reparam_diagonal_not_degree_two
No real coefficient c exists such that the reparameterized analytic combiner diagonal equals any degree-two RCL diagonal for every local coordinate s. Workers on the Recognition Science functional equation would cite this to refute the claim that analyticity at the origin forces polynomial degree at most two. The proof assumes the existence of c, evaluates the equality at s=1 and s=2 to produce two inconsistent linear equations for c, and obtains a contradiction by linear arithmetic.
claimThere does not exist a real number $c$ such that for all real $s$, $4s + 18s^2 + 16s^3 + 4s^4 = 4(s + s^2) + c(s + s^2)^2$.
background
The AnalyticCounterexample module shows that the Phase 6 conjecture, which posits that a real-analytic combiner at the origin must have polynomial degree at most two, is false. It starts from the RCL variable $K = cosh(t) - 1$ and applies the analytic reparameterization $a = s + s^2$. This produces the diagonal $4s + 18s^2 + 16s^3 + 4s^4$ in the local coordinate $s$ (MODULE_DOC). The degree-two family with boundary condition $Phi(a,0) = 2a$ has diagonal $4(s + s^2) + c(s + s^2)^2$ in the same coordinate. Upstream definitions supply the explicit forms of these diagonals, while structures from ledger factorization and phi-forcing calibrate the underlying J-cost.
proof idea
The proof proceeds by contradiction. Assume there exists $c$ such that the two diagonals agree for all $s$. Specialize the equality to the points $s=1$ and $s=2$. After unfolding the definitions and normalizing the arithmetic, the first specialization gives the equation $42 = 8 + 4c$ while the second gives $168 = 24 + 36c$. Linear arithmetic on these two equations yields an immediate contradiction because they force incompatible values for $c$.
why it matters in Recognition Science
This result is invoked directly by reparam_coefficients_obstruct_degree_two, analytic_reparameterization_not_counted_once, and analytic_composition_not_enough. The last two theorems conclude that analytic reparameterization alters the polynomial degree of the combiner and that analytic composition is insufficient to enforce the degree-two RCL form. In the Recognition Science framework it supplies the algebraic core of the counterexample to the corrected Phase 6 conjecture, confirming that the RCL composition law alone does not restrict analytic diagonals to quadratic polynomials. It leaves open the question of which additional regularity conditions would restore the polynomial structure.
scope and limits
- Does not address the behavior of the combiner away from the local coordinate origin.
- Does not incorporate the full Recognition Composition Law beyond the diagonal.
- Does not connect to physical constants such as the fine-structure constant or the phi-ladder.
- Does not prove existence of the reparameterized combiner, only its diagonal obstruction.
Lean usage
theorem analytic_reparameterization_not_counted_once : ¬ ∃ c : ℝ, ∀ s : ℝ, reparamDiagonal s = degreeTwoDiagonal c s := reparam_diagonal_not_degree_twoformal statement (Lean)
38theorem reparam_diagonal_not_degree_two :
39 ¬ ∃ c : ℝ, ∀ s : ℝ, reparamDiagonal s = degreeTwoDiagonal c s := by
proof body
Term-mode proof.
40 intro h
41 rcases h with ⟨c, hc⟩
42 have h1 := hc 1
43 have h2 := hc 2
44 unfold reparamDiagonal degreeTwoDiagonal at h1 h2
45 norm_num at h1 h2
46 -- h1: 42 = 8 + 4c, so c = 17/2.
47 -- h2: 168 = 24 + 36c, so c = 4.
48 linarith
49
50/-- Equivalently, no coefficient `c` makes
51`4s + 18s^2 + 16s^3 + 4s^4 = 4(s+s^2) + c(s+s^2)^2` as a function of `s`. -/