IndisputableMonolith.Foundation.LogicAsFunctionalEquation.QuarticLogCounterexample
This module constructs the quartic-log comparison as an explicit counterexample to degree bounds on analytic combiners satisfying the Recognition Composition Law. Researchers examining analytic extensions of the RCL cite the construction. It proceeds by reparameterizing the J-cost coordinate and verifying algebraic deviations from the RCL family via direct comparison.
claimThe quartic-log comparison is the object obtained by substituting the reparameterization $f(s)=s+s^2$ into the cost coordinate $K=\\cosh(t)-1$, producing a quartic term outside the RCL family.
background
The module resides in the LogicAsFunctionalEquation section and imports DirectProof, which isolates the finite pairwise polynomial closure hypothesis as the regularity condition needed to force the Recognition Composition Law family. The RCL is the functional equation $J(xy)+J(x/y)=2J(x)J(y)+2J(x)+2J(y)$ where $J(x)=\cosh(\log x)-1$. DirectProof supplies the operative comparison wrapper around the translation theorem.
proof idea
The module defines quarticLogComparison together with quarticCombiner and its symmetric and nonnegativity variants. It proves the quartic combiner lies outside the RCL family by algebraic verification of the defining equation. Symmetry and nonnegativity on nonnegative inputs are shown by term rewriting and case analysis on the reparameterized coordinate.
why it matters in Recognition Science
This module supplies the algebraic core fed into AnalyticCounterexample, which demonstrates that the corrected Phase 6 conjecture (real-analytic combiner at the origin implies polynomial degree ≤2) is false. It exhibits an explicit quartic deviation from the RCL family in the Recognition framework.
scope and limits
- Does not establish the full analytic reparameterization counterexample; that is completed in the downstream AnalyticCounterexample module.
- Does not address physical constants, the phi-ladder, or spatial dimension D=3.
- Does not claim that every quartic function violates the RCL; the counterexample is specific to the log-reparameterized form.