[petalas]

Summary

Construct a direct integer-index bijection for every invertible finite-field map g satisfying g(A) subset C and g(B) subset D, with no prescribed injection. Decoding uses O(1+n^5) field operations and O(1+n^4) exact integer operations. Uniform indices give exactly uniform completions.

Claim type: scoped constructive proof with exact finite corroboration. The written construction covers every finite field; the executable covers n<=12, prime q<=31 and F4. No new kappa or selected-frontier change.

index
  common injection into C intersect D
  weighted intersection subspace and graph
  independent second-capacity image and compatible lifts
  ambient basis extension
  standard-coordinate map

Ali Romman's #6539 supplies the cited quotient/lift count and the sampling question. The new step directly decodes those factors for U=0. Its count theorem is not re-audited. The older #6607 already samples the general prescribed-injection contract through an exponential candidate scan; its self-reduction and code are not used here. Classical Gaussian, subspace-graph and basis-extension methods retain credit. This contribution was produced by Nick Petalas.

Evidence

• Before: the existing general sampler scans candidate vectors. After: the U=0 direct construction has a polynomial operation bound, explicit inverse proof and no ambient-vector scan.
• Complete endpoint-set equality and no collisions pass for 674 complete small contracts, with 919 decoded endpoints, and 69 further selected fixtures, with 3,301 decoded endpoints. The fixtures include F4 and a two-branch crossed-plane case. Selected fixtures can repeat a contract; these totals count executed fixtures.
• Component enumeration covers 115 Grassmann endpoints and 197,660 ordered-injection endpoints. Sixteen selected large indices through dimension twelve satisfy their complete invertibility/capacity endpoints.
• All 64 pairs of three-bit entropy blocks for GL2(F2) give ten occurrences of each of six matrices and four exhausted pairs. Singleton and zero-attempt cases pass. Fifteen invalid-input controls reject.
• CPython 3.12.15, macOS arm64, standard library. Warning-strict direct normal and optimized commands pass in 11.27 s and 11.22 s with identical generated hash-bound receipts. These durations check the replay deadline; no measured speedup is claimed.
• Both registered commands, package metadata/source pins, committed bytes/modes and declared source-history checks pass at bb5f5c31f1114b45ea144df6f3eb7c3d8d996992. The official export passes integrity and declared-input coverage.

From the worktree root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/two-capacity-direct-unranking
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/two-capacity-direct-unranking --check-source-history --check-modes
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/two-capacity-direct-unranking/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -O -B research/two-capacity-direct-unranking/verify.py

Another CPython 3.12 installation may replace the absolute interpreter path. Expected markers are PASS_TWO_CAPACITY_DIRECT_UNRANKING, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES. Each registered command has a 240-second timeout. Receipts are generated outside the checkout and bind package and declared-input hashes.

Limitations and unrun obligations

The result is U=0 unrestricted GL generation. Nonzero prescribed images, native/symplectic realization, signed phase, an actual contact, legal access, physical randomness, dirty restoration, precision, stock, tapes, compiler and complete outer costs remain open. All five selected assumptions remain. No complete multiplication theorem or independent human/formal-kernel acceptance is claimed.

The bounded sampling API accepts at most 64 supplied entropy blocks and reports exhaustion. For N completions and b=ceil(log2 N), R independent uniform b-bit blocks exhaust with probability (1-N/2^b)^R. Its output is exactly uniform conditional on success. Provisioned bits and the returned acceptance position are distinguished. The proof's expected online bit bound is not a measured cost of this batch API.

The all-dimension theorem rests on the written proof and cited count. Large examples are selected indices, not complete large censuses. The current selected-baseline binding check passed. No predecessor scientific suite, selected cold replay, Lean or Docker ran. No local registered obligation is deferred to CI.

Credit ToikY #6535 for coupled feasibility, Ali Romman #6533/#6539 for capacity counting, Nick Petalas #6489/#4919/#6607 for related prior work, Brian/JoyboyBrian #5020, Cao Yifan/intqwq #4610, codyw912 #4735, and the retained Cinna Davis, Dugongue, zxv and Jacob Alan Sussman lineage.

Open next questions

1. Extend the direct polynomial decoder to a nonzero prescribed injection. Begin with the common lift subject to Count all prescribed two-capacity finite-field completions #6539's quotient discrepancy; compatible affine representatives and their unique inverse decomposition remain.
2. Bind one actual shared native contact to these samples and price its complete caller. Exact bases, signed/symplectic restrictions, chronology and physical costs are required.

Related open work

No matching open PR studying this two-capacity sampling contract was found in the refreshed complete 545-body corpus or the latest live PR/claim checks. Relevant predecessors #6539, #6535, #6533 and #6607 are merged. Sparse local notes, arbitrary historical diffs and unpublished work remain coverage limits.

Completion refresh inspected all 61 open account PRs, including the nine other open submissions created since the advance CLAIM. None studies the unprescribed two-capacity sampling contract or computes its distribution. No additional related-open-work line was needed, and no extra scientific computation checked those contributions.

The target and main step come from Ali Romman's #6539. The older account sampler #6607 is related background. Canonical main e54523517d0f03e58352e647c403a5af3e525a57, priority revision 8 and selected #7000 were refreshed before committing. Advance CLAIM.

CI notes

Hosted CI did not certify this package at unchanged head bb5f5c31f1114b45ea144df6f3eb7c3d8d996992. In run 43024464841, plan job 130535527661 failed in "Onboarding helper tests". The package replay job was skipped because plan failed.

I read the checks once and confirmed the job log. Exactly one of 760 tests failed, with 24 skipped: test_research_current_status.CurrentCiNavigationTests.test_current_ci_section_matches_selected_record. Its assertion was that 'make frontier7000-verify' was not found in docs/ci-verification.md.

This is inherited from canonical main e54523517d0f03e58352e647c403a5af3e525a57: the selected record names make frontier7000-verify, while the CI document's current section still describes #6030 and make frontier6030-verify. Both files are unchanged by this package. I also inspected the failed logs of 42905094314, 42903313790, 43023514635, 43024009248, and 43024227699. Each reports the same single named failure, missing command, 760-test total and 24 skips.

All registered package checks, direct normal/optimized checks, source pins, committed bytes/modes/history, official export integrity and Git-free exported replay passed locally at this head. Those remain local evidence. No hosted scientific replay completed. No fix, retrigger commit or workflow rerun was made; the repository-level documentation repair remains with maintainers.

Merge Danger

Door: two-way. Blast radius: isolated. Eight new files form one opt-in research package; the selected record and existing implementations retain their bytes.