Foundation Phi Forcing Unconditional Posting Closure At Base

The base closure step

The declaration posting_closure_at_base is a small, precise statement about sequences. It says: if a sequence obeys the rule that every term (from the third onward) is the sum of the two preceding terms, then the third term is the sum of the first two. In symbols, if s(n+2) = s(n+1) + s(n) for every n, then s(2) = s(1) + s(0). This is the n = 0 case of the rule, so the theorem is immediate from the assumption; its role is to make the base of a ladder explicit rather than assumed.

The statement belongs to a larger argument about the golden ratio. A sequence built by adding adjacent terms, such as 1, 1, 2, 3, 5, 8, has ratios between consecutive terms that settle toward phi, about 1.618. The full theorem ratio_tendsto_phi proves this convergence for any positive starting values. The base step recorded here is the first rung of that ladder: it confirms that the recurrence holds at the very start, so the later convergence argument has a firm footing.

What the declaration does not claim is just as important. It does not assert that the recurrence itself is true for any physical or economic process. It only says that if the recurrence holds, then this one base equality follows. The origin of the recurrence, why a posting operation should obey adjacent additivity, is a separate question left open by this theorem. The declaration also does not say that the ratio reaches phi exactly at any finite step; convergence is asymptotic, not exact at the base.

In the framework's library, this step is part of a chain that promotes the golden ratio from a chosen constant to a forced outcome of additive composition. The base closure is the hinge that connects the abstract recurrence to its first concrete instance. Without it, the ladder has no starting point; with it, the convergence theorem can proceed from a verified base.

/-- The falsifier's base instance "levels 2 = levels 1 + levels 0" is the
n = 0 case of the adjacent recurrence. -/
theorem posting_closure_at_base (hrec : ∀ n, s (n + 2) = s (n + 1) + s n) :
    s 2 = s 1 + s 0 := hrec 0

/-- **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

What this page does not claim

The declaration does not prove that any actual posting operation obeys the adjacent additive recurrence. The declaration does not show that any finite ratio equals phi exactly. The declaration does not address the origin or provenance of the recurrence rule itself.

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)

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Derived articles

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