[petalas]

Summary

Scoped all-field proof with exact finite corroboration. For a typed-copy cache tree whose terminals are coordinate Lagrangian charts in one common symplectic basis, arbitrary intermediate charts cannot lower its rank-only cost.

L_v -> echelon pivots in self-dual coordinate order -> R(L_v)
terminals fixed; d(R(L),R(M)) <= d(L,M)
any nondecreasing paid edge tariff therefore cannot increase

Every variable vertex can be restricted to 2^n coordinate charts. This holds for arbitrary finite trees and includes the power tariff r^(1-kappa). No new kappa, selected-frontier change or complete multiplication theorem is claimed.

This contribution was produced by Nick Petalas. Credit zxv's PR5283 for the snapshot-tree contract and paid Lagrangian metric, Matarener's PR5060, Dugongue's PR4534 and the retained native/helper/compiler/supplier lineage. Elementary pivot and initial-subspace facts are background; no literature-priority claim is intended. The predecessor constructions and counts are cited rather than replayed or audited.

Evidence

Before, the cited cache model permits arbitrary intermediate Lagrangians. After, the explicit retraction fixes all common-coordinate terminals and contracts every edge distance. The written proof supplies the universal quantifiers.

The standalone checker enumerates 2,379 row-reduced planes in seven small-field domains, retains 359 charts, and checks all 44,447 ordered pairs using literal vector-set intersections independently of the pivot calculation. It records 37,780 strict contractions and 416 monotone tariff/distance checks. Another 675 labeled trees cover three fixed four-terminal topologies; restricted and unrestricted minima agree at 3,3,7/2 under prices (0,1,3/2). Seven malformed-chart rejections and four scope controls pass.

Locally on CPython 3.12.15, warning-strict normal and optimized commands pass in 0.067 and 0.065 seconds respectively. These are timeout measurements, not a runtime improvement claim. Registered metadata/replay, tabnanny, function-length bounds, whitespace, exact committed bytes and source ancestry all pass. The official committed export passes integrity checking, and both direct modes pass from its read-only Git-free extraction with empty PATH and all seven exported files unchanged. The generated stdout receipt binds all six package files by SHA256.

From the repository root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/coordinate-cache-retraction/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -O -W error -B research/coordinate-cache-retraction/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/coordinate-cache-retraction
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/coordinate-cache-retraction --check-source-history

Any CPython 3.12 interpreter can replace the local path. Expected markers are PASS_COORDINATE_CACHE_RETRACTION, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES. Standard library only; replay writes nothing. Add --emit-result to the direct command for its generated hash-bound receipt.

Limitations

Every terminal must use one common coordinate basis. Negative internal-edge credits, decreasing prices and unproved intermediate feasibility constraints are outside the reduction. The theorem prices rank-dependent edges only. It does not supply a source-bound current caller, ordinary monomial costs, absolute phases, native realization, ownership, dirty restoration, snapshot lifetimes, precision/tape bounds or a complete signed invoice. All five selected framework assumptions remain. Selected cold replay, Lean and Docker are unrun. Independent human and formal proof-kernel review are unclaimed. Ruff is unavailable locally; warning-strict interpreter checks and tabnanny pass.

Open next questions

Find one actual repeated source snapshot whose required read charts share a coordinate basis. Pin its exact operator row and lifetime, then price one complete retracted tree including ordinary work and restoration. The physical consumer and complete costs remain blockers; no follow-on search is reserved.

Related open work

No open PR for this coordinate-terminal cache-tree reduction was found in the complete 310-PR overlap snapshot, supplemented by the refreshed latest 100 open PRs and current claims. Coverage excludes unpublished work and arbitrary historical diffs.

• Explore joint frame retiming after carried swaps (ongoing; no new κ) #4993 explores joint frame retiming after carried swaps. Its changed word and boundary costs differ from this rank-only cache-tree restriction; no numerical comparison is claimed.

• Prove strict traps for exact rank-one projection cuts (no new κ) #6194 constructs strict local traps for rank-one projection optimization. This gives a simultaneous global search restriction for common-coordinate Lagrangian terminals, rather than a statement about improving local moves.

• Prove odd cyclotomic Clifford rank cap, sharp at degree four (no κ) #6192 bounds odd-cyclotomic whole-bank Clifford responses using Pauli symplectic defect and native B rank. Those are different quantities from this cache-tree edge distance; no common numerical result is compared.

Completion refresh inspected all twelve other account PRs still open and created since the CLAIM at 2026-10-29T16:22:26Z. Apart from the adjacent #6192 and #6194 described above, they concern cyclic reversal boundary ranks, coefficient recovery, XOR snapshot scheduling, source-specific P104 birth costs, E8 spans, draft/export/replay performance, cubic readers and ordered-shear primes. None studies this coordinate-terminal cache-tree minimum or reports a conflicting same-instance number. No extra computations were run to validate another submission.

Related merged PR4866 gives a separate three-contact rule over all subspaces and explicitly excludes Lagrangian transfer. This theorem concerns any finite tree with common-coordinate Lagrangian terminals. Canonical main was refreshed before claiming and committing and remains a5bff3b. Sparse local-note coverage was incomplete; distinctive main README/formula searches and all-state/merged PR searches supplemented it.

CI notes

Both hosted checks passed at exact head 4ee826b9cdfa2587ec96a00ba0cf80faa2173d12 in run 40912691207: plan in 3m43s and replay (research/coordinate-cache-retraction) in 1m23s. The completed check list was read once; run metadata confirms the successful conclusion and matching head. Hosted replay certifies the registered normal and optimized finite checks. The listed local checks also pass. This does not establish independent human or formal proof-kernel acceptance of the all-field proof or physical caller.

Advance claim: #1166 (comment)

Merge Danger

Door: two-way. Blast radius: package. Six new files in one scoped research directory. Shared implementations, frozen evidence and the selected record are unchanged.