IndisputableMonolith.Foundation.DAlembert.Counterexamples
This module supplies explicit counterexamples showing that symmetry, normalization, C² smoothness, calibration, and existence of some combiner P fail to force the d'Alembert or RCL structure. Researchers building the four-gate inevitability argument cite it to justify the remaining gates. The module consists of definitions of quadratic forms and related functions that meet the weak conditions but violate the target functional equation.
claimCounterexample functions $F_{ m quad}$, $H_{ m quad}$, and $P_{ m add}$ that satisfy symmetry, normalization, $C^2$, calibration, and existence of a combiner yet violate $F(xy)+F(x/y)=2F(x)F(y)+2F(x)+2F(y)$.
background
The module sits inside the DAlembert development that isolates the conditions needed for the Recognition Composition Law. It imports the Cost module and the FunctionalEquation helpers that supply lemmas for the T5 cost uniqueness proof. The setting assumes the J-cost functional equation and the phi-ladder conventions from the upstream Cost layer.
proof idea
This is a definition module, no proofs. It introduces concrete quadratic and additive forms (Gquad, Fquad, Padd, Hquad and their variants) that serve as counterexamples.
why it matters in Recognition Science
The counterexamples feed directly into NecessityGates, which states that symmetry plus normalization plus C² plus calibration plus some combiner P does not force the d'Alembert/RCL structure. They are also imported by CurvatureGate, EntanglementGate, FourthGate, and TriangulatedProof to complete the four-gate argument.
scope and limits
- Does not claim the listed conditions are sufficient for RCL.
- Does not construct counterexamples outside the Cost setting.
- Does not address the full four-gate closure.
- Does not provide numerical evaluations of the counterexamples.