Foundation Dalembert Right Affine From Factorization Polynomial Consistency Impl

Right-affine as a consequence

A function is affine when it graphs as a straight line. The Recognition Science framework works with a two-variable combiner P(u,v) that describes how two recognition costs combine. The declaration polynomial_consistency_implies_right_affine proves that if P is a symmetric quadratic polynomial, and the underlying cost function F satisfies the framework's standard hypotheses, then for each fixed first argument u, the function v ↦ P(u,v) is affine in v. In plain terms, once you assume P has polynomial shape, its behavior in the second variable is forced to be linear.

The proof is short and algebraic. The framework's Inevitability module already proves that under its hypotheses, a symmetric quadratic polynomial P must have the form P(u,v) = 2u + 2v + c·uv for some constant c. Substituting that form, for fixed u the expression becomes (2 + c·u)·v + 2u, which is visibly affine in v with slope (2 + c·u) and intercept 2u. The declaration simply packages this observation as a theorem.

This matters because an earlier module, FactorizationForcing.lean, assumed right-affineness as a hypothesis in its gate structure. This declaration shows that under the stronger assumption that P is polynomial, that hypothesis is redundant. It is a consequence, not an independent premise. The framework also builds the full gate from polynomial consistency alone, and proves the main RCL forcing theorem without separately assuming right-affineness.

What the declaration does not claim is that P must be polynomial in the first place. Starting only from continuity or smoothness of F, deriving that P has polynomial shape remains open. That is a separate, genuinely open step involving classical Aczél theory on functional equations. The framework's own summary flags this explicitly. There is also an alternative route, rcl_without_gate, that avoids the polynomial question entirely by using surjectivity of the cost function J, but that is a different theorem with different hypotheses.

/-- Right-affine follows from polynomial consistency with a cost functional `F`.

This theorem takes the Inevitability hypotheses (F normalized, consistent with
a symmetric quadratic polynomial P, non-trivial, continuous) and concludes
that P is right-affine. -/
theorem polynomial_consistency_implies_right_affine
    (F : ℝ → ℝ) (P : ℝ → ℝ → ℝ)
    (hNorm : Inevitability.IsNormalized F)
    (hCons : Inevitability.HasMultiplicativeConsistency F P)
    (hPoly : ∃ (a b c d e f : ℝ), ∀ u v, P u v = a + b*u + c*v + d*u*v + e*u^2 + f*v^2)
    (hSymP : ∀ u v, P u v = P v u)
    (hNonTriv : ∃ x : ℝ, 0 < x ∧ F x ≠ 0)
    (hCont : ContinuousOn F (Set.Ioi 0)) :
    ∀ u, ∃ α β, ∀ v, P u v = α * v + β := by
  obtain ⟨c, hc, _⟩ := Inevitability.bilinear_family_forced F P hNorm hCons hPoly hSymP hNonTriv hCont
  exact bilinear_implies_right_affine P c hc

/-- Bilinear form implies right-affine. -/
theorem bilinear_implies_right_affine
    (P : ℝ → ℝ → ℝ) (c : ℝ)
    (h_bilinear : ∀ u v, P u v = 2*u + 2*v + c*u*v) :
    ∀ u, ∃ α β, ∀ v, P u v = α * v + β := by
  intro u
  refine ⟨2 + c*u, 2*u, ?_⟩
  intro v
  rw [h_bilinear u v]
  ring

/-- **Main Theorem of this Module**: RCL follows from polynomial consistency
without separately assuming right-affine.

This closes Gap 4 in the direction of: if we're willing to assume P is
polynomial, then right-affine is a theorem, not a hypothesis. -/
theorem polynomial_consistency_forces_rcl
    (F : ℝ → ℝ) (P : ℝ → ℝ → ℝ)
    (hNorm : Inevitability.IsNormalized F)
    (hCons : Inevitability.HasMultiplicativeConsistency F P)
    (hPoly : ∃ (a b c d e f : ℝ), ∀ u v, P u v = a + b*u + c*v + d*u*v + e*u^2 + f*v^2)
    (hSymP : ∀ u v, P u v = P v u)
    (hNonTriv : ∃ x : ℝ, 0 < x ∧ F x ≠ 0)
    (hCont : ContinuousOn F (Set.Ioi 0))
    (hP11 : P 1 1 = 6)
    (hP0 : ∀ u, P u 0 = 2 * u) :
    ∀ u v, P u v = 2 * u * v + 2 * u + 2 * v :=
  gate_forces_rcl P
    (gate_from_polynomial_consistency F P hNorm hCons hPoly hSymP hNonTriv hCont hP11 hP0)

What this page does not claim

The declaration does not prove that P must be polynomial from continuity or smoothness of F. The declaration does not claim right-affineness follows without any assumption on P's form. The declaration does not address the alternative gate-free route via surjectivity of J.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Foundation/DAlembert/RightAffineFromFactorization.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

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