[petalas]

Result

Scoped proof with an exact checker. No new κ, supplier or construction claim.

PR #5788's general-rank power-variation certificate (scripts/compare_power_variation.py, docs/research/recursive-power-variation.md) bounds all real roots of F(t)=Σ c_r r^t by the number V of coefficient sign changes. It spends that budget only at t=0, t=1 and inside (0,1). When roots lie at t<0 or t>1, the budget stays unspent and the report is inconclusive. That document's Open next question asks for a proof of such an outside-domain root.

This package certifies those roots exactly and subtracts them before the interior count:

• the sign of F as t→−∞ is the sign of the smallest-rank coefficient, and as t→+∞ the sign of the largest;
• F(k) is an exact rational at each integer k;
• each exact zero at a listed integer and each strict sign change between adjacent points on (−∞,0) or (1,∞) is a distinct root outside [0,1].

PR #5788's closure rule then runs on the reduced budget V − e_0 − e_1 − W. The proof is in the package README.

Example: 4 − 8·2^t − 3^t + 2·6^t = 2(2^t − 1/2)(3^t − 4) has V=2 and equal endpoint values −3, and #5788 reports inconclusive. An exact zero at t=−1 and a sign change on (1,2) use the whole budget, so the extended check proves the difference is uniformly lower on 0<t<1.

Evidence

research/power-variation-outside-roots/verify.py, about 13 s:

• two fixtures move from inconclusive under Certify all-exponent order for general-rank recursive invoices #5788 to uniformly_lower, with the stated witnesses;
• factored oracle census: every product of one to three factors q·a^t − p with a ∈ {2,3,5} and 15 thresholds p/q, 4,455 invoices in total. Roots are known in closed form. Interior membership, integer roots and the sign at t=1/2 are exact integer tests. Results:
  • Certify all-exponent order for general-rank recursive invoices #5788 certifies 391;
  • this extension certifies 2,987, of which 2,596 are newly settled;
  • 0 false certificates, 0 budget overruns and 0 changed Certify all-exponent order for general-rank recursive invoices #5788 verdicts.
• failure controls: two mutations are rejected by the same oracle.
  • Swapped asymptotic signs: 887 false certificates, 2,026 budget overruns and 94 changed verdicts.
  • Double-counting zero endpoints: 463 false certificates, 491 overruns and 104 changed verdicts.

The verifier fails closed without assert, so it also checks under -O. The shared script is imported unchanged and pinned as a live input.

Reproduce from the repository root. The local runs used CPython 3.12.14 at ~/git/integer-mult-bounds/.venv/bin/python3.12.

python3.12 -B research/power-variation-outside-roots/verify.py
python3.12 -I -O -B research/power-variation-outside-roots/verify.py
python3.12 -B scripts/research_package.py --package research/power-variation-outside-roots
python3.12 -B scripts/check_research_commit.py --package research/power-variation-outside-roots --check-source-history

All four passed locally. The registered check reported PASS_REGISTERED_RESEARCH_PACKAGE ... outcome=completed, and the commit check reported PASS_RESEARCH_COMMIT_BYTES.

Limitations

• This change covers invoice arithmetic only. It certifies no construction, source extraction, native operator, compiler endpoint, supplier admission or κ, and the selected result is unchanged.
• The census is a finite exact check of the implementation. It does not prove completeness.
• Two outside roots in the same unit interval are not witnessed. The report then stays inconclusive, which is sound.
• Touching (even-multiplicity) interior roots remain unresolved, as in Certify all-exponent order for general-rank recursive invoices #5788.
• scripts/compare_power_variation.py is not modified; folding the rule into it is a maintainer decision.
• No source invoice from the selected ledger was run.
• Independent human and formal proof-kernel review are unclaimed.

Open next questions

• Finer outside brackets: extend directed enclosures to r^(k+s) so that two roots in one outside unit interval can be separated. Test case: (2^t − 5/2)(3^t − 4), with both roots in (1,2).
• Run the extension on the pinned P104 ledger exported by the selected-invoice tool (Export the selected P104 recursive invoice from authenticated source #5735) and on its declared edits. Does any committed invoice that Certify all-exponent order for general-rank recursive invoices #5788 leaves inconclusive become certified?
• Find an exact certificate for touching interior roots in general-rank exponential sums.

Related open work

No open PR studies the power-variation root budget or the #5788 comparator. Searches covered the title and body terms "power variation", "outside-domain", "root budget" and "Descartes". Rechecked after CI: #5977 (plane-port clique savings) and #5978 (address-twist syzygies), opened since this claim, study different objects and compute no shared quantity.

CI notes

All checks passed on head cf1504a968ab242c99c0907a722593b490ce8a39: plan and replay (research/power-variation-outside-roots), run https://github.com/CrocSwap/integer-mult-bounds/actions/runs/40564990746.

Credit

The root budget, endpoint multiplicities, interior brackets, invoice schema and directed power engine come from Ali Romman's #5788 and #5746 and their credited predecessors: zxv's signed-invoice interface, woklerx's T16 unique-crossing proof and mightyOil's moment-order audit. The generalized Descartes rule is classical. The outside-domain witness rule, its proof, the oracle census and the controls are new and were produced by Nick Petalas.