Foundation Coherence Exponent Uniqueness Disagreement At 1

The first disagreement

In the Recognition Science framework, a central quantity is the coherence exponent, a natural number that characterizes how a system's structure scales with dimension. The framework considers two independent routes for computing this exponent. The first route, the Fibonacci deficit, is defined as k_fib(D) = 2^D - D. The second route, the integration measure, is defined as k_int(D) = D + 2. The declaration disagreement_at_1 is a machine-checked theorem stating that these two routes give different values when the dimension D equals 1: k_fib(1) = 1 and k_int(1) = 3, so 1 ≠ 3.

This disagreement is not a flaw but a feature. The framework's library of formal theorems proves that the two routes also disagree at D = 2 and D = 4, with values (2, 4) and (12, 6) respectively. They agree only at D = 3, where both equal 5. The master theorem exponent_unique_at_D3 states that, among the dimensions {1, 2, 3, 4}, D = 3 is the unique dimension where the two routes agree. This uniqueness is what forces the coherence exponent to be exactly 5, a value that then connects to other framework constants such as the reduced Planck constant and the Einstein coupling.

The declaration disagreement_at_1 establishes a precise negative fact: the two routes are not identical in general. It is the first of three such disagreement proofs that, together with the agreement at D = 3, form the uniqueness argument. The theorem itself is a simple computation, proved by the 'decide' tactic, which means the Lean kernel verified it without any axioms or unresolved proofs. It does not, however, make any claim about dimensions beyond 4, nor does it assert that the coherence exponent is physically meaningful in those dimensions. It is a statement about the internal consistency of two formulas within the framework, not a claim about the physical world.

/-- Disagreement at D = 1. -/
theorem disagreement_at_1 : k_fib 1 ≠ k_int 1 := by decide

/-- Disagreement at D = 2. -/
theorem disagreement_at_2 : k_fib 2 ≠ k_int 2 := by decide

/-- Disagreement at D = 4. -/
theorem disagreement_at_4 : k_fib 4 ≠ k_int 4 := by decide

/-- D = 3 is the unique dimension in {1,2,3,4} where both routes agree. -/
theorem exponent_unique_at_D3 :
    ∀ D ∈ ({1, 2, 3, 4} : Finset ℕ), k_fib D = k_int D ↔ D = 3 := by
  decide

What this page does not claim

That the two routes disagree in all dimensions other than three, since the theorem only covers dimensions 1, 2, 3, and 4. That the coherence exponent has any physical meaning in dimensions other than three. That the agreement at D = 3 alone proves the physical existence of three spatial dimensions.

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/CoherenceExponentUniqueness.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.

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