Encyclopedia Foundation Foundation Phi Forcing Unconditional Ratio Tendsto Phi

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Foundation Phi Forcing Unconditional Ratio Tendsto Phi

Any sequence where each term is the sum of the two before it has consecutive ratios that settle toward the golden ratio, no matter where it starts.

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The asymptotic ratio theorem

The golden ratio φ, about 1.618, is the positive solution of r² = r + 1, equivalently r = 1 + 1/r. It appears throughout mathematics, most famously as the limit of ratios of consecutive Fibonacci numbers: 1, 2, 1.5, 1.667, 1.6, and so on, drawing closer to φ each step. The Fibonacci sequence itself begins 1, 1, 2, 3, 5, 8, where each term is the sum of the two preceding terms, and this additive rule, not any special starting point, drives the ratios toward φ.

The theorem ratio_tendsto_phi in the Recognition Science framework's machine-checked library of formal theorems states this fact in full generality. For any sequence of positive numbers s(0), s(1), s(2), ... that obeys the adjacent additive recurrence s(n+2) = s(n+1) + s(n), the ratio s(n+1)/s(n) converges to φ as n grows without bound. The proof runs through a contraction: writing r_n for the ratio at step n, the recurrence gives r_{n+1} = 1 + 1/r_n, and since φ = 1 + 1/φ, the error r_n − φ shrinks by a factor of about 0.618 at each step. No geometric ladder, no uniform ratio, and no closure hypothesis is assumed; geometricity is demoted from premise to asymptotic corollary.

The result matters because it removes a hidden assumption from the framework's derivation of φ as the unique self-similar scaling. Earlier versions assumed a uniform ratio r across levels and derived r = φ. This theorem shows the same conclusion follows from the additive recurrence alone, for any positive seeds. Within the framework, the recurrence is the arithmetic of the ledger, a discrete record of recognition events where composed work is the sum of posted work. The theorem itself is unconditional in its hypotheses: positivity and the recurrence are sufficient.

What the theorem does not claim is just as important. It does not assert that any particular physical system obeys the recurrence; that is a separate premise, carried in the framework by posting-closure certificates whose provenance is the ledger's posting operation. The theorem also does not prove that the golden ratio is the only possible asymptotic ratio for all sequences, only for those satisfying the additive recurrence. And it says nothing about the rate of convergence beyond the exponential bound implied by the contraction factor, nor about the values of the early ratios, which can be arbitrary positive numbers.

In plain terms, the theorem is a pure statement about sequences: add the two previous terms, and the ratios inevitably approach φ. It is the mathematical core of the framework's claim that φ is forced, not chosen, but the forcing happens only once the recurrence itself is granted. The reader can now see that the golden ratio's appearance in the framework is not an artifact of assuming a geometric ladder; it is a property of the additive rule itself.

THEOREM ratio_tendsto_phi · IndisputableMonolith/Foundation/PhiForcingUnconditional.lean
ratio_tendsto_phi · IndisputableMonolith/Foundation/PhiForcingUnconditional.lean:166 
/-- **T6, promoted: the golden ratio is the asymptotic ratio of every additive
posting ladder.** Any positive sequence obeying the adjacent additive
recurrence s(n+2) = s(n+1) + s(n) has inter-level ratio converging to φ.
No geometric ladder, no uniform ratio, and no closure hypothesis is assumed:
geometricity is demoted from premise to asymptotic corollary. -/
theorem ratio_tendsto_phi (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) := by
  have hp : 0 < phi := by linarith [one_lt_phi]
  have hq0 : 0 ≤ (1:ℝ) / phi := by positivity
  have hq1 : (1:ℝ) / phi < 1 := by
    rw [div_lt_one hp]; exact one_lt_phi
  set C := phi * (|s 1 / s 0 - phi| + |s 2 / s 1 - phi|) with hC
  have hbound : ∀ n, |s (n + 1) / s n - phi| ≤ C * (1 / phi) ^ n := fun n =>
    ratio_bound_all hpos hrec n
  have htend : Tendsto (fun n => C * (1 / phi) ^ n) atTop (nhds 0) := by
    have h := (tendsto_pow_atTop_nhds_zero_of_lt_one hq0 hq1).const_mul C
    simpa using h
  have habs : Tendsto (fun n => |s (n + 1) / s n - phi|) atTop (nhds 0) :=
    squeeze_zero (fun n => abs_nonneg _) hbound htend
  have hsub : Tendsto (fun n => s (n + 1) / s n - phi) atTop (nhds 0) := by
    have hneg : Tendsto (fun n => -|s (n + 1) / s n - phi|) atTop (nhds 0) := by
      have h := habs.neg
      simpa using h
    exact tendsto_of_tendsto_of_tendsto_of_le_of_le hneg habs
      (fun n => neg_abs_le _) (fun n => le_abs_self _)
  have h : Tendsto (fun n => (s (n + 1) / s n - phi) + phi) atTop (nhds (0 + phi)) :=
    hsub.add tendsto_const_nhds
  simpa using h
THEOREM ratio_tendsto_phi · IndisputableMonolith/Foundation/PhiForcingUnconditional.lean
ratio_tendsto_phi · IndisputableMonolith/Foundation/PhiForcingUnconditional.lean:166 
/-- **T6, promoted: the golden ratio is the asymptotic ratio of every additive
posting ladder.** Any positive sequence obeying the adjacent additive
recurrence s(n+2) = s(n+1) + s(n) has inter-level ratio converging to φ.
No geometric ladder, no uniform ratio, and no closure hypothesis is assumed:
geometricity is demoted from premise to asymptotic corollary. -/
theorem ratio_tendsto_phi (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) := by
  have hp : 0 < phi := by linarith [one_lt_phi]
  have hq0 : 0 ≤ (1:ℝ) / phi := by positivity
  have hq1 : (1:ℝ) / phi < 1 := by
    rw [div_lt_one hp]; exact one_lt_phi
  set C := phi * (|s 1 / s 0 - phi| + |s 2 / s 1 - phi|) with hC
  have hbound : ∀ n, |s (n + 1) / s n - phi| ≤ C * (1 / phi) ^ n := fun n =>
    ratio_bound_all hpos hrec n
  have htend : Tendsto (fun n => C * (1 / phi) ^ n) atTop (nhds 0) := by
    have h := (tendsto_pow_atTop_nhds_zero_of_lt_one hq0 hq1).const_mul C
    simpa using h
  have habs : Tendsto (fun n => |s (n + 1) / s n - phi|) atTop (nhds 0) :=
    squeeze_zero (fun n => abs_nonneg _) hbound htend
  have hsub : Tendsto (fun n => s (n + 1) / s n - phi) atTop (nhds 0) := by
    have hneg : Tendsto (fun n => -|s (n + 1) / s n - phi|) atTop (nhds 0) := by
      have h := habs.neg
      simpa using h
    exact tendsto_of_tendsto_of_tendsto_of_le_of_le hneg habs
      (fun n => neg_abs_le _) (fun n => le_abs_self _)
  have h : Tendsto (fun n => (s (n + 1) / s n - phi) + phi) atTop (nhds (0 + phi)) :=
    hsub.add tendsto_const_nhds
  simpa using h
THEOREM ratio_contract · IndisputableMonolith/Foundation/PhiForcingUnconditional.lean
ratio_contract · IndisputableMonolith/Foundation/PhiForcingUnconditional.lean:88 
/-- 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 theorem does not assert that any particular physical system obeys the recurrence; that is a separate premise. The theorem does not prove φ is the only possible asymptotic ratio for all sequences, only for those satisfying the additive recurrence. The theorem says nothing about the rate of convergence beyond the exponential bound, nor about the values of early ratios.

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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)

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