reparam_coefficients_obstruct_degree_two
No real coefficient c makes the reparameterized diagonal 4s + 18s² + 16s³ + 4s⁴ coincide with the degree-two RCL form 4(s + s²) + c(s + s²)² for all s. Researchers examining analytic counterexamples to the Phase 6 conjecture would cite this to establish the coefficient obstruction. The proof is a one-line wrapper that feeds the pair (c, assumed equality) into the existential negation theorem reparam_diagonal_not_degree_two.
claimFor every real number $c$, the identity $4s + 18s^2 + 16s^3 + 4s^4 = 4(s + s^2) + c(s + s^2)^2$ fails to hold for all real $s$.
background
The module constructs an analytic counterexample to the claim that a real-analytic combiner at the origin must have polynomial degree at most two. It starts from the standard RCL variable $K = cosh(t) - 1$ and reparameterizes the cost coordinate by $f(s) = s + s^2$. In the new local coordinate $a = s + s^2$, the induced diagonal becomes the quartic $4s + 18s^2 + 16s^3 + 4s^4$ (reparamDiagonal).
proof idea
The proof assumes for contradiction that a fixed $c$ makes the two diagonals identical for every $s$, then feeds the pair $(c, h)$ directly into the upstream existential negation theorem reparam_diagonal_not_degree_two.
why it matters in Recognition Science
This theorem supplies the coefficient-wise obstruction that completes the algebraic core of the counterexample, showing that the reparameterized diagonal cannot arise from any degree-two RCL-family combiner with boundary form $Phi(a,0)=2a$. It thereby falsifies the corrected Phase 6 conjecture that real-analytic combiners imply polynomial degree ≤ 2. The construction relies on the Recognition Composition Law and the specific reparameterization $f(s)=s+s^2$.
scope and limits
- Does not address whether higher-degree polynomials can match the diagonal.
- Does not consider non-analytic or non-polynomial combiners.
- Does not prove the full analytic counterexample outside the diagonal restriction.
- Applies only to the specific reparameterization $f(s)=s+s^2$.
formal statement (Lean)
52theorem reparam_coefficients_obstruct_degree_two
53 (c : ℝ) :
54 ¬ (∀ s : ℝ, reparamDiagonal s = degreeTwoDiagonal c s) := by
proof body
Term-mode proof.
55 intro h
56 exact reparam_diagonal_not_degree_two ⟨c, h⟩
57
58end LogicAsFunctionalEquation
59end Foundation
60end IndisputableMonolith