[petalas]

Summary

Scoped proof with finite corroboration: three fixed subspace contacts admit an exact four-candidate minimum for every nonnegative, nondecreasing concave rank-distance tariff, over every field and inside a common mandatory-subspace/receiver-cap interval.

d(U,V) = dim(U)+dim(V)-2dim(U intersect V)
M = (A intersect B)+(B intersect C)+(C intersect A)
min_X sum_i phi_i(d(X,A_i)) = min_{X in {A,B,C,M}} sum_i phi_i(d(X,A_i))

This is a new local optimization rule for the explicitly schematic distance in Chrisbryan17's #2844, with Cinna Davis's #4447 supplying the separate span/intersection contact motivation. Standard metric-median and concavity ingredients are credited as such. The predecessor constructions and numbers are cited, not replayed or audited. No new kappa is claimed.

This contribution is produced by Nick Petalas. Credit Chrisbryan17 #2844/#2820, Ali Romman #1125, the #968 frame-pair lineage, Cinna Davis #4447/#4440/#4441, zxv, Jacob Alan Sussman and their retained native/controller/bit/transfer contributors. Independent human and formal proof-kernel review are unclaimed.

Evidence

• Before: the cited schematic model permits searching every intermediate subspace. After: the written proof reduces the exact optimum to four explicitly constructed frames. The median can be essential, and a specified terminal can be uniquely optimal.
• Full finite corroboration covers 57,361 unordered contact triples and 3,637,645 competing frames over F2 in dimensions two, three and four and F3 in dimensions two and three. Lossless grouping by complete attainable-distance sets gives 388,002 exact optimum comparisons over independent concave integer tariffs. The interval check adds 66 intervals, 1,462 contact triples and 15,802 frame comparisons. These are separate coverage counts.
• Four invalid-tariff controls reject. A convex-price counterexample costs 23 outside the four candidates versus 28 within them. A fixed-dimension/Lagrangian counterexample prevents transfer to that restricted domain.
• Warning-strict direct normal and optimized checks pass locally on CPython 3.12.15/macOS arm64 in 3.405s and 3.415s. Both registered commands pass, with the combined package replay taking 7.009s. These are timeout measurements, not performance claims.
• Metadata/source pins and exact committed-byte/source-history preflight pass for all eight package/input files. The official committed export passes integrity checking. An optimized Git-free export replay passes, and changed-source/changed-receipt controls reject. RESULT.json is generated and bound to all five other package files.

Exact reproduction from the repository root:

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

Expected direct output: PASS_THREE_CONTACT_PRICING with contact_triples=57361, frame_comparisons=3637645, exact_optimum_comparisons=388002. The registered replay ends with PASS_REGISTERED_RESEARCH_PACKAGE. Standard library only; a clean CPython 3.12 interpreter can replace the exact local path.

Canonical main was refreshed before claiming and committing and remains 624de9dc66858af0945e5f8a0942488d62280d43; guidance and the selected record are unchanged. The formal-interface-lemma card is explicitly changed to this original schematic optimization question. Cached overlap covered 381 open PRs and 3,862 Discussion records; sparse local coverage omitted 6,202 tracked notes. Distinctive all-main README searches, merged PR searches, live open PRs and current claims were also checked. This is dated coverage, not a novelty certificate.

Limitations

This is a scoped mathematical proof with finite corroboration, not a complete multiplication theorem. The allowed frames must contain the whole stated subspace interval. No actual Clifford tariff is identified with the schematic distance. No new scalar word, native realization, physical caller, ownership scheme, compiler, precision/tape bound, supplier admission or exponent improvement is supplied. A graph with several variable frames is outside the claim. Native, Lean, Docker, historical suites and the selected cold replay were not run; the current selected-source metadata control passed. All five package files and the generated receipt are isolated in one new directory.

Open next questions

• Find a finite candidate rule for four-contact junctions in the same metric. Start with all quadruples in F2^3 and identify essential distance vectors. The three-coordinate proof does not settle that problem.
• Bind a physical P104 carrier and three literal contacts to these exact prices, with a fully paid preparation/restoration word. The native realization gap in Finite research construction: signed P104 mixed-carrier banks and E8 root-line gate (no new kappa) #2844 remains a blocker.

Related open work

• Add phase-exact native Clifford compiler and complete contact neighborhoods #4841 studies phase-exact native Clifford compilation and Lagrangian chart neighborhoods. This theorem concerns all subspaces in the schematic symmetric-difference dimension metric; its fixed-dimension counterexample explicitly prevents treating it as a Lagrangian compiler rule.

• Count and index shortest binary chart words (no new κ) #4859 counts and indexes shortest words between binary Lagrangian charts. This package optimizes a three-contact junction in the different full-subspace domain and compares no competing numerical quantity.

• Prove exact cycle-constrained chart neighborhoods (no new κ) #4864 classifies cycle-constrained simultaneous Lagrangian chart neighborhoods. Its restricted chart fibers differ from this full-subspace junction optimum; the Lagrangian limitation here prevents transferring the four-candidate rule to that problem.

• Prove exact mixed-direction Clifford batching criteria (no new κ) #4861 classifies mixed-direction full-space Clifford batching under reciprocal encoders. This package supplies a schematic subspace-distance optimum, without a native encoder or competing batch value.

Before DONE, this account's nine open PRs created since the CLAIM at 2026-10-22 22:18:31 UTC were inspected, including this PR. The nearby chart-word, chart-neighborhood and mixed-direction submissions are distinguished above. The remaining submissions concern permutation workspace, QQ16 parked volume, root-code floors, dirty-echo equality and cyclic checkpoints. None computes the same three-contact optimum; no additional predecessor computations were run.

CI notes

Both hosted checks passed on exact head 49fb8b2c3d5abbcba16f819f2d52d84ee735c7a9 in Registered research packages run 39864794588: plan in 3m29s and the registered package replay in 1m43s. Run metadata confirms that head, completed status and successful conclusion. The replay executes both warning-strict normal and optimized VERIFY.json commands. Hosted CI certifies execution of these bounded checks; independent human and formal proof-kernel acceptance of the all-field theorem remain unclaimed.

Merge danger

Door: two-way. Blast radius: package. All changes are confined to the new research package; frozen sources and the selected frontier are unchanged.