[petalas]

Summary

Scoped written proof with exact finite certificates, no κ change. On the complete #7000 invoice, balanced refinement gives the largest first crossing on 0 ≤ κ ≤ 0.02 when each original helper may have at most m final pieces. The m=2 crossing is approximately 0.004689286593684114 and the full rank-one crossing is approximately 0.011115356648105199. Paid jobs and all main-origin targets stay fixed.

one original rank-r helper, repeated in 18 LOCALs
  replace r with a positive integer partition (a_i), sum a_i = r
  raw margin gain = 18 * (sum a_i^(1-kappa) - r^(1-kappa))
  maximize pointwise with balanced min(m,r) pieces
  certify the first root with outward integer intervals

The new source-specific results are the nine-row ceiling ladder, all 87 nontrivial partitions of ranks 2–9, their per-helper exchange rates and the exact minimum touched-helper counts at two comparison prices. At the fixed complex wrapper price 0.00376669, the minima are 43 with arbitrary partitions and 137 with at most two final pieces. At paired BIT ordinary price 0.00385248 they are 68 and 218. Rank-nine stock suffices in all four relaxed plans.

The scope is deliberately the first crossing on [0,0.02]. A global single-crossing extension would be false: the fully split margin is positive again at κ=1. This qualification was posted in the progress finding, following the original claim.

Evidence

• Before: the pinned invoice has raw crossing 0.003630058174688367… and rank surplus 1,520.
• After: exact integer intervals certify every ladder and single-split root with a rational bracket of width 2×10^-45. The exchange table proves each integer threshold by opposite signs at n−1 and n. The minimum-helper argument compares every helper rank and verifies available stock.
• Independent combinations enumeration agrees with all rank-1…9 partitions; 436 exact majorization cases corroborate the balancing proof. Integer 100th powers check every rank enclosure at κ=1/100. Six controls pass in normal and optimized modes.
• CPython 3.12.15 direct normal and optimized commands pass locally, about 2.4 seconds each. The registered package replay passes both commands. Static compilation passes. PASS_RESEARCH_COMMIT_BYTES passes at 777b642e85eb81564d1774e4747cb2313f0b54ec, including source-history ancestry and executable modes.

From the repository root, using the standard library:

~/git/integer-mult-bounds/.venv/bin/python3.12 -W error -B research/helper-split-headroom-ladder/verify.py
~/git/integer-mult-bounds/.venv/bin/python3.12 -W error -O -B research/helper-split-headroom-ladder/verify.py
~/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/helper-split-headroom-ladder
~/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/helper-split-headroom-ladder --check-source-history --check-modes

Expected scientific marker is PASS_HELPER_SPLIT_HEADROOM_LADDER, with pin_any=43 pin_binary=137 paired_any=68 paired_binary=218 controls=6; the launcher prints PASS_REGISTERED_RESEARCH_PACKAGE.

Limitations and credit

This is an optimistic invoice relaxation, not a word, a legal split, supplier admission or a multiplication theorem. Pivot/cut/boundary bases, independence, ADD/SCALE costs, the ordinary bill, compiler/native admission and outer gates remain separate obligations. No selected-source cold replay, Lean or Docker run was performed. Independent human and formal proof-kernel review are unclaimed. In particular, the paired BIT comparison price is not automatically available to the complex wrapper.

Produced by Nick Petalas. Credit Jerry Chen / CubeJerry's #7000 invoice and #6980 caller, DreamingOfClouds's #6936/#6937, Ali Romman's helper-splitting method, dillona, ToikY, Rohan Arun and the retained P104 lineage. Classical balanced-partition optimality is already recorded in #6651 and is not claimed as new. zxv's #5554 concerns a different fixed-local-bill path model. Exact sources and hashes are in SOURCES.json. Predecessor files and notices are unchanged.

Open next questions

• Can legal rank-nine 5+4 splits with no added paid rank approach the 137-helper wrapper threshold? Begin with an exact prefix/suffix witness and complete correction/restoration costs.
• How does a particular successor paid profile alter the ladder? Authenticate its helper-origin decomposition before substituting it into this model.

Related open work

These are source or invoice neighbors, not construction claims replayed here. This verifier computes neither their changed invoices nor their proposed κ values, so it makes no numerical agreement/disagreement claim about them.

• #7164: application review of a successor invoice; this result does not transfer its source admission or fixed-profile ceiling.
• #7161: funded rank-eight exit; it changes a source word or complete invoice, while this theorem fixes the selected Conditional κ = 0.00363005817 (+0.0838% over #6936): shared wires and two helper splits #7000 paid profile and main targets.
• #7158: application review of a successor invoice; this result does not transfer its source admission or fixed-profile ceiling.
• #7156: temporal chart306 continuation; it changes a source word or complete invoice, while this theorem fixes the selected Conditional κ = 0.00363005817 (+0.0838% over #6936): shared wires and two helper splits #7000 paid profile and main targets.
• #7155: compose the chart-237 and chart-595 splits; it changes a source word or complete invoice, while this theorem fixes the selected Conditional κ = 0.00363005817 (+0.0838% over #6936): shared wires and two helper splits #7000 paid profile and main targets.
• #7148: temporal chart-595 split; it changes a source word or complete invoice, while this theorem fixes the selected Conditional κ = 0.00363005817 (+0.0838% over #6936): shared wires and two helper splits #7000 paid profile and main targets.
• #7147: temporal chart237 continuation; it changes a source word or complete invoice, while this theorem fixes the selected Conditional κ = 0.00363005817 (+0.0838% over #6936): shared wires and two helper splits #7000 paid profile and main targets.
• #7145: endpoint and temporal composition; it changes a source word or complete invoice, while this theorem fixes the selected Conditional κ = 0.00363005817 (+0.0838% over #6936): shared wires and two helper splits #7000 paid profile and main targets.
• #7143: temporal chart-120 split; it changes a source word or complete invoice, while this theorem fixes the selected Conditional κ = 0.00363005817 (+0.0838% over #6936): shared wires and two helper splits #7000 paid profile and main targets.
• #7142: endpoint-funded split; it changes a source word or complete invoice, while this theorem fixes the selected Conditional κ = 0.00363005817 (+0.0838% over #6936): shared wires and two helper splits #7000 paid profile and main targets.
• #7140: causal-history buffer split; it changes a source word or complete invoice, while this theorem fixes the selected Conditional κ = 0.00363005817 (+0.0838% over #6936): shared wires and two helper splits #7000 paid profile and main targets.
• #7131: compose output rechart; it changes a source word or complete invoice, while this theorem fixes the selected Conditional κ = 0.00363005817 (+0.0838% over #6936): shared wires and two helper splits #7000 paid profile and main targets.
• #7130: nonunit-pivot helper split; it changes a source word or complete invoice, while this theorem fixes the selected Conditional κ = 0.00363005817 (+0.0838% over #6936): shared wires and two helper splits #7000 paid profile and main targets.
• #7124: global output rechart; it changes a source word or complete invoice, while this theorem fixes the selected Conditional κ = 0.00363005817 (+0.0838% over #6936): shared wires and two helper splits #7000 paid profile and main targets.
• #7123: recharted helper split; it changes a source word or complete invoice, while this theorem fixes the selected Conditional κ = 0.00363005817 (+0.0838% over #6936): shared wires and two helper splits #7000 paid profile and main targets.
• #7122: global input rechart; it changes a source word or complete invoice, while this theorem fixes the selected Conditional κ = 0.00363005817 (+0.0838% over #6936): shared wires and two helper splits #7000 paid profile and main targets.
• #7121: newly available rank-eight split; it changes a source word or complete invoice, while this theorem fixes the selected Conditional κ = 0.00363005817 (+0.0838% over #6936): shared wires and two helper splits #7000 paid profile and main targets.
• #7120: compose ToikY chart-120 pattern; it changes a source word or complete invoice, while this theorem fixes the selected Conditional κ = 0.00363005817 (+0.0838% over #6936): shared wires and two helper splits #7000 paid profile and main targets.
• #7119: reciprocal helper split; it changes a source word or complete invoice, while this theorem fixes the selected Conditional κ = 0.00363005817 (+0.0838% over #6936): shared wires and two helper splits #7000 paid profile and main targets.
• #7118: residual helper split; it changes a source word or complete invoice, while this theorem fixes the selected Conditional κ = 0.00363005817 (+0.0838% over #6936): shared wires and two helper splits #7000 paid profile and main targets.
• #7117: paid helper meeting; it changes a source word or complete invoice, while this theorem fixes the selected Conditional κ = 0.00363005817 (+0.0838% over #6936): shared wires and two helper splits #7000 paid profile and main targets.
• #7116: compose signed caller; it changes a source word or complete invoice, while this theorem fixes the selected Conditional κ = 0.00363005817 (+0.0838% over #6936): shared wires and two helper splits #7000 paid profile and main targets.
• #7115: concrete successor construction; it changes a source word or complete invoice, while this theorem fixes the selected Conditional κ = 0.00363005817 (+0.0838% over #6936): shared wires and two helper splits #7000 paid profile and main targets.
• #7113: concrete successor construction; it changes a source word or complete invoice, while this theorem fixes the selected Conditional κ = 0.00363005817 (+0.0838% over #6936): shared wires and two helper splits #7000 paid profile and main targets.
• #7112: concrete successor construction; it changes a source word or complete invoice, while this theorem fixes the selected Conditional κ = 0.00363005817 (+0.0838% over #6936): shared wires and two helper splits #7000 paid profile and main targets.
• #7110: cocharted rank-nine helper split; it changes a source word or complete invoice, while this theorem fixes the selected Conditional κ = 0.00363005817 (+0.0838% over #6936): shared wires and two helper splits #7000 paid profile and main targets.
• #7109: rank-eight helper split; it changes a source word or complete invoice, while this theorem fixes the selected Conditional κ = 0.00363005817 (+0.0838% over #6936): shared wires and two helper splits #7000 paid profile and main targets.
• #7108: concrete successor construction; it changes a source word or complete invoice, while this theorem fixes the selected Conditional κ = 0.00363005817 (+0.0838% over #6936): shared wires and two helper splits #7000 paid profile and main targets.
• #7107: concrete successor construction; it changes a source word or complete invoice, while this theorem fixes the selected Conditional κ = 0.00363005817 (+0.0838% over #6936): shared wires and two helper splits #7000 paid profile and main targets.
• #7105: concrete successor construction; it changes a source word or complete invoice, while this theorem fixes the selected Conditional κ = 0.00363005817 (+0.0838% over #6936): shared wires and two helper splits #7000 paid profile and main targets.
• #7004: peer MOVE retiming; it changes a source word or complete invoice, while this theorem fixes the selected Conditional κ = 0.00363005817 (+0.0838% over #6936): shared wires and two helper splits #7000 paid profile and main targets.
• #7003: five MOVE fusions; it changes a source word or complete invoice, while this theorem fixes the selected Conditional κ = 0.00363005817 (+0.0838% over #6936): shared wires and two helper splits #7000 paid profile and main targets.
• #5606: earlier P104 helper constructions on different source invoices; their paid profiles are not used here.
• #5553: earlier P104 helper constructions on different source invoices; their paid profiles are not used here.
• #3729: an earlier fixed-inventory packing ceiling; its historical invoice and optimization variables differ from the Conditional κ = 0.00363005817 (+0.0838% over #6936): shared wires and two helper splits #7000 helper-refinement model.

Related-work refresh before DONE checked all open PRs from this account created since the claim. The following newer contributions study different objects; no shared computed quantity was found, and no extra numerical comparison was run.

• #7167: frame-indicator hosting costs, rather than fixed-invoice helper-target refinement.
• #7168: Gram-class minima under root-line deletion, rather than the Conditional κ = 0.00363005817 (+0.0838% over #6936): shared wires and two helper splits #7000 helper inventory.
• #7169: a one-sided masked COPY completion on a fixed puncture, rather than helper-split moment gains.
• #7170: scalar COPY perturbations at a fixed puncture, rather than this paid/target invoice.
• #7172: doubled centre Gram stabilization, rather than the helper-split first crossings.
• #7173: a five-stage coefficient-space rank floor, rather than the fixed Conditional κ = 0.00363005817 (+0.0838% over #6936): shared wires and two helper splits #7000 caller and helper credit.
• #7174: realization of a specific E8 line complement, rather than productive-helper partitions.
• #7176: endpoint chain windows in a six-source characteristic-six schedule, rather than this invoice.

CI notes

CI did not certify this package. At head 777b642e85eb81564d1774e4747cb2313f0b54ec, run 42984445251, plan job 130402173676, failed in "Onboarding helper tests". I read the checks once and the job log directly. The log reports 760 tests, exactly one failure and 24 skipped:

test_research_current_status.CurrentCiNavigationTests.test_current_ci_section_matches_selected_record
AssertionError: '`make frontier7000-verify`' not found

The package replay job was skipped because plan failed. Local checks described above are the package's verification evidence; hosted CI supplied no scientific certification.

The mismatch is present in canonical base e54523517d0f03e58352e647c403a5af3e525a57: certificates/selected-result.json names make frontier7000-verify, while the current section of docs/ci-verification.md still describes #6030 and make frontier6030-verify. I read both base files and confirmed that this PR changes neither file nor the failing test. No unrelated fix, retrigger commit or job rerun was made.

I also read each of these failed plan logs. Every one reports the same exact failing test and assertion, 760 tests, one failure and 24 skipped:

• Run 42905094314.
• Run 42903313790.
• #7167, run 42983769381.
• #7168, run 42983916692.
• #7170, run 42984005087.
• #7174, run 42984295238.
• #7176, run 42984461381.

Merge Danger

Door: two-way. Blast radius: package. All seven files are new and confined to research/helper-split-headroom-ladder/.