Foundation Dalembert Right Affine From Factorization Bilinear Implies Right Affi

The affine simplification

In mathematics, a function of two variables is called bilinear if, when you hold one variable fixed, the function is a straight line in the other variable. The Recognition Science declaration bilinear_implies_right_affine proves a specific instance of this: if a function P(u, v) has the exact form 2u + 2v + c·uv, where c is any constant, then for each fixed u, P(u, v) is a straight line in v. The slope of that line is 2 + c·u and the intercept is 2u. This is a theorem in the machine-checked library of formal theorems, meaning the proof has been verified step by step by a computer.

The result matters because of a larger argument in the framework. The framework's core forcing chain aims to show that a certain cost function, which measures the price of recognition events, must have a unique form. A key step in that chain involves a combiner function P that takes two costs and produces a new cost. Earlier work had assumed as a hypothesis that P is right-affine, meaning it is a straight line in its second argument. This declaration removes that assumption in one special case: if P is known to be a symmetric quadratic polynomial, then right-affineness follows automatically as a theorem, not as an extra assumption.

In Recognition Science, the framework models how a discrete record of events, called the ledger, updates when two events are combined. The combiner P is the rule for that update. The declaration bilinear_implies_right_affine is the algebraic core of a larger result, polynomial_consistency_implies_right_affine, which states that if P is a symmetric quadratic polynomial and the cost function F satisfies the standard normalization, consistency, non-triviality, and continuity conditions, then P is right-affine. This means the right-affine hypothesis in the original gate is redundant under the stronger assumption that P is polynomial.

What the declaration does not claim is equally important. It does not prove that P must be a polynomial in the first place. That step, deriving polynomial form from continuity or smoothness of P, remains open in the formal library. The declaration also does not prove the full RCL forcing result on its own; it only establishes the affine property for a polynomial of a specific shape. The stronger gate-free result, which proves P equals 2uv + 2u + 2v without any polynomial or right-affine assumption, lives in a separate module and uses a different argument based on the surjectivity of the cost function J.

The practical consequence is that the framework's forcing chain does not rest on an unproved right-affine assumption in the polynomial case. If one is willing to assume P is polynomial, right-affineness is a proved consequence, not a hypothesis. This tightens the logical structure of the framework and clarifies which assumptions are genuinely needed. The remaining open question, whether polynomial form itself can be derived from smoothness, is a separate concern that does not affect the validity of the gate-free path to the same conclusion.

/-- 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

/-- 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

/-- Build `FactorizationAssociativityGate` from polynomial consistency.

This packages: symmetric (from the symmetric polynomial P), zeroBoundary
(supplied as hypothesis, derivable from F's normalization via
`symmetry_and_normalization_constrain_P`), unitDiagonal (supplied as
calibration hypothesis), and right-affine (derived via bilinear_family_forced). -/
theorem gate_from_polynomial_consistency
    (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) :
    FactorizationAssociativityGate P :=
  { symmetric := hSymP
    rightAffine := polynomial_consistency_implies_right_affine F P
      hNorm hCons hPoly hSymP hNonTriv hCont
    zeroBoundary := hP0
    unitDiagonal := hP11 }

What this page does not claim

This declaration does not prove that P must be a polynomial in the first place. This declaration does not prove the full RCL forcing result on its own. This declaration does not cover the gate-free path that uses surjectivity of J instead of polynomial assumptions.

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)

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