IndisputableMonolith.Verification.UnitNormalizationZeroCert

   1import Mathlib
   2import IndisputableMonolith.Cost.FunctionalEquation
   3
   4namespace IndisputableMonolith
   5namespace Verification
   6namespace UnitNormalizationZero
   7
   8open IndisputableMonolith.Cost.FunctionalEquation
   9open Real
  10
  11/-!
  12# Unit Normalization → Zero at Origin Certificate
  13
  14This certificate packages the proof that if F(1) = 0, then G_F(0) = 0.
  15
  16## Key Result
  17
  18If F : ℝ → ℝ satisfies F(1) = 0, then G_F(0) = F(exp(0)) = F(1) = 0.
  19
  20## Why this matters for the certificate chain
  21
  22This result connects unit normalization in the multiplicative domain to the
  23log-coordinate representation:
  24
  251. **Multiplicative domain**: F(1) = 0 means the cost at "unity" is zero
  26   - No cost when there's no deviation from the reference point
  27
  282. **Log-coordinates**: G_F(0) = 0 means the log-coordinate cost at t = 0 is zero
  29   - The origin of log-coordinates (t = 0 ↔ x = exp(0) = 1) has zero cost
  30
  313. **Initial condition for ODE**: Combined with H = G + 1, this gives H(0) = 1,
  32   which is the initial condition for the ODE uniqueness theorem
  33
  34This is a simple but crucial link in the chain:
  35  F(1) = 0 → G_F(0) = 0 → H(0) = 1 → ODE initial conditions
  36
  37## Mathematical Content
  38
  39The proof is immediate from the definitions:
  40  G_F(0) = F(exp(0)) = F(1) = 0
  41-/
  42
  43structure UnitNormalizationZeroCert where
  44  deriving Repr
  45
  46/-- Verification predicate: unit normalization F(1) = 0 implies G_F(0) = 0.
  47
  48This certifies that the log-coordinate representation respects unit normalization. -/
  49@[simp] def UnitNormalizationZeroCert.verified (_c : UnitNormalizationZeroCert) : Prop :=
  50  ∀ (F : ℝ → ℝ), F 1 = 0 → G F 0 = 0
  51
  52/-- Top-level theorem: the certificate verifies. -/
  53@[simp] theorem UnitNormalizationZeroCert.verified_any (c : UnitNormalizationZeroCert) :
  54    UnitNormalizationZeroCert.verified c := by
  55  intro F hUnit
  56  exact G_zero_of_unit F hUnit
  57
  58end UnitNormalizationZero
  59end Verification
  60end IndisputableMonolith
  61