Foundation Dalembert Right Affine From Factorization Gate From Polynomial Consis

Polynomial consistency and the gate

The declaration gate_from_polynomial_consistency is a theorem in the framework's machine-checked library of formal theorems. It shows that a certain structural condition, called right-affineness, need not be assumed separately. If a function P is a symmetric quadratic polynomial, and another function F meets a set of standard hypotheses, then P is right-affine. Right-affine means that for each fixed first input, P behaves like a straight line in its second input: P(u, v) = αv + β for some α and β that may depend on u.

The theorem builds a complete gate, a collection of structural conditions, from polynomial consistency alone. The gate includes symmetry, zero boundary, unit diagonal, and the derived right-affine property. This matters because an earlier proof of the framework's core forcing result had assumed right-affineness as a hypothesis. This theorem shows that under the stronger assumption that P is a symmetric quadratic polynomial, that hypothesis is redundant: it follows as a theorem rather than being an independent assumption.

The main theorem of the module, polynomial_consistency_forces_rcl, goes further. Under the same hypotheses, it proves that P must equal the RCL polynomial 2uv + 2u + 2v. This is the same polynomial that the framework's central cost function J forces through other routes. The module also documents a gate-free alternative: if F equals the cost function J, then for any function P satisfying the consistency relation, P equals the RCL polynomial on the nonnegative quadrant, with no assumption on P's form at all.

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 shape from continuity or smoothness of P, remains open. The gate-free path avoids this question entirely, but the polynomial-consistency route explicitly assumes the polynomial form as a hypothesis. The theorem also does not claim to prove the full forcing chain from scratch; it relies on the Inevitability hypotheses and the bilinear family forcing result as inputs.

/-- 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

/-- **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

What this page does not claim

It does not prove that P must be a polynomial; that step remains open. It does not prove the full forcing chain from scratch; it relies on the Inevitability hypotheses and bilinear family forcing. It does not claim the gate-free route is the only alternative; the polynomial-consistency route is an additional path.

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: