[petalas]

Result

Scoped proof with an exact finite check. No new κ; the selected frontier is unchanged.

For the merged binary query-tree bill (#6348, research/balanced-query-tree, conditional κ 0.00314933), let F(t) = Σ_r (T_r − P_r) r^t be the complete signed raw moment. Its 19 nonzero coefficients change sign 13 times, so a Descartes count allows up to 13 real roots. Their tail sums B_k = c_k + … + c_n change sign exactly once. By Laguerre's classical tail-sum rule of signs, F has at most one zero on all of t > 0. With F(0) = −56309 < 0 and F(1) = 808 > 0, it has exactly one, in (0,1), and changes sign there.

Consequence: for every saving C < 1, the bill's complete raw target moment beats its paid moment exactly when C lies below the raw crossing. The source package proves uniqueness only for C ≤ 0.0034, by a derivative bound. This certificate rules out a second favourable window on the rest of the range, using exact integers only.

PROOF.md states the lemma and reduces it to the variation-diminishing property of the kernel e^{ut} (Pólya–Szegő Part V; Karlin, Total Positivity). That property is cited, not reproved.

Evidence and reproduction

Interpreter: CPython 3.12 (~/git/integer-mult-bounds/.venv/bin/python3.12 locally). From the repository root:

python3.12 -I -B research/tail-sum-crossing-uniqueness/verify.py
python3.12 -I -O -B research/tail-sum-crossing-uniqueness/verify.py
python3.12 -B scripts/research_package.py --package research/tail-sum-crossing-uniqueness
python3.12 -B scripts/check_research_commit.py --package research/tail-sum-crossing-uniqueness --check-source-history

Local results: each verifier run printed PASS_TAIL_SUM_CROSSING_UNIQUENESS in under a second; research_package.py printed PASS_REGISTERED_RESEARCH_PACKAGE … outcome=completed; check_research_commit.py printed PASS_RESEARCH_COMMIT_BYTES. Add --json to print the certificate.

The checker rebuilds the bill from the pinned research/balanced-query-tree/RESULT.json (19 calls of the local profiles, 16 × 104 thin queries, main targets and whole idle jobs). It checks 14214 fields, 70523 paid calls and surplus 808, the Descartes budget 13, the single tail-sum sign change and F(0) < 0 < F(1). As a consistency control only, it runs a 60-digit decimal sign check at t = 0.99685 and t = 0.99686. Three rejection controls fail closed: the two-root invoice 6 − 4·2^t − 3·3^t + 2·6^t, the reversed invoice, and the bill without its rank-153 targets.

Limitations

• Only invoice arithmetic is proved. The construction, native maps, compiler, outer ledger and the five retained framework assumptions are untouched.
• The crossing's location is the source's enclosure; it is not recomputed here.
• The classical lemma is cited; there is no proof-kernel check. Independent human and formal proof-kernel review are unclaimed.

Open next questions

1. Does this rule settle invoices that scripts/compare_power_variation.py reports as inconclusive? That script's Descartes budget also counts roots at t ≤ 0. First step: run certify() on those invoices. This PR does not modify the shared script.
2. Is there a valid count for (0,1) alone? Tail sums of c_r r bound roots on t > 1, but subtracting two upper bounds gives no bound.

Related open work

• Prove the exact nineteen-leaf query-tree topology optimum (no κ) #6380 studies the nineteen-leaf query-tree topology. It concerns the tree shape, not root counting; I computed no shared number.
• Compose native input lifts with the query tree at conditional κ 0.00314980 #6350 composes native input lifts with the query tree, which gives a different bill. This certificate is not applied to it.
• Prove exact signed idle completion for overlapping stage outputs (no κ) #6391 concerns idle completion for overlapping stage outputs on the same caller family. It does not price the moment's root count.

Credit

Produced by Nick Petalas. The invoice and source pricing are Ali Romman's #6348, with the predecessor credits recorded there. The general-rank Descartes certificate is Ali Romman's #5788. Laguerre's rule is classical.

CI notes

CI has not certified this package. In run https://github.com/CrocSwap/integer-mult-bounds/actions/runs/41671056402 at head 001cf13d783978714563971bc2657f5abf4a574c, the plan job reached the 10-minute job timeout and was cancelled during the repository-wide "Onboarding helper tests" step (16:08:18Z to 16:18:40Z). Checkout and setup-python succeeded. The log shows those tests passing up to the cancellation, with no test error. Package selection and the package replay never ran (replay was skipped). This PR changes only files under research/tail-sum-crossing-uniqueness, which that step does not test. Another run from the same period (41670552412) was cancelled the same way, so this looks like runner slowness. I pushed no commit to retrigger CI.

The local checks listed under Evidence and reproduction passed at this commit.