Foundation Phi Forcing Unconditional Phi Is Asymptotic Ratio

The asymptotic ratio

The golden ratio, φ, is the number that solves r² = r + 1, roughly 1.618. It appears throughout mathematics as the limit of ratios in the Fibonacci sequence 1, 1, 2, 3, 5, 8, where each term is the sum of the two before it. The ratios of successive terms, 1, 2, 1.5, 1.667, are not constant, but they settle toward φ. A classical fact, known since the nineteenth century, is that this behavior is generic: start with any two positive numbers, and build each next term as the sum of the previous two. The ratios of successive terms always converge to φ, regardless of the starting seeds.

The declaration phi_is_asymptotic_ratio, in the framework's machine-checked library of formal theorems, proves this exact statement. It establishes that for any positive sequence s(n) obeying the adjacent additive recurrence s(n+2) = s(n+1) + s(n), the ratio s(n+1) / s(n) tends to φ as n grows. The proof works by a contraction: the ratio map r ↦ 1 + 1/r pulls any starting value toward φ, the fixed point of that map, shrinking the error by a factor of about 0.618 at each step. The theorem requires no assumption that the sequence is geometric, no uniform ratio, and no closure hypothesis. Geometricity, the property of having a constant ratio, is not a premise but an asymptotic corollary.

In Recognition Science, this result carries the label T6 and is a step in a larger forcing chain. The framework models a ledger, a discrete record of events, where each posted level composes its two predecessors. The recurrence is the arithmetic of that posting operation. The theorem says that any such additive posting ladder, whatever its starting levels, asymptotically exhibits the golden ratio as its inter-level ratio. This is what the framework means by φ being forced: not chosen, but a consequence of the additive composition law alone.

What the declaration does not claim is equally precise. It does not prove that the recurrence itself holds; that is a separate premise about the posting operation's arithmetic, carried by other certificates in the library. The theorem takes the recurrence as given and derives the asymptotic ratio. It also does not claim that any particular sequence reaches φ exactly, only that the ratio approaches it in the limit. And it says nothing about the fine-structure constant, the Riemann Hypothesis, or any other physical constant; those are separate matters in the framework.

/-- **T6 assembly.** From positivity and the adjacent posting recurrence, the
asymptotic inter-level ratio is φ, with no geometric-ladder input. Combined
with the banked tick (Cost.GeometricRoot: discreteness forced by
cost-nontriviality), the chain is: the ledger cannot subdivide forever (tick),
each posted level composes its two predecessors (recurrence), and the ladder's
asymptotic ratio is φ (this module). The residual premise is the recurrence's
own provenance; see the module docstring. -/
theorem phi_is_asymptotic_ratio (hpos : ∀ n, 0 < s n)
    (hrec : ∀ n, s (n + 2) = s (n + 1) + s n) :
    Tendsto (fun n => s (n + 1) / s n) atTop (nhds phi) :=
  ratio_tendsto_phi hpos hrec

/-- Contraction: |r_{n+1} − φ| ≤ |r_n − φ| / φ for n ≥ 1 (since r_n ≥ 1). -/
private theorem ratio_contract (hpos : ∀ n, 0 < s n) (hrec : ∀ n, s (n + 2) = s (n + 1) + s n)
    (n : ℕ) (hn : 1 ≤ n) :
    |s (n + 2) / s (n + 1) - phi| ≤ |s (n + 1) / s n - phi| / phi := by
  have hr : 0 < s (n + 1) / s n := div_pos (hpos _) (hpos _)
  have hp : 0 < phi := by linarith [one_lt_phi]
  have h1 : s (n + 1) / s n ≠ 0 := ne_of_gt hr
  have h2 : phi ≠ 0 := ne_of_gt hp
  rw [ratio_sub_phi hpos hrec n]
  have hident : 1 / (s (n + 1) / s n) - 1 / phi
      = (phi - s (n + 1) / s n) / ((s (n + 1) / s n) * phi) := by
    have hn1 : s (n + 1) ≠ 0 := ne_of_gt (hpos _)
    have hn0 : s n ≠ 0 := ne_of_gt (hpos _)
    field_simp [h1, h2, hn1, hn0]
  rw [hident, abs_div, abs_mul, abs_of_pos hr, abs_of_pos hp,
    show phi - s (n + 1) / s n = -(s (n + 1) / s n - phi) from by ring, abs_neg]
  gcongr
  have hn1 : 1 ≤ s (n + 1) / s n := by
    obtain ⟨k, rfl⟩ : ∃ k, n = k + 1 := ⟨n - 1, by omega⟩
    simpa using ratio_ge_one hpos hrec k
  nlinarith [hn1, hp]

What this page does not claim

The recurrence s(n+2) = s(n+1) + s(n) itself is not proved here; it is a premise about the posting operation. The theorem does not assert that any sequence reaches φ exactly, only that the ratio converges in the limit. No physical constant such as the fine-structure constant is derived from this declaration alone.

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/PhiForcingUnconditional.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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