[petalas]

Summary

Scoped proof with an exact finite certificate. At block four, every correct width-five program in the declared shift/diagonal/coordinate-permutation game obeys

Target leakage demand:       448 per array
Direction-v call leakage:   112 wt(v), exactly
Therefore:                  sum wt(v) >= 4 W
Coordinate calls only:      minimum 4 W, attained

Arbitrary complex register mixing, zero scratch, copies, erasures and arbitrary diagonal phases are allowed. Inverse calls have the same weight. General XOR-linear address maps are excluded. No new kappa or selected-frontier change is claimed.

The new fifth-exterior-power face space has dimension 448 inside dimension 4368. Its target leakage Gram is exactly (15/16) I_448. The direction-weight ranks are 112, 224, 336 and 448. This also settles the obvious certificate's limitation: if every nonzero direction has unit price, its demand-to-cap ratio is exactly one. A better unrestricted-direction obstruction needs a changed certificate.

Credit Jacob Alan Sussman's #4628 for the kept-subspace and block-three face methods, and eumemic #288 and Swapnil Jain's P1/P2 as credited there. The new proof and checker are self-contained; no predecessor executable or reported rank was audited. This contribution was produced by Nick Petalas.

Evidence

Before this contribution, #4628 supplied the block-three face certificate and separate block-four bounds in other games. The new block-four certificate checks every one of 200,704 target-compression minors over the integers, all fifteen nonzero call directions over Gaussian integers with modular rank lower bounds and analytic upper bounds, and nine rejection controls. Both registered modes print PASS_BLOCK4_COORDINATE_FACE_TARIFF, and the runner prints PASS_REGISTERED_RESEARCH_PACKAGE.

From the repository root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -B research/block4-coordinate-face-tariff/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -O -W error -B research/block4-coordinate-face-tariff/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/block4-coordinate-face-tariff
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/block4-coordinate-face-tariff --check-source-history

CPython 3.12.15, macOS arm64, standard library only. The direct ordinary and optimized checks each take about two seconds; registered replay takes about four seconds total. The exact committed-byte/source-history check passes for all five files. A read-only, Git-free committed export passes optimized isolated replay and preserves every exported byte. Its scientific output equals the generated hash-bound publication receipt.

The historical card baseline stopped with KeyError: 'complex_profile'; current baseline orientation stopped at sparse-omitted research/fresh-helper-thinned/PROOF.md. Neither is a new package failure or claimed successful replay. The package has no live external mathematical inputs. Canonical main was refreshed unchanged at 624de9dc66858af0945e5f8a0942488d62280d43 before publication. The overlap helper covered 367 open PR bodies and 3,823 Discussion records; sparse local-note coverage was incomplete, supplemented by distinctive searches across main and merged PRs.

Limitations

The program theorem is a written scoped proof supported by the exact finite identities. Independent human and formal proof-kernel review are unclaimed. The finite ranks concern block four only. This does not exclude unrestricted-direction small units, supply a physical implementation, price compilation, settle rounding, or prove a complete multiplication theorem. Those obligations were not run or claimed.

Related open work

• Research: lower bounds for small complex units: m = 2 never saves, m = 3 needs 14W/5 calls (Lean, engine game), m = 4 needs 41W/13, and the smallest open cases (no new κ) #4628 studies small-unit lower bounds and supplies the predecessor method. This adds a new block-four certificate and a different weighted game, without rechecking its bounds.
• feat(research): exact 3-array 10-call block-4 unit schedule on C^16 (κ = 0.1315 > 0.10) #4818 proposes a finite block-four three-array schedule. The present theorem concerns the stated width-five free-step contract, so it does not test that schedule or compare its observation ranks.
• feat(research): exact 13-array 41-call block-4 boundary schedule on C^16 (κ = 0.1714) #4819 proposes a finite block-four thirteen-array schedule. The same contract distinction applies; no schedule or paid-word audit is performed here.
• feat(research): exact 3-array 9-call block-5 unit schedule on C^32 (κ = 0.3174) #4826 studies a block-five schedule, outside this block-four certificate.
• Add phase-exact native Clifford compiler and complete contact neighborhoods #4841 supplies phase-exact native compilation and contact-neighborhood results. Native admission and general address maps are outside this theorem.
• Certify complete block-cascade conditioning and audit paid-word admission (no new κ) #4842 studies complete observation-matrix conditioning and paid-word admission for the proposed schedules. Its matrices and numerical norms are separate from the exterior-face leakage operators here.

The pre-DONE refresh inspected all nine other open submissions from this account created since the advance CLAIM: #4844, #4845, #4846, #4847, #4849, #4850, #4851, #4852 and #4854. They concern different objects from block-four exterior-face leakage; no new same-object numerical comparison is made.

Open next questions

• Find a kept space with demand/cap above 7/2 for unit-priced arbitrary directions and diagonal phases. Test a weight-four call first; this space cannot supply that bound.
• Determine whether coordinate-face compression yields useful weighted bounds at another block size. Begin with the exact compression Gram; no new-block rank is claimed here.

CI notes

Both hosted checks passed on exact head 8b8de0281f429b592932b424987cea3be178b213 in run 39861300537: plan and registered package replay. The replay runs both warning-strict ordinary and optimized commands. Local registered replay, publication-byte/source-history checks and the read-only Git-free export also passed. Hosted execution is bounded automated verification, not independent human or formal proof-kernel acceptance.

Merge danger

Door: two-way. Blast radius: one isolated research package. The five added files contain the proof, verifier and metadata.