Foundation Coherence Exponent Uniqueness Coherence Exponent Eq 5

The coherence exponent

A coherence exponent is a whole number that appears in the Recognition Science framework's account of how a discrete ledger of events, a record that keeps one entry per recognized event, scales with the number of dimensions it lives in. The framework's library, a machine-checked collection of formal theorems, defines two separate routes to this exponent and proves that they agree in exactly one case. The first route, the Fibonacci deficit, counts how far the number of cells in a D-dimensional grid, 2^D, falls short of the dimension itself, so k_fib(D) = 2^D - D. The second route, the integration measure, simply adds two to the dimension, k_int(D) = D + 2.

For one dimension the two routes give 1 and 3, for two dimensions 2 and 4, for four dimensions 12 and 6. Only at three dimensions do they both return 5, since 2^3 - 3 = 5 and 3 + 2 = 5. The theorem exponent_unique_at_D3 states this agreement is unique within the checked range: for D in the set {1, 2, 3, 4}, the two routes agree if and only if D equals 3. The declaration coherenceExponent_eq_5 records the outcome as a definitional fact: the coherence exponent is 5. The library also certifies the whole package in a single structure, CoherenceExponentCert, that bundles the agreement, the disagreement at the other dimensions, and the uniqueness statement, all checked with no gaps and no extra axioms.

Within the framework, this value of 5 is not an isolated number. The same library connects it to the golden ratio phi, the classical proportion where a whole relates to its larger part as the larger part relates to the smaller, by setting the reduced Planck constant hbar to phi^(-5) in framework-native units. It also links the exponent to an Einstein coupling kappa = 8 * phi^5, tying the exponent to an eight-tick period, the framework's unit cycle of recognition events. These connections are part of the framework's internal chain, not claims about measured physics.

What the declaration does not claim is just as important as what it proves. It does not assert that three dimensions are physically forced; the theorem is about a counting agreement within the framework's own definitions, not about the world. It does not claim that the coherence exponent is 5 for any dimension outside the checked set {1, 2, 3, 4}. And it does not say that the two routes are the only possible ways to define such an exponent. The uniqueness is relative to these two definitions and this finite range, nothing more.

/-- 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

theorem coherenceExponent_eq_5 : coherenceExponent = 5 := rfl

/-- Both equal 5 at D = 3. -/
theorem both_equal_5_at_3 : k_fib 3 = 5 ∧ k_int 3 = 5 := by decide

def coherenceExponentCert : CoherenceExponentCert where
  agree_at_3 := agreement_at_3
  both_five := both_equal_5_at_3
  disagree_1 := disagreement_at_1
  disagree_2 := disagreement_at_2
  disagree_4 := disagreement_at_4
  unique_at_3 := exponent_unique_at_D3
  k5_forced := k5_forced_at_D3

What this page does not claim

The theorem does not prove that three spatial dimensions are physically necessary; it proves a counting agreement within the framework's definitions. The uniqueness statement covers only the dimensions 1, 2, 3, and 4, not all natural numbers. The two routes are not claimed to be the only possible definitions of a coherence exponent.

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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