Give a closed Lucas formula for quinary repair ranks (no κ)

research/quinary-repair-lucas-formula/PROOF.md

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+# A Lucas formula for the resonant disjointness repair
+
+## Result and dependency boundary
+
+Let F be a field of characteristic five, h=4k with k>=1, and A_h the
+disjointness matrix on the nonempty proper subsets of [h]. Let L_0=2,
+L_1=1 and L_(n+2)=L_(n+1)+L_n be the Lucas numbers. Then
+
+```text
+rank_F(I-A_h) = (4*2^h - 2*L_h - 5)/5.                 (1)
+```
+
+This is an explicit evaluation of Brian / JoyboyBrian's PR5221 theorem,
+not a new proof of its matrix or endpoint-deletion claims. The precise
+premises used from that package are:
+
+1. Its ordinary integer multiplicities c_(h,j) start at (1,0,0,0,0).
+2. One step applies the replacements
+   `1 -> 2; 2 -> 1+3; 3 -> 2+4; 4 -> 3+5; 5 -> 5+5`.
+3. For h=4k>=4, its rank is `2^h-1-b_h`, where `b_h=sum_j c_(h,j)`.
+
+The new step is the arithmetic identity
+
+```text
+b_h = (2^h + 2*L_h)/5  for every even h>=0.            (2)
+```
+
+Combining the cited rank theorem with (2) gives (1). Consequently (1) is
+also the exact number of additive scalar rank-one corrections, under the
+same algebraic repair model as PR5221. No physical correction cost is inferred.
+
+## Evaluate the recurrence
+
+Write c_h=(a_h,b_h',d_h,e_h,p_h), using b_h' to distinguish the second
+multiplicity from the total block count b_h. The cited replacements give
+
+```text
+c_(h+1) = (b_h', a_h+d_h, b_h'+e_h, d_h, e_h+2*p_h).
+```
+
+At even h, only the first, third and fifth multiplicities can be nonzero.
+At odd h, only the second, fourth and fifth can be nonzero. This follows
+inductively from the displayed update, starting at h=0.
+
+Let F_0=0, F_1=1 and F_(n+2)=F_(n+1)+F_n. For every even h>=2,
+
+```text
+a_h=F_(h-1),  d_h=F_h,  b_h'=e_h=0.                  (3)
+```
+
+Indeed c_2=(1,0,1,0,0). Two steps send the first and third
+multiplicities (a,d) to (a+d,a+2d). The Fibonacci recurrence sends
+(F_(h-1),F_h) to (F_(h+1),F_(h+2)) by exactly that map. Induction
+proves (3). This Fibonacci pattern is classical; it is not new here.
+
+The weighted dimension identity is
+
+```text
+sum_(j=1)^5 j*c_(h,j) = 2^h.                         (4)
+```
+
+For completeness, each replacement doubles its label's weight:
+2=2*1, 1+3=2*2, 2+4=2*3, 3+5=2*4 and 5+5=2*5.
+The initial weight is one, so induction gives (4) as an integer identity.
+
+For even h>=2, subtract (4) from five times the block count and use (3):
+
+```text
+5*b_h - 2^h = 4*F_(h-1) + 2*F_h
+             = 2*(F_(h-1)+F_(h+1))
+             = 2*L_h.
+```
+
+The identity `L_h=F_(h-1)+F_(h+1)` for h>=1 follows from the two
+recurrences and their initial values. At h=0, (2) holds separately since
+b_0=1 and L_0=2. This proves (2) for all even h, including its exact
+divisibility by five. Substitution in PR5221's rank formula proves (1).
+
+For example (1) gives 9 at h=4 and 185 at h=8, the inherited examples.
+Those evaluations are not new independent matrix-rank checks.
+
+## Credit and scope
+
+The source is PR5221 at `ad83478022fed6c1d6a6014096bd8b443e8c9d9a`,
+retained in this package's base. Its Jordan rules and endpoint theorem are
+cited premises. Its predecessors include Ali Romman / PR4962, Nick
+Petalas / PR5173, mikey1201 / PR172, and icekylinx / PR144.
+
+The modular Jordan/Fibonacci description is classical. See David Benson,
+[Commutative Banach algebras, representations of finite groups, and
+Hilbert space](https://homepages.abdn.ac.uk/d.j.benson/pages/papers/b/benson/banach-book.pdf),
+Section 5.1, page 83, for the same tensor-product rules and Fibonacci
+multiplicities. The proof above is self-contained arithmetic once PR5221's
+premises are granted. The contribution here is the explicit Lucas-number
+specialization to the repository's resonant proper-subset repair rank.
+No general literature-priority claim is made.
+
+This scoped written proof was produced by Nick Petalas. Independent human
+and formal proof-kernel review are unclaimed. Apache-2.0.
+
+This proof does not decide nonresonant ranks, validate PR5221's algebraic
+premises, supply a physical caller, or reduce the number of required repairs.
+It replaces the recurrence presentation by a closed formula for the same
+quantity. All five selected framework assumptions remain. No selected-frontier
+change or complete multiplication theorem is claimed.

research/quinary-repair-lucas-formula/README.md

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+# A closed Lucas formula for the resonant repair rank
+
+For every field F of characteristic five and every positive h divisible by
+four, the proper-subset disjointness matrix A_h has
+
+```text
+rank_F(I-A_h) = (4*2^h - 2*L_h - 5)/5,
+L_0=2, L_1=1, L_(n+2)=L_(n+1)+L_n.
+```
+
+Claim type: scoped written arithmetic proof with exact finite corroboration,
+depending on Brian / JoyboyBrian's [PR5221](https://github.com/CrocSwap/integer-mult-bounds/pull/5221)
+rank theorem. The new contribution evaluates its five-state integer recurrence
+in closed form. It proves `total_blocks=(2^h+2*L_h)/5` for every even h.
+The inherited rank and additive rank-one correction count are unchanged.
+See [PROOF.md](PROOF.md) for the complete argument and premise boundary.
+
+The Fibonacci multiplicities and modular Jordan theory are classical. David
+Benson's [Section 5.1, page 83](https://homepages.abdn.ac.uk/d.j.benson/pages/papers/b/benson/banach-book.pdf)
+describes the same tensor/Fibonacci pattern. The claimed contribution is its
+explicit Lucas specialization to the repository's resonant repair endpoint;
+no literature-priority claim is made for the classical ingredients.
+
+## Evidence and reproduction
+
+The direct verifier applies the quoted replacements as integer bookkeeping;
+it does not import or replay a predecessor scientific verifier. Iterative fast
+doubling and a separate binomial sum agree on all 513 Lucas values through
+index 512. The literal recurrence agrees with 512 parity/multiplicity formulas,
+257 even block-count formulas and 128 resonant rank substitutions. Fifteen
+invalid inputs and five named formula mutations are rejected. A generated
+stdout receipt binds every package file and the exact recurrence transcript.
+No receipt is handwritten or committed.
+
+From the worktree root, with the local interpreter or any CPython 3.12:
+
+```sh
+/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -X dev -B research/quinary-repair-lucas-formula/verify.py
+/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -O -W error -X dev -B research/quinary-repair-lucas-formula/verify.py
+/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/quinary-repair-lucas-formula
+/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/quinary-repair-lucas-formula --check-source-history
+```
+
+Expected direct marker: `PASS_QUINARY_REPAIR_LUCAS_FORMULA`. The registered
+commands run both warning-strict normal and optimized modes, each with a
+120-second limit. Direct verification needs only these six files and the
+standard library. Repository metadata validation additionally needs reachable
+Git provenance from [SOURCES.json](SOURCES.json).
+
+## Limits and implementation bounds
+
+The all-dimensional conclusion follows from the written arithmetic proof and
+the cited rank premise. Bounded replay is corroboration. This contribution
+does not audit PR5221's Jordan rules, deletion argument or matrix ranks.
+Nonresonant dimensions, physical realization, native calls, routing, dirty
+restoration, precision, stock, compiler and outer admission remain outside
+scope. All five selected assumptions remain. No new kappa, selected-frontier
+change or complete multiplication theorem is claimed. No predecessor scientific
+suite, selected cold replay, Lean or Docker run is required or claimed.
+
+The verifier has no recursion or subprocesses. Its outer domain is 0..512,
+fast doubling uses at most ten bit steps and each binomial sum at most 256
+terms. The bookkeeping has five integer counts; its extra terminal update
+has total dimension 2^513. Each formula mutation checks at most 128 indices.
+Explicit checks remain active under optimization. Power of Ten's fixed-memory
+rule has a bounded CPython allocation deviation: arithmetic, six source reads
+and JSON serialization allocate managed objects during this terminating run.
+All functions are below 60 physical lines. Warning-strict execution and
+in-memory syntax checks are the available diagnostics; no separate static
+analyzer or formal proof kernel is claimed.
+
+## Open next questions
+
+Can a nonadditive physical repair avoid this additive rank obstruction?
+First specify its arbitrary-input identity and full preparation, routing and
+dirty-restoration invoice. The missing physical caller blocks any supplier
+conclusion. No continuation is reserved.
+
+## Credit and coordination
+
+This contribution was produced by Nick Petalas. Independent human and formal
+proof-kernel review are unclaimed. Apache-2.0.
+
+Credit Brian / JoyboyBrian PR5221 for the recurrence and rank theorem; Ali
+Romman PR4962, Nick Petalas PR5173, mikey1201 PR172 and icekylinx PR144 for
+the retained tensor, endpoint and compiler lineage; and David Benson for the
+classical modular-representation context. Predecessor files and credits are
+unchanged. Exact base-file Git and SHA256 pins are provenance, not live inputs.
+
+[Advance claim](https://github.com/CrocSwap/integer-mult-bounds/discussions/1166#discussioncomment-19370178).
+Base `7c49fe32c55de02c2c873655c804b16e4c3fef0a`, priorities revision 8,
+selected PR6030. This changes the coframe card to a historical arithmetic
+side result, as recorded in [TASK.json](TASK.json). PR6469 studies the speed
+of a bounded rank evaluator. No evaluator, matrix corpus or implementation
+question from that work is extended here. Live handoffs, open PR bodies,
+merged formula/object searches and main READMEs were checked. Arbitrary
+historical diffs and unpublished work remain coverage limits.

research/quinary-repair-lucas-formula/SOURCES.json

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+{
+  "schema_version": 1,
+  "base_commit": "7c49fe32c55de02c2c873655c804b16e4c3fef0a",
+  "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/quinary-resonant-disjointness-repair/PROOF.md",
+      "usage": "provenance",
+      "git_blob": "8cf640e2063512c24d9fa9c02a77249b3ad852dc",
+      "sha256": "a0a821bfbbd772e04efd8cfbbde44d8a7ddfbd3986738ea54e70dc0d599b42ef"
+    },
+    {
+      "path": "research/quinary-resonant-disjointness-repair/README.md",
+      "usage": "provenance",
+      "git_blob": "f0ee79a05770a50004032c4b8536182e19264d73",
+      "sha256": "d8dfd60d785fc353f02d2cfaf20de86a922e98b1e36b9b18fe3d2b1deb8650c8"
+    }
+  ]
+}

research/quinary-repair-lucas-formula/TASK.json

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+{
+  "schema_version": 1,
+  "task_id": "complex-rank-shape-decoder",
+  "priority_revision": 8,
+  "base_commit": "7c49fe32c55de02c2c873655c804b16e4c3fef0a",
+  "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": "Closed Lucas-number evaluation of the ordinary integer recurrence in PR5221, specialized to its resonant characteristic-five proper-subset repair rank.",
+  "scope_change_reason": "A scoped arithmetic side result replaces a supplier coframe experiment. It cites the existing Jordan and endpoint theorem instead of auditing or extending its matrix implementation.",
+  "bottleneck_link": "Makes an existing exact additive repair obstruction explicit in the dimension. This is a historical algebraic side result, not a current supplier improvement.",
+  "budget_minutes": 60,
+  "stop_rule": "Ship the first exact closed form; do not extend to nonresonant ranks, physical realization or new characteristic families.",
+  "outcome": "completed",
+  "evidence": "PROOF.md supplies the all-even recurrence evaluation and cited rank corollary; verify.py corroborates exact arithmetic through index 512 without predecessor scientific replays.",
+  "author": "Nick Petalas",
+  "ai_assistance": "Independent human and formal proof-kernel review are unclaimed.",
+  "open_questions": [
+    {
+      "question": "Can a nonadditive physical repair avoid this additive rank obstruction?",
+      "why_it_matters": "An explicit count of additive corrections does not bound a changed operator contract.",
+      "first_step": "Write an arbitrary-input identity and the full paid preparation, routing and dirty-restoration bill.",
+      "blocked_by": "No physical caller or complete invoice is supplied."
+    }
+  ],
+  "predecessor": {
+    "pr": 5221,
+    "author": "JoyboyBrian",
+    "head": "ad83478022fed6c1d6a6014096bd8b443e8c9d9a",
+    "premises": "Five-state integer recurrence and rank = 2^h-1-total_blocks for h divisible by four."
+  },
+  "related_account_work": {
+    "pr": 6469,
+    "distinction": "A bounded rank-evaluator implementation and replay-time improvement. No evaluator, matrix corpus, performance question or implementation is extended here."
+  },
+  "claim_url": "https://github.com/CrocSwap/integer-mult-bounds/discussions/1166#discussioncomment-19370178",
+  "environment": {
+    "python": "3.12.15",
+    "implementation": "cpython",
+    "dependencies": "standard library"
+  },
+  "overlap": {
+    "canonical_main": "7c49fe32c55de02c2c873655c804b16e4c3fef0a",
+    "priority_revision": 8,
+    "selected_pr": 6030,
+    "limitations": "Live metadata, main READMEs and source-specific formula searches do not cover arbitrary historical diffs or unpublished work."
+  }
+}

research/quinary-repair-lucas-formula/VERIFY.json

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+{
+  "schema_version": 1,
+  "status": "ready",
+  "python": "3.12",
+  "requirements": [],
+  "inputs": [],
+  "commands": [
+    {
+      "argv": [
+        "{python}",
+        "-I",
+        "-W",
+        "error",
+        "-X",
+        "dev",
+        "-B",
+        "{package}/verify.py"
+      ],
+      "timeout_seconds": 120,
+      "checks": "Exact recurrence/Lucas/binomial corroboration, domain rejections and five formula mutations."
+    },
+    {
+      "argv": [
+        "{python}",
+        "-I",
+        "-O",
+        "-W",
+        "error",
+        "-X",
+        "dev",
+        "-B",
+        "{package}/verify.py"
+      ],
+      "timeout_seconds": 120,
+      "checks": "Exact recurrence/Lucas/binomial corroboration, domain rejections and five formula mutations."
+    }
+  ]
+}