IndisputableMonolith.Foundation.LogicAsFunctionalEquation.QuarticLogCounterexample
The QuarticLogCounterexample module supplies the algebraic counterexample showing that real-analytic combiners at the origin need not have polynomial degree at most 2. Researchers examining regularity conditions for the Recognition Composition Law would cite this construction. It proceeds by reparameterizing the standard J-cost with a quadratic map and verifying the resulting quartic-log terms violate expected closure.
claimReparameterize the RCL variable $K = cosh(t)-1$ by the quadratic map $f(s)=s+s^2$ to obtain a real-analytic combiner whose quartic-log expansion lies outside the RCL family.
background
In the LogicAsFunctionalEquation setting the Recognition Composition Law is the functional equation $J(xy)+J(x/y)=2J(x)J(y)+2J(x)+2J(y)$ with $J(x)=cosh(log x)-1$. The upstream DirectProof module isolates the finite pairwise polynomial closure hypothesis as the precise regularity condition that forces the RCL family and proves the strict bi-affine subcase directly. This module introduces the quartic-log comparison as the explicit algebraic object that breaks the degree bound under analytic reparameterization.
proof idea
This is a definition module, no proofs. It assembles the quarticLogComparison and quarticCombiner objects together with their symmetric and non-negative variants, then records the auxiliary facts that the square-root term fails bilinearity and that the full combiner lies outside the RCL family.
why it matters in Recognition Science
The module supplies the algebraic core for the AnalyticCounterexample parent, which establishes that the corrected Phase 6 conjecture is false. It therefore sharpens the open question of the exact regularity conditions needed to force the RCL family and directly feeds the downstream claim that real-analyticity at the origin does not imply polynomial degree ≤2.
scope and limits
- Does not establish analyticity or convergence of the reparameterized combiner.
- Does not treat the original uncorrected Phase 6 conjecture.
- Does not supply numerical checks or plots of the counterexample.
- Does not extend the construction beyond the positive-ratio domain.