Foundation Hierarchy Forcing Min Max Achieved

The simplest choice

The declaration min_max_achieved proves the smallest possible value of the expression max(a, b), where a and b are positive whole numbers, is 1, and that the only way to reach it is a = 1 and b = 1. This is pure arithmetic: max(1, 1) = 1, and any other positive pair gives a maximum of 2 or more. The statement is so simple it might seem empty, but in the Recognition Science framework it is the final step in a chain that selects the simplest possible rule for building a hierarchy.

The chain starts with the idea of a ledger, a discrete record of events. The framework models a multilevel composition as a sequence of levels, each level a positive real number, with the requirement that the ratio between adjacent levels is the same everywhere. The theorem uniform_scaling_forced proves that if this ratio is not allowed to vary, then all adjacent ratios must be equal to a single constant σ. A separate condition, that each level is the sum of the two below it, then forces σ to be the golden ratio, about 1.618.

The role of min_max_achieved is to justify the additive rule itself. Among all possible recurrence rules of the form level(n+2) = a·level(n+1) + b·level(n), with a and b positive integers, the pair (1, 1) uniquely minimizes the larger coefficient. The theorem additive_composition_is_minimal proves this uniqueness, and min_max_achieved shows that the minimum is actually attained. The framework's library of machine-checked formal theorems then uses this to argue that the simplest possible composition rule is the one nature follows.

What the declaration does not claim is that this arithmetic fact alone forces the golden ratio. The full derivation requires the uniformity condition, the positivity of all levels, and the additive recurrence. The theorem min_max_achieved is only the final arithmetic step in that longer argument. It also says nothing about what the levels physically represent; that identification is a separate modeling choice, not a consequence of the arithmetic.

/-- The pair (1,1) achieves max = 1. -/
theorem min_max_achieved : max 1 1 = 1 := by simp

/-- **Theorem (Phase 3)**: Among recurrence coefficients (a, b) with
a ≥ 1 and b ≥ 1, the pair (1, 1) uniquely minimizes max(a, b).
No axiom needed — this is pure arithmetic. -/
theorem additive_composition_is_minimal (a b : ℕ) (ha : 1 ≤ a) (hb : 1 ≤ b) :
    max a b = 1 → a = 1 ∧ b = 1 := by
  intro h
  constructor
  · exact Nat.le_antisymm (by omega) ha
  · exact Nat.le_antisymm (by omega) hb

/-- **Theorem**: No free scale parameters forces uniform adjacent ratios.

The canonical derivation now uses `HierarchyRealization.realized_uniform_ratios`
which derives uniform ratios from the `RealizedHierarchy.ratio_self_similar`
field, with `no_continuous_moduli` as backup
(`HierarchyRealization.no_moduli_forces_uniform_ratios`). -/
theorem uniform_scaling_forced
    (M : NontrivialMultilevelComposition)
    (no_free_scale : ∀ j k,
      M.levels (j + 1) / M.levels j = M.levels (k + 1) / M.levels k)
    (ratio_gt_one : 1 < M.levels 1 / M.levels 0) :
    ∃ σ : ℝ, 1 < σ ∧ ∀ k, M.levels (k + 1) = σ * M.levels k := by
  use M.levels 1 / M.levels 0
  refine ⟨ratio_gt_one, fun k => ?_⟩
  have hk := M.levels_pos k
  have h0 := M.levels_pos 0
  have hratio := no_free_scale k 0
  rw [div_eq_div_iff (ne_of_gt hk) (ne_of_gt h0)] at hratio
  have : M.levels (k + 1) = M.levels 1 / M.levels 0 * M.levels k := by
    field_simp; linarith
  exact this

/-- The forced hierarchy yields σ = φ. -/
theorem hierarchy_forced_gives_phi
    (M : NontrivialMultilevelComposition)
    (no_free_scale : ∀ j k,
      M.levels (j + 1) / M.levels j = M.levels (k + 1) / M.levels k)
    (ratio_gt_one : 1 < M.levels 1 / M.levels 0)
    (additive : M.levels 2 = M.levels 1 + M.levels 0) :
    (hierarchy_forced M no_free_scale ratio_gt_one).ratio = PhiForcing.φ :=
  hierarchy_emergence_forces_phi
    (hierarchy_forced M no_free_scale ratio_gt_one)
    additive

What this page does not claim

This declaration alone forces the golden ratio. The levels represent any specific physical quantity. The arithmetic result is unique to Recognition Science.

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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Derived articles

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