Foundation Pair Kernel Affine Weyl Event Action Finite Weyl Clock Occupation Cos

The clock cost's guarantee

The declaration finiteWeylClockOccupationCost27_nonneg is a theorem in the Recognition Science framework's machine-checked library of formal theorems. It proves that a particular bookkeeping quantity, the clock occupation cost, is always greater than or equal to zero. Think of this cost as a kind of energy bill for how much a proposed configuration of events deviates from a clock's regular ticking pattern. The theorem says this bill can never be a negative number, no matter what configuration you feed it.

The proof works by unpacking the definition of the cost and showing it is a sum of squared terms. Since a square of any real number is never negative, and a sum of non-negative numbers is non-negative, the result follows. The same argument also proves the companion shift occupation cost is non-negative. These two facts combine into a larger theorem, affineWeylEventAction27_nonneg, which shows the total action, the combined cost of a shift and a clock together, is also never negative.

The non-negativity of the clock cost is what makes a later uniqueness result possible. When the clock cost is strictly positive, the framework proves there is exactly one way to balance the shift and clock costs against each other by adjusting a dilation coordinate. That uniqueness would not hold if the clock cost could be zero or negative in a degenerate way. So this small theorem is a foundation stone for the framework's claim that its model does not carry a free shift-to-clock ratio.

In Recognition Science, this declaration is part of a larger construction called the affine Weyl event action, which models how events might be assigned costs under a dynamical rescaling. The framework is explicit that this is a model, not a forced consequence of its core principles. The declaration does not claim that the clock cost is ever positive, only that it is never negative. It does not claim that the model describes physical time, that it attaches to SI units, or that it forces Fourier exchange. Those remain open targets in the framework, not results of this theorem.

theorem finiteWeylClockOccupationCost27_nonneg
    (center : WeylEventSite27)
    (ψ : WeylEventAmplitude27) :
    0 ≤ finiteWeylClockOccupationCost27 center ψ := by
  unfold finiteWeylClockOccupationCost27
  apply Finset.sum_nonneg
  intro axis _
  apply Finset.sum_nonneg
  intro site _
  exact mul_nonneg (Complex.normSq_nonneg _) (sq_nonneg _)

theorem finiteWeylClockOccupationCost27_nonneg
    (center : WeylEventSite27)
    (ψ : WeylEventAmplitude27) :
    0 ≤ finiteWeylClockOccupationCost27 center ψ := by
  unfold finiteWeylClockOccupationCost27
  apply Finset.sum_nonneg
  intro axis _
  apply Finset.sum_nonneg
  intro site _
  exact mul_nonneg (Complex.normSq_nonneg _) (sq_nonneg _)

theorem affineWeylEventAction27_nonneg
    (event : RealizedPostingEvent3 3)
    (dilation : ℝ)
    (ψ : WeylEventAmplitude27) :
    0 ≤ affineWeylEventAction27 event dilation ψ := by
  unfold affineWeylEventAction27
  exact add_nonneg
    (mul_nonneg (sq_nonneg _)
      (finiteWeylShiftOccupationCost27_nonneg ψ))
    (mul_nonneg (sq_nonneg _)
      (finiteWeylClockOccupationCost27_nonneg _ ψ))

/-- When the clock cost is positive, the positive on-shell dilation is unique.
Thus this MODEL action does not retain a free shift/clock ratio. -/
theorem affineWeylDilationCritical27_unique_of_clock_pos
    (event : RealizedPostingEvent3 3)
    (ψ : WeylEventAmplitude27)
    (hclock :
      0 < finiteWeylClockOccupationCost27
        (realizedPostingWeylClockCenter27 event) ψ)
    (dilation₁ dilation₂ : EventDilationCoord)
    (hcritical₁ :
      AffineWeylDilationCritical27 event ψ dilation₁)
    (hcritical₂ :
      AffineWeylDilationCritical27 event ψ dilation₂) :
    dilation₁ = dilation₂ := by
  have hbalance₁ :=
    (affineWeylDilationCritical27_iff_balance
      event ψ dilation₁).mp hcritical₁
  have hbalance₂ :=
    (affineWeylDilationCritical27_iff_balance
      event ψ dilation₂).mp hcritical₂
  have hpow : dilation₁.1 ^ 4 = dilation₂.1 ^ 4 := by
    nlinarith
  apply Subtype.ext
  exact
    (pow_left_inj₀ dilation₁.2.le dilation₂.2.le
      (by norm_num : 4 ≠ 0)).mp hpow

/-- Reciprocal affine pairing of the shift and clock costs. -/
def affineWeylEventAction27
    (event : RealizedPostingEvent3 3)
    (dilation : ℝ)
    (ψ : WeylEventAmplitude27) : ℝ :=
  dilation⁻¹ ^ 2 * finiteWeylShiftOccupationCost27 ψ +
    dilation ^ 2 *
      finiteWeylClockOccupationCost27
        (realizedPostingWeylClockCenter27 event) ψ

What this page does not claim

The clock occupation cost is ever positive, only that it is never negative. The model describes physical time or attaches to SI units. The theorem forces Fourier exchange or molecular identity, which remain open in the framework.

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/PairKernelAffineWeylEventAction.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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