Foundation Phi Closure Selection Plastic Ladder Exists

The plastic ladder

The plastic constant is the real number r greater than 1 that solves 1 + r = r³. Its value is about 1.3247. It is the classical cousin of the golden ratio, which solves the analogous equation at the second power, 1 + r = r². Both numbers describe self-similar growth, where each new stage combines the two previous ones, but the plastic constant does so with a longer memory: it is the limiting ratio of the Padovan and Perrin sequences, just as the golden ratio is the limiting ratio of the Fibonacci sequence. The plastic constant was studied by the Dutch monk and architect Hans van der Laan in the 1920s, who used it as a system of proportions for buildings and furniture.

In Recognition Science, the framework's machine-checked library of formal theorems proves that this third-order ladder actually exists. The declaration plastic_ladder_exists establishes, by a kernel-checked argument, that there is a real number r greater than 1 satisfying 1 + r = r³. The proof is not a numerical approximation; it is an exact existence theorem, built from the intermediate value theorem: the function x³ - x - 1 is continuous, negative at x = 1, and positive at x = 2, so it must cross zero somewhere between. That crossing point is the plastic constant, and the theorem records it as a formal fact.

The existence of this ladder matters because it closes off a tempting route. One might hope that the framework's cost function, which measures the expense of recognition events, would select the golden ratio by simple minimization: among all possible closure levels k, the cheapest one wins. The theorem plastic_cheaper_than_phi shows that this hope fails. The plastic ladder at k = 3 exists and its closure cost is strictly lower than the golden ladder's cost at k = 2. In fact, the per-closure cost strictly decreases as k rises, so cost-minimization alone selects no finite level at all. The golden ratio is not chosen because it is cheap; it is chosen for a structural reason, because its second rung can be composed from earlier rungs, a property the plastic ladder lacks.

What the declaration does not claim is equally important. It does not assert that the plastic constant is the framework's preferred scaling, nor does it assign any physical role to the plastic ladder. It does not claim that cost-minimization is a valid selection principle, only that it fails to pick out a finite level. The theorem is a negative result dressed as an existence proof: it clears away one plausible explanation so that the structural account can stand. The plastic ladder remains a mathematical possibility, formally certified, but it is not the one the framework's structure selects.

/-- The `k = 3` closure ladder exists: `1 + r = r³` has a root above 1
(the plastic constant), by the intermediate value theorem. -/
theorem plastic_ladder_exists : ∃ r : ℝ, 1 < r ∧ 1 + r = r ^ 3 := by
  have hcont : ContinuousOn (fun x : ℝ => x ^ 3 - x - 1) (Set.Icc (1 : ℝ) 2) :=
    ((continuous_pow 3).sub continuous_id |>.sub continuous_const).continuousOn
  have hmem : (0 : ℝ) ∈ Set.Ioo ((1 : ℝ) ^ 3 - 1 - 1) ((2 : ℝ) ^ 3 - 2 - 1) := by
    constructor <;> norm_num
  have hivt := intermediate_value_Ioo (by norm_num : (1 : ℝ) ≤ 2) hcont hmem
  obtain ⟨r, hr, hfr⟩ := hivt
  refine ⟨r, hr.1, ?_⟩
  have h0 : r ^ 3 - r - 1 = 0 := hfr
  linarith

/-- The concrete witness for the route-kill: the plastic ladder exists and
its closure is strictly cheaper than the φ closure. -/
theorem plastic_cheaper_than_phi :
    ∃ r : ℝ, 1 < r ∧ 1 + r = r ^ 3 ∧
      Cost.Jcost (1 + r) < Cost.Jcost (1 + phi) := by
  obtain ⟨r, hr, hrc⟩ := plastic_ladder_exists
  refine ⟨r, hr, hrc, ?_⟩
  have hphi2 : 1 + phi = phi ^ 2 := by linarith [phi_sq_eq]
  have h := closure_cost_strictly_decreasing (k := 0) one_lt_phi
    (by simpa using hphi2) hr (by simpa using hrc)
  exact h

/-- **Route-kill measurement.** The per-closure cost `J(1 + r)` is strictly
decreasing as the closure level rises: for any two closure ladders at
consecutive levels, the higher-level ladder has the cheaper closure.
Cost-minimization therefore does not select `k = 2`; it selects nothing
finite. This kills the sub-route "the J-cost excess selects the closure
level by minimization." -/
theorem closure_cost_strictly_decreasing {r s : ℝ} {k : ℕ}
    (hr : 1 < r) (hrc : 1 + r = r ^ (k + 2))
    (hs : 1 < s) (hsc : 1 + s = s ^ (k + 3)) :
    Cost.Jcost (1 + s) < Cost.Jcost (1 + r) := by
  have hsr : s < r := by
    by_contra h
    push_neg at h
    rcases lt_or_eq_of_le h with hlt | heq
    · -- r < s: strict monotonicity of the closure polynomial gives
      -- f_{k+3}(r) < f_{k+3}(s) = 0, but f_{k+3}(r) = r² - 1 > 0.
      have h1 : r ^ (k + 3) - r - 1 < s ^ (k + 3) - s - 1 :=
        closurePoly_strictMono (le_of_lt hr) hlt (by omega)
      have hfs : s ^ (k + 3) - s - 1 = 0 := by
        have e : s ^ (k + 3) = 1 + s := by rw [← hsc]
        linarith
      have hfr : r ^ (k + 3) - r - 1 = r ^ 2 - 1 := by
        have e0 : k + 3 = (k + 2) + 1 := by omega
        have e : r ^ (k + 3) = r * r ^ (k + 2) := by rw [e0, pow_succ']
        rw [e, ← hrc]
        ring
      nlinarith [h1, hfs, hfr, hr,
        mul_pos (sub_pos.mpr hr) (by linarith : (0 : ℝ) < r + 1)]
    · -- r = s: then r^{k+2} = r^{k+3}, impossible.
      rw [← heq] at hsc
      have e1 : r ^ (k + 2) = r ^ (k + 3) := by rw [← hrc, ← hsc]
      have := pow_inj_right hr e1
      omega
  have hlog : Real.log (1 + s) < Real.log (1 + r) :=
    Real.log_lt_log (by linarith) (by linarith)
  have hc : Real.cosh (Real.log (1 + s)) < Real.cosh (Real.log (1 + r)) :=
    cosh_strictMono_on_nonneg (Real.log_nonneg (by linarith)) hlog
  have e1 : Cost.Jcost (1 + s) = Real.cosh (Real.log (1 + s)) - 1 :=
    Cost.GeometricRoot.jcost_eq_cosh_log_sub_one (by linarith)
  have e2 : Cost.Jcost (1 + r) = Real.cosh (Real.log (1 + r)) - 1 :=
    Cost.GeometricRoot.jcost_eq_cosh_log_sub_one (by linarith)
  linarith [e1, e2, hc]

What this page does not claim

The plastic ladder is not claimed to be the framework's preferred scaling or to have any physical role. Cost-minimization is not claimed to be a valid selection principle; the theorem shows it fails to pick a finite level. The declaration does not claim that the plastic constant is irrational or transcendental.

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/PhiClosureSelection.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

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