[petalas]

Summary

Prove a sharp mixed-rank power-invoice envelope, including exact interior touching multiplicity, and supply a portable integer certificate checker. Claim type: scoped algebraic proof with finite corroboration. The examples are synthetic invoices; no physical supplier or new kappa is claimed.

D(t) = epsilon + lambda product_i(a_i^(q*t)-a_i^p), even positive factor count
minimum D = epsilon, uniquely at t=p/q; zero multiplicity = factor count if epsilon=0
6 - 3*4^t - 2*9^t + 36^t = (4^t-2)(9^t-3)

The four-rank example has a unique double zero at 1/2. Adding one H1 gives sharp minimum gap one. The checker compares the full new-paid minus new-targets minus old-paid plus old-targets coefficients with the supplied factor expansion, including rank collisions and target debits.

New proof application, integer invoice family and checker produced by Nick Petalas. Credit Ali Romman #5788 for the explicit touching-root continuation and general-rank comparison context, zxv for signed-invoice accounting, mightyOil for moment-order context, and woklerx for the specialized crossing predecessor. Their results are cited, not audited or replayed. Elementary algebra and calculus are classical methods; no literature priority for them is claimed.

Evidence

Before: #5788 explicitly leaves touching-root multiplicity outside its simple-interior-witness closure. This PR changes that premise for a declared synchronized-factor family; it does not retest the predecessor.

After: the written proof gives the sharp envelope for every real exponent and arbitrary finite positive even factor count. Local CPython 3.12.15 warning-strict normal and optimized checks pass all 696 contexts, 4,176 certificates, 49,944 expanded support entries, 23,472 exact rational-power evaluations and 26 controls. The direct registered commands took 1.095 and 1.086 seconds locally, below their 60-second limits. These are replay timings, not a performance improvement claim.

Exact reproduction from the repository root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -B research/mixed-rank-touching-envelope/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -O -W error -B research/mixed-rank-touching-envelope/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/mixed-rank-touching-envelope
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/mixed-rank-touching-envelope --check-source-history --check-modes
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B research/mixed-rank-touching-envelope/envelope.py research/mixed-rank-touching-envelope/EXAMPLE.json

The verifier prints a generated receipt bound to package/input hashes and PASS_MIXED_RANK_TOUCHING_ENVELOPE. Registered replays, committed-byte/mode/source-history preflight and official export integrity pass at 0e80c6baa599d5b110401e1fd041fe45005984b7. Both normal and optimized replays also pass from a read-only export with Git absent from PATH; every exported byte and mode is unchanged. Standard library only. The checker and verifier write no files.

Canonical main was refreshed before work and submission and remains de5ec2e7a6074068452f45b1a191b5c72832df4a, priority revision 8. Live overlap covered 437 open PRs and 5,361 Discussion records with complete pagination; distinctive all-main README and merged-PR searches supplement the sparse local-note search. This coverage is not proof of novelty.

Limitations

The checker accepts 2,4,6,8 factors, bases 2..64, 2<=q<=16, 1<=p<q, integer positive scale and nonnegative offset through 10^60, expanded ranks through 10^9 and counts through 10^60. The proof has no such finite capacities. Certificate discovery and arbitrary exponential-polynomial classification remain open.

No actual changed supplier's bill is supplied. Source extraction, physical target independence, native implementation, restoration, ownership, compiler/supplier admission and a complete multiplication theorem are unclaimed. The selected cold replay, Lean, independent human review and formal proof-kernel review were not run. Selected-frontier files and frozen evidence are untouched.

Open next questions

• Find one actual source-bound old/new complete invoice pair whose exact difference has this certificate after a justified common bill cancels. Start with sparse coefficient divisibility; physical bill completeness is the blocker.
• Extend the checker to several supplied rational touching points, grouped by exact rational equality. Root recognition outside that specified rational domain is not settled here.

Related open work

• Certify all-exponent order for general-rank recursive invoices #5788 supplies the general-rank all-exponent comparator and this question. This PR adds sharp synchronized touching-point envelopes using factor identity certificates; it does not change or validate that implementation.

• Certify best-invoice regions across common-base power costs #5768 certifies winner regions for common-base invoices using Sturm methods. This PR handles a restricted mixed-rank family, including ranks 4 and 9, with a different exact certificate. No common numerical fixture was recomputed for comparison.

• Prove one-axis transported coframe optimization is NP-hard #5888 proves hardness for transported coframe rank-power optimization. It uses the same power-tariff quantity on a different finite-word family; it neither computes these synchronized invoice envelopes nor supplies a shared numerical fixture.

CI notes

At commit 0e80c6baa599d5b110401e1fd041fe45005984b7, run 40495697539 failed in plan at Onboarding helper tests. This was the only failed step. All ten failure/error entries are in unchanged tests/test_research_current_assignment.py, with the final summary FAILED (failures=8, errors=2, skipped=22).

The two errors are KeyError: 'follow_up_steps' and KeyError: 'catalog_first_step'. The eight failure entries concern the frozen-card count (0 != 16), missing-helper/reference rejection and sparse-reference receipts. All ten entries match PR #5886's plan log exactly. This confirms the shared onboarding failure on main de5ec2e7a6074068452f45b1a191b5c72832df4a; this PR changes only its new research package. No shared code or tests were changed to address it.

replay was skipped. Hosted replay has not certified this package; only the local package checks reported above have run. No hosted scientific success is claimed.

Merge Danger

Door: two-way. Blast radius: isolated. The diff adds one opt-in research package and changes no shared implementation or selected result.