[petalas]

Summary

Scoped proof with exact finite corroboration: exhaustive exact rank-one projection moves can stop at a strict local minimum, while two commuting projections improve the same complete paid frame-graph objective. This holds over every field and every tariff r^t with 0<t<1.

For the specified graph with fixed boundaries 0 and K^2 and one variable frame, the exact minimum edge count is 2 floor(1/(2-2^t))+1. At t=997/1000 it is 481:

initial K^2: 241 * 2^t
any single-projection image: 481, strictly worse
joint zero image: 240 * 2^t, strictly better

The 481-edge example remains a strict trap for 0<t<=997/1000, including the pinned selected complex supplier's power range. Minimality is asserted at t=997/1000 and only within this two-boundary class. The universal claim is proved in research/projection-local-trap/PROOF.md; the finite checks corroborate it. No new kappa is claimed.

David Ganz / Dugongue retains PR556's binary projection-cut theorem, frame-graph contract and implementation. The new contribution is this strict local trap, sharp multiplicity formula and exact example. Standard concavity and rank-nullity are background; no general literature-priority claim is made for local-search traps. This contribution was produced by Nick Petalas. The predecessor theorem and implementation were cited, not audited or replayed.

Advance CLAIM. Base main a5bff3bf133e482541c09eba4e773884bdb2b1b7, priority revision 8; the card explicitly changes to a frame-search side result. The delivered scope matches the CLAIM.

Evidence

Before, an exact optimum for each binary projection move gave no explicit sharp example of a joint escape. After, the written three-value proof gives an all-field construction, a sharp integer threshold, and exact integer power inequalities certifying the 481-edge witness.

Local CPython 3.12.15, macOS 27.0.1 arm64:

• 323 rational tariffs, 516800 multiplicity pairs, 125663 strict traps, 614 ties with cheaper zero, and 28047 smaller-graph pairs.
• All 17764 two-by-two matrices over F2,F3,F5,F7,F11; 236 rank-one projectors; all 21640 ordered projector pairs; 236 complementary joint escapes. Literal profiles retain 20683 edge occurrences across all 43 subspaces.
• Exact integer powers exclude all 28680 pairs with incoming multiplicity below 241 at t=997/1000. Fifteen named domain, strict-boundary, map and receipt controls pass.
• Warning-strict normal and optimized direct checks agree exactly. The two registered commands took 1.629 s and 1.664 s locally, below their 120 s limits. These are timeout observations, not performance comparisons. Registered package replay passes.
• Committed bytes, executable modes and declared source-history ancestry pass at d297eb399c5b2457b2028130c0752234fad9ccba for all seven package files. Official export integrity/input coverage passes. Normal and optimized read-only Git-free export replays pass; all eight exported files retain their hashes and modes. Source-byte and typed-receipt mutations reject in both modes.

Exact reproduction commands, from the repository root:

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

A reviewer may substitute a local CPython 3.12 interpreter. No dependencies beyond the standard library. The checker recomputes the generated hash-bound RESULT.json and writes nothing. SOURCES.json records the exact predecessor and selected-record provenance.

Limitations

The free vertex has 481 incident edges and is not identified with an ordinary bounded-arity scalar instruction. This is an abstract frame-graph obstruction, not a scalar word, physical independent-target construction, supplier admission or complete multiplication theorem. No global minimum over arbitrary frame graphs is claimed. Selected observations, dirty restoration, productive credits, native phases, owner/stock/precision/tape costs, uniform compilation and a full signed caller invoice remain separate. The selected frontier and all five framework assumptions are unchanged.

No selected scientific cold replay, Lean, Docker, independent human acceptance or formal proof-kernel check was run. The unbounded claim rests on the written proof; finite fields are not extrapolated to arbitrary fields.

Open next questions

1. Can one actual selected scalar-word neighborhood realize the coordinated projection advantage after every productive target and restoration fee? Extract its fixed observations and compare both complete signed invoices. No physical data paths are supplied here.
2. Can a useful class of two-projection moves be optimized with an exact certificate? Derive the mixed inclusions for two specified commuting projections and four image choices per event, retaining the full paid and productive objective.

Related open work

• Draft: Conditional κ = 7.10436534421701e-4: concave-descent frame retiming of 25 gates on #251's source527 banks #260 performs concave-descent retiming on an actual frozen scalar word. It does not determine this all-field rank-one projection trap or its sharp two-boundary threshold; its source spans and complete supplier costs remain its own contract.

• Two successor hand-offs on the P104 word: conditional κ 0.0030428242 #5553 changes P104 successor hand-offs and productive boundaries. This package instead studies a separate paid-only abstract graph and makes no comparison of its conditional kappa.

• Conditional κ 0.00309871 from a prepared P104 split helper #5606 studies a prepared P104 split helper. It supplies an actual source-specific supplier candidate, whereas this result supplies no source embedding or new helper split.

• Exclude birth-lowered P104 markets without paid deletions (no κ) #6197 bounds a source-specific P104 market-birth change under an unchanged paid inventory. It shares concave recursive accounting, but does not study this projection-move graph or compute its strict-trap threshold. No numerical comparison is asserted.

Completion refresh inspected this account's nine other open PRs created since the CLAIM: #6192, #6193, #6195, #6196, #6197, #6198, #6199, #6200 and #6201. Apart from #6197 discussed above, they concern cyclotomic native ranks, cyclic reversal boundary ranks, sum-assisted recovery, XOR snapshots, odd-characteristic root spans, draft-creation performance, cubic-reader repair and ordered-shear primes. None computes this graph's strict local-trap minimum; no common-instance numerical disagreement was identified. No extra predecessor computations were run.

The refreshed open-PR and claim searches found no submission computing the same strict-trap threshold. Main README/formula and merged-PR searches were also checked. Sparse local-note coverage is incomplete; these searches are not proof of novelty. No extra computations were run to validate other submissions.

CI notes

At exact head d297eb399c5b2457b2028130c0752234fad9ccba, both plan and replay (research/projection-local-trap) passed in run 40911437343. The completed checks were read once, and the run's successful conclusion and exact head were confirmed. Hosted replay ran the registered package checks. Local scientific, metadata, publication and read-only export checks listed above also pass. This certifies those replay commands; independent human and formal proof-kernel acceptance remain unclaimed.

Merge Danger

Door: two-way. Blast radius: isolated. Only research/projection-local-trap/ is added; shared code, frozen sources and selected-frontier files are unchanged.