Foundation Pair Kernel Scale Bearing Self Dual Posting Law Scale Bearing Posting

The fixed point of dilation

In the Recognition Science framework, a fundamental physical process is modeled as an action, a number assigned to a configuration that the process then minimizes. The relevant action here is the sum of two parts: a cost associated with changing the scale, or dilation, of an event, and a cost associated with the event's internal field configuration. The theorem scaleBearingPostingDilationResponse27_eq_zero_iff concerns the derivative of this total action with respect to the dilation. It states that this derivative is zero if and only if the dilation equals one.

This is a statement about a fixed point. A derivative of zero means the action is at a stationary point, neither increasing nor decreasing with a small change in scale. The theorem proves that the only such stationary point occurs at the unit dilation, the scale where no stretching or shrinking has been applied. The function in question is defined as (1 - dilation⁻¹²) / 2, and the theorem establishes that this expression equals zero exactly when dilation is one. This is a purely mathematical fact, proved within the framework's machine-checked library of formal theorems.

The significance is that this fixed point is unique. The theorem does not merely show that unit dilation is a stationary point; it shows that no other dilation can be. This uniqueness is what allows the framework to identify the unit scale as physically special, the scale at which the action is minimized. Other theorems in the same module build on this, showing that the action at unit dilation is strictly less than at any other dilation, and that the joint ground state of the system, the configuration minimizing the total action, must occur at this unit dilation.

What the theorem does not claim is that this unit scale corresponds to any particular physical size, such as a meter or a Planck length. The framework's native unit of length, the voxel, is a separate definition. The theorem also does not claim that the unit dilation is the only minimum of the action, only that it is the only point where the derivative with respect to dilation vanishes. The action could in principle have other stationary points that are not minima, though the framework's other results rule this out for the full action. Finally, this theorem is about the mathematical structure of the action, not about any empirical measurement of a physical scale.

/-- The dilation response vanishes exactly at the native event extent. -/
theorem scaleBearingPostingDilationResponse27_eq_zero_iff
    (dilation : PostingDilationCoord) :
    scaleBearingPostingDilationResponse27 dilation.1 = 0 ↔
      dilation.1 = 1 := by
  have hd : dilation.1 ≠ 0 := dilation.2.ne'
  constructor
  · intro h
    unfold scaleBearingPostingDilationResponse27 at h
    have hsquare : dilation.1 ^ 2 = 1 := by
      field_simp [hd] at h
      nlinarith
    nlinarith [dilation.2]
  · intro h
    rw [h]
    norm_num [scaleBearingPostingDilationResponse27]

/-- First variation in the dilation coordinate. -/
def scaleBearingPostingDilationResponse27
    (dilation : ℝ) : ℝ :=
  (1 - dilation⁻¹ ^ 2) / 2

/-- The dilation response vanishes exactly at the native event extent. -/
theorem scaleBearingPostingDilationResponse27_eq_zero_iff
    (dilation : PostingDilationCoord) :
    scaleBearingPostingDilationResponse27 dilation.1 = 0 ↔
      dilation.1 = 1 := by
  have hd : dilation.1 ≠ 0 := dilation.2.ne'
  constructor
  · intro h
    unfold scaleBearingPostingDilationResponse27 at h
    have hsquare : dilation.1 ^ 2 = 1 := by
      field_simp [hd] at h
      nlinarith
    nlinarith [dilation.2]
  · intro h
    rw [h]
    norm_num [scaleBearingPostingDilationResponse27]

What this page does not claim

The theorem does not claim that the unit dilation corresponds to any particular physical size, such as a meter. The theorem does not claim that the unit dilation is the only minimum of the action, only the only stationary point. The theorem does not claim any empirical measurement of a physical scale.

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/PairKernelScaleBearingSelfDualPostingLaw.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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