IndisputableMonolith.Foundation.DAlembert.Counterexamples
This module supplies explicit quadratic counterexamples showing that symmetry, normalization, C² regularity, calibration, and existence of some combiner P fail to force the d'Alembert structure or RCL. It defines forms such as Fquad and Hquad that meet the premises yet violate the interaction property F(xy) + F(x/y) ≠ 2F(x) + 2F(y). Researchers proving RCL inevitability cite it to justify the additional gates in the triangulated argument.
claimThere exist $C^2$ functions $F$ and combiners $P$ satisfying symmetry, normalization, and calibration such that $F(xy) + F(x/y) = 2F(x) + 2F(y)$ fails for some $x,y > 0$, so these conditions alone do not imply the Recognition Composition Law.
background
The module imports Cost and Cost.FunctionalEquation, the latter supplying lemmas for the T5 cost uniqueness proof. It operates inside the DAlembert hierarchy that tests minimal assumptions on the cost functional J before deriving the RCL $J(xy) + J(x/y) = 2J(x)J(y) + 2J(x) + 2J(y)$. The local setting is the search for necessary conditions that close loopholes in the forcing chain from T5 onward.
proof idea
This is a counterexample module. It defines quadratic forms Gquad, Fquad, Hquad together with supporting lemmas Fquad_on_exp, Fquad_symm, Hquad_simp, and Hquad_not_dAlembert; each lemma verifies that the listed assumptions hold while the interaction gate fails.
why it matters in Recognition Science
The module feeds NecessityGates and TriangulatedProof by exhibiting the explicit counterexample that demonstrates why symmetry plus normalization plus C² plus calibration plus some P is insufficient. It thereby motivates the four-gate structure (interaction, entanglement, curvature, d'Alembert) required for full RCL inevitability.
scope and limits
- Does not claim these conditions can never force RCL under stronger hypotheses.
- Does not address non-quadratic function classes.
- Does not incorporate curvature or entanglement gates.
- Does not connect to the phi-ladder or mass formula.