[petalas]

Summary

Scoped written proof with exact finite corroboration. This settles the nested-target case of JoyboyBrian / Brian's PR5020 Open next question 2: one prescribed partial injection extends to an invertible map meeting arbitrarily overlapping late sources and any finite flag of target capacities exactly when the cumulative-source compatibility and quotient-capacity conditions hold. The proof constructs a map and counts every completion over each finite field.

S_i = A_1 + ... + A_i
h(U intersect S_i) <= C_i
dim(U+S_i)-dim U <= dim C_i-dim(C_i intersect h(U))

Separate tests can pass while the shared map is impossible. The new binary fixture has individual completion counts 4 and 12 and joint count zero. This contribution was produced by Nick Petalas. Credit JoyboyBrian / PR5020 for the question and one-capacity contract, Cao Yifan / intqwq / PR4610 for the native-map caller, Nick Petalas / PR4919 for the different three-subspace correspondence problem, codyw912 / PR4735, and the retained Cinna Davis, Dugongue, zxv and Jacob Alan Sussman lineage. Basis extension and finite-field independent-vector counts are classical; no general literature-priority claim is made. No predecessor scientific verifier was replayed.

Evidence

• Before: the one-capacity interface leaves simultaneous late capacities open. After: an all-field cumulative-source theorem, exact finite-field count and bounded reference constructor cover nested targets.
• Literal vector-set and complete-GL enumeration checks every source pair, nested target pair and prescribed partial injection in F2 dimensions zero through three and F3/F5 dimension two: 9,539,317 exact count comparisons pass. Separate row-reduction reference calls pass 14,141 count comparisons and 3,119 constructor calls. Every returned census matrix is checked by complete vector permutations and literal inclusions.
• Another 112 attaining cases cover dimensions through six and flags through eight stages. Fourteen malformed boundary cases reject in both entry points. The mixed prescribed-relation obstruction is counted again by a separate literal matrix loop.
• Warning-strict direct normal and optimized CPython 3.12.15 runs pass in 2.52 and 2.47 seconds. Both registered commands pass locally; the whole package runner takes 5.20 seconds. Normal and optimized generated receipts agree byte for byte. These are replay-deadline measurements, not performance comparisons.
• Package metadata/source pins, in-memory syntax compilation, function-size bounds and staged whitespace pass. Exact committed bytes, executable modes and declared source ancestry pass for all seven files at fdb7b8808ce89cd1738ea49dcd65c19ae1dddedd. The official export passes integrity/input-coverage checks; optimized execution in its read-only Git-free tree with an empty PATH reproduces the same receipt byte for byte.

From the repository root, exactly:

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

Any CPython 3.12 executable may replace that local interpreter path. Expected direct marker: PASS_NESTED_LATE_CAPACITY_COMPLETION. Each registered command has a 120-second limit. The verifier emits a generated JSON receipt bound to all seven package-file hashes and writes no files. There are no external live inputs or third-party dependencies.

Limitations

This is unrestricted GL completion with nested target capacities. Arbitrary nonnested targets, symplectic constraints, absolute phases, actual selected-word contacts, native realization, causal access, dirty restoration, basis preparation, delivery, productive target debits and the complete paid invoice remain open. Precision, stock, tape, compiler, outer admission and all five selected framework assumptions remain. There is no new kappa or selected-frontier change. The reference software covers prime q in {2,3,5,7}, dimension at most six and at most eight contacts; the written proof has the stated all-field/all-length scope. No native build, selected scientific replay, Lean or Docker run was performed. Independent human and formal proof-kernel review are unclaimed. No locally unrun registered check is deferred to CI.

Open next questions

• Extend the prescribed-map theorem to two nonnested target capacities. Start with intersecting binary planes and an oblique prescribed subspace; coupled image compatibility is the missing step.
• Export one actual same-operator contact family and test whether its capacities form a flag. Native/phase realization and a complete signed invoice are required before any helper or kappa claim.

Related open work

• PR6381 studies fixed-schedule stop deletion under variable helper endpoints on a P104 word. This theorem concerns existence and counts of one prescribed GL extension, and does not check that word or price a deletion.
• PR6388 prices a carrier itinerary through several distinct entrance lines. This result concerns simultaneous map constraints, rather than transport tariffs, and computes no competing route value.
• PR6494 proves a transvection-length minimum for binary symplectic partial-map completion with rank-one residual form. This theorem instead counts unrestricted GL extensions under nested late capacities; no shared numerical quantity is computed.

The refreshed corpus covers 438 open PR bodies and 6,420 Discussion records with complete reported remote pagination; local note coverage is 130 of 8,917 notes. The live open-PR corpus and current claims were searched for prescribed-map, nested-capacity and simultaneous-subspace work. No other open PR stating this nested-capacity completion theorem was found in that coverage; sparse historical notes, arbitrary diffs and unpublished work remain limits. The current account submissions above are context, not targets for a counterpart or follow-up.

CI notes

Both hosted checks, plan and replay (research/nested-late-capacity-completion), passed at exact head fdb7b8808ce89cd1738ea49dcd65c19ae1dddedd in run 41555972847. The completed checks were read once: plan took 4m23s and replay took 1m36s. Separate run metadata confirms completed success and the matching commit. The registered replay executes both normal and optimized scientific checks. Local scientific, metadata, committed-byte/source-history and read-only export checks also passed. Hosted execution certifies the bounded registered execution, not independent scientific acceptance or formal proof-kernel verification of the universal proof.

The completion refresh inspected all fourteen other open account PRs created since the advance CLAIM and added PR6494 above. No other inspected submission states this nested-capacity GL completion count; no additional computations were run to audit another result or compare unshared quantities.

Advance CLAIM. The publication base is 7c49fe32c55de02c2c873655c804b16e4c3fef0a, priority revision 8, selected PR6030. The original coverage-law card is explicitly changed to this algebraic shared-map prerequisite.

Merge Danger

Door: two-way. Blast radius: isolated package. Seven new files are added under research/nested-late-capacity-completion/. Shared code, predecessor packages and the selected frontier are unchanged.