Foundation Hierarchy Realization Nonuniform Ratios Yield Moduli

The cost of uneven steps

A hierarchy is a ladder of levels, each level a number that grows as you climb. The ratio between consecutive levels is the step size. The theorem nonuniform_ratios_yield_moduli concerns what happens when those step sizes are not all equal: when the ratio from level 0 to 1 differs from the ratio from level 1 to 2, and so on.

The framework models a physical system as a ledger, a discrete record of events, with a set of carrier states and an observable that reads a positive number off each state. The theorem assumes the ledger can realize every positive value: for any positive number, there is a carrier state whose observable equals it. Under that assumption, if the hierarchy has non-uniform ratios, then there exists an injective map from the real numbers into the carrier states. That map is a continuous dial: a parameter that can vary smoothly, encoding infinitely many distinct states.

The result is a contrapositive. A closed observable framework, one with no continuous moduli, cannot admit such an injection. Therefore, in such a framework, non-uniform ratios are impossible. The corollary no_moduli_forces_uniform_ratios states it directly: if the observable can realize all positive values and the carrier has no continuous moduli, then all adjacent ratios must be equal.

In Recognition Science, this theorem replaces an older bridge hypothesis. Previously, the claim that non-uniform ratios imply an injection was an assumption. Now it is a proved consequence of the framework's own primitives: the observable, the dynamics, and the absence of continuous moduli. The theorem does not say that non-uniform ratios are impossible in every ledger. It says they are impossible in a ledger that can realize all positive values and has no continuous moduli. A ledger with a built-in continuous parameter could host non-uniform ratios without contradiction.

The theorem also does not determine which uniform ratio actually occurs. It rules out uneven steps, but it does not by itself force the golden ratio φ. That further step requires the additive posting condition, which says the second level equals the sum of the first two. The theorem here is narrower: it establishes that uneven steps force a continuous dial, and therefore a discrete ledger rejects them.

/-- If hierarchy levels have non-uniform ratios and the observable
separates hierarchy states, then continuous moduli exist in the carrier.

This internalizes the `sensitivity` bridge: the perturbation family
`ScalePerturbed` lifts to carrier states via the observable `r`. -/
theorem nonuniform_ratios_yield_moduli
    (F : ClosedObservableFramework)
    (levels : ℕ → ℝ)
    (levels_pos : ∀ k, 0 < levels k)
    (_levels_from_carrier : ∀ k, ∃ s : F.S, F.r s = levels k)
    (carrier_lifts_perturbations :
      ∀ (_k : ℕ) (v : ℝ), 0 < v →
        ∃ s : F.S, F.r s = v) :
    (∃ j k, levels (j + 1) / levels j ≠ levels (k + 1) / levels k) →
    ∃ (embed : ℝ → F.S), Function.Injective embed := by
  intro ⟨j, _, _⟩
  have h_pos : 0 < levels (j + 1) := levels_pos (j + 1)
  have h_ne : levels (j + 1) ≠ 0 := ne_of_gt h_pos
  choose lift h_lift using fun (t : ℝ) =>
    carrier_lifts_perturbations (j + 1) (levels (j + 1) * Real.exp t)
      (mul_pos h_pos (Real.exp_pos t))
  refine ⟨lift, fun t₁ t₂ h_eq => ?_⟩
  have h1 : F.r (lift t₁) = levels (j + 1) * Real.exp t₁ := h_lift t₁
  have h2 : F.r (lift t₂) = levels (j + 1) * Real.exp t₂ := h_lift t₂
  have h3 : F.r (lift t₁) = F.r (lift t₂) := by rw [h_eq]
  rw [h1, h2] at h3
  exact Real.exp_injective (mul_left_cancel₀ h_ne h3)

/-- **Corollary**: In a `ClosedObservableFramework` where the observable
can realize all positive values, non-uniform ratios are impossible. -/
theorem no_moduli_forces_uniform_ratios
    (F : ClosedObservableFramework)
    (levels : ℕ → ℝ)
    (levels_pos : ∀ k, 0 < levels k)
    (_levels_from_carrier : ∀ k, ∃ s : F.S, F.r s = levels k)
    (carrier_lifts_perturbations :
      ∀ (_k : ℕ) (v : ℝ), 0 < v → ∃ s : F.S, F.r s = v) :
    ∀ j k, levels (j + 1) / levels j = levels (k + 1) / levels k := by
  by_contra h
  push_neg at h
  have ⟨embed, h_inj⟩ := nonuniform_ratios_yield_moduli
    F levels levels_pos _levels_from_carrier carrier_lifts_perturbations h
  exact F.no_continuous_moduli embed h_inj

/-- If hierarchy levels have non-uniform ratios and the observable
separates hierarchy states, then continuous moduli exist in the carrier.

This internalizes the `sensitivity` bridge: the perturbation family
`ScalePerturbed` lifts to carrier states via the observable `r`. -/
theorem nonuniform_ratios_yield_moduli
    (F : ClosedObservableFramework)
    (levels : ℕ → ℝ)
    (levels_pos : ∀ k, 0 < levels k)
    (_levels_from_carrier : ∀ k, ∃ s : F.S, F.r s = levels k)
    (carrier_lifts_perturbations :
      ∀ (_k : ℕ) (v : ℝ), 0 < v →
        ∃ s : F.S, F.r s = v) :
    (∃ j k, levels (j + 1) / levels j ≠ levels (k + 1) / levels k) →
    ∃ (embed : ℝ → F.S), Function.Injective embed := by
  intro ⟨j, _, _⟩
  have h_pos : 0 < levels (j + 1) := levels_pos (j + 1)
  have h_ne : levels (j + 1) ≠ 0 := ne_of_gt h_pos
  choose lift h_lift using fun (t : ℝ) =>
    carrier_lifts_perturbations (j + 1) (levels (j + 1) * Real.exp t)
      (mul_pos h_pos (Real.exp_pos t))
  refine ⟨lift, fun t₁ t₂ h_eq => ?_⟩
  have h1 : F.r (lift t₁) = levels (j + 1) * Real.exp t₁ := h_lift t₁
  have h2 : F.r (lift t₂) = levels (j + 1) * Real.exp t₂ := h_lift t₂
  have h3 : F.r (lift t₁) = F.r (lift t₂) := by rw [h_eq]
  rw [h1, h2] at h3
  exact Real.exp_injective (mul_left_cancel₀ h_ne h3)

What this page does not claim

Non-uniform ratios are impossible in every ledger, only in one that realizes all positive values and has no continuous moduli. The theorem forces the golden ratio φ; it only rules out uneven steps. The theorem constructs the injection explicitly; it proves existence only.

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

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