Holography Moving Recognition Horizon Cut Moving Step Heat Extend Cut

The moving cut

In the Recognition Science framework, a recognition horizon is a discrete boundary that separates what an observer has recorded from what remains exterior. The framework models physical change as a ledger: a record of events, each entry a bit, where the cost of recognition is forced by a proved uniqueness theorem. A fixed horizon has a fixed set of exterior apertures, channels through which bits can be posted. The declaration movingStepHeat_extendCut concerns what happens when that horizon moves: one new exterior aperture is appended while every old exterior and seam reading is preserved.

The theorem states an exact identity. If you take a cut, a complete assignment of bits across the horizon, and extend it by exposing one new bit, then the heat across that moving step, defined as the change in active posted bits, equals the integer value of the newly exposed bit. If the bit is 1, the heat is 1; if the bit is 0, the heat is 0. The extension also increases the aperture count by one and the one-sided horizon record count by one. These three facts bundle together in a single certificate: area capacity grows, the record grows, and the heat is exactly the new bit.

The theorem is proved in a machine-checked library of formal theorems. It is a discrete moving-boundary result, derived from the old context's one-sided record law, with the Rindler rate, the boost parameter, inherited unchanged. On a fixed context, the moving-step heat reduces to the already proved exterior-step heat, so the new theorem is a genuine generalization to varying contexts, not a separate postulate.

What the theorem does not claim is as important as what it proves. It does not supply a smooth interpolation of the moving horizon. It does not introduce null covectors, Raychaudhuri's equation, Ricci curvature, or a sourced field equation. The result is purely discrete and combinatorial, a statement about finite sums of bits, not about differential geometry. It is a first certificate for a varying-context recognition move, not a theory of dynamical horizons in general relativity.

The consequence is that a moving boundary in this framework has a precise, local accounting: the heat is not spread out or renormalized, it is exactly the bit that crosses. This gives a clean test for any future extension: if a proposed smooth limit or continuum version changes this identity, it is not a limit of the discrete theory.

/-- Exact moving-cut heat: only the newly exposed bit is posted. -/
theorem movingStepHeat_extendCut {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 := by
  unfold movingStepHeat
  rw [activeExteriorBits_extendCut]
  ring

/-- 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]⟩

/--
On a fixed context, varying-context heat reduces to the already proved
`exteriorStepHeat`.
-/
theorem movingStepHeat_eq_exteriorStepHeat
    {a s b r : ℕ} {kappa : ℝ}
    {H : LocalHorizonContext a s b r kappa}
    (c c' : LocalCut H) :
    movingStepHeat c c' = exteriorStepHeat c c' := by
  unfold movingStepHeat
  rw [activeExteriorBits_eq_exteriorPotential,
    activeExteriorBits_eq_exteriorPotential,
    exteriorStepHeat_eq_potential]

What this page does not claim

The theorem does not provide a smooth interpolation or any differential-geometric structure for the moving horizon. The theorem does not introduce null covectors, Raychaudhuri's equation, Ricci curvature, or a sourced field equation. The theorem does not claim that all moving boundaries in general relativity satisfy this discrete identity.

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)

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.

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