+
+This extends JoyboyBrian's [PR6088](https://github.com/CrocSwap/integer-mult-bounds/pull/6088)
+from two added ADDs to arbitrary linear bank maps. Increasing the scalar
+correction count in these regions cannot repair the split. A changed transport,
+window, prefix or storage contract is needed. No new kappa is claimed.
+
+## Result and evidence
+
+[PROOF.md](PROOF.md) specifies the complete contract and proves the theorem.
+The first bank has six fields. Only field 572 moves to the second bank before
+field 709 enters. Six protected output equations force the transmitted form to
+recover the displaced old helper value; a single linear channel then cannot
+also carry the new independent input.
+
+The exact standard-library certificate derives eight independent source forms
+from the prefix, with a minor determinant `1/32`, and a six-row protected suffix
+minor of determinant `8`. It checks the complete original-input boundary
+identities, every public response of the erased 778 and V-final 740 columns,
+and the source chronology. The proof uses these finite facts to cover every
+finite correction length and arbitrary rational coefficients.
+
+The verifier also enumerates 722 rank-one products over the fields of orders
+2, 3 and 5, checks a two-channel positive scope control and a singular minor,
+and rejects four changed schedule/original-input fixtures and changed source
+bytes. These controls corroborate the rank argument; the universal rational
+statement is the written proof. No predecessor census or theorem is replayed.
+The generated JSON receipt binds the package files and exact source hash.
+Default execution writes nothing.
+
+## Reproduce
+
+From the repository root, using the standard-library CPython 3.12 interpreter:
+
+```sh
+/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/p104-unrestricted-572-seam/verify.py
+/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -O -W error -B research/p104-unrestricted-572-seam/verify.py
+/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/p104-unrestricted-572-seam
+/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/p104-unrestricted-572-seam --check-source-history --check-modes
+```
+
+Another CPython 3.12 path may be substituted. Expected scientific marker:
+`PASS_P104_UNRESTRICTED_572_SEAM`. The registered normal and optimized checks
+have separate 60-second limits. Metadata and committed-byte checks require a
+Git checkout. The direct verifier requires only its package and declared word.
+
+## Scope and sources
+
+Scientific base: canonical main `a5bff3bf133e482541c09eba4e773884bdb2b1b7`,
+priority revision 8. The selected record gives conditional kappa `0.003125804`.
+The current source-binding orientation passes; it is not a selected cold replay.
+[SOURCES.json](SOURCES.json) pins the exact selected word and historical task
+provenance. [TASK.json](TASK.json) records the changed scope and next questions.
+
+The immediate question source is JoyboyBrian's PR6088 at
+`6e6236ec763d2aa3eac40201c9634051c583320c`,
+`research/p104-noncarrier-two-add-seam/PROOF.md`. Its finite pair result is cited
+context. The general carrier exclusions of apoursh PR6067 and Jonathan Capalbo
+PR6069 are distinct because they preserve original rows at each atomic update.
+JosieElliston PR6108 documents the remaining cross-time gap. The earlier PR6040
+unit-injection census is retained as lineage. This contribution changes the
+correction model; it does not retarget that census to another word or group.
+
+[Advance claim](https://github.com/CrocSwap/integer-mult-bounds/discussions/1166#discussioncomment-19279693).
+Live handoffs, open/account PRs and local reservations were checked. Merged PR
+and main-tree searches included helper 572, unrestricted recoding, two-chart
+seams and arbitrary correction lengths. The overlap helper reported complete
+remote pagination with 338 open PRs and 5,959 Discussion records, but only
+78 of 8,623 local note paths were materialized. Direct main-tree searches
+supplemented that sparse coverage. Unpublished work and historical code diffs
+remain outside this bounded check; no global novelty certificate is claimed.
+
+No selected native/compiler/caller/outer cold replay, Lean or Docker run is
+needed for this scalar obstruction. No physical construction, cost improvement,
+new supplier or complete multiplication theorem is supplied. All five selected
+framework assumptions remain. Independent human and formal proof-kernel review
+are unclaimed.
+
+## Open next questions
+
+- Can a second independent transferred form repair this split at favorable
+  complete cost? First specify its legal source/destination frames and solve
+  all original-input equations. The missing transport, normalization,
+  restoration and full paid/target invoice are blockers.
+- Can a pre-cut save or changed chronological window avoid the obstruction?
+  First pin that changed boundary and its owned lifetime, then derive its
+  independent source forms and protected-output minor. The present frozen
+  two-region certificate does not transfer automatically.
+
+## Engineering limits
+
+The verifier uses Build the Lever and Prove It Works: the two source-bound
+minors and all boundary equations are reproducible from one declared input.
+No recursive call exists. Loops are bounded by the fixed 22,803 events, 799
+fields, 771 input columns, eight response columns, 722 finite-field products
+and at most `8!` determinant summands. Every function is below 60 lines.
+Explicit failure checks remain enabled under optimization.
+
+Power-of-Ten rule 3 has a managed-runtime deviation in `verify.py`: Python's
+Fraction, JSON and dictionary operations allocate during execution. The scope
+is this offline fixed-input certificate. The authenticated dimensions bound
+state and snapshots by fewer than ten `799*771` coefficient entries, plus the
+fixed word and eight-column suffix. No concurrent work or growing input stream
+is accepted. Warning-strict normal/optimized registered replay is the diagnostic
+check. A fixed-capacity rational representation would be required before reuse
+in a runtime that forbids dynamic allocation.
+
+Credit JoyboyBrian for the seam and question, apoursh, Jonathan Capalbo,
+JosieElliston, Matarener, flyingviper, Ali Romman, DreamingOfClouds and their
+retained P104/native/caller predecessors. The fixed-word chronology and rank
+factorization methods are standard; the new contribution is this source-bound
+unbounded-length obstruction and its compact certificate. Apache-2.0.

research/p104-unrestricted-572-seam/SOURCES.json

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+{
+  "schema_version": 1,
+  "base_commit": "a5bff3bf133e482541c09eba4e773884bdb2b1b7",
+  "files": [
+    {
+      "path": "certificates/selected-result.json",
+      "usage": "provenance",
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+      "sha256": "9b8d0eea95b3bf0dfcb7938a732a6439b87a2685dec2a5ab9e525b3b34debbfa"
+    },
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+      "usage": "provenance",
+      "git_blob": "3404fbdd5e7529a0674eef26114b5071e4586cec",
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+    },
+    {
+      "path": "research/p104-helper470-twin-rung-removal/WORD.json.gz",
+      "usage": "input",
+      "git_blob": "22f9ab02d1b73ea0aa4b07ce445a14cc2602c760",
+      "sha256": "7834725d9e34ee846a9ccc615221ea81e0201613938afa185d2eda0c9a6c232f"
+    },
+    {
+      "path": "research/p104-noncarrier-two-add-seam/PROOF.md",
+      "usage": "provenance",
+      "git_blob": "089d6b55e1e02b85afd7467582b5dcaa2b0d9b2a",
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+}