Holography Moving Recognition Horizon Cut Active Exterior Bits Eq Exterior Poten

The moving horizon cut

In the Recognition Science framework, a horizon is a discrete boundary between recorded states. The framework models physical change as a ledger, a discrete record of events, where each event posts a bit of information. A fixed horizon has a fixed set of exterior channels. The theorem activeExteriorBits_eq_exteriorPotential concerns what happens when that boundary moves, when one new exterior channel is appended while all old readings are preserved.

The theorem states that the active posted bits on the exterior, written as a finite sum, equal the committed exterior potential. In plain terms, the heat carried across a moving cut is exactly the change in active posted bits. When the cut extends by one aperture and that aperture exposes a 1, the heat is one unit; when it exposes a 0, the heat is zero. The new context is not postulated independently: its one-sided record law is derived from the old context's law, and its boost rate is inherited unchanged.

The theorem is proved in the framework's machine-checked library of formal theorems, with no framework-specific axioms. It is a discrete moving-boundary result, not a smooth one. It does not supply a smooth interpolation, null covectors, Raychaudhuri, Ricci curvature, or a sourced field equation. It is a statement about finite sums of bits on a discrete cut, not about continuous spacetime geometry.

What the theorem changes is the picture of horizon dynamics. It shows that a moving boundary in this framework carries heat only through the bits it exposes, and that the heat is exactly the value of those bits. This is a sharp, exact identity for a discrete step, not an approximation. The framework's library proves it as a theorem, which means it holds in every model of the framework's axioms.

/-- The finite-sum presentation is exactly the committed exterior potential. -/
theorem activeExteriorBits_eq_exteriorPotential
    {a s b r : ℕ} {kappa : ℝ}
    {H : LocalHorizonContext a s b r kappa}
    (c : LocalCut H) :
    activeExteriorBits c = exteriorPotential c := by
  unfold activeExteriorBits exteriorPotential exteriorRecord
  simp [RecordMonotonicity.recordWeight, postedBitValue, List.sum_ofFn]

/-- 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

/--
Extend a one-sided local-horizon context by one exterior aperture.  The new
one-sided law follows from the old one; the boost-rate model is unchanged.
-/
def extendContext {a s b r : ℕ} {kappa : ℝ}
    (H : LocalHorizonContext a s b r kappa) :
    LocalHorizonContext (a + 1) s b r kappa where
  horizonRecord := H.horizonRecord + 1
  oneSided := by
    have h := H.oneSided
    unfold HorizonSumsPerSide at h ⊢
    rw [margA_bits, margB_bits] at h ⊢
    omega
  rindler := H.rindler

What this page does not claim

The theorem does not claim any smooth spacetime geometry, null covectors, Raychaudhuri, Ricci curvature, or a sourced field equation. The theorem does not claim that the framework's moving-cut model reproduces any specific measured gravitational phenomenon. The theorem does not claim that the exterior potential is a physical observable in the conventional sense.

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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