Extract complete raw finite-field pencil blocks and exact maps

research/raw-pencil-decomposition/PROOF.md

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+# Raw finite-field pencil decomposition
+
+Ali Romman, with substantial OpenAI Codex assistance. Apache-2.0.
+
+## Claim and interface
+
+Let K be a supplied finite field. For any two m by n matrices A and B over K,
+the algorithm returns the full Kronecker block list and invertible matrices
+R and Q such that A Q = R A0 and B Q = R B0. Thus the strict action is
+(R^-1, Q); the slice map is the identity. Zero rows, zero columns, zero-index
+blocks, dependent slices and the zero pencil are allowed. There is no need
+for an invertible slice over K.
+
+The output uses these models for s A0 + t B0:
+
+- Right index e: e by e+1, A0 has a unit diagonal and B0 a unit superdiagonal.
+- Left index e: the transpose of that right block.
+- Finite primary (f,e): (companion(f^e), I), with monic irreducible f.
+- Infinity of length e: (I, companion(x^e)).
+
+The code gives a deterministic algorithm polynomial in m+n for each fixed K.
+It uses field tables and exact row reduction. A coarse bound is
+O_K((m+n+1)^10) field operations and O_K((m+n+1)^4) live field entries.
+No bound polynomial in log(|K|), fast canonical-form bound, or numerical
+stability claim is made. The executable shares the usual finite memory and
+Python recursion limits. The written algorithm has no dimension bound.
+
+This completes the raw-input boundary for the supplied-model character
+interface. It does not compute the character formulas of that interface.
+It does not prove a physical supplier, native map, route, restoration schedule,
+precision bound, repeated preparation cost or complete kappa improvement.
+The selected .00310045 result and its main-transform bottleneck are unchanged.
+
+## 1. Shortest chains and singular blocks
+
+A polynomial right kernel vector of degree e has coefficients v0,...,ve with
+A v0=0, A vi+B v(i-1)=0 and B ve=0. Solve this homogeneous linear system in
+increasing e. If a right block exists, the first nonzero solution occurs at
+its least minimal index. Search through min(m,n-1); a right block of index e
+uses e rows and e+1 columns, so this range is complete.
+
+Use the classical Kronecker decomposition to justify the splitting step.
+A shortest chain has zero component on every regular or left block, and on
+every right block with larger index. There is no smaller right block. On each
+right block with the least index, it is a scalar multiple of its generator:
+a nonconstant multiplier would increase the degree. At least one scalar is
+nonzero. A change among equal-index copies makes this chain one direct
+summand. This also proves that its e+1 domain vectors and e image vectors
+are independent. No search through all vectors of K^n is required.
+
+Set ui=(-1)^i v(e-i), and wi=A ui for 0<=i<e. The chain equations give
+A ui=wi and B u(i+1)=wi, with the two endpoint zeros. These are precisely the
+stated right block. Let U and W contain these columns. Recover its complement
+by the linear equations
+
+```text
+F U = I, G W = I, G A = Ablock F, G B = Bblock F.
+```
+
+A solution exists because the embedding splits. The bases [U,ker F] and
+[W,ker G] put both coefficient matrices in block diagonal form. The code
+checks this identity before it recurses. For index zero, W has zero columns
+and G has zero rows; the same equations split an isolated zero column.
+
+After all right blocks are removed, apply the same step to the transpose.
+If At Qt=Rt Ct, then A Rt^-t=Qt^-t Ct^t. This gives the correct row and
+column maps for each left block, including an isolated zero row. The residue
+has no left or right polynomial kernel, hence is square and regular by
+Kronecker theory. This reasoning credits that theorem; it is not a new proof
+of its existence or uniqueness.
+
+## 2. Infinity without a rational invertible slice
+
+For the regular residue, start V0=0 and form V(i+1)={v:Bv is in A Vi}.
+Each step is a nullspace of [B,-A Vi] followed by projection onto the v
+coordinates. The increasing sequence stabilizes in at most n steps.
+
+On a finite block (C,I), every Vi is zero. On an infinity block (I,N),
+Vi=ker(N^i). Thus its limit is exactly the whole infinity space. A is
+injective there and its image W has the same dimension. B V is in W.
+Use W=A V to obtain the local pair (I,N), then solve the same linear
+projection equations to split this direct summand. The residual B is
+invertible. Normalize it to (B^-1 A,I). This proves the step even when each
+of the q+1 rational directions gives a singular matrix. No scalar search
+is used to select a global chart.
+
+## 3. Finite-field factorization and primary cyclic summands
+
+Find the minimal polynomial of T=B^-1 A from the first linear relation among
+I,T,...,T^n. Factor it with the classical Berlekamp method. For a squarefree
+polynomial f, the kernel of the q-power map minus the identity on K[x]/f
+has one dimension per irreducible factor. Its elements have values in K
+on each field factor. Successive gcds with b-c, for a basis b and all c in K,
+separate all factors. The code checks the final number of factors against
+that kernel dimension.
+
+For non-squarefree f, recurse on gcd(f,f') and its quotient. If f'=0,
+extract the p-th root of each coefficient with the inverse Frobenius power
+q/p and multiply the returned multiplicities by p. Each call reduces the
+degree. This includes extension fields and inseparable multiplicities.
+The verification uses a separate bounded monic-divisor enumeration, in
+addition to multiplying all factors back to the input polynomial.
+
+For each irreducible f with maximal exponent e, take ker(f(T)^e). These
+primary spaces have direct sum equal to the full space. On one primary space,
+find its nilpotency length e and a basis vector v with f(T)^(e-1)v nonzero.
+Its annihilator is exactly f^e. Its Krylov columns through degree deg(f)*e-1
+are independent and give the stated companion block. A cyclic submodule of
+maximal length in a direct sum of K[x]/f^j is a direct summand: a maximal
+component of v is a unit modulo f, so elementary module changes make it a
+generator in that component. Solve L U=I and L T=C L for its projection.
+Its kernel is a T-invariant complement. Repeat. Apply the same procedure
+with f=x to the infinity operator.
+
+All these steps are exact linear algebra or polynomial arithmetic with
+degrees bounded by m+n. The largest dense splitting system has O(N^2)
+unknowns and equations, with N=m+n+1. The stated coarse polynomial bound
+covers all degree searches, all splits, all primary steps and fixed-q
+Berlekamp splitting. It does not assert that this dense implementation is
+competitive with fast canonical-form algorithms.
+
+## 4. Certificate and independent acceptance boundary
+
+Both original equations A Q=R A0 and B Q=R B0 must hold and both maps must be
+invertible. Classification also requires irreducible primary labels and the
+complete block multiset; map equality alone would not justify a malformed
+primary label. The algorithm obtains those labels by the factorization above.
+The public `check` function checks maps to the requested model; it is not a
+standalone proof that arbitrary caller-supplied labels are canonical.
+
+The separate orbit oracle enumerates elementary invertible row and column
+operations on every raw pencil in five complete domains. Elementary additions
+and nonzero row/column scalings generate GL; swaps are products of these.
+It checks that the block type is constant on each orbit and differs between
+orbits. It does not use polynomial null chains, module decompositions or
+factorization to build its orbits. Every raw input also has both coefficient
+identities checked. Finite enumeration corroborates the algorithm; the
+all-dimension claim uses the arguments above and the classical module theorem.
+
+## Attribution and prior work
+
+[Nick Petalas PR #5250](https://github.com/CrocSwap/integer-mult-bounds/pull/5250)
+supplied the invertible-slice character formula and canonical-form reduction.
+[apoursh PR #5672](https://github.com/CrocSwap/integer-mult-bounds/pull/5672)
+supplied repeated singular blocks and explicit slice lifts.
+[PR #5683](https://github.com/CrocSwap/integer-mult-bounds/pull/5683) and
+[PR #5689](https://github.com/CrocSwap/integer-mult-bounds/pull/5689) supplied the
+mixed singular and complete finite-field model character interfaces. They
+explicitly leave raw decomposition separate. David Ganz's A030/A026/A029
+atlas lineage supplies the broader tensor-orbit motivation.
+
+The exact field and row-reduction module is retained from
+[PR #5664 source](https://github.com/aliromman/integer-mult-bounds/blob/8f7773e700d468396cbf893dfb18bf3ae6c20f66/research/residue-product-tensor-recognition/vendor/local.py).
+Its notice is unchanged. SOURCES.json pins its retained bytes. The remainder
+of this package is new integration, proof of this implementation and checks.
+
+Kronecker decomposition, primary module decomposition and Berlekamp
+factorization are classical. See [P. Van Dooren, 1979](https://perso.uclouvain.be/paul.vandooren/publications/VDooren79.pdf)
+for prior arbitrary-pencil extraction, and
+[Berlekamp's polynomial-factorization account](https://math.berkeley.edu/~berlek/poly.html).
+The preimage sequence is a Wong sequence; see
+[The Quasi-Kronecker Form for Matrix Pencils](https://epubs.siam.org/doi/abs/10.1137/110826278).
+We use their mathematical lineage, not their numerical implementations or
+complexity bounds. No global literature priority, human independent review,
+formal-kernel acceptance or new canonical-form theorem is claimed.

research/raw-pencil-decomposition/README.md

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+# Raw finite-field pencil decomposition
+
+This package extracts every singular and regular Kronecker block from two
+arbitrary rectangular finite-field matrices. It returns exact row and column
+maps and does not need an invertible base-field slice. It supplies the raw
+input boundary left by the supplied-model character packages.
+
+Read [PROOF.md](PROOF.md) for the algorithm, conventions, scope and credit.
+Read [RESULT.json](RESULT.json) for exact evidence and [SOURCES.json](SOURCES.json)
+for retained bytes. `decompose.decompose(A,B,m,n,field)` returns `(blocks,R,Q)`
+with `A Q=R A0` and `B Q=R B0`. Field elements use the retained module's
+base-p digit encoding. The slice map is the identity.
+
+```sh
+python3.12 -B scripts/research_package.py --package research/raw-pencil-decomposition
+```
+
+Expected: `PASS_RAW_PENCIL_DECOMPOSITION` and `PASS_REGISTERED_RESEARCH_PACKAGE`.
+Python 3.12 and the standard library suffice. No physical supplier or new
+kappa is claimed. The selected frontier is unchanged. Ali Romman with
+substantial OpenAI Codex assistance; no subagents. Apache-2.0.
+
+## Open next questions
+
+Connect the extracted blocks and maps to the supplied-model characters in
+PR5689, with an explicit convention check. For a physical candidate, price
+all maps, storage, preparation, routes and restoration before a kappa claim.
+The dense exact implementation is a reference algorithm; faster extraction
+can use established canonical-form methods with the same certificate boundary.

research/raw-pencil-decomposition/RESULT.json

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+{
+  "coefficient_equations": 22812,
+  "complete_domains": [
+    {
+      "pencils": 256,
+      "q": 2,
+      "shape": [
+        2,
+        2
+      ],
+      "strict_orbits": 16
+    },
+    {
+      "pencils": 6561,
+      "q": 3,
+      "shape": [
+        2,
+        2
+      ],
+      "strict_orbits": 24
+    },
+    {
+      "pencils": 65536,
+      "q": 4,
+      "shape": [
+        2,
+        2
+      ],
+      "strict_orbits": 34
+    },
+    {
+      "pencils": 4096,
+      "q": 2,
+      "shape": [
+        2,
+        3
+      ],
+      "strict_orbits": 20
+    },
+    {
+      "pencils": 4096,
+      "q": 2,
+      "shape": [
+        3,
+        2
+      ],
+      "strict_orbits": 20
+    }
+  ],
+  "constructed": 96,
+  "factor_polynomials": 1776,
+  "failure_controls": 183,
+  "no_invertible_slice": 12,
+  "source_inputs": 1,
+  "status": "PASS_RAW_PENCIL_DECOMPOSITION"
+}

research/raw-pencil-decomposition/SOURCES.json

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+{
+  "schema_version": 1,
+  "base_commit": "df0137efd35e11b4445734bf612bdd4c7ebc330c",
+  "files": [
+    {
+      "path": "certificates/selected-result.json",
+      "usage": "provenance",
+      "git_blob": "581f8d87008354eb5b869f7ed9de1317ad837ce8",
+      "sha256": "2a0220e866417774606e0f991e41b9c1d26b1f43b97857238cf478dbc7d330b5"
+    },
+    {
+      "path": "docs/research/agent-task-catalog.json",
+      "usage": "provenance",
+      "git_blob": "f126dd2e2dad2b669e75d1d0d0bd05c600725602",
+      "sha256": "f7a48bfa2659b88378282a8eee7d9fe52cdf505594124c4d1686b58510fffd72"
+    },
+    {
+      "path": "research/raw-pencil-decomposition/field_source.py",
+      "usage": "input",
+      "git_blob": "1066bb74c19d180bfef7d9dd1c0650ec6bbd7780",
+      "sha256": "6daa160346d04691c6ca3c9c820b3df8982de4320ec01e5db10e3cb932f42c00",
+      "pin": "head:c5612cbefef23f70f305542b99ce161d985c953b:research/raw-pencil-decomposition/field_source.py"
+    }
+  ]
+}

research/raw-pencil-decomposition/TASK.json

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+{
+  "schema_version": 1,
+  "task_id": "bit-new-register-model",
+  "priority_revision": 7,
+  "base_commit": "df0137efd35e11b4445734bf612bdd4c7ebc330c",
+  "original_card": {
+    "id": "bit-new-register-model",
+    "question": "Can an explicitly larger or nonabelian register model escape the scoped three-register obstruction?",
+    "starting_points": [
+      "docs/research/research-family-status.md",
+      "research/bit-three-register-lower-bound/README.md",
+      "certificates/selected-result.json",
+      "research/bit-gl3-mixed-targets-length10/README.md",
+      "docs/research/community-efficient1000-review.md"
+    ],
+    "methods": [
+      "Construct the smallest exact identity in a newly specified legal operation alphabet.",
+      "Derive a target-specific relaxation bound for that same explicit alphabet."
+    ],
+    "acceptance_gate": "Name the changed hypothesis, all counted roles and a uniform identity; finite column tests alone do not certify it. A scoped local result may be retained without a complete supplier. Expanding it into further construction rounds requires the family-level budget and obligation gate in research-family-status.md; an optimistic passing relaxation alone is not acceptance.",
+    "first_step": {
+      "command": [
+        "python3",
+        "-B",
+        "scripts/research_baseline.py"
+      ],
+      "expected": "PASS_RESEARCH_BASELINE_BINDINGS_ONLY",
+      "inspect": "Read Model section 1 of the three-register bound and identify the exact excluded free operations.",
+      "scope": "Positive control / source-bound orientation only; this is not a new construction or full cold replay."
+    },
+    "first_experiment": "First compare the alphabet, role count and length with the fully replayed #1974 census and #2027 mixed-root tariff. Identify a changed premise and a legal native implementation; then price one smallest completed instance before searching larger words.",
+    "bottleneck_link": "A small-width completed saving changes the bit primitive; a finite-address identity alone is insufficient.",
+    "allowed_variations": [
+      "Choose a fixed larger register count or a specified nonabelian operation family."
+    ],
+    "scope_change_triggers": [
+      "Remaining inside the already-excluded three-register abelian model.",
+      "Repeating the two-register GL3 census or a covered three-register length without changing a hypothesis; a formal group-algebra word without native conversion is a partial result."
+    ],
+    "overlap_terms": [
+      "three-register",
+      "nonabelian",
+      "group algebra"
+    ]
+  },
+  "scope_status": "changed",
+  "actual_scope": "Complete raw finite-field rectangular matrix-pencil decomposition with explicit maps.",
+  "scope_change_reason": "The live tensor-atlas handoff leaves arbitrary-input extraction separate from the completed supplied-model characters.",
+  "bottleneck_link": "Provide canonical blocks and maps to evaluate two-slice tensor candidates; physical costs and the complete kappa bill remain open.",
+  "budget_minutes": 150,
+  "stop_rule": "Stop after full extraction proof and independent complete-domain map/type checks, or a counterexample.",
+  "outcome": "completed",
+  "evidence": "PROOF.md; RESULT.json; verify.py",
+  "author": "Ali Romman with substantial OpenAI Codex assistance",
+  "ai_assistance": "Substantial OpenAI Codex assistance; no subagents.",
+  "open_questions": [
+    {
+      "question": "Can a physical caller use an admissible raw pencil with a profitable complete bill?",
+      "why_it_matters": "Algebraic recognition alone does not improve kappa.",
+      "first_step": "Apply the supplied-model characters to these blocks and price every native map and route.",
+      "blocked_by": "No physical caller is supplied."
+    }
+  ]
+}

research/raw-pencil-decomposition/VALIDATION.md

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+# Validation record
+
+Python3.14 normal replay and Python3.12.14 registered normal replay pass.
+A Python3.12.14 isolated optimized replay of the sealed science export also
+passes with no Git checkout and empty PATH. Ten exported files retain their
+exact bytes and read-only modes. An actual appended change to the retained
+field source fails authentication before mathematical checks.
+
+RESULT.json records 80545 raw pencils in five complete domains, 114 strict
+orbits, 1776 polynomial factorization cases, 96 constructed pencils, 22812
+constructed coefficient equations, 12 cases with no invertible rational slice
+and 183 failure controls. Every exhaustive raw pencil also has both maps
+checked. The independent GL orbit oracle covers all inputs in its stated
+domains. Larger dimensions are constructed evidence, not exhaustive evidence.
+
+The exact committed-byte and official export checks pass. The final export
+adds this record and the corrected 600-second manifest cap; the mathematical
+code and result are the sealed replay's bytes. Full-history ancestry preflight
+is omitted in this shallow checkout. Immediate source ancestry, the original
+remote field-source blob and all retained bytes were checked.
+
+The final overlap refresh covers all 7678 local records, 376 open PRs and
+5006 Discussion records with complete remote pagination, through
+2026-10-26T05:53:29Z. Canonical main remains
+`df0137efd35e11b4445734bf612bdd4c7ebc330c`. No competing raw decomposition
+claim was found. This does not cover unpublished work or every historical diff.
+
+The PR draft is checked with GitHub's Markdown renderer and its pinned file
+targets before publication. The published body and rendered HTML must then
+match that reviewed draft. Hosted CI, independent human review and proof-kernel
+acceptance remain separate. No selected-frontier edit, merge or closure of
+another contributor's work is part of this result.

research/raw-pencil-decomposition/VERIFY.json

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+{
+  "schema_version": 1,
+  "status": "ready",
+  "python": "3.12",
+  "requirements": [],
+  "inputs": [
+    "research/raw-pencil-decomposition/field_source.py"
+  ],
+  "commands": [
+    {
+      "argv": [
+        "{python}",
+        "-B",
+        "{package}/verify.py"
+      ],
+      "timeout_seconds": 600,
+      "checks": "Complete strict GL orbits, literal maps, factor divisor oracle, mixed primaries, infinity and failure controls."
+    }
+  ]
+}