[petalas]

Summary

Scoped written proof with exact finite corroboration, produced by Nick Petalas. For coordinate stage masks over every field, one fixed common noncoordinate basis has exactly the same feasible idle-degree multisets as the coordinate basis. The result also covers fixed globally independent subfamilies.

invertible basis matrix
  -> choose a supported determinant permutation
  -> replace every direction by its matched coordinate
  -> preserve every idle stage set and degree

This extends the applicability of CubeJerry's #7002 coordinate coalescing obstruction to the fixed globally independent subclass. Its coordinate optimum and numerical shortfall are cited premises. The new result is the allocation reduction, not a replay or certification of that predecessor. No new κ or physical caller is claimed.

Package: research/fixed-basis-idle-coalescing. Base: e54523517d0f03e58352e647c403a5af3e525a57. Changed joint-query-controller scope. Advance CLAIM.

Evidence

• Before: Construct a connected 18-stage root and bound its native completion (no new κ) #7002 explicitly excludes arbitrary noncoordinate shared-direction decompositions from its coordinate-role result.
• After: a determinant/support matching proves that fixed globally independent directions cannot improve that relaxation. The proof retains full stage sets, so no concavity assumption is needed for the equality of degree-only objectives.
• Local CPython 3.12 checks enumerate 85,833 matrices, 31,617 bases and 1,251,947 eligible stage/subset pairs. Every basis is compared with all supported permutations. A five-stage fixture preserves degrees (3,2,3). Six invalid inputs reject; two controls expose dependence and unsupported matching.
• Warning-strict normal and optimized runs pass with identical generated hash-bound receipts. The two registered commands take 1.03 seconds each locally. Registered metadata/replay and exact committed-byte/mode/source-history checks pass.

From the repository root, exact reproduction commands are:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/fixed-basis-idle-coalescing/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -O -B research/fixed-basis-idle-coalescing/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/fixed-basis-idle-coalescing
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/fixed-basis-idle-coalescing --check-source-history

Any CPython 3.12 interpreter can replace the absolute interpreter path. Expected markers are PASS_FIXED_BASIS_IDLE_COALESCING, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES. The standalone verifier uses only the standard library and writes nothing.

Limits and credit

Globally dependent direction families and changing bases are outside the theorem. The reduction maps optimistic allocations, not physical HOST operators. Native compatibility, paid basis changes, chronological transport, dirty restoration, ordinary costs, precision/tape, stock, compiler, supplier admission and the complete multiplication theorem remain unproved here. No predecessor scientific verifier, selected cold replay, Lean or Docker was run. Ruff and mypy were unavailable on PATH. Independent human and formal proof-kernel review are unclaimed.

Credit CubeJerry, Jerry Chen, #7002 for the source question and coordinate relaxation, DreamingOfClouds #6937 for the connected-mask lineage and the retained #6936/#6912 native/root authors. Determinant expansion and Hall matching are classical. SOURCES.json pins historical context on main; there are no predecessor runtime inputs. Predecessor and selected files are unchanged. Apache-2.0.

Related open work

• Scope rank-two native and decoder obstructions (no new κ) #7098 constructs and prices a normalized noncoordinate projector relocation for Construct a connected 18-stage root and bound its native completion (no new κ) #7002. This contribution proves an obstruction for every fixed globally independent family and does not recompute that schedule or price.
• Bound fixed caller queries and two-ADD HOST repairs #7106 extends that projector scheduling work, enforcing compatibility on overlapping intervals. The present theorem requires global independence across the whole schedule, a stricter premise. No numerical agreement or disagreement is claimed.
• Bound lane-wise BIT IDLE regrouping (0% κ change) #7188 bounds lane-wise BIT IDLE regrouping with a fixed BIT productive profile. This theorem concerns a coordinate-mask allocation model and the complex Construct a connected 18-stage root and bound its native completion (no new κ) #7002 application, without a BIT invoice comparison.

Open next questions

• Can globally dependent directions improve a fully specified idle bill while each stage still has independent directions? Begin with three lines in a two-dimensional space and a short coordinate-mask sequence. A favorable allocation with paid preparation and restoration is missing.
• Can a changing basis improve a complete connected-root caller? Begin with two explicit bases and one paid transition, including HOST compatibility. No legal conversion or favorable complete bill is supplied here.

CI notes

CI did not certify this package at fd4f5e506977527983c96871ee912ee55a31d81c. In run 43012902583, plan job 130496673962 failed in "Onboarding helper tests" on exactly one of 760 tests, with 24 skipped:

test_research_current_status.CurrentCiNavigationTests.test_current_ci_section_matches_selected_record

The AssertionError reports that 'make frontier7000-verify' was not found in the CI verification doc. Canonical base e54523517d0f03e58352e647c403a5af3e525a57 names that command in certificates/selected-result.json, while docs/ci-verification.md still describes #6030 and make frontier6030-verify. Those files and the failing test are unchanged by this PR.

The same sole failure and summary of 760 tests with 24 skipped were verified directly in the logs of run 42905094314, run 42903313790, #7203 run 43012497455, #7204 run 43012524632, #7205 run 43012595145 and #7206 run 43012862401.

The package replay job was skipped because plan failed. Local registered replay, normal/optimized direct checks, metadata and committed-byte/mode/source-history checks passed as described above; they are not hosted certification. No repair, retrigger commit, push or job rerun was made. The repository-level navigation repair is left to maintainers.

Merge danger

Reversible isolated research package. Blast radius: package-local. No shared code, selected frontier or predecessor evidence changes.