IndisputableMonolith.Foundation.LogicAsFunctionalEquation.QuarticLogCounterexample
The QuarticLogCounterexample module supplies the algebraic definitions and verifications for a quartic-log combiner used to construct a counterexample in the Recognition Composition Law setting. Researchers examining regularity conditions for the RCL family in Phase 6 analysis would cite it. The module proceeds by defining the combiner and its symmetry and non-negativity properties, then performing direct algebraic checks that it lies outside the RCL family.
claimDefine the quartic combiner via reparameterization $f(s)=s+s^2$ of the standard RCL cost coordinate $K=\cosh t-1$; the resulting object satisfies symmetry and non-negativity on the positive domain but violates the RCL functional equation.
background
The module imports the DirectProof module, whose doc-comment states it supplies the paper-facing operative comparison wrapper around the translation theorem and isolates the finite pairwise polynomial closure hypothesis as the regularity condition needed to force the RCL family. The RCL itself is the functional equation $J(xy)+J(x/y)=2J(x)J(y)+2J(x)+2J(y)$ with $J(x)=\cosh(\log x)-1$. This module introduces the quartic-log comparison as the concrete algebraic object that tests the boundary of that family.
proof idea
This module consists of definitions (quarticLogComparison, quarticCombiner and its symmetric and non-negative variants) together with direct algebraic verifications; the key result quarticCombiner_not_rcl_family is obtained by explicit substitution into the RCL equation and term-by-term comparison showing mismatch.
why it matters in Recognition Science
The module is imported by AnalyticCounterexample, whose doc-comment states it proves the algebraic core of the counterexample showing that the corrected Phase 6 conjecture (real-analytic combiner at the origin implies polynomial degree ≤2) is false. It therefore supplies the concrete counterexample construction needed to delimit the precise regularity conditions that force the RCL family in the Recognition framework.
scope and limits
- Does not contain the full analytic reparameterization; that argument lives in the downstream AnalyticCounterexample module.
- Does not claim every quartic object violates RCL; only the specific log-based combiner defined here.
- Does not address the original Phase 6 conjecture before the correction mentioned in the downstream doc-comment.