Prove fixed-rank symmetric-phase rigidity of the PR4610 round-three word (no new κ)

research/symmetric-phase-rigidity/README.md

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+# Exact symmetric-phase rigidity of the PR4610 round-three word
+
+**Scoped exact linear certificate; no new kappa.** The unchanged-paid-rank
+mechanism in [Dugongue's #4654 phase-form grammar](https://github.com/CrocSwap/integer-mult-bounds/pull/4654)
+has only the zero solution on Cao Yifan's frozen round-three native word under
+the boundary conditions below. All **310,640** free binary symmetric-form
+coordinates are forced zero by **559,016** exact homogeneous equations. Both
+boundaries of all 408 independent helpers are allowed to vary.
+
+This answers the concrete next question of whether that new grammar supplies
+more productive helper rank at unchanged paid ranks on the strongest local
+word checked in this session. It does not. It does not rule out phases that
+increase some paid ranks, changed subspaces/maps, changed scalar order, relaxed
+adapter/owned interfaces, or a new controller. No global phase obstruction or
+optimal kappa is claimed.
+
+## Frozen object and exact scope
+
+The input is `research/kappa-round3-warp-20261022/ACTUAL-WORD.json.gz` at PR4610
+head `1eb8da6cfb35512c0cc2533413e5ca42f48e13c6`. Its exact bytes are vendored
+with original commit/path/blob provenance and an exact Git pin for the retained
+copy in [SOURCES.json](SOURCES.json). Its local
+paid profile is `{1:2201,2:412,3:510,4:127,5:40,8:8}` and its productive helper
+profile is the retained `{4:28,7:7,8:47,9:326}`. This package neither changes
+that word nor claims its proposed integer-multiplication exponent as reviewed.
+
+At every scalar contact or chart boundary with subspace U, assign an arbitrary
+symmetric bilinear form b on U, with dim(U)(dim(U)+1)/2 coordinates. This is the
+full local symmetric-form space, not a search of 511 ambient rank-one forms.
+The actual fixed conditions are:
+
+* Keep all original subspaces, scalar chronological operations and shared G maps.
+* Freeze both main endpoints, every explicit adapter boundary, and every scalar
+  contact involving the owned C field to the original zero form.
+* Preserve every paid Gaussian rank and every monomial native warp rank.
+* Use one common form for every operand at each scalar contact. Both helper
+  entrance and exit forms may vary, with their actual difference tested.
+
+In particular the result does not cover a redesigned owned-center protocol.
+The construction represents 19,199 nodes, including two inert sentinels, with
+957 frozen nodes. Zero-dimensional contacts carry no form coordinates.
+
+## Why the equations are sufficient and complete for this model
+
+For a nested paid edge U subset W, the phase-form grammar gives
+
+    new_rank = dim(W)-dim(U) + rank((b_U+b_W)|U).
+
+Its old rank is unchanged exactly when the restricted forms agree. For each
+pair of source basis vectors x,y, emit b_U(x,y)+b_W(x,y)=0. On a fixed logical
+warp G:U->W, rank zero is preserved exactly when
+
+    b_U(x,y) = b_W(Gx,Gy).
+
+The verifier checks that the actual shared G maps the source chart onto the
+target chart before emitting its equations, using G inverse on negative
+occurrences. Scalar contacts share one node. Frozen node coordinates are zero.
+These are the original conventions of #4654; no F2 reinterpretation of the
+rational BIT geometry is involved. Only the complex nine-bit frame word is used.
+
+In coordinates x=B a, y=B c, the coefficient of S_ii is a_i c_i, and the
+coefficient of S_ij for i<j is a_i c_j+a_j c_i. The verifier computes these
+coordinates by exact binary elimination. It emits 554,922 nonzero paid-edge
+equations and 4,094 nonzero warp equations, rather than trusting a solver or
+merely checking a saved numerical rank.
+
+An elementary substitution certificate suffices. Each one-variable equation
+sets that coordinate to zero; each two-variable equation identifies their
+classes. Longer equations are revisited after substitution. The closure merges
+all 310,640 coordinates into the zero class and discharges every equation.
+There is no residual system requiring numerical elimination or a rank estimate.
+Since each substitution follows an original homogeneous equation, every
+solution of the original system must be zero. Conversely zero satisfies every
+equation. This proves uniqueness.
+
+For each helper, the verifier also evaluates all symmetric coefficient
+functionals of (b_start+b_end) on its entrance space. All 80 nontrivial queries
+are forced zero, so the productive rank cannot increase. This latter statement
+would still be useful if some irrelevant form coordinates survived; here the
+stronger all-coordinate rigidity is certified.
+
+## Reproduce and verification limits
+
+Standard-library CPython 3.12, from the repository root:
+
+```sh
+python3.12 -B scripts/research_package.py --package research/symmetric-phase-rigidity
+python3.12 -I -O -B research/symmetric-phase-rigidity/verify.py
+```
+
+The replay authenticates the source, reconstructs the graph and all equations,
+discharges them exactly, and compares [RESULT.json](RESULT.json). Replacing
+every frame basis by an independently changed basis gives the identical result.
+Three small exact controls distinguish a surviving free direction, a forced
+zero direction and a higher-arity equation that becomes decisive after other
+substitutions. The discovery calculation was also performed with a separate
+C++ equality/elimination implementation; the registered proof uses Python only.
+Both implementations were prepared by the same assisted author, not independent
+human reviewers.
+
+This checks the frozen word's geometric phase constraints. It does not rerun
+its arbitrary-column native endpoint proof, absolute Clifford phases, the
+uniform compiler, BIT admission, precision, fixed-tape/Word RAM transfer or the
+outer theorem. Those are unnecessary to the stated no-solution certificate and
+are not claimed as verified here. The selected conditional kappa remains
+0.00302015 at canonical base `b58ad9ef4d4e861b0eaa5dba41742bdc072dca51`.
+
+## Open next questions
+
+A useful phase construction must escape at least one of the stated constraints.
+The next bounded experiment is to allow explicitly costed rank increases at
+paid or warp edges and compare the complete helper reward with that added bill.
+Alternatively a changed chronological graph or a separately realized owned
+center gauge can escape this certificate. A larger unchanged-rank nullspace
+search on these exact bytes cannot. See [TASK.json](TASK.json).
+
+## Credit
+
+Prepared by Cinna Davis with substantial OpenAI Codex assistance, Apache-2.0.
+Credit Dugongue for the source-convention-aware symmetric-form grammar and
+phase-flow construction in #4654; Cao Yifan for PR4610's round-three word and
+native G/helper interface design; Cinna Davis's #4440/#4441/#4447/#4470 supplier
+and controller lineage; zxv, Jacob Alan Sussman and the retained scalar, E8,
+Clifford, native, precision and compiler contributors. Classical bilinear-form
+coordinates and exact equality substitution are not claimed as new mathematics.

research/symmetric-phase-rigidity/RESULT.json

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+{
+  "all_coordinates_forced_zero": true,
+  "contact_nodes_including_two_sentinels": 19199,
+  "controls": [
+    "free_direction_detected",
+    "zero_boundary_propagation",
+    "higher_arity_substitution",
+    "all_frame_basis_changes_preserve_result"
+  ],
+  "equations": {
+    "paid": 554922,
+    "warp": 4094
+  },
+  "frame_registry": 4429,
+  "free_binary_form_coordinates": 310640,
+  "frozen_nodes": 957,
+  "helpers": 408,
+  "independent_equalities": 310640,
+  "native_roles": 632,
+  "nontrivial_helper_coefficient_queries": 80,
+  "scope": "Fixed subspaces/logical word/maps, unchanged paid ranks, zero main/adapter/owned gauges; both helper boundaries may vary.",
+  "status": "EXACT_FIXED_PAID_RANK_PHASE_RIGIDITY_NO_NEW_KAPPA",
+  "unresolved_equations": 0
+}

research/symmetric-phase-rigidity/SOURCES.json

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+{
+  "schema_version": 1,
+  "base_commit": "fcdf5ce4682beb29e6729f9a41eb971a1d35d0fa",
+  "files": [
+    {
+      "path": "certificates/selected-result.json",
+      "usage": "provenance",
+      "git_blob": "c35482cd9035843b15a634cc8276a7e1e40e8395",
+      "sha256": "6fe0a099b89aa2aed796a13c3a21dafc63f3388ebf397464e44e0451893b9375"
+    },
+    {
+      "path": "docs/research/agent-task-catalog.json",
+      "usage": "provenance",
+      "git_blob": "f126dd2e2dad2b669e75d1d0d0bd05c600725602",
+      "sha256": "f7a48bfa2659b88378282a8eee7d9fe52cdf505594124c4d1686b58510fffd72"
+    },
+    {
+      "path": "research/symmetric-phase-rigidity/sources/round3-native.json.gz",
+      "usage": "input",
+      "git_blob": "46d929df83817b85a5387bcdb9745e0627d6366e",
+      "sha256": "7679daa4ec6a40f56fc7f0202379a6f454310e94945aba4f48beaaaafb838d05",
+      "pin": "base_commit:research/symmetric-phase-rigidity/sources/round3-native.json.gz",
+      "origin": {
+        "commit": "1eb8da6cfb35512c0cc2533413e5ca42f48e13c6",
+        "path": "research/kappa-round3-warp-20261022/ACTUAL-WORD.json.gz",
+        "git_blob": "46d929df83817b85a5387bcdb9745e0627d6366e"
+      }
+    }
+  ]
+}

research/symmetric-phase-rigidity/TASK.json

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+{
+  "schema_version": 1,
+  "task_id": "complex-new-helper-inventory",
+  "priority_revision": 7,
+  "base_commit": "fcdf5ce4682beb29e6729f9a41eb971a1d35d0fa",
+  "original_card": {
+    "id": "complex-new-helper-inventory",
+    "question": "Can a new local cost interface change the deficit-to-recursive-cost ratio beyond current packing?",
+    "starting_points": [
+      "docs/research/research-family-status.md",
+      "research/complex-local-word-ceiling/README.md",
+      "research/heterogeneous-owner-assembly/README.md",
+      "certificates/selected-result.json",
+      "research/selected-k8-unrestricted-ends/README.md"
+    ],
+    "methods": [
+      "Propose a changed helper inventory and derive an optimistic exact moment.",
+      "Design a small completed instance and independently verify its endpoint and rank bill."
+    ],
+    "acceptance_gate": "A new physically realizable contract outside the frozen catalogue whose per-call profile, screened with ceiling.ceiling(), beats the selected word's packing ceiling; a lower cap does not establish identical circuitry or order attainable optima, and is insufficient to pass the stronger-budget target under this model. 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",
+        "research/complex-local-word-ceiling/verify.py"
+      ],
+      "expected": "PASS_COMPLEX_LOCAL_WORD_CEILING",
+      "inspect": "Read the selected word's per-call local profile, its packing ceiling and the lever table; run levers.py for the ideal budget of the lever you target.",
+      "scope": "Positive control and planning budget only; this is not a new construction or full cold replay."
+    },
+    "first_experiment": "Before any synthesis, write the candidate per-call local profile (and centre charge / unpaid coordinates if changed) and screen it with ceiling.ceiling(); reject it if even the ideal ceiling does not beat the selected word's by your stated margin.",
+    "bottleneck_link": "Packing of the selected word is capped at +0.87% (ceiling model); only a lower local excess spend or a changed coverage law raises the cap.",
+    "allowed_variations": [
+      "Narrow to one changed boundary or helper type with explicit counts.",
+      "If the coverage law, template or guard changes, derive an applicable fully charged replacement bound instead of applying the incumbent ceiling outside its scope."
+    ],
+    "scope_change_triggers": [
+      "Mixing only providers already inside the pinned catalogue.",
+      "Only changing helper terminal subspaces and monotone frames within the fixed scalar word, entrances and cleanup contract already covered by #1995/#2013.",
+      "Continuing a partial cost improvement while leaving the same unpriced consumer, restoration or central-readout obligation unresolved; refresh the family expansion decision first. Local product-retiming evidence has a narrow scope and is not a ban on changed graphs or contracts."
+    ],
+    "overlap_terms": [
+      "helper inventory",
+      "heterogeneous-owner",
+      "mutable leaver"
+    ]
+  },
+  "scope_status": "changed",
+  "actual_scope": "Exactly decide all fixed-paid-rank symmetric phase forms on PR4610 round-three under frozen main, adapter and owned contacts, allowing both helper boundaries to vary.",
+  "scope_change_reason": "User requests a structural kappa breakthrough. PR4654 supplies a new symmetric-form grammar; test its unchanged-paid-rank productive-helper route on the strongest checked native word, outside the preceding subspace optimization.",
+  "bottleneck_link": "A nonzero helper phase difference could raise target ranks at unchanged paid costs. All 310640 variables instead are forced zero in this exact model.",
+  "budget_minutes": 60,
+  "stop_rule": "Certify the exact constraint system and its solution space; preserve paid-rank and native/outer scope, and do not infer a universal phase or exponent ceiling.",
+  "outcome": "completed",
+  "evidence": "RESULT.json, README.md proof and independently implemented standard-library verify.py",
+  "author": "Cinna Davis, with substantial OpenAI Codex assistance",
+  "ai_assistance": "Substantial OpenAI Codex assistance in derivation, C++ discovery, Python exact verification and writing; no subagents.",
+  "open_questions": [
+    {
+      "question": "Can explicitly paid phase changes yield a strictly profitable full target/paid moment?",
+      "why_it_matters": "The unchanged-paid-rank phase space is rigid on this word, so a positive phase construction must change a stated premise.",
+      "first_step": "Allow specified paid/warp rank increases, emit the actual phase gauges and children, and certify their full moment against the actual helper rewards.",
+      "blocked_by": "No positive complete costed phase witness has been established."
+    }
+  ],
+  "upstream_checkpoint": {
+    "main": "b58ad9ef4d4e861b0eaa5dba41742bdc072dca51",
+    "source_pr": 4610,
+    "source_head": "1eb8da6cfb35512c0cc2533413e5ca42f48e13c6",
+    "grammar_pr": 4654,
+    "grammar_head": "dca175fed8ebde8980635fd3b4508b8cba89a1b0",
+    "selected_kappa": "0.00302015"
+  },
+  "overlap_coverage": {
+    "local": {
+      "head": "fcdf5ce4682beb29e6729f9a41eb971a1d35d0fa",
+      "records": 5862,
+      "head_records": 5862,
+      "complete_head_inventory": true,
+      "unsearched_head_paths": []
+    },
+    "open_prs": {
+      "fetched_at": "2026-10-22T14:45:51.046622+00:00",
+      "complete_pagination": true,
+      "records": 434
+    },
+    "discussions": {
+      "fetched_at": "2026-10-22T14:46:46.485113+00:00",
+      "complete_pagination": true,
+      "records": 3593
+    }
+  },
+  "overlap_scope": "Builds directly on the new phase-form grammar in PR4654, which does not report a useful nonzero solution. No competing exact rigidity certificate surfaced in searched metadata; this is not proof of novelty."
+}

research/symmetric-phase-rigidity/VERIFY.json

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+{
+  "schema_version": 1,
+  "status": "ready",
+  "python": "3.12",
+  "requirements": [],
+  "inputs": [
+    "research/symmetric-phase-rigidity/sources/round3-native.json.gz",
+    "research/symmetric-phase-rigidity/RESULT.json"
+  ],
+  "commands": [
+    {
+      "argv": [
+        "{python}",
+        "-B",
+        "{package}/verify.py"
+      ],
+      "timeout_seconds": 120,
+      "checks": "Reconstruct all 559016 exact constraints, force all 310640 symmetric-form coordinates to zero, check all 80 helper functionals, basis invariance and exact free/forced direction controls."
+    }
+  ]
+}