Foundation Dalembert Necessity Gates Fquad Additive

The additive branch

The d'Alembert equation, written by Jean le Rond d'Alembert in 1747, describes waves that keep their shape as they travel. In Recognition Science, a similar equation governs how the cost of comparing two things changes when you compose comparisons. The framework asks: if the cost of a combined comparison is the sum of the individual costs, what cost function can satisfy that rule? The answer, derived by hand and then checked by machine, is a quadratic in the logarithm: F(x) = (log x)² / 2.

This additive branch is internally consistent. The machine-checked library of formal theorems proves the lemma Fquad_additive: for any positive x and y, the cost of the product comparison plus the cost of the quotient comparison equals twice the cost of x plus twice the cost of y. In symbols, F(xy) + F(x/y) = 2F(x) + 2F(y). That identity holds for every positive pair, with no exceptions. The branch is a fully working solution to the composition law under the additivity assumption.

But the additive branch fails a test that the real cost function passes. The framework defines a property called interaction: a cost has interaction if there exists at least one pair of comparisons whose combined cost is not purely additive in the two individual costs. The quadratic-log function has no interaction at all, by the very identity above. The machine-checked theorem Fquad_noInteraction states this failure directly: no such pair exists for F(x) = (log x)² / 2.

In Recognition Science, the actual cost function J(x) = (x + 1/x)/2 - 1 does have interaction. The theorem Jcost_hasInteraction exhibits a concrete witness: x = y = 2 gives J(4) + J(1) = 9/8, while 2J(2) + 2J(2) = 1, and those are not equal. So the additive branch is not merely unusual; it is ruled out by a gate that the true cost passes. The framework's necessity gates exist precisely to separate branches like this one from the forced structure.

What the declaration does not claim is broader. It does not say the additive branch is impossible as a mathematical object; the lemma proves it is perfectly consistent. It does not say the quadratic-log function fails any other test, only the interaction gate. And it does not, by itself, prove that J is the unique cost function; that larger claim lives elsewhere in the framework's chain of theorems. The additive branch is a counterexample that sharpens the boundary: it shows why one extra condition, non-additivity, is needed to force the real structure.

lemma Fquad_additive (x y : ℝ) (hx : 0 < x) (hy : 0 < y) :
    Fquad (x * y) + Fquad (x / y) = 2 * Fquad x + 2 * Fquad y := by
  -- This is exactly `Fquad_consistency` with `Padd u v = 2u+2v`.
  simpa [Padd] using (Fquad_consistency x y hx hy)

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 additive branch is mathematically impossible; it is a consistent solution that fails one specific gate. The quadratic-log function fails any test other than the interaction gate. Fquad_additive alone proves that J is the unique cost function; that conclusion requires the full forcing chain.

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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