Foundation Hierarchy Forcing Scale Perturbed Pos

The perturbation lemma

A ledger in the Recognition Science framework is a discrete record of levels, written as a sequence of positive real numbers. The declaration scale_perturbed_pos concerns a specific operation on such a sequence: choose a position j, then multiply every level above that position by the factor exp(t), where t is any real number. The lemma states that if the original sequence has all positive entries, then every perturbed sequence also has all positive entries, for every choice of t and every position j.

This is a pure arithmetic fact, proved in the framework's machine-checked library of formal theorems. It does not depend on any physical assumption or axiom beyond the standard rules of real arithmetic. The proof is short: for levels at or below position j, the value is unchanged and therefore positive by the original hypothesis; for levels above j, the value is the original positive number multiplied by exp(t), and the exponential of any real number is strictly positive, so the product is positive.

The lemma matters because it is part of a larger construction. The same file defines a family of perturbed sequences, one for each parameter t, and proves that different parameters yield different sequences. That family provides a test: if a hierarchy of levels claims to have no free scale parameters, then the perturbation shows how one could insert a continuous knob, and the framework's main theorem (uniform_scaling_forced) proves that such a knob is incompatible with the zero-parameter condition. The positivity lemma is the basic hygiene that keeps the whole construction well-defined.

What the lemma does not claim is just as important. It does not assert that any particular hierarchy is physical, that the perturbation preserves any structural property beyond positivity, or that the golden ratio emerges from this lemma alone. The positivity fact is a necessary condition for the perturbation family to be usable, not a statement about the real world. It is a small gear in a larger machine, and its content is entirely about the arithmetic of positive sequences.

/-- Every perturbed level is positive. -/
theorem scale_perturbed_pos (levels : ℕ → ℝ) (j : ℕ)
    (h_pos : ∀ k, 0 < levels k) (t : ℝ) (k : ℕ) :
    0 < ScalePerturbed levels j t k := by
  unfold ScalePerturbed
  split
  · exact h_pos k
  · exact mul_pos (h_pos k) (Real.exp_pos t)

/-- Every perturbed level is positive. -/
theorem scale_perturbed_pos (levels : ℕ → ℝ) (j : ℕ)
    (h_pos : ∀ k, 0 < levels k) (t : ℝ) (k : ℕ) :
    0 < ScalePerturbed levels j t k := by
  unfold ScalePerturbed
  split
  · exact h_pos k
  · exact mul_pos (h_pos k) (Real.exp_pos t)

What this page does not claim

No claim that any particular hierarchy is physically realized. No claim that the perturbation preserves any property beyond positivity. No claim that the golden ratio emerges from this lemma alone.

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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