[petalas]

Result

Finite exact certificate on an accounting relaxation. No new κ; the selected frontier is unchanged. This result is produced by Nick Petalas.

On the authenticated selected #5599 P104 invoice (scripts/export_selected_invoice.py at 19fdb086e, invoice SHA-256 ad826c82…3e4c1), the complex bill fails at the admitted BIT saving C = 17/5000 by exactly 72.892623830207, or 4.049590212789 per FULL. Every local paid job and local helper target appears exactly 18 times, once per FULL. With all other paid work and targets fixed:

Local change per FULL	Passes at 17/5000?
delete one paid job of rank ≤ 4	no (rank 4 gives 3.98)
delete one paid job of rank 5, 6 or 7	yes
fewest deletions using only rank ≤ 1 / 2 / 3 / 4	5 / 3 / 2 / 2
free full refinement of rank-9 helpers into rank-1 targets	needs 61 of 412
free balanced 4+5 split of rank-9 helpers	needs 194 of 412
refine every local helper target fully	first crossing in [0.0094772838495, 0.0094772838496]
the same at C = 1/100	fails

Full refinement maximizes a helper's target moment at every C, since Σ p^(1-C) ≤ r for any partition of r. So no split of local helper targets alone, with paid work fixed, lets the selected invoice pass at 1/100. For comparison, #5599 refines 13 helpers, and #5702 needs 15 fully refined H9 sources to fund one added H1 per FULL.

Evidence and reproduction

verify.py checks the invoice bytes against the exporter's hash and the 18-fold replication. It evaluates every r^(1-C) with correctly rounded 80-digit decimal ln/exp, so the total error is far below 1e-50 and every sign clears 1e-30. A float evaluation corroborates the deficit. Each boundary case is rerun on the actual changed invoice: rank 4 versus 5, 4 versus 5 rank-one deletions, 60 versus 61 and 193 versus 194 H9 splits, and the 1/100 failure.

From the repository root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -B -W error research/selected-invoice-bit-cap-thresholds/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -O -B -W error research/selected-invoice-bit-cap-thresholds/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/selected-invoice-bit-cap-thresholds
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/selected-invoice-bit-cap-thresholds --check-source-history

Local results: both verifier runs print PASS_SELECTED_INVOICE_BIT_CAP_THRESHOLDS in about 0.2 s, research_package.py prints PASS_REGISTERED_RESEARCH_PACKAGE … outcome=completed in 0.65 s, and check_research_commit.py prints PASS_RESEARCH_COMMIT_BYTES at d9d9f7c66. On a full checkout, export_selected_invoice.py reproduces the packaged invoice hash.

Limitations

These deletions and splits are free in an optimistic accounting relaxation. No restoration, native alignment, precision, compiler, stock or outer cost is charged, and no physical word is supplied. A passing row keeps a route open. It is not a supplier. Once complex passes at 17/5000, the admitted BIT saving binds, but this package does not prove the outer κ. It certifies only the signs at the stated C values and the root bracket. Uniqueness of the refined crossing is not claimed beyond the separate failure at 1/100. Independent human and formal proof-kernel review are unclaimed.

Open next questions

1. Which physical window of the selected local word can delete one paid job of rank 5, 6 or 7, or two of rank 3 or 4, while keeping every target and restoration? Under this accounting, any one of them lets complex pass at the admitted BIT saving.
2. Helper splitting alone stops below 1/100. Which paid-side or main-target changes, priced with the same moment, would let complex pass at 1/100? A target above .0034 also needs BIT improvement.

Related open work

• Save twenty P104 scalar units and two events through dirty-output sharing (no new κ) #5999: saves P104 scalar units through dirty-output sharing. That is scalar work, not the paid histogram priced here.
• Compress the ordinary events of the PR5907 P104 word: 25,998 to 25,861, same invoice #5975 and Exhaustion census: no PR5599-type helper split remains on the PR5907 word (no new kappa) #5994: study the successor Conditional κ .003125282 (+5.22e-7 over #5827): remove helper 468's frame-593 rung with rerouted reads #5907 word, not the selected Raise conditional kappa to 0.00310045 with independent helper splits #5599 invoice. I did not compute their invoices.
• Conditional κ 0.00309871 from a prepared P104 split helper #5606 and Two successor hand-offs on the P104 word: conditional κ 0.0030428242 #5553: propose P104 helper-split κ values on other words. Those are distinct invoices; no number is compared.
• Prove the exact 406-merge threshold for the frozen joint invoice #5932: finds the minimum number of cross-side paid-job merges (406) that let the frozen Joint bit+complex supplier cannot beat the average of the two p16 invoices under the whole-job premise (negative result) #5298 joint BIT/complex ledger pass at C = 7/2000. It uses the same signed moment test at a fixed C, but on a different ledger and at a different C. This PR computes nothing at 7/2000 on that ledger, so no number is compared.
• Certify the pooled R437 carrier moment ceiling: 0.00173678297, unreachable 17/5000 below width 28571 (no new κ) #5992: certifies the moment ceiling of the pooled R437 carrier profile (crossing near 0.00173678297) and shows 17/5000 is unreachable there. It uses the same kind of fixed-C moment test, and also tests at 17/5000, but on a different invoice. This PR's deficit at 17/5000 is for the selected Raise conditional kappa to 0.00310045 with independent helper splits #5599 P104 invoice only, so no number is compared.
• Other open PRs from this account opened since the CLAIM (Prove exact joint Gaussian transition cost with dirty workspace (no new κ) #6001–Prove the char-2 jet obstruction on jf0..2 holds at every order (no new κ) #6009, Prove an integer-shear target and its inverse need eight calls: no cross-role sharing (no new κ) #6011) study different objects and do not price this invoice.

Credit

The P104 word is by Pierre-Loic Doulcet (hexapode), with Ali Romman's caller and helper accounting and exporter, Cinna Davis's controller, Jacob Alan Sussman's productive recursion and zxv's invoice interface. theon199's #5702, Clear1nsight's #5614 and the merged productive-lever-reprice package price related refinements and an older-invoice gap. They are cited, not re-checked. Claim: #1166 (comment)

CI notes

Both hosted checks passed at head d9d9f7c66bde26375ea79d1357897acc226b906d: plan and replay (research/selected-invoice-bit-cap-thresholds), in run https://github.com/CrocSwap/integer-mult-bounds/actions/runs/40585594796. The check list was read once and the run head matches the PR head.