[petalas]

Summary

Scoped mathematical proof: the fixed-boundary one-axis transported binary coframe rank-moment problem is NP-hard, and its constructed-family decision problem is NP-complete. The reduction retains both endpoint adapters and every scalar phase. It holds simultaneously for every power tariff r^t, 0<t<=1.

F = [[1,1],[1,-1]]/(1+i), F^2=-iI, F^-1=iF
transported edge rank = 1[endpoint choices agree]
M = 2n+1
B_G(x) = M (|E|-cut_G(x)) + 2|x|
maxcut(G) = |E| - floor(min_x B_G(x)/M)

This explicit reduction is the new step. Credit Ali Romman #5783 for the submodular-table interface and its unsupported-law boundary, David Ganz #5693 for coframe conjugation and fixed transports, and Garey, Johnson and Stockmeyer for classical Simple Max Cut hardness. Predecessor instances and verifiers are not audited or replayed. The contribution was produced by Nick Petalas.

Advance claim. Base: de5ec2e. Head: e17980b.

Evidence

Before, #5783 certifies submodular tables while the arbitrary fixed-transport extension remains outside its solver. This package gives a one-axis unswitchable triangle and a full polynomial hardness reduction, including nonzero boundary rank charges.

Local CPython 3.12.15 normal and optimized checks both report PASS_TRANSPORTED_COFRAME_HARDNESS: 1,099 simple labeled graphs through five vertices, all 33,866 assignments, 7,523 decision thresholds, 420 complete-word basis columns and 2,440 scalar output equalities over Gaussian rationals. Seven invalid graphs, six hypothesis mutants and eight triangle complements are checked. Complete-graph minimum bills are 0,2,9,22,48. The finite census corroborates the universal written proof; it does not establish complexity hardness by testing.

Exact reproduction from the repository root:

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

Any CPython 3.12 standard-library interpreter can replace the absolute path. Both registered commands pass locally, timed separately at 0.15 and 0.14 seconds; the package runner took 0.60 seconds. These are timeout-budget observations, not a speedup claim. Metadata, committed bytes, executable modes and full-history source ancestry pass at the stated head. Official export integrity with declared-input coverage and both read-only Git-free normal/optimized replays pass. Generated stdout receipts bind all six package-file hashes and agree across these replays.

Portable export:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/export_research_package.py export --package research/transported-coframe-hardness --output /tmp/transported-coframe-hardness.tar.gz
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/export_research_package.py check /tmp/transported-coframe-hardness.tar.gz --check-inputs

Limitations and unrun obligations

The objective is the explicitly declared Gaussian rank moment. Ordinary work, precision, storage and tape costs are separate and unpriced. This is a constructed finite-word family; no hardness is claimed for a selected source word, common-anchor identity transports or unrestricted resynthesis of the final linear map. The repeated logical shears are part of the frozen family, not a lower bound against compressing them in another model.

No supplier admission, new kappa, selected-frontier change, full physical invoice or multiplication theorem is claimed. Independent human and formal proof-kernel review are unclaimed. No Lean, Docker or heavy supplier replay was run. The original card scope change is explicit in TASK.json.

Open next questions

Which actual transported coframe graphs have small signed-cycle deletion sets? First extract and aggregate one named source word's complete pair tables, including every fixed transport and target debit. A source-bound table instance and full physical legality contract are prerequisites. The worst-case reduction does not preclude structured exact cut algorithms.

Related open work

Refreshed the titles and bodies of all nine of this account's open PRs created since the CLAIM: #5886 through #5893 and #5895, including this PR. No extra mathematical computations were run to compare those results.

• Prove sharp mixed-rank invoice envelopes with touching roots (no κ) #5887 studies mixed-rank power-invoice envelopes with touching roots. It shares the rank-power accounting quantity but has a different synthetic invoice family, with no shared numerical fixture or conflicting computed value.

• Reduce bounded NTT temporary storage from 17,920 to 1,282 (no new κ) #5886 studies the bounded NTT temporary arena, Prove all-precision dyadic affine feasibility (no new κ) #5889 dyadic affine feasibility, Prove the sharp R-unit Gaussian bulk denominator (no new κ) #5890 Gaussian bulk denominator debt, Exclude scratch-reading two-FULL completions (no new κ) #5891 two-FULL scratch-reading obstructions, Prove sharp divergence windows for observed coupled bills (no new κ) #5892 observed coupled recurrences, Prove sharp migration tariffs for two reusable resources (no new κ) #5893 two-resource migration tariffs, and Prove exact weighted five-node clean scratch (no new κ) #5895 five-node clean scratch. None computes this transported-coframe optimum or supplies a common numerical instance.

• Certify global optima for submodular binary frame cost tables #5783 certifies global optima for submodular binary tables; this result identifies a transported rank-table family beyond that contract and makes no objection to its solver.

• Classify every canonical P104 block chart and audit PR5756 #5772 classifies canonical charts for a fixed P104 scalar block; this constructed one-axis incidence family is a different optimization domain and asserts no change to its computed tariff or chart optimum.

• P104 #5718 cache delivery cannot save the tested centre charges #5765 bounds delivery on a fixed P104 query word; this result concerns arbitrary constructed transported coframe graphs and supplies no alternative delivery word or same-instance numerical comparison.

CI notes

At commit e17980b2457bd1815d8ff37c0a3e2f47878f2ef6, run 40495767334 failed in plan at Onboarding helper tests. This was the only failed step. All ten failure/error entries are in unchanged tests/test_research_current_assignment.py, ending FAILED (failures=8, errors=2, skipped=22).

The two errors are KeyError: 'follow_up_steps' and KeyError: 'catalog_first_step'. The eight failure entries concern the frozen-card count (0 != 16), missing-helper/reference rejection and sparse-reference receipts. All ten failure/error headers match PR #5886's same-step log exactly. This confirms the reported shared onboarding failure on main de5ec2e7a6074068452f45b1a191b5c72832df4a. No shared code, tests or files outside the new package were changed to address it; maintainers will repair that failure.

replay was skipped. Hosted replay has not certified this package; only the local package checks reported above have run. No hosted scientific success is claimed.

Merge Danger

Door: two-way. Blast radius: package-only. The six added files are isolated under research/transported-coframe-hardness; no shared code, predecessor artifact or selected record changes.