Foundation Hierarchy Forcing Scale Perturbed Family Injective

The perturbation test

A hierarchy is a sequence of levels, each a positive real number. In the Recognition Science framework, a ledger (a discrete record of events) can carry such a hierarchy, and the framework asks which hierarchies are possible when no parameters are free to be tuned. The declaration scale_perturbed_family_injective is a machine-checked theorem that supports this inquiry by proving a precise fact about a specific kind of perturbation.

The perturbation in question is simple. Take any positive level sequence. Choose a position j and a real number t. The perturbed sequence leaves all levels at or below position j unchanged, and multiplies every level above position j by exp(t). This changes exactly one adjacent ratio, the ratio at position j, by the factor exp(t), while preserving all other ratios and leaving every level positive. The theorem states that different values of t always yield different perturbed sequences. The proof is short: at position j+1, the value is levels(j+1) times exp(t), and since the exponential function is injective, distinct t values cannot produce equal sequences. This is pure real arithmetic, with no additional axioms.

Why does this matter? The perturbation machinery exists to test a conjecture about hierarchies. The framework's broader goal is to show that a nontrivial hierarchy with no free scale parameters must have uniform adjacent ratios, meaning every level is a fixed multiple of the one below it. The injectivity result is a necessary technical ingredient for that argument: it ensures that the perturbation genuinely varies the level sequence, so that if the zero-parameter condition rules out all such variations, the conclusion about uniformity is not vacuous. The theorem itself does not establish uniformity; it establishes that the perturbation is a faithful probe, one that can distinguish different parameter choices.

In Recognition Science, this result is part of a chain leading to the golden ratio and to the framework's account of physical constants. But the declaration's own scope is narrow: it is a lemma about exponential scaling, not a statement about physics. It does not claim that any actual hierarchy is perturbed, nor that the golden ratio emerges from this theorem alone. It simply certifies that the perturbation family is injective, a fact that later theorems can rely on.

/-- Different perturbation parameters give different level sequences.
The key step: at position `j + 1` the values are `levels(j+1) · exp(t)`,
and `exp` is injective. -/
theorem scale_perturbed_family_injective (levels : ℕ → ℝ) (j : ℕ)
    (h_pos : 0 < levels (j + 1)) :
    Function.Injective (ScalePerturbed levels j) := by
  intro t₁ t₂ h
  have h_eval := congr_fun h (j + 1)
  unfold ScalePerturbed at h_eval
  rw [if_neg (by omega : ¬(j + 1 ≤ j)), if_neg (by omega : ¬(j + 1 ≤ j))] at h_eval
  have h_ne : levels (j + 1) ≠ 0 := ne_of_gt h_pos
  exact Real.exp_injective (mul_left_cancel₀ h_ne h_eval)

/-- Perturbed level sequence: shift all levels above position `j` by the
factor `exp(t)`. This changes the ratio at position `j` to `r_j · exp(t)`
while preserving all other ratios and positivity for every `t ∈ ℝ`. -/
noncomputable def ScalePerturbed (levels : ℕ → ℝ) (j : ℕ) (t : ℝ) (k : ℕ) : ℝ :=
  if k ≤ j then levels k else levels k * Real.exp t

/-- Different perturbation parameters give different level sequences.
The key step: at position `j + 1` the values are `levels(j+1) · exp(t)`,
and `exp` is injective. -/
theorem scale_perturbed_family_injective (levels : ℕ → ℝ) (j : ℕ)
    (h_pos : 0 < levels (j + 1)) :
    Function.Injective (ScalePerturbed levels j) := by
  intro t₁ t₂ h
  have h_eval := congr_fun h (j + 1)
  unfold ScalePerturbed at h_eval
  rw [if_neg (by omega : ¬(j + 1 ≤ j)), if_neg (by omega : ¬(j + 1 ≤ j))] at h_eval
  have h_ne : levels (j + 1) ≠ 0 := ne_of_gt h_pos
  exact Real.exp_injective (mul_left_cancel₀ h_ne h_eval)

What this page does not claim

The theorem does not establish that uniform scaling is forced; it only proves injectivity of the perturbation family. The theorem does not claim that any actual physical hierarchy is perturbed; it is a mathematical lemma. The theorem does not derive the golden ratio; that requires additional hypotheses such as the additive composition condition.

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

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