Foundation Posting Extensivity Posting Coefficients Minimal

The minimal posting coefficients

The declaration posting_coefficients_minimal proves the arithmetic fact that the maximum of 1 and 1 is 1. This is a trivial statement in isolation, but it carries weight in the Recognition Science framework because it is the final step in a chain that forces the golden ratio φ to appear as the unique self-similar scaling. The chain starts with the Recognition Composition Law, a forced equation governing how the cost of a compound comparison decomposes into the costs of its constituents. From that law, the framework derives a d'Alembert identity for a shifted cost function called the posting potential, and then shows that a geometric scale sequence closed under additive composition must satisfy the golden equation σ² = σ + 1.

The role of posting_coefficients_minimal is to pin down the coefficients in the additive recurrence ℓ₂ = αℓ₁ + βℓ₀. The framework's zero-parameter posture forbids introducing arbitrary constants, so the coefficients must be minimal. Since they count sub-events in a countable carrier, they are natural numbers, and the minimal positive pair with maximum equal to 1 is (1, 1). This yields the Fibonacci recurrence ℓ₂ = ℓ₁ + ℓ₀, which is exactly the additive closure condition that forces the golden ratio. The declaration thus completes the bridge from the forced combiner structure to φ without assuming linearity or additive composition as axioms.

What the declaration does not claim is broader significance on its own. It does not assert that the golden ratio exists, that the Fibonacci recurrence is physically real, or that the posting potential is a measurable quantity. Those claims belong to other theorems in the chain, such as posting_extensivity_forces_phi, which requires a uniform scale ladder and the closure condition as hypotheses. The minimality fact is a lemma about natural numbers, not a statement about the physical world. It earns its place only as one link in a longer derivation, and its proof is a simple simplification in the machine-checked library of formal theorems.

/-- The unit pair (1, 1) is minimal: max(1, 1) = 1. -/
theorem posting_coefficients_minimal : max 1 1 = 1 := by simp

/-- 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

/-- **End-to-end theorem**: From the forced RCL combiner structure
(specifically, the d'Alembert identity on posting potentials),
a geometric scale sequence closed under additive posting, with
discrete minimal coefficients, forces φ.

This chains the entire derivation:
  RCL → posting d'Alembert → additive closure → golden equation → φ -/
theorem posting_extensivity_forces_phi
    (L : UniformScaleLadder)
    (closure : L.levels 2 = L.levels 1 + L.levels 0) :
    L.ratio = PhiForcing.φ :=
  hierarchy_emergence_forces_phi L closure

What this page does not claim

The declaration does not assert that the golden ratio exists in the physical world. The declaration does not claim that the Fibonacci recurrence is physically real or measurable. The declaration does not establish that the posting potential is a measurable quantity.

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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Derived articles

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