Prove every continuous P112 Walsh basis gauge costs 82 (no new κ)

research/p112-walsh-gl2-strata/PROOF.md

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+# Continuous Walsh-plane gauge obstruction
+
+This is a scoped computer-assisted proof. Its universal step is the support
+partition below; its finite geometric statements are exact rank certificates
+computed by `verify.py`. It is not formal-kernel or independent human review.
+
+## Model and theorem
+
+Use the P112 matrix `X` and nine-bit source labels `p_i` and reader labels `q_i`
+from Mihir Srivastava's `p112-single-lift/geometry.py`, following
+DreamingOfClouds' PR717 geometry. The 112 rows of X have eight coordinates.
+For two real 112-vectors define the support tariff
+
+```
+beta(d,t) = rank_F2 ((q_i dot p_j)_(d_i != 0, t_j != 0)).
+```
+
+This is the inherited pairing-rank tariff, not a physical execution invoice.
+For `a,b` in F2^3 set `chi_a(b)=(-1)^(a dot b)`, with b in binary order.
+Choose any two distinct characters for each ordered pair of columns of the
+8-by-2 matrices U and V. For every M in GL2(R), set
+
+```
+U' = U M,        V' = V M^(-T),
+D  = [X, XU'],   T  = [X(I - V'U'^T), XV'].
+```
+
+**Theorem.** Every such ten-term factorization has the pairing-rank profile
+`{8:8, 9:2}`. Its total tariff is exactly 82. In particular no rational or
+real invertible basis change in these fixed Walsh planes improves PR2042's
+finite-domain profile. The exact power tariff is `8*8^s + 2*9^s` for every
+real exponent s, since the whole rank multiset is unchanged.
+
+This extends Cinna Davis's PR2042 family from five matrices to all GL2(R).
+It does not decide the arbitrary overlapping rank-two problem posed in
+mightyOil's PR1197: U and V must still have the specified Walsh column spaces
+and the rank-two product UV^T must stay fixed during the basis change.
+
+## Algebraic identity
+
+The inverse transpose gives `U'V'^T = UV^T`. Therefore
+
+```
+DT^T = X(I - U'V'^T)X^T + XU'V'^T X^T = XX^T.
+```
+
+The eight original reader columns are the columns of X. Their source columns
+are independent of M, since `V'U'^T = VU^T`. The remaining reader columns
+are nonzero linear combinations of the two columns of XU; the remaining
+source columns are nonzero combinations of the two columns of XV. Invertibility
+of M and the full rank of the character pairs ensure nonzero coefficient
+vectors on both sides.
+
+## The complete support partition
+
+Write u and v for the two columns of XU. For each row i, the vanishing equation
+for a combination `z u + w v` is `z*u_i + w*v_i = 0`. If both coefficients
+are zero it vanishes for every direction. Otherwise it has exactly one zero
+direction in the real projective line. The verifier enumerates these directions
+as primitive integer pairs `(v_i,-u_i)`, with a fixed overall sign.
+
+For each of the 56 ordered character pairs, the entire set of zero directions
+is exactly
+
+```
+(1,0), (0,1), (1,1), (1,-1).
+```
+
+Each exceptional point has its own support. At every other real projective
+point the support is exactly `supp(u) union supp(v)`. Thus the four points
+and the generic representative `(1,2)` are exhaustive, including irrational
+slopes. They are not a bounded coefficient search. The same statement applies
+to the V planes. There is no assumption that the four entries of M have small
+height or that its determinant is a unit.
+
+## Finite certificates
+
+The standalone verifier imports only the exact pinned geometry constructor.
+It uses a separate nine-bit elimination routine with nine explicitly bounded
+pivot steps per input row. It checks:
+
+1. For each of the eight coordinate reader supports, the span of q_i has
+   dimension 8.
+2. For every ordered character pair and each of its five support strata,
+   both the p_i span and q_i span on that support have dimension 9. This is
+   280 support certificates, each with both ranks checked.
+3. For all 56 choices of U and 56 choices of V, each of the eight corrected
+   source columns `X_j - U_(j,0)*(XV)_0 - U_(j,1)*(XV)_1` has p_i span of
+   dimension 9. This is 25,088 corrected-source certificates.
+
+All rows and labels are reconstructed from the pinned constructor. The support
+receipt is hashed in deterministic order and printed with hashes of package
+files. No stored predecessor rank receipt is consumed.
+
+The standard binary dot product is nondegenerate. If the source labels span
+all F2^9, the restricted pairing has rank equal to the dimension of the
+reader-label span. Statement 1 and statement 3 therefore give rank 8 to
+each original term. Statement 2 gives rank 9 to each added term, for every
+nonzero real coefficient direction on either side. This proves the theorem.
+
+The checker also verifies literal 112-by-112 factor identities for a
+determinant-minus-two gauge and a determinant-one gauge. These are corroborating
+endpoint fixtures, not the justification of the infinite quantifier. A
+singular gauge and a dropped added term are rejected. A wrong inverse is
+separated from the required inverse transpose.
+
+## Scope and cost boundary
+
+The result rules out further basis searches while these Walsh planes and their
+fixed product are retained. It gives no exclusion for different planes,
+simultaneous mixing with the original eight channels, changed geometry,
+masked endpoints, or a different tariff. It does not supply preparation,
+source access, frame transport, dirty restoration, hosting, native operations,
+precision, stock, a recursive compiler, or a complete multiplication theorem.
+No historical packing ceiling is transferred to the selected P104 supplier.

research/p112-walsh-gl2-strata/README.md

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+# Every continuous Walsh two-total basis gauge still costs 82
+
+Scoped computer-assisted proof, produced by Nick Petalas. Apache-2.0.
+
+For all 3,136 ordered choices of two distinct Walsh characters on each side
+of Cinna Davis's PR2042 P112 ten-term factorization, every invertible real
+or rational 2-by-2 basis change `U -> UM, V -> VM^(-T)` has exactly the
+pairing-rank profile `{8:8,9:2}` and total tariff 82. The power tariff is
+therefore always `8*8^s + 2*9^s`. This replaces a finite five-basis search
+with an exact classification of the entire continuous basis orbit.
+
+The new proof is in [PROOF.md](PROOF.md). Every Walsh plane has exactly four
+projective support-zero directions; all other real directions share one
+support. The verifier checks both label spans on all 280 plane/support cases,
+the eight original reader spans, and all 25,088 corrected source-column spans.
+The nondegenerate binary pairing then gives every term's exact rank. No
+predecessor rank receipt or verifier is replayed as evidence for this result.
+
+This is a decoder side result, with no new kappa or selected-frontier change.
+It shows that further basis searches in these fixed Walsh planes cannot lower
+their invoice. Arbitrary changed planes, changed rank-two products and physical
+supplier realization remain open. The historical P112 packing ceiling is not
+an applicable ceiling on the current selected P104 construction.
+
+## Reproduction
+
+From the repository root, with CPython 3.12 and the standard library:
+
+```sh
+/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/p112-walsh-gl2-strata/verify.py
+/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -O -W error -B research/p112-walsh-gl2-strata/verify.py
+/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/p112-walsh-gl2-strata
+/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/p112-walsh-gl2-strata --check-source-history
+```
+
+Other machines can substitute their CPython 3.12 executable. The verifier prints
+`PASS_P112_WALSH_GL2_STRATA` in JSON with the complete certificate counts,
+support digest and package hashes. It writes no files. Both registered checks
+have a 120-second limit. `SOURCES.json` pins the sole executable external input,
+Mihir Srivastava's exact P112 geometry constructor, to canonical main
+`721c03822c421b7f7fcbb2598993b947c4d88419`. Source hashes are checked before import.
+
+Finite fixtures also check complete 112-by-112 identities under determinant
+minus two and determinant one. Singular gauges, a missing total, empty support,
+a missing label direction and the wrong inverse are rejected or separated by
+explicit controls. The universal claim rests on the written support partition
+and complete finite rank certificate, rather than these sample matrices.
+
+## Relation to prior work
+
+- Cinna Davis, [PR2042](https://github.com/CrocSwap/integer-mult-bounds/pull/2042),
+  supplies the asymmetric Walsh family and its finite five-basis result.
+  This package extends the basis domain to all GL2(R).
+- DreamingOfClouds, PR717, and Mihir Srivastava, PR845 and
+  `p112-single-lift/geometry.py`, supply the P112 coordinates and pairing tariff.
+- Philipp / mightyOil, PR1197, poses the general coupled overlapping-lift case.
+  This result closes only the specified Walsh-plane basis orbit.
+- Gonzalo Mier, [PR4020](https://github.com/CrocSwap/integer-mult-bounds/pull/4020),
+  uses projective support strata for arbitrary rational pair gauges on a
+  different 34-channel factorization. The support-stratum method is credited;
+  this package's new result concerns the two-added-total Walsh family.
+- DreamingOfClouds, PR2338, establishes broader baseline decoder floors under
+  its stated format. A floor of 64 does not determine the exact value 82 here.
+
+The source files and predecessor credit remain unchanged. No claim of a new
+support-stratification method or independent human review is made.
+
+## Limitations and next decision
+
+Source preparation, adapters, frame transport, dirty restoration, native
+execution, hosting, precision, stock, uniform compilation and outer recovery
+are unpriced. No full supplier replay, Lean proof or Docker run is claimed.
+The result is a scoped exact exclusion, not a search failure or a complete
+integer multiplication theorem. Python uses managed allocations, bounded here
+by 112 geometry rows, 56 plane records, nine-bit rank workspaces and a fixed
+25,088-case loop. No unbounded search, recursion or external solver is used.
+
+## Open next questions
+
+Can changing the rational planes and their product `UV^T`, rather than only
+their bases, lower the profile? Start with one explicitly different overlapping
+plane pair and its complete support arrangement. This matters because the
+arbitrary rank-two branch of PR1197 remains open. A favorable example and
+its physical source/frame/restoration bill are still missing. The structured
+question is retained in `TASK.json`; this package stops at the classification.

research/p112-walsh-gl2-strata/SOURCES.json

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+{
+  "schema_version": 1,
+  "base_commit": "721c03822c421b7f7fcbb2598993b947c4d88419",
+  "files": [
+    {
+      "path": "certificates/selected-result.json",
+      "usage": "provenance",
+      "git_blob": "d59b5035e7b4cbaa9d688a1d32dd0d6eb79c466a",
+      "sha256": "9b8d0eea95b3bf0dfcb7938a732a6439b87a2685dec2a5ab9e525b3b34debbfa"
+    },
+    {
+      "path": "docs/research/agent-task-catalog.json",
+      "usage": "provenance",
+      "git_blob": "3404fbdd5e7529a0674eef26114b5071e4586cec",
+      "sha256": "ead6cdc9d0af6d58e8a49744d8ec8c7c15e13d2ba93c276c434bcf9b51b93a81"
+    },
+    {
+      "path": "research/p112-single-lift/geometry.py",
+      "usage": "input",
+      "git_blob": "34c1abf059d7be5fe5e5044cf01a09b843071c68",
+      "sha256": "3857c11ec731a602b8059d85d58fbb3b641bab687bb33016a0e54f7567752c8a"
+    },
+    {
+      "path": "research/p112-walsh-two-total-seed6ed83d/README.md",
+      "usage": "provenance",
+      "git_blob": "97193e7f6fe31fd3b96d1ff543ea12a3352de9ca",
+      "sha256": "ffcfee6e621118d244326023589589b01440648f077e07ee793ee7738630e458"
+    }
+  ]
+}

research/p112-walsh-gl2-strata/TASK.json

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+{
+  "schema_version": 1,
+  "task_id": "complex-rank-shape-decoder",
+  "priority_revision": 8,
+  "base_commit": "721c03822c421b7f7fcbb2598993b947c4d88419",
+  "original_card": {
+    "id": "complex-rank-shape-decoder",
+    "question": "Can a legal decoder or coframe change improve the full signed moment, including changed helper targets?",
+    "starting_points": [
+      "docs/research/current-priorities.md",
+      "docs/research/research-family-status.md",
+      "certificates/selected-result.json",
+      "docs/research/pr-intake-evidence/pr5599-completion/PROOF-REVIEW.md",
+      "docs/research/recursive-invoice-preflight.md"
+    ],
+    "methods": [
+      "Construct and independently check one source-bound instance of the stated first experiment.",
+      "Seek an exact counterexample or target-specific complete-budget obstruction to that same proposed change."
+    ],
+    "acceptance_gate": "A source-bound arbitrary-input identity and legal producer/consumer boundary with the complete paid and productive target invoice, including restoration, ownership, native alignment and extra charges. Price under the actual current model; historical ceiling.py assumptions must be established before use. A favorable optimistic relaxation keeps a route open but is not a completed supplier. A scoped local result remains useful; expansion requires the family-level budget gate.",
+    "first_step": {
+      "command": [
+        "python3",
+        "-B",
+        "scripts/research_selected_baseline.py"
+      ],
+      "expected": "PASS_RESEARCH_BASELINE_BINDINGS_ONLY",
+      "inspect": "Read revision 8, the selected review and live exact-source handoffs. Then follow this card's bounded first_experiment.",
+      "scope": "Source-bound orientation only, not a supplier replay or theorem proof."
+    },
+    "first_experiment": "Price one complete changed itinerary with both paid and productive histograms. Check #5693/#5698 for overlap and their exact hypotheses before another coframe search. Use the signed invoice preflight where its boundary contract agrees; an inconclusive concave comparison is not a power crossing. Stop if an optimistic complete budget fails at the chosen target or native delivery remains unspecified.",
+    "bottleneck_link": "Judge the actual current supplier contract and complete signed target/paid bill; frozen-profile ceilings require an applicability proof.",
+    "allowed_variations": [
+      "One exact source head, helper/cut region or quantified proposition and changed hypothesis; check live handoffs first. Different seeds are not different scopes."
+    ],
+    "scope_change_triggers": [
+      "Changing source head, helper/cut domain, operation alphabet or caller boundary without recording the new scope.",
+      "Treating an archived proposal or local numerical pass as complete supplier admission."
+    ],
+    "overlap_terms": [
+      "coframe",
+      "signed moment",
+      "5693",
+      "5698"
+    ],
+    "existing_checks": [
+      {
+        "scope": "Selected PR5599: complete helper-subspace construction and independent caller replay; five framework assumptions remain.",
+        "evidence": "docs/research/pr-intake-evidence/pr5599-completion/PROOF-REVIEW.md",
+        "remaining": "The next experiment must identify uncovered source-specific work. Successor archive intake does not transfer this admission.",
+        "status": "reviewed",
+        "source_commit": "09dd0965711ca7eeb169057a420abab1c9052137",
+        "selected_record_sha256": "2a0220e866417774606e0f991e41b9c1d26b1f43b97857238cf478dbc7d330b5"
+      }
+    ]
+  },
+  "scope_status": "changed",
+  "actual_scope": "Classify every real or rational GL2 basis change on every ordered pair of the PR2042 Walsh lift planes.",
+  "scope_change_reason": "Replace the current supplier-itinerary experiment with the explicitly open continuous coupled-lift side question in PR2042. This is an infinite-domain extension, not a replay or audit of its finite census.",
+  "bottleneck_link": "Scoped decoder exclusion: changing bases inside these Walsh planes cannot lower their pairing-rank invoice. A useful replacement must change the planes, their product, or the channel family. This does not claim a selected P104 improvement.",
+  "budget_minutes": 60,
+  "stop_rule": "Stop at the first verified complete support classification; do not search changed planes or optimize downstream bills.",
+  "outcome": "negative-result",
+  "evidence": "PROOF.md and exact support certificate computed by verify.py; no predecessor result receipt is read.",
+  "author": "Nick Petalas",
+  "ai_assistance": "Independent human and formal proof-kernel review are unclaimed.",
+  "open_questions": [
+    {
+      "question": "Can a changed rational two-plane product UV^T, outside the paired Walsh family, improve the ten-term pairing-rank profile?",
+      "why_it_matters": "The present obstruction fixes the product; the general overlapping rank-two branch of PR1197 stays open.",
+      "first_step": "Enumerate exact support-zero strata for one explicitly different rational plane pair and price all ten terms.",
+      "blocked_by": "A favorable example and complete physical source/frame/restoration costs are not supplied."
+    }
+  ],
+  "source_question": {
+    "pr": 2042,
+    "author": "1337isnot1337",
+    "claim_url": "https://github.com/CrocSwap/integer-mult-bounds/discussions/1166#discussioncomment-19246923"
+  },
+  "environment": {
+    "python": "3.12.15",
+    "implementation": "CPython",
+    "platform": "Darwin",
+    "machine": "arm64"
+  },
+  "claim_type": "scoped computer-assisted proof"
+}

research/p112-walsh-gl2-strata/VERIFY.json

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+{
+  "schema_version": 1,
+  "status": "ready",
+  "python": "3.12",
+  "requirements": [],
+  "inputs": [
+    "research/p112-single-lift/geometry.py"
+  ],
+  "commands": [
+    {
+      "argv": [
+        "{python}",
+        "-I",
+        "-W",
+        "error",
+        "-B",
+        "{package}/verify.py"
+      ],
+      "timeout_seconds": 120,
+      "checks": "Complete real-projective support strata, exact binary ranks, literal endpoints and rejecting controls."
+    },
+    {
+      "argv": [
+        "{python}",
+        "-I",
+        "-O",
+        "-W",
+        "error",
+        "-B",
+        "{package}/verify.py"
+      ],
+      "timeout_seconds": 120,
+      "checks": "Same complete certificate with assertions disabled."
+    }
+  ]
+}