Exhibit a field-uniform three-block signature missed by every generic support

research/three-block-cancellation-signature/PROOF.md

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+# Cancellation adds a three-block signature
+
+This is a scoped algebraic proof. It answers Open next question 1 of
+SHIN09-git's [#6708](https://github.com/CrocSwap/integer-mult-bounds/pull/6708).
+The new proof and construction were produced by Nick Petalas. Apache-2.0.
+
+## 1. The signature and the generic model
+
+Partition the rows of an invertible square matrix P into nonempty blocks.
+Message j has source mask consisting of the blocks on which column j of P
+is nonzero. Its reader mask consists of the blocks on which row j of
+P inverse is nonzero. This is the source/reader convention of #6708,
+Gonzalo Mier's #3990/#3997 and Nick Petalas's #6691.
+
+A generic support matrix has one separate algebraically independent
+indeterminate in each allowed entry, zero elsewhere. Invertibility and
+inverse supports are taken over the rational-function field. The generic
+census is the union of signatures over all such structurally invertible
+supports, not just the support of one selected numerical witness.
+
+**Result.** On three blocks of two rows there is an integer unimodular P
+whose source masks are all ABC and whose reader masks are
+
+    BC, BC, AC, AC, AB, AB.
+
+The same matrices realize the same signature over every field. No generic
+support matrix has that signature, over any coefficient field. In fact,
+for b >= 2 blocks and arbitrary positive block sizes, if every generic
+source mask is full then at least b reader masks are full. This generic
+lower bound is sharp.
+
+The counterexample settles the stated six-port existence question. It
+does not enumerate all three-block signatures or price a decoder.
+
+## 2. A field-uniform integer witness
+
+All displayed blocks are two by two. Put
+
+    T = [ 0 -1 ]      T^2 + T + I = 0,      det T = 1.
+        [ 1 -1 ]
+
+Define the two six-by-six integer matrices
+
+        [ 0 I I ]                [  T^2 -T^2 -T ]
+    B = [ I 0 I ],           P = [  -T    T -T ].
+        [ I T 0 ]                [ -T^2  -T  T ]
+
+Direct block multiplication using T^2 + T + I = 0 gives
+
+    P B = B P = I_6.
+
+For example the first block row of P B is
+(-T^2-T, T^2-T^2, T^2-T^2) = (I,0,0).
+The remaining block rows follow by the same identity. Both matrices
+therefore have integer inverses and are unimodular. This establishes the
+identity over every field, including characteristic two; no division by
+two is used.
+
+Every block of P is one of T, -T, T^2, -T^2 and is invertible over every
+field. Every column of P consequently has a nonzero restriction to each
+of the three row blocks. The diagonal blocks of B are zero and every
+off-diagonal block is I or T, also invertible. Each row in block A of B
+therefore reads exactly B and C, and likewise for the other blocks.
+This proves the claimed signature over every field.
+
+## 3. Generic inverse supports from a determinant matching
+
+Here is the classical matching/path description, with conventions fixed
+to avoid transposing source and reader masks.
+
+Choose a determinant matching of a structurally invertible generic
+matrix. Write r(j) for the row matched to column j. Relabel column j by
+r(j), so the matched entries become diagonal in a matrix Q. On the row
+vertices put a directed edge i -> r(j) for every supported P[i,j].
+
+**Path fact.** The generic inverse entry P inverse[j,i] is nonzero if and
+only if i is reachable from r(j) in this directed graph, allowing the
+length-zero path.
+
+To prove this, first use Q. A simple path from u to v supplies a matching
+in Q with row v and column u removed: use the path's edges and the
+diagonal matching at the other vertices. Conversely any matching of that
+minor, together with the removed row/column, decomposes into a directed
+path from u to v and disjoint cycles. Thus the corresponding cofactor
+has a determinant matching exactly when the path exists. Each supported
+entry is a distinct indeterminate, so different matchings have different
+monomials. They cannot cancel, even in characteristic two. A cofactor is
+therefore nonzero exactly in this case. Undoing the column relabeling
+gives the stated convention for P inverse. The u=v case uses the
+diagonal matching in the principal minor.
+
+This uses the standard structural determinant/cofactor matching method;
+it is not a new matching theorem. The new use is the block-mask bound
+below.
+
+## 4. The generic full-reader floor and its sharpness
+
+Suppose every source mask of P contains all b blocks. Take a source
+strongly connected component C of the directed graph above, meaning one
+with no edge into it from another component. Such a component exists
+because the component graph is a finite directed acyclic graph.
+
+For any column j with r(j) in C, every block supplies a supported entry
+P[i,j]. Each such entry gives an edge i -> r(j). The source property
+forces i into C. Thus C meets every block and has at least b vertices.
+
+For each vertex r(j) in C, strong connectivity gives a path to every
+vertex of C. By the path fact, row j of P inverse has a nonzero entry in
+each block. The distinct vertices of C correspond to distinct messages.
+At least |C| >= b messages therefore have full reader masks. This proves
+the bound for every structurally invertible support, irrespective of the
+matching chosen.
+
+For sharpness choose one representative row in each block. Support every
+entry of those b rows, and support only the diagonal entry of each other
+row. There is a diagonal matching. The representatives form one source
+strongly connected component; they reach every row. Every other vertex
+has only its self-loop. The path fact gives full reader masks exactly
+at the b representatives. All columns have full source masks. Because
+b >= 2, the other rows' singleton reader masks are not full. The lower
+bound is attained for all positive block sizes.
+
+For the six-port witness b=3, but no reader mask is full. Hence its
+signature is absent from the union over all generic support matrices.
+Merely noting cancellation in its own entry support would not prove
+this stronger exclusion; the universal bound does.
+
+## 5. Scope and finite corroboration
+
+The all-field conclusion and the exclusion of all six-port generic
+supports use the proof above. The verifier independently inverts the
+integer matrices, checks both products and signatures, and specializes
+to F2, F3, F5, F7 and F11. It checks every full-source support for block
+sizes (1,1,1), (2,1,1) and (2,2,1), comparing literal cofactor matchings
+with graph reachability. It also checks the sharp six-port generic
+example and the generic signature of the actual witness support.
+These are finite corroborations, not an exhaustive six-port census.
+
+The predecessor's two-block classification is a cited result, not
+replayed or audited. Its theorem remains within its two-block scope.
+The new result shows why its generic-signature equality cannot simply
+be extended to three blocks. No paid physical preparation, restoration,
+delivery, native word, supplier, kappa or multiplication theorem follows.
+Independent human and formal proof-kernel review are unclaimed.

research/three-block-cancellation-signature/README.md

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+# Cancellation adds a three-block decoder signature
+
+A six-by-six integer unimodular basis realizes a source/reader signature
+absent from every generic support pattern. It works over every field,
+including F2. This answers Open next question 1 of SHIN09-git's
+[#6708](https://github.com/CrocSwap/integer-mult-bounds/pull/6708).
+The new contribution was produced by Nick Petalas. Apache-2.0.
+
+Claim type: scoped written proof with exact finite corroboration.
+No decoder price, native word, supplier admission or new kappa is claimed.
+
+## Result
+
+Divide the six port rows into A={0,1}, B={2,3}, C={4,5}. Every column of
+the explicit basis P uses all three blocks. The six rows of its inverse
+use exactly BC, BC, AC, AC, AB, AB. All entries are 0, 1 or -1 and both
+inverse products are integer identities.
+
+In contrast, a generic support matrix on b >= 2 nonempty row blocks,
+with every column using all b blocks, must have at least b inverse rows
+using all b blocks. The bound is sharp for every choice of block sizes.
+The proof chooses a source strongly connected component in the directed
+graph of a determinant matching. It excludes every generic support,
+not just the entry support of P. [PROOF.md](PROOF.md) gives the matrices
+and the full argument.
+
+This is the first improvement being delivered. The remaining signature
+classification and any physical use are left below.
+
+## Reproduce
+
+From the repository root, with CPython 3.12 and its standard library:
+
+```sh
+/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/three-block-cancellation-signature
+/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/three-block-cancellation-signature/verify.py
+/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -O -W error -B research/three-block-cancellation-signature/verify.py
+/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/three-block-cancellation-signature --check-source-history
+```
+
+Another CPython 3.12 path may replace the local interpreter. Direct replay
+prints a generated receipt bound to the package hashes, followed by
+`PASS_THREE_BLOCK_CANCELLATION_SIGNATURE`. It writes no files. No receipt
+is handwritten or stored as an asserted output.
+
+The verifier checks both integer products, independently solves both
+inverses with rational elimination, and checks the signature over F2, F3,
+F5, F7 and F11. It rejects all 27 single-nonzero-entry omissions from P.
+It enumerates all 59,131 full-source supports at block sizes (1,1,1),
+(2,1,1) and (2,2,1). The 58,081 structurally invertible supports pass
+independent cofactor-matching and graph-reachability checks. Two six-port
+patterns corroborate sharpness and the difference between the numerical
+witness and its generic support. Five invalid-input controls pass.
+The all-size theorem and all-field witness use the written proof;
+this is not an exhaustive six-port census.
+
+## Scope, sources and decision
+
+The `coverage-law-change` card changes scope to this square-basis
+algebraic question. Base main is
+`e54523517d0f03e58352e647c403a5af3e525a57`, priority revision 8,
+selected #7000. The selected source-binding orientation passes; no
+selected scientific replay is performed. The result does not change
+the selected record or any of its five inherited assumptions.
+
+The [advance claim](https://github.com/CrocSwap/integer-mult-bounds/discussions/1166#discussioncomment-19538530)
+states the same target. Live handoffs, current PRs, recent account work,
+local reservations, main README searches, merged searches and the overlap
+helper were consulted. The source-reference search found no recent
+account PR using #6708. The overlap helper's working-note inventory is
+sparse; explicit main-tree searches supplement it. Arbitrary historical
+diffs, nested notes and unpublished work remain coverage limits.
+
+SHIN09-git's two-block classification is cited without auditing or
+replaying its scientific checker. It proves rational generic equality
+for two blocks and explicitly asks the three-block question settled here.
+Its proof and README are pinned in [SOURCES.json](SOURCES.json). The
+source/reader model and earlier screens retain Gonzalo Mier's #3990/#3997
+and Nick Petalas's #6691 credit. Structural determinant/cofactor matchings
+and the inverse-path description are classical. The new steps are the
+sharp block-mask consequence and the field-uniform six-port separation.
+
+No complete physical producer or consumer is supplied. Preparation,
+delivery, arbitrary dirty-state restoration, routing, precision, stock,
+compiler closure and a signed invoice remain unpaid obligations. Unknown
+costs are not zero. Independent human and formal proof-kernel review,
+Lean, Docker and supplier admission are unclaimed. The decision is to
+retain this settled algebraic question and stop.
+
+## Open next questions
+
+1. Which other three-block signatures lie outside the generic census?
+   Start with supports of inverse bases having a zero diagonal block;
+   a complete cancellation-sensitive classification is the blocker.
+2. Can a physical caller exploit this signature after all preparation,
+   delivery and restoration costs? Start with these literal matrices and
+   one source/reader schedule. No favorable full invoice is supplied.
+
+## Implementation bounds
+
+All matrix dimensions are at most six. Matching lists have at most 720
+terms; each cofactor has at most 120. The replay enumerates exactly the
+three stated support domains and two six-port patterns. There is no
+recursion, subprocess, network access or dynamic source generation.
+All functions are below sixty physical lines and internal checks remain
+active under optimized Python. The only module dispatch loads the fixed
+local model file; the invalid-input table has five fixed call targets.
+
+Power-of-Ten rule 3 deviation: Python and Fraction allocate during
+execution. Fixed dimensions, coefficient fixtures, one support processed
+at a time and bounded matching lists limit live memory. A fixed local
+module loader is the rule 9 dispatch deviation needed by isolated Python;
+the target is `model.py`, with lifetime equal to this process. Rule 10
+uses warning-strict normal/optimized execution; no additional Python
+static analyzer is claimed. These are mathematical checks, not physical
+memory or tape tariffs.

research/three-block-cancellation-signature/SOURCES.json

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+{
+  "schema_version": 1,
+  "base_commit": "e54523517d0f03e58352e647c403a5af3e525a57",
+  "files": [
+    {
+      "path": "certificates/selected-result.json",
+      "usage": "provenance",
+      "git_blob": "75d77b328ea02ded535e7f23aee5b1e55d0bb1a8",
+      "sha256": "154c3d30455d778fd3343b6b4851f0fa4665f0f7625b0adb3675e2bafbbf6bf1"
+    },
+    {
+      "path": "docs/research/agent-task-catalog.json",
+      "usage": "provenance",
+      "git_blob": "3404fbdd5e7529a0674eef26114b5071e4586cec",
+      "sha256": "ead6cdc9d0af6d58e8a49744d8ec8c7c15e13d2ba93c276c434bcf9b51b93a81"
+    },
+    {
+      "path": "research/two-block-signatures/PROOF.md",
+      "usage": "input",
+      "git_blob": "51cb134d77f700c031084311b43cb4c02e17ba68",
+      "sha256": "47094558499d02e81a26aeb7969efbdfc8d91bc719537e2c36d6cca0cc629179"
+    },
+    {
+      "path": "research/two-block-signatures/README.md",
+      "usage": "input",
+      "git_blob": "7e568edb6c84a6ec3575cd420f42cc66cd7d93c0",
+      "sha256": "b96207a9dbc79a1b3ff228a8409d7e960425b59a92d5afbe84ff1fef7cfd2096"
+    }
+  ]
+}