[petalas]

Summary

A scoped written proof computes the exact source-action crossover for Tekalign / tekuuu's PR4077 shared atom powers when three I-bit exponents have prescribed total E. The predecessor's implementation and vector tariffs are cited premises, not replay or audit deliverables.

fixed total E
  maximize complete counter/index overhead with 24 carry states per bit
  retain an attaining exponent triple
  certify every placement iff (E-1)rho > maximum overhead

For E>=2, at most 192I bit-mask trials give the sharp threshold. At I=4,w=1,E=10, the maximum overhead is 257, attained by (4,4,2). Integral radix invoice rho>=29 is necessary and sufficient uniformly over those triples; separate bounds on bitlength and positive count only certify rho>=30. The minimum saving at 29 is four source actions, while the attaining triple loses five at 28. These are parametric action tariffs; no validator of cost 29 was executed. The zero-total case ties and total one always loses.

The actual child geometry has E=2(n-c), so the result gives a sufficient test independent of placement. Sharpness concerns all sum-E triples, which may exceed a particular source's legal geometry. No new kappa is claimed. This contribution was produced by Nick Petalas.

Evidence

The new carry maximum agrees with direct applications of the cited tariffs on all 149,792 ordered triples at I=1..5 and w in {1,2,3,7}, giving 704 exact maximum comparisons. Strict boundaries, attaining witnesses, zero/one totals, an undercut maximum, separate-statistic incompatibility, larger widths and twelve invalid arguments pass. The written carry invariant supplies the all-size proof; finite enumeration corroborates it.

Normal and optimized warning-strict direct checks, both registered commands, metadata/provenance validation and seven-file committed-byte/source-history preflight pass. Each registered command took about 0.21 seconds locally on CPython 3.12.15, Darwin 27.0.0. The official export integrity check and a read-only, Git-free optimized replay pass with all eight exported files unchanged. No third-party dependency is required. Every direct run prints a generated receipt bound to the seven package-file hashes.

From the repository root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/shared-atom-crossover
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -B research/shared-atom-crossover/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -O -W error -B research/shared-atom-crossover/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/shared-atom-crossover --check-source-history

A different CPython 3.12 installation may replace the interpreter path. Expected markers are PASS_SHARED_ATOM_CROSSOVER, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES.

Limitations and credit

The theorem assumes PR4077's successful-input tariff contract and counts each record/residue/index read, write and unit move equally. It does not supply a physical exchange rate, a runtime selector, a realized radix invoice, narrower-geometry sharpness or a complete caller/supplier. Context/payload composition, allocator/control, retained backing, stock, precision and outer admission remain separate. All five selected framework assumptions remain. No predecessor verifier, selected cold replay, Lean or Docker run was used. Independent human and formal proof-kernel review are unclaimed.

Credit Tekalign / tekuuu's PR4077 construction, exact costs and explicit affordability question; David Arias / PR2753, Jose Manuel / jmbarrancoml, ToikY / PR4025, ywz65 / PR2590, Chris R / PR4062 and their retained lineage. Binary digit dynamic programming is classical. Original sources and credit are unchanged. Canonical base is 7c49fe3, priority revision 8, selected PR6030. The formal-interface-lemma card explicitly changes to this written cost theorem.

Related open work

No open PR studying this shared atom-power crossover was identified in the refreshed search. PR4077 and the related caller/preparation work #4062/#4176 and quotient work #4854 are merged; their respective implementation and cost claims are cited without replay. Other open rounding, carry-state and native-geometry contributions do not compute this quantity.

The completion refresh inspected all 125 account open PRs, including the 12 created since the CLAIM. The other eleven, #6572–#6583 excluding this PR, concern derived phases, native rounding, ADDRESS margins, parity fanout, algebra automorphisms, E8 basis/equality tariffs, local factors and replay runtime. None studies this fixed-total shared-power cost maximum or reports a competing value. No additional computations were run to compare their results.

Open next questions

• Restrict the maximum to one actual child family's legal placements and external gap constraints.
• Supply a source-bound lower bound for rho and price an encoded selector in one complete caller.

CI notes

Both hosted checks passed at exact head d0d8ae85a23b338f437ab0402126c7e5e1fc7ff7 in run 41617293792: plan in 3m46s and replay (research/shared-atom-crossover) in 1m39s. The completed checks were read once; run metadata confirms this head, completed status and success. Local registered replay, direct normal/optimized checks, committed bytes/source history and read-only export checks also passed. Hosted execution certifies the declared finite checks and source bindings, not independent mathematical acceptance.

Advance CLAIM.

Merge Danger

Door: two-way. Blast Radius: isolated research package.

Only research/shared-atom-crossover is added. Shared code, predecessor evidence and the selected frontier are unchanged.