Holography Moving Recognition Horizon Cut Extend Cut Aperture Count

The moving cut

In the Recognition Science framework, a recognition event is a discrete record of a bit being read, and a cut is a chosen boundary that separates exterior aperture channels, which are the positions where bits can be read, from interior ones. The declaration extendCut_apertureCount is a machine-checked theorem about what happens when this boundary moves outward to expose one new exterior channel. It proves that the total number of exterior channels, called the aperture count, increases by exactly one when the cut is extended by a single bit.

The theorem is part of a larger discrete moving-boundary result. It states that if you take a valid cut and append one new bit to its exterior reading, the new cut has an aperture count exactly one greater than the old one. This is not an approximation or a limiting statement; it is an exact identity in the formal system. The proof is a direct calculation using the definition of the aperture count as the sum of two natural numbers, and it is verified by the machine-checked library of formal theorems.

The same library also proves a companion result about the heat, which is the change in active posted bits across a step. When the newly exposed bit is a 1, the moving-step heat is exactly 1; when it is a 0, the heat is 0, even though the aperture count still grows by one. This distinction is the content of two further theorems, posted_one_extension_bundle and posted_zero_extension_decoy. The framework models a moving horizon as a sequence of such discrete steps, each adding one channel and possibly one unit of heat.

What the theorem does not claim is as important as what it proves. It is a statement about a discrete, finite extension of a cut. It does not provide a smooth interpolation between cuts, nor does it introduce null covectors, Ricci curvature, Raychaudhuri's equation, or a sourced field equation. Those are concepts from continuous differential geometry that the framework does not derive here. The result is a discrete combinatorial identity, not a statement about the smooth structure of spacetime.

/-- One exterior extension adds exactly one aperture channel. -/
theorem extendCut_apertureCount {a s b r : ℕ} {kappa : ℝ}
    {H : LocalHorizonContext a s b r kappa}
    (c : LocalCut H) (x : ZMod 2) :
    apertureCount (extendCut c x) = apertureCount c + 1 := by
  unfold apertureCount
  omega

/-- Exposing a posted `1` adds one aperture, one horizon-record unit, and one heat unit. -/
theorem posted_one_extension_bundle
    {a s b r : ℕ} {kappa : ℝ}
    {H : LocalHorizonContext a s b r kappa}
    (c : LocalCut H) :
    apertureCount (extendCut c 1) = apertureCount c + 1 ∧
      (extendContext H).horizonRecord = H.horizonRecord + 1 ∧
      movingStepHeat c (extendCut c 1) = 1 :=
  ⟨extendCut_apertureCount c 1, rfl, by
    rw [movingStepHeat_extendCut, postedBitValue_one]⟩

/-- Exposing a `0` still grows area capacity but posts no active heat. -/
theorem posted_zero_extension_decoy
    {a s b r : ℕ} {kappa : ℝ}
    {H : LocalHorizonContext a s b r kappa}
    (c : LocalCut H) :
    apertureCount (extendCut c 0) = apertureCount c + 1 ∧
      movingStepHeat c (extendCut c 0) = 0 :=
  ⟨extendCut_apertureCount c 0, by
    rw [movingStepHeat_extendCut, postedBitValue_zero]⟩

/-- Certificate for the first varying-context recognition-cut move. -/
structure MovingRecognitionHorizonCutCert : Prop where
  capacity_step :
    ∀ {a s b r : ℕ} {kappa : ℝ}
      {H : LocalHorizonContext a s b r kappa}
      (c : LocalCut H) (x : ZMod 2),
      apertureCount (extendCut c x) = apertureCount c + 1
  record_step :
    ∀ {a s b r : ℕ} {kappa : ℝ}
      (H : LocalHorizonContext a s b r kappa),
      (extendContext H).horizonRecord = H.horizonRecord + 1
  heat_step :
    ∀ {a s b r : ℕ} {kappa : ℝ}
      {H : LocalHorizonContext a s b r kappa}
      (c : LocalCut H) (x : ZMod 2),
      movingStepHeat c (extendCut c x) = postedBitValue x
  zero_decoy :
    ∀ {a s b r : ℕ} {kappa : ℝ}
      {H : LocalHorizonContext a s b r kappa}
      (c : LocalCut H),
      apertureCount (extendCut c 0) = apertureCount c + 1 ∧
        movingStepHeat c (extendCut c 0) = 0

What this page does not claim

The theorem does not provide a smooth interpolation between cuts. It does not introduce null covectors, Ricci curvature, Raychaudhuri's equation, or a sourced field equation.

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

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