Foundation Recurrence Bridge Phi Of Posting Locality Additivity

The recurrence bridge

The golden ratio φ ≈ 1.618 is the limit of the ratio of successive terms in the Fibonacci sequence, where each term is the sum of the two before it. The declaration phi_of_posting_locality_additivity proves that this same ratio emerges from two much more general conditions on any sequence of positive numbers. First, each new term is built from the two immediately preceding terms by some fixed operation. Second, the size of the result of that operation is exactly the sum of the sizes of its two inputs. Under these two conditions alone, the ratio of consecutive terms must tend to φ.

The result is a theorem in the framework's machine-checked library of formal theorems. It states that for any positive sequence s and any operation comp, if s(n+2) = comp(s(n+1), s(n)) and comp(x, y) = x + y for all x and y, then the ratio s(n+1)/s(n) approaches φ as n grows. The proof is immediate: the two conditions together force the adjacent recurrence s(n+2) = s(n+1) + s(n), and a separate theorem shows that any positive sequence obeying this recurrence has ratios converging to φ.

The declaration names the two premises explicitly. The first, posting locality, is the requirement that each new rung of the ladder is composed only from its two immediate predecessors, not from any distant rungs. The second, ledger additivity, is the requirement that the size of a composed rung is the sum of the sizes of the parts. The theorem shows that these two properties are sufficient to produce the golden-ratio rhythm.

In Recognition Science, this theorem is a bridge. It connects the abstract forcing chain that derives φ as the unique self-similar scaling to a concrete picture of how a ledger of recognition events might grow. The framework models recognition as a discrete record of events, and the theorem shows what kind of record produces the golden ratio: one where each new entry is built from its two neighbors and where sizes add. The theorem does not, however, show that any real ledger actually satisfies these conditions.

The theorem's limits are as precise as its statement. It does not claim that the operation comp must be addition; it assumes it. It does not claim that posting locality and ledger additivity are necessary for the golden ratio; other routes may exist. It does not claim that the ratio reaches φ exactly at any finite step, only that it approaches it in the limit. And it does not claim that the integer ladder s(n) = n+1, which satisfies closure and generation but fails the recurrence, is a counterexample to this theorem; that ladder fails the ratio floor premise, which this theorem does not require.

/-- Posting locality plus ledger additivity suffice for the φ rhythm. -/
theorem phi_of_posting_locality_additivity (s : ℕ → ℝ) (comp : ℝ → ℝ → ℝ)
    (hpos : ∀ n, 0 < s n)
    (hgen : ∀ n, s (n + 2) = comp (s (n + 1)) (s n))
    (hadd : ∀ x y, comp x y = x + y) :
    Tendsto (fun n => s (n + 1) / s n) atTop (nhds phi) :=
  ratio_tendsto_phi hpos (recurrence_of_adjacent_generation_additive s comp hgen hadd)

What this page does not claim

The theorem does not claim that the operation comp must be addition; it assumes it as a premise. The theorem does not claim that posting locality and ledger additivity are necessary for the golden ratio, only sufficient. The theorem does not claim that any actual physical ledger satisfies these conditions.

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/RecurrenceBridge.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.

Derived articles

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: