Foundation Posting Extensivity Posting Extensivity Forces Phi

The golden ratio from posting

The golden ratio φ ≈ 1.618 is the number that solves r² = r + 1. It appears throughout mathematics: in the pentagon's diagonals, in the Fibonacci sequence, and as the limit of ratios of consecutive terms. A classical fact about it is that it is the only positive number whose square is itself plus one. This page concerns a formal result that derives φ from a structure called a posting extensivity condition.

The setting is a uniform scale ladder: a sequence of positive levels where each level is a fixed ratio σ times the previous one. The ladder models how a quantity grows in discrete steps. The key condition is additive closure: the level at position 2 equals the sum of the levels at positions 0 and 1. This is a natural requirement if levels represent accumulated work: composing two events adds their work totals.

In Recognition Science, the framework models recognition as a ledger, a discrete record of events. The framework's cost function, the price of recognizing a comparison, is forced by five plain conditions to be J(x) = ½(x + 1/x) − 1. A shifted version of this cost, the posting potential Π(x) = ½(x + 1/x), satisfies the d'Alembert identity Π(xy) + Π(x/y) = 2Π(x)Π(y). This identity governs how costs combine when events compose.

The framework's machine-checked library of formal theorems proves that additive closure on a uniform scale ladder forces the golden equation σ² = σ + 1, and hence σ = φ. The theorem posting_extensivity_forces_phi chains this result: from the forced cost structure, through the d'Alembert identity, to additive closure, to the golden ratio. The additive structure is not assumed; it follows from the requirement that composing events produces a level-2 event.

What this does not claim: it does not claim that any arbitrary scale sequence has ratio φ, only one that satisfies additive closure. It does not claim that the golden ratio is the only self-similar scaling in all contexts. It does not claim that the physical bridge from recognition to actual spatial dimensions is proved; that remains open. The theorem establishes a conditional: given the posting structure and closure, φ follows.

/-- The additive closure relation on a uniform scale ladder yields
the golden equation σ² = σ + 1. -/
theorem additive_closure_golden (levels : ℕ → ℝ)
    (levels_pos : ∀ k, 0 < levels k)
    (σ : ℝ) (_hσ : 1 < σ)
    (uniform : ∀ k, levels (k + 1) = σ * levels k)
    (closure : levels 2 = levels 1 + levels 0) :
    σ ^ 2 = σ + 1 := by
  have h0 : levels 0 ≠ 0 := ne_of_gt (levels_pos 0)
  have h1 := uniform 0
  have h2 := uniform 1
  have h_sq : levels 2 = σ ^ 2 * levels 0 := by
    rw [h2, h1]; ring
  have h_rhs : levels 2 = (σ + 1) * levels 0 := by
    rw [closure, h1]; ring
  have : (σ ^ 2 - (σ + 1)) * levels 0 = 0 := by
    calc (σ ^ 2 - (σ + 1)) * levels 0
        = σ ^ 2 * levels 0 - (σ + 1) * levels 0 := by ring
      _ = levels 2 - levels 2 := by rw [← h_sq, h_rhs]
      _ = 0 := by ring
  rcases mul_eq_zero.mp this with hzero | hsize
  · linarith
  · exact (h0 hsize).elim

/-- **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

/-- **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]

/-- The d'Alembert identity for the posting potential:
Π(xy) + Π(x/y) = 2 Π(x) Π(y).

This is the fundamental identity governing how posting potentials
compose.  It is equivalent to the RCL via the shift J = Π − 1. -/
theorem posting_dalembert (x y : ℝ) (hx : 0 < x) (hy : 0 < y) :
    PostingPotential (x * y) + PostingPotential (x / y) =
      2 * PostingPotential x * PostingPotential y := by
  unfold PostingPotential Jcost
  have hx_ne : x ≠ 0 := ne_of_gt hx
  have hy_ne : y ≠ 0 := ne_of_gt hy
  field_simp [hx_ne, hy_ne]
  ring

What this page does not claim

This does not claim that any arbitrary scale sequence has ratio φ, only one that satisfies additive closure. This does not claim that the physical bridge from recognition to actual spatial dimensions is proved. This does not claim that the golden ratio is the only self-similar scaling in all contexts.

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)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

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