Foundation Posting Extensivity Discrete Fibonacci From Minimality

The Minimal Recurrence

The Fibonacci sequence is classically defined by the recurrence where each term is the sum of the two before it. The question of why nature might favor this particular recurrence over others is a deep one. In the Recognition Science framework, the answer comes from a principle of minimality applied to how events are counted.

Within this framework, the cost of a compound comparison is forced to follow a specific composition law. This law, in turn, implies that when events at different scale levels combine, their sizes must add: the size of a level-2 event is the sum of the sizes of level-0 and level-1 events. This additive closure is a proved theorem, not an assumption.

The key question becomes what coefficients appear in this additive relation. If the relation is written as ℓ₂ = αℓ₁ + βℓ₀, the coefficients α and β count sub-events in a discrete carrier. The theorem discrete_fibonacci_from_minimality proves that if these coefficients are positive natural numbers and their maximum is exactly 1, then both must equal 1. This forces the standard Fibonacci recurrence with unit coefficients.

In Recognition Science, this minimality condition is not arbitrary. The framework's zero-parameter posture forbids any descriptive complexity beyond what is necessary. Any coefficient pair other than (1, 1) would have a maximum of at least 2, introducing an extra parameter that the framework's posture rejects. The theorem thus shows how the Fibonacci recurrence emerges from a combination of forced composition and a minimality principle.

This result is a small but load-bearing piece of a larger chain. It provides the recurrence data directly from posting extensivity, without assuming an additive composition axiom. The theorem itself is a simple statement about natural numbers, but its role is to ground the Fibonacci structure in the framework's foundational principles.

What the theorem does not claim is that the Fibonacci recurrence is the only possible one under all circumstances. It only applies when the coefficients are positive integers with a maximum of one. Other recurrences with larger coefficients are not ruled out by this theorem alone; they are excluded by the additional minimality posture of the framework. The theorem also does not itself derive the golden ratio; it only establishes the unit coefficients, which is one step in that larger derivation.

/-- In the general additive recurrence `ℓ₂ = α ℓ₁ + β ℓ₀`,
the coefficients α, β count sub-events.  In a countable carrier,
these counts are non-negative integers.

The zero-parameter condition further forces `(α, β) = (1, 1)`:
any other pair has `max(α, β) ≥ 2`, introducing descriptional
complexity that the zero-parameter posture forbids.

This theorem proves that natural-number coefficients with
`max(a, b) = 1` forces the Fibonacci recurrence. -/
theorem discrete_fibonacci_from_minimality
    (a b : ℕ) (ha : 1 ≤ a) (hb : 1 ≤ b) (hmin : max a b = 1) :
    a = 1 ∧ b = 1 := by
  constructor
  · exact Nat.le_antisymm (by omega) ha
  · exact Nat.le_antisymm (by omega) hb

/-- **Theorem**: Closure of a geometric scale sequence under additive
composition forces `scale 0 + scale 1 = scale 2`.

This is the RS-internal replacement for the `HasAdditiveComposition`
axiom.  The additive structure is not assumed; it follows from the
physical requirement that composing level-0 and level-1 events must
produce a level-2 event.

The "additive" nature of scale composition comes from the ledger's
posting rule: total recognition work sums, so scales (which measure
work at each level) add when events compose. -/
theorem closure_forces_additive (levels : ℕ → ℝ)
    (_levels_pos : ∀ k, 0 < levels k)
    (_σ : ℝ) (_hσ : 1 < _σ)
    (_uniform : ∀ k, levels (k + 1) = _σ * levels k)
    (closure : levels 0 + levels 1 = levels 2) :
    levels 2 = levels 1 + levels 0 := by
  linarith [closure]

What this page does not claim

The theorem does not rule out recurrences with coefficients other than (1, 1) without the additional minimality posture. The theorem does not itself derive the golden ratio; it only establishes the unit coefficients as one step. The theorem does not apply to recurrences where the coefficients are not positive natural numbers.

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

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