[petalas]

Summary

Scoped proof with finite exact corroboration: variable productive boundaries can strictly improve a normalized signed tensor-frame invoice. This package determines the exact optimum and threshold for one explicit abstract graph, rather than treating failure of an averaging argument as a construction.

J_t = sum_i dim(S_i)^t - sum_(i<j) d(S_i,S_j)^t
min J_1 in K^m = -floor(3m/2), over every field K
min J_t / 2^t at m=2 = min(-1, 3/2^t - 3)
one-copy optimum = -1; strict coupling advantage iff 2^t > 3/2

Every linear minimizer is classified: the spaces are pairwise disjoint and have balanced dimensions. The proof constructs attaining graph subspaces over every field. At the linear tariff, two copies improve the normalized value from -1 to -3/2. This is a scoped rank-graph result, not a legal scalar word, supplier, complete multiplication theorem or new kappa.

The new result addresses the variable-boundary limitation and second open question of Ali Romman's PR6171, at 343232cdf5273d1f77569de390f0977b46916644. It goes beyond that package's arithmetic negative-credit control by proving an exact attainable graph optimum, equality cases and power threshold. Its fixed-boundary theorem is cited, not audited or replayed. Cinna Davis retains the projection inequality and icekylinx the subspace-frame construction; all P104/native/helper/caller/compiler predecessor credit is retained. This contribution was produced by Nick Petalas. Standard dimension and affine-envelope facts are background; no literature-priority claim.

Package: research/signed-triangle-tensor-gap/. Base main 3f0152a32eeebac0df11946dbd50cabe5a7f7839, priority revision 8. Selected conditional kappa remains .003125804. Advance claim. The card is explicitly changed to an abstract signed-boundary side result; the physical-caller obligation is not claimed complete.

Evidence

Before, the fixed-boundary theorem left variable productive credits outside its conclusion. After, the written dimension proof and two-dimensional table give a sharp counterexample to extending its no-coupling conclusion to every signed variable-boundary graph.

All local registered checks pass on CPython 3.12.15, macOS 27.0.1 arm64:

• 305,721 complete ordered subspace triples over F2 in dimensions 0 through 4 and F3/F5 in dimension 2. Every equality case is checked. The binary RREF inventory is checked independently by closure under adjoining vectors.
• 64 explicit attaining constructions and 27 exact affine-envelope endpoint checks. Missing/spurious table rows, intersecting-plane equality and strict-threshold controls pass.
• Warning-strict normal and optimized direct checks take 0.240s and 0.247s respectively. These are timeout measurements, not a performance comparison; both are well below their 120s limits. Both VERIFY.json commands and the registered replay pass.
• At head 0821ba9ff3b55204042b007212f04d764b5d3921, all seven committed files match working bytes; the declared base passes source-history ancestry. The official export passes integrity/input-coverage checks. Optimized replay passes in a read-only Git-free export with empty PATH; all eight exported files keep their bytes and modes. Altered proof bytes, an altered count and a Boolean receipt mutation are rejected.
• The generated RESULT.json binds all six source files and records per-domain minima, minimizer counts and ordered trace hashes. No receipt was handwritten. No predecessor scientific computation was repeated.

Exact reproduction from the repository root:

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

The marker is PASS_SIGNED_TRIANGLE_TENSOR_GAP ordered_triples=305721 constructions=64 power_endpoint_checks=27. A clean CPython 3.12 interpreter can replace the exact local path; standard library only.

Canonical main, live open PRs, current claims and distinctive merged/main README searches were refreshed before committing. Cached overlap covered 317 open PRs and 5,857 Discussion records, with complete remote pagination; its sparse local search omitted 8,491 note paths, so separate searches used main's README tree. These are bounded overlap checks, not proof of novelty.

Limitations and unrun obligations

Only canonical subspace charts and the stated six-edge signed graph are optimized. The negative edges have no supplied realization as independent targets on arbitrary dirty inputs. A legal scalar contact graph, native phases, physical target delivery, restoration, ownership, adapters, movement, precision/tape, compiler closure, BIT and outer admission remain open and unpriced. The result does not change a paid supplier histogram or the selected frontier. The all-field conclusions rest on the written proof, not finite extrapolation. No independent human or formal proof-kernel review, selected cold replay, Lean or Docker run is claimed.

Open next questions

• Realize this signed graph with a legal dirty-input scalar circuit, or prove a contact/path obstruction. First assign the three productive pairs to distinct logical lifetimes and charge every paid transition; independent target delivery and restoration are blockers.
• Determine the all-width optimum for 0<t<1, starting with unequal dimensions and nonzero intersections. The linear theorem and exact two-copy table do not resolve that optimization; a graph improvement would still need a complete physical invoice.

Related open work

• #6171 proves no tensor advantage with fixed repeated productive boundaries. This package changes that hypothesis and gives an exact signed graph gap; it does not dispute the theorem or compare a shared numerical optimum.

• #4993 explores variable-boundary retiming on a particular carried-swap P104 graph. This package studies an abstract triangle of productive pairs and supplies no source-compatible word or competing P104 price.

• #6173 computes two-channel support in a specified P104 root code. This package varies abstract productive subspace boundaries and computes no code-support value.

• #6174 realizes bipartite rank-distance metrics in rational shear operators. This package optimizes a signed subspace triangle, without a rational-shear construction or competing distortion value.

The completion refresh read all four open account PRs and both other submissions created since the CLAIM at 2026-10-29T13:00:19Z. Their distinct quantities are described above; no extra computations were run to check them. Neither supplies a competing signed-triangle optimum. PR6171 and PR4993 remain open.

CI notes

Both hosted checks passed at exact head 0821ba9ff3b55204042b007212f04d764b5d3921 in Registered research packages run 40879241242: plan in 4m18s and package replay in 1m35s. The checks were read once at completion; run metadata confirms the same head and successful completed conclusion. The registered replay executes both normal and optimized VERIFY.json commands. All required local commands also passed. Hosted execution certifies these bounded checks; independent human or proof-kernel acceptance of the scoped theorem and physical supplier admission remain unclaimed.

Merge danger

Door: two-way. Blast radius: one isolated research package. Seven files are added; shared code, frozen sources and selected-frontier records are unchanged.