[aliromman]

A concave-order screen can be inconclusive even when a proposed invoice is worse for every power tariff. This adds an exact comparator for differences whose uncancelled ranks are powers of one integer. With x=b^t, it classifies the signed polynomial over the whole open interval1<x<b. It distinguishes strict order, isolated equality and actual crossings, including repeated and endpoint roots.

This completes the supported all-exponent question; it does not sample t. For example, the inconclusive difference1+4^t-22^t is strictly positive for every0<t<1. The related difference9+44^t-122^t has one interior equality, while22^t-3 crosses zero. Paid terms and target debits use the existing strict invoice schema, with exact input-byte hashes.

python3 -B scripts/compare_power_polynomial.py OLD.json NEW.json
python3 -B -m unittest discover -s tests -p 'test_research_power_polynomial.py' -v

Implementation, proof and limits, independent tests.

Validation: all343 quadratics with coefficients-3..3 agree with an independent extrema/monotone-interval oracle;288 factored polynomials check known multiplicities and endpoints. Further controls cover irrational roots, degree29, target debits, cancellation, large counts and unsupported inputs. Seven standard-library tests pass on Python3.14.7 and registered3.12.14 with isolated optimized execution. An optional SymPy1.14.0 root-isolation oracle checks64 more cases, including24 dense degree29 polynomials near the count limit; all eight tests pass in that temporary environment. The optional oracle is skipped when SymPy is absent; the tool has no such dependency. All ten existing comparator tests and git diff --check pass. A declared edit of the complete selected invoice also demonstrates the inconclusive-to-strict transition.

This is classical Sturm theory with an invoice reduction, not a new root-count theorem. Credit Philippe Pebay, J. Maurice Rojas and David C. Thompson's endpoint treatment, and the existing comparator, zxv, mightyOil's T16 audit and credited predecessors. PR5732 handles arbitrary ranks at one rational exponent; PR5731 provides concave witnesses. They are complementary, unmerged dependencies are not imported, and their source files are unchanged. New implementation, proof and checks: Ali Romman with substantial OpenAI Codex assistance, Apache-2.0.

Scope: ranks must share an integer base after cancellation. The existing rank limit10^9 bounds degree by29. This checks declared arithmetic; it does not authenticate a circuit, detect a missing source charge, prove ownership/native legality or admit a compiler or kappa. No selected-frontier change. Independent human/kernel review and full selected theorem replay were not run. Full-history preflight is unavailable in this shallow checkout. Hosted CI is pending; existing main navigation failures are already covered by PR5659/5703.

Base df0137e, priority7, selected conditional.00310045. Canonical main and overlap were refreshed before work and submission; latest search covered419 open PRs,5,094 Discussion records and all7,676 main notes, with only this scope matching. Canonical claim.