[petalas]

Summary

Prove the exact rational asymptotic of the Lagrangian projection recurrence from Jacob Alan Sussman #4628 and Maximilian Lutz #4643. Claim type: scoped mathematical proof with exact finite corroboration.

cited block-four seed (0,27456,48576,68320,86592)
    -> normalized coherent Lagrangian distance laws
    -> lim a_m(m)/a_m(1) = 439277/149175

The new theorem gives a finite closed-form limit for every nonnegative seed profile at any fixed seed block, and an explicit convergence bound. The block-four value is approximately 2.9447092341209986. This extends the predecessors' decimal tail description; their hook-rank inputs and engine premises remain cited assumptions. No new kappa or selected-frontier change.

Only research/projection-recursion-limit/ is added. The proof uses a monotone coupling and a summable comparison of reduction kernels. The package contains its standalone exact checker, generated hash-bound receipt and source/replay metadata. This result was produced by Nick Petalas. Sussman, Lutz, eumemic #288 and Swapnil Jain retain their predecessor credit.

Evidence

Before: the cited packages describe the tail numerically and bound it by a comparison chain. After: PROOF.md identifies the exact rational limit and proves a general seed formula with an explicit error estimate.

At head e6c8e945a47dcc4a6f08974682090fc7f5e221a3, local CPython 3.12.15 on macOS arm64 passes:

• Both registered warning-strict normal/optimized commands and both direct commands. Direct commands took 0.20 seconds each; the registered runner took 0.72 seconds. These are local observations, not performance claims.
• 3,320 kernel identities/error bounds, 63 coherent-law reductions, 80 stationary-chain indices, 73 original-seed enclosures, 54 general-seed enclosures, 38 raw-integer/probability-law comparisons and six failure controls.
• Exact committed-byte and declared source-history preflight.
• Official package export/integrity checks and normal/optimized replay from a read-only Git-free export. All eight payload files and the export manifest remain byte-identical.

Exact reproduction commands from the repository root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/projection-recursion-limit
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -X dev -W error -B research/projection-recursion-limit/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -O -X dev -W error -B research/projection-recursion-limit/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/projection-recursion-limit --check-source-history

Another CPython 3.12 interpreter may replace the local absolute path. Expected markers are PASS_PROJECTION_RECURSION_LIMIT, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES.

Limitations and unrun obligations

The theorem is unconditional for its stated arithmetic recurrence. Interpreting it as an engine bound inherits the hook profile and representation argument, including the two unresolved stored-rank certificates identified by #4643. Those ranks were not recomputed. The optional triangle/max strengthening is outside the theorem. No finite-m infimum or monotonicity theorem is claimed. Physical realization, native admission, complete cost, selected cold replay and formal proof-kernel review were not run or supplied. Independent human review is unclaimed.

Open next questions

• Find a finite-m closed form or improve the conservative convergence rate, starting with the coherent-law comparison.
• Characterize seed profiles whose top ratios are monotone and whose infimum equals the limit, starting with adjacent-block differences as linear seed functionals.

Related open work

• Research: lower bounds for small complex units: m = 2 never saves, m = 3 needs 14W/5 calls (Lean, engine game), m = 4 needs 41W/13, and the smallest open cases (no new κ) #4628 studies the complex small-block hook bounds and supplies the cited block-four seed/recurrence context. This PR adds an exact arithmetic limit theorem and leaves its engine and hook-rank obligations unchanged.

Completion refresh read the nine other open contributions from this account created since the CLAIM. None studies this projection recurrence or its asymptotic ratio, so no new same-quantity comparison is required.

CI notes

Both hosted checks passed at e6c8e945a47dcc4a6f08974682090fc7f5e221a3 in run 39943559940: plan in 4m50s and the registered package replay in 1m12s. The completed run has conclusion success and its head SHA matches this contribution. The replay runs both declared normal and optimized commands. All local obligations listed above also passed; none was deferred to CI.

Merge danger

Door: reversible. Blast radius: one research package. Frozen predecessors, shared code, pinned evidence and the selected frontier are unchanged.