IndisputableMonolith.Verification.OneLtPhiCert

   1import Mathlib
   2import IndisputableMonolith.PhiSupport.Lemmas
   3
   4namespace IndisputableMonolith
   5namespace Verification
   6namespace OneLtPhi
   7
   8open IndisputableMonolith.PhiSupport
   9
  10/-!
  11# One Less Than Phi Certificate
  12
  13This certificate proves that `1 < Constants.phi`, i.e., φ > 1.
  14
  15## Key Result
  16
  17`1 < Constants.phi`
  18
  19## Why this matters for the certificate chain
  20
  21This is a **basic bound** on the golden ratio:
  22
  231. **Definition**: φ = (1 + √5)/2
  242. **Lower bound**: φ > 1
  253. **Exact value**: φ ≈ 1.618
  26
  27This bound is used throughout Recognition Science:
  28- Ensures φ-powers grow exponentially
  29- Guarantees J-cost has proper behavior at φ
  30- Ensures the geometric series based on 1/φ converges
  31
  32## Mathematical Content
  33
  34Since φ = (1 + √5)/2 and √5 > 2:
  35```
  36φ = (1 + √5)/2 > (1 + 2)/2 = 1.5 > 1
  37```
  38
  39More precisely, √5 ≈ 2.236, so:
  40```
  41φ = (1 + 2.236)/2 = 3.236/2 ≈ 1.618 > 1
  42```
  43
  44## Physical Significance
  45
  46The golden ratio being greater than 1 ensures:
  47- Ledger balances grow (φ > 1) rather than shrink
  48- Cost minimization selects φ uniquely (J(φ) is a minimum, not a maximum)
  49- The discrete spectrum of energy levels is well-ordered
  50
  51This is part of what makes φ the "natural unit" for Recognition Science.
  52
  53## Relationship to Other Properties
  54
  55This bound works with:
  56- `phi_pos` (implicit): φ > 0
  57- `phi_ne_zero`: φ ≠ 0
  58- `phi_squared`: φ² = φ + 1 (implies φ > 1 since φ > 0)
  59-/
  60
  61structure OneLtPhiCert where
  62  deriving Repr
  63
  64/-- Verification predicate: phi is greater than 1. -/
  65@[simp] def OneLtPhiCert.verified (_c : OneLtPhiCert) : Prop :=
  66  1 < Constants.phi
  67
  68/-- Top-level theorem: the certificate verifies. -/
  69@[simp] theorem OneLtPhiCert.verified_any (c : OneLtPhiCert) :
  70    OneLtPhiCert.verified c := by
  71  exact one_lt_phi
  72
  73end OneLtPhi
  74end Verification
  75end IndisputableMonolith
  76