IndisputableMonolith.Foundation.DAlembert.Counterexamples
This module constructs explicit counterexamples showing that symmetry, normalization, C² smoothness, calibration, and existence of some combiner P do not force the d'Alembert or RCL structure. Researchers establishing the four-gate inevitability argument cite it to justify adding curvature and entanglement conditions. The module proceeds by defining quadratic forms such as Fquad and Gquad that meet the weak hypotheses yet violate the target functional equation.
claimCounterexamples demonstrate that symmetry + normalization + C² + calibration + existence of a combiner $P$ do not imply the d'Alembert structure $J(xy) + J(x/y) = 2J(x)J(y) + 2J(x) + 2J(y)$.
background
This module sits in the Foundation.DAlembert section, which develops the minimal gates needed to force the Recognition Composition Law from the T5 cost uniqueness result. It imports the Cost module and the FunctionalEquation helpers that supply lemmas for the T5 proof. The counterexamples target the weak conditions listed in NecessityGates: symmetry, normalization, C² regularity, calibration, and existence of some P, showing they permit non-interacting or non-hyperbolic solutions.
proof idea
This is a counterexample module. It defines quadratic forms (Gquad, Fquad, Hquad) together with auxiliary lemmas (Fquad_on_exp, Fquad_symm, Hquad_not_dAlembert) that verify the forms satisfy the weak conditions while failing the full RCL or d'Alembert property.
why it matters in Recognition Science
The module supplies the counterexample referenced in NecessityGates, which states that (symmetry + normalization + C² + calibration + existence of some combiner P) does not force the d'Alembert/RCL structure. It is imported by CurvatureGate, EntanglementGate, FourthGate, NecessityGates, and TriangulatedProof to support the four-gate argument that additional conditions (constant nonzero curvature in the log metric and nonzero cross-derivative of P) are required for inevitability.
scope and limits
- Does not prove that the listed weak conditions suffice for the RCL.
- Does not address the curvature or entanglement gates.
- Does not classify all possible combiners P.
- Does not claim uniqueness of J without the full gate set.