Foundation Cost Floor Boundary Jcost Plastic Certified Bounds

The certified cost floor

The plastic constant is the real number r that solves r³ = r + 1, approximately 1.3247. In the Recognition Science framework, every scale ladder that records recognition events carries a per-step cost measured by the function J(x) = (x + 1/x)/2 - 1, calibrated so that J(e) = 1. The declaration jcost_plastic_certified_bounds is a machine-checked theorem stating that this plastic constant lies between 13/10 and 133/100, and that its J-cost lies between 1/30 and 41/1000, roughly 3 to 4 percent of the cost unit. The theorem establishes a precise numerical window: any ladder whose ratio grows faster than plastic must have each rung step costing more than about 0.033 in calibrated units.

The significance is not the number itself but what it marks. The framework's earlier results force the golden ratio φ as the unique self-similar scaling for any ladder that grows faster than plastic, but they do not force the growth rate itself. The certified bounds turn that residual premise into a concrete cost threshold: each rung must cost more than J(plastic), the certified floor. This is the exact boundary the framework's kernel can certify but cannot cross on its own.

What the declaration does not claim is equally important. It does not prove that any actual ladder in the framework must exceed this floor. The banked structure, which packages all the kernel's forced properties about scale ladders, admits two distinct instances: the φ-ladder with ratios constantly φ, and the integer ladder with ratios converging to 1, whose per-rung costs shrink to zero. The theorem banked_independence shows these two ladders coexist, so no theorem about the banked structure alone can force the φ rhythm or any positive cost floor. The independence is proved: the cost floor is logically independent of everything the kernel has established about ladders.

This independence is the declaration's real content. It separates what the framework has proved from what remains a genuine open premise. The positive assembly theorem banked_plus_floor_gives_phi shows that if a banked ladder does satisfy a ratio floor above plastic, then its ratios converge to φ; the certified bounds give the cost reading of that floor. But the floor itself must come from structure outside the abstract kernel, such as mass-spectrum stability physics, not from the kernel's forced properties. The declaration thus marks a precise boundary: the framework's abstract core forces the golden ratio conditionally, and the cost floor is the single certified but unforced premise that remains.

/-- **The certified threshold.** The plastic constant lies in
(13/10, 133/100), and its J-cost lies in (1/30, 41/1000) — about 3 to 4
percent of the calibrated cost unit (J(e) = 1). The floor T6 needs is:
each rung step costs more than this. -/
theorem jcost_plastic_certified_bounds :
    ∃ r : ℝ, 1 < r ∧ 1 + r = r ^ 3 ∧
      (1 / 30 < Cost.Jcost r ∧ Cost.Jcost r < 41 / 1000) := by
  have hcont : ContinuousOn (fun x : ℝ => x ^ 3 - x - 1)
      (Set.Icc (13 / 10) (133 / 100)) :=
    ((continuous_pow 3).sub continuous_id |>.sub continuous_const).continuousOn
  have hmem : (0 : ℝ) ∈
      Set.Ioo ((13 / 10 : ℝ) ^ 3 - 13 / 10 - 1)
        ((133 / 100 : ℝ) ^ 3 - 133 / 100 - 1) := by
    constructor <;> norm_num
  have hivt := intermediate_value_Ioo
    (by norm_num : (13 / 10 : ℝ) ≤ 133 / 100) hcont hmem
  obtain ⟨r, hr, hfr⟩ := hivt
  have hr3 : 1 + r = r ^ 3 := by
    have h0 : r ^ 3 - r - 1 = 0 := hfr
    linarith
  have hr1 : 1 < r := by linarith [hr.1]
  refine ⟨r, hr1, hr3, ?_, ?_⟩
  · have hmono := jcost_strictMono_one_lt (a := (13 : ℝ) / 10) (b := r)
      (by norm_num) hr.1
    have hval : Cost.Jcost (13 / 10 : ℝ) = 9 / 260 := by norm_num [Cost.Jcost]
    linarith [hmono, hval]
  · have hmono := jcost_strictMono_one_lt (a := r) (b := (133 : ℝ) / 100)
      (le_of_lt hr1) hr.2
    have hval : Cost.Jcost (133 / 100 : ℝ) = 1089 / 26600 := by
      norm_num [Cost.Jcost]
    linarith [hmono, hval]

/-- **The boundary theorem.** The banked structure admits two ladders with
different asymptotic ratios (φ and 1). Hence the ratio floor above plastic —
and a fortiori the per-rung cost floor — is logically independent of
everything the kernel has forced about scale ladders. -/
theorem banked_independence :
    ∃ L₁ L₂ : BankedLadder,
      Tendsto (fun n => L₁.s (n + 1) / L₁.s n) atTop (nhds phi) ∧
      Tendsto (fun n => L₂.s (n + 1) / L₂.s n) atTop (nhds 1) ∧ phi ≠ 1 :=
  ⟨phiLadderBanked, integerLadderBanked, phiLadder_ratio_tendsto,
    integerLadder_ratio_tendsto, ne_of_gt one_lt_phi⟩

/-- **The positive assembly.** A banked ladder with a ratio floor above the
plastic constant has ratio converging to φ. The floor is exactly the
residual premise: everything else is banked. -/
theorem banked_plus_floor_gives_phi (L : BankedLadder)
    {ρ : ℝ} (hρ : 1 < ρ) (hρ3 : ρ + 1 < ρ ^ 3)
    (hfloor : ∀ n, ρ * L.s n ≤ L.s (n + 1)) :
    Tendsto (fun n => L.s (n + 1) / L.s n) atTop (nhds phi) :=
  phi_of_floor_above_plastic L.pos hρ hρ3 hfloor L.closure

/-- A ratio floor ρ is exactly a per-rung J-cost floor J(ρ): the cost reading
of the recurrence's residual premise. -/
theorem ratio_floor_gives_cost_floor {s : ℕ → ℝ} (hpos : ∀ n, 0 < s n)
    {ρ : ℝ} (hρ : 1 < ρ) (hfloor : ∀ n, ρ * s n ≤ s (n + 1)) :
    ∀ n, Cost.Jcost ρ ≤ Cost.Jcost (s (n + 1) / s n) := by
  intro n
  have hratio : ρ ≤ s (n + 1) / s n := by
    rw [le_div_iff₀ (hpos n)]
    exact hfloor n
  have hlog : Real.log ρ ≤ Real.log (s (n + 1) / s n) :=
    Real.log_le_log (by linarith [hρ]) hratio
  rw [Cost.GeometricRoot.jcost_eq_cosh_log_sub_one (by linarith [hρ] : (0 : ℝ) < ρ),
    Cost.GeometricRoot.jcost_eq_cosh_log_sub_one (div_pos (hpos (n + 1)) (hpos n))]
  exact sub_le_sub_right
    (Cost.GeometricRoot.cosh_mono_on_nonneg (Real.log_nonneg (le_of_lt hρ)) hlog) 1

What this page does not claim

The declaration does not prove that any actual ladder must exceed the certified cost floor. The declaration does not derive the golden ratio φ unconditionally from the kernel's forced properties. The declaration does not establish the ratio floor itself; it only certifies its cost value if such a floor is assumed.

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/CostFloorBoundary.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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