[petalas]

Summary

Prove an effective matching lower bound for the multiplicative-shell theorem of JoyboyBrian's #6086. Claim type: scoped written mathematical proof with exact finite corroboration.

#6086: J_l = O((l+1)^(r-1))
Here:  J_l = Theta((l+1)^(r-1)) for every fixed finite rank alphabet containing M
New:   J_l >= binom(floor(l/H)+r-1,r-1) at every level

Here r is the rational prime-valuation rank, and H is the maximum ceiling log-base-M of generators extending v(M) to a valuation basis. A bounded exponent-tuple injection gives the lower bound even when other generators are dependent. The same argument gives an exact shell identity for independent generators. Thus a smaller full-monoid shell exponent cannot improve this adaptive width estimate. Actual reachable-set restrictions and LCM cancellation remain separate possibilities. No compiler-width lower bound or new kappa is claimed.

This contribution was produced by Nick Petalas. Credit JoyboyBrian for #6086's upper theorem and adaptive application, codyw912 for #4638's level-LCM recurrence, Nick Petalas for #5962, and Mihir Srivastava, zxv, Jacob Alan Sussman, Ryanson Jonathan, Cinna Davis and their retained compiler/native lineage. The upper theorem and reported p16 rank eleven are cited premises, without predecessor scientific replay. Unique factorization and stars and bars are classical; general literature priority is unclaimed.

Evidence

Before, the available shell estimate supplied an upper exponent without a matching general lower theorem. After, PROOF.md proves the effective all-level lower bound and hence sharpness for every fixed finite alphabet.

At head 426d1f55263d91ccfedd85a80b059659a634091b, local CPython 3.12.15 checks pass:

• 360 alphabet/base fixtures, 1,800 shells and 4,016 constructed lower-witness occurrences. Multiplicative closure agrees with an independent integer-divisibility oracle in every fixture.
• 49 exact independent-generator shell identities with 297 point occurrences, plus nine domain and semantic controls.
• Warning-strict normal and optimized executions, both registered commands, metadata and source pins, AST inspection, and committed-byte/source-history/executable-mode checks. The timed commands each completed in about 0.22 seconds, below their 120-second bounds.

Each direct run prints PASS_MULTIPLICATIVE_SHELL_SHARPNESS inside a generated JSON receipt bound to all six package files. The registered runner prints PASS_REGISTERED_RESEARCH_PACKAGE; publication preflight prints PASS_RESEARCH_COMMIT_BYTES. No receipt is handwritten.

Exact reproduction from the worktree root:

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

Any CPython 3.12 path can replace the local interpreter path. The standard library is sufficient.

Limitations and unrun obligations

The universal claim rests on the written injection proof and the cited upper theorem. Finite enumeration corroborates it. Constants depend on the fixed alphabet and height base. Varying alphabets/bases, restricted reachable subsets and monoids without M are outside the theorem. The p16 degree-ten consequence uses #6086's reported rank as a premise.

This is an isolated counting obstruction in the alternative adaptive compiler. No actual width lower bound, physical memory/runtime claim, native construction, selected assumption discharge, supplier admission, multiplication theorem or frontier update follows. No predecessor scientific verifier, full p16 compiler, selected cold replay, Lean or Docker was run. Independent human and formal proof-kernel review are unclaimed.

Open next questions

1. Derive a smaller actual reachable plateau language for one nonhomogeneous compiler, retaining every chronological cut. A proved reachability invariant and full LCM bound remain missing.
2. Prove an all-prime source-count cancellation bound along actual child chains in one nonhomogeneous profile. Sharp shell counts alone do not control the LCM.

Related open work

No other open PR studying this multiplicative-shell lower theorem was found in the refreshed open list and targeted searches for multiplicative shells, semigroups and the adaptive compiler. The source-specific intake found zero references to #6086 among this account's 38 PRs created since November 4, across fully paginated open and closed corpora. Text searches are not a novelty certificate and omit unpublished work.

Merged #6086 is the direct predecessor. Merged #5962 concerns default-height rank counting; merged #6612 exploits cancellation in a homogeneous width recurrence. Neither is rechecked or extended as this deliverable. Advance CLAIM.

The completion refresh read the four other open contributions created since the CLAIM, #7179 through #7182. They study A2 target tariffs, Gaussian inverse norms, Clifford response sectors and cube-zero lifts. None studies this shell-count quantity or computes a conflicting shared value. No extra scientific comparison computations were run.

CI notes

At exact head 426d1f55263d91ccfedd85a80b059659a634091b, run 42992088329, plan job 130427170517, 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
AssertionError: '`make frontier7000-verify`' not found
FAILED (failures=1, skipped=24)

The job log shows that the CI verification document still describes selected #6030 and make frontier6030-verify. Direct inspection of main commit e54523517d0f03e58352e647c403a5af3e525a57 confirms that certificates/selected-result.json names make frontier7000-verify, while the current section of docs/ci-verification.md has the older command. This PR changes only its six package files and changes neither file involved in that mismatch.

The same exact failing test, missing command, 760-test total, one failure and 24 skips were independently confirmed from the failed-step logs of all these runs:

• Other contributors: 42905094314, 42903313790.
• Lower P104 frame-0 hosting from 40 to 14 monomials (no κ) #7167: 42983769381.
• Prove the exact E8 Gram-class minimum through 24 deletions (no κ) #7168: 42983916692.
• Bound scalar COPY savings at one 93-line E8 puncture (no κ) #7170: 42984005087.
• Exclude the proposed 96-line E8 complement at inner dimension 8k (no κ) #7174: 42984295238.
• Prove sharp c=6 endpoint chain windows (no κ) #7176: 42984461381.

CI did not certify this package. Its replay job was skipped because plan failed. The local registered normal/optimized, metadata and publication checks recorded above remain the available automated package evidence. No repair, retrigger commit or job rerun was made; the repository-level mismatch is left for maintainers.

Merge Danger

Door: two-way. Blast radius: isolated.

Six new files in research/multiplicative-shell-sharpness/. Shared code, frozen predecessor evidence and selected records are unchanged.