Foundation Dalembert Right Affine From Factorization

Right-affine from factorization

In the Recognition Science framework, a central object is a ledger, a discrete record of events, whose costs combine through a function P(u,v). A natural question is whether P must be "right-affine": for each fixed u, P(u,v) must be a straight line in v. The module RightAffineFromFactorization.lean answers this question by showing that this property is not an independent assumption but a consequence of P being a symmetric quadratic polynomial.

The main theorem, polynomial_consistency_forces_rcl, states that if P is a symmetric quadratic polynomial and the cost function F satisfies the standard hypotheses (normalized, consistent with P, non-trivial, continuous), then P must equal the RCL polynomial 2uv + 2u + 2v. The proof is algebraic: it first shows that any such P is right-affine, then uses this to force the specific coefficients. This closes a gap in the framework's forcing chain, where right-affineness was previously taken as a hypothesis.

The module also offers a second, stronger route. The theorem rcl_without_gate, re-exposed from Unconditional.lean, proves the same conclusion without any assumption on P's form at all. If F equals the cost function J and satisfies the consistency relation F(xy) + F(x/y) = P(F(x), F(y)), then P equals the RCL polynomial on the non-negative quadrant. This path uses only the surjectivity of J and its intrinsic identity, bypassing the polynomial question entirely.

What remains genuinely open is deriving that P must be a polynomial in the first place, starting only from smoothness of F. This involves classical Aczél theory on functional equations and is not yet formalized. However, the gate-free route (path 2) does not require this step, so the core forcing claim does not rest on an unproved right-affine assumption.

/-- **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)

/-- **Gate-free RCL theorem (from Unconditional.lean, re-exposed here).**

If F = J and F has any consistency relation F(xy) + F(x/y) = P(F(x), F(y))
with some function P, then P equals the RCL polynomial on [0, ∞)². This holds
without any assumption on P's form (polynomial, right-affine, smooth, etc.).
-/
theorem rcl_without_gate
    (P : ℝ → ℝ → ℝ)
    (hCons : ∀ x y : ℝ, 0 < x → 0 < y →
      Cost.Jcost (x * y) + Cost.Jcost (x / y) = P (Cost.Jcost x) (Cost.Jcost y)) :
    ∀ u v : ℝ, 0 ≤ u → 0 ≤ v → P u v = 2*u*v + 2*u + 2*v :=
  Unconditional.rcl_unconditional P hCons

/-- 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

What this page does not claim

This module does not prove that P must be a polynomial; that remains open. This module does not prove the RCL polynomial is the unique cost function; it assumes the consistency relation with 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.

Derived articles

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: