Foundation Dalembert Necessity Gates Has Interaction

The interaction gate

The d'Alembert equation, named for Jean le Rond d'Alembert's 1747 work on vibrating strings, describes wave propagation. In the Recognition Science framework, a related structure governs how the cost of recognition events combines. The central theorem of the framework proves that a cost function satisfying five conditions must equal J(x) = (x + 1/x)/2 - 1. But a counterexample shows those five conditions alone do not force the result: the function F(x) = (log x)²/2 also satisfies symmetry, normalization, smoothness, calibration, and the existence of some combination rule. This quadratic-log branch would make costs purely additive under composition, and it does not produce the framework's structure.

To rule out that branch, the framework adds a minimal gate called HasInteraction, meaning there exists at least one pair of positive comparisons whose combined cost is not purely additive in the two individual costs. In symbols, the gate requires that for some x, y > 0, F(x·y) + F(x/y) ≠ 2F(x) + 2F(y). The quadratic-log function fails this test: for every x and y, its combined cost is exactly additive, a fact proved as a theorem in the machine-checked library of formal theorems. The framework's own cost function J passes the gate, with the concrete witness x = y = 2, where J(4) + J(1) = 9/8 differs from 2J(2) + 2J(2) = 1.

The interaction gate is the weakest possible anti-quadratic condition: it merely asserts that additivity fails somewhere, without prescribing where or how. This weakness is deliberate. The gate does not claim that interaction must be large, that it must follow a particular functional form, or that it must hold for every pair of comparisons. It only excludes the purely additive branch, and it is satisfied by the framework's cost function. What the gate changes is the status of the inevitability claim: with it, the five original conditions plus non-additivity do force the d'Alembert structure; without it, the quadratic-log impostor remains a live alternative.

theorem Fquad_noInteraction : ¬ HasInteraction Fquad := by
  intro h
  rcases h with ⟨x, y, hx, hy, hneq⟩
  exact hneq (Fquad_additive x y hx hy)

theorem Jcost_hasInteraction : HasInteraction Jcost := by
  -- Concrete witness: x = y = 2.
  refine ⟨2, 2, by norm_num, by norm_num, ?_⟩
  -- Evaluate numerically: J(4) + J(1) ≠ 2J(2) + 2J(2).
  -- J(1)=0, J(2)=1/4, J(4)=9/8.
  norm_num [Jcost]

What this page does not claim

The gate does not force the framework's cost function by itself; it only excludes the quadratic-log branch when combined with the five base conditions. The gate does not require interaction for every pair of comparisons, only for at least one pair. The gate does not specify the magnitude or functional form of the non-additivity.

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/NecessityGates.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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