IndisputableMonolith.Foundation.LogicAsFunctionalEquation.AnalyticCounterexample
AnalyticCounterexample module defines the diagonal of the analytically reparameterized quartic-log combiner in the local coordinate s where a equals s plus s squared. Researchers studying functional equations for logic would cite this when testing whether continuous symmetric combiners can violate the affine structure needed for counted-once comparisons. The module organizes definitions of the reparameterized diagonal together with lemmas that compare it to degree-two forms and exhibit coefficient obstructions.
claimDiagonal of the reparameterized combiner $C(x,y)= (log(x/y))^4$ in the local coordinate $s$ satisfying $a=s+s^2$.
background
This module belongs to the LogicAsFunctionalEquation section of the Recognition Science foundation. It imports the quartic-log counterexample, which states that $C(x,y)=(log(x/y))^4$ admits a continuous symmetric combiner on the nonnegative range. The local coordinate change $a=s+s^2$ supplies an analytic reparameterization of that combiner. Sibling objects include the reparameterized diagonal, its degree-two counterpart, and lemmas establishing that the reparameterized form is not degree two while its coefficients obstruct the degree-two property. The downstream CountOnceComparison module requires any counted-once combiner to be affine in each argument separately, taking the explicit form $a + b u + c v + d u v$.
proof idea
This is a definition module, no proofs. It introduces the reparameterized diagonal via the coordinate $s$ with $a=s+s^2$, defines the corresponding degree-two diagonal, and records the two lemmas that the reparameterized diagonal fails to be degree two and that its coefficients obstruct the degree-two property.
why it matters in Recognition Science
The module supplies the analytic counterexample that feeds the CountOnceComparison module. That parent formalizes the requirement that each constituent comparison is counted once, which algebraically forces the combiner to be affine in each variable separately. It realizes the algebraic heart of the counterexample from the Logic Functional Equation paper by exhibiting an analytic reparameterization whose diagonal fails the degree-two condition.
scope and limits
- Does not claim uniqueness of the quartic-log form among continuous symmetric combiners.
- Does not address the Recognition Composition Law or J-uniqueness.
- Does not derive spatial dimensions, the eight-tick octave, or the phi-ladder.
- Does not connect to mass formulas, Berry thresholds, or the alpha band.