Holography Moving Recognition Horizon Cut Moving Recognition Horizon Cut Cert

The moving cut certificate

In the Recognition Science framework, a horizon is a boundary that separates exterior apertures, which can be read, from interior ones, which cannot. A fixed horizon stays put; a moving one gains a new exterior aperture while preserving every old exterior reading and seam reading. The declaration movingRecognitionHorizonCutCert is a certificate, a machine-checked theorem, that this first varying-context move obeys exact discrete identities.

The certificate proves three things happen together when one new exterior-private aperture is appended. First, the exterior aperture capacity increases by exactly one. Second, the one-sided horizon record count increases by exactly one. Third, the active posted-record heat across the varying cut is exactly the value of the newly exposed bit: posting a 1 adds one unit of heat, while posting a 0 adds none. The new context is not postulated independently; its one-sided record law is derived from the old context's horizon sums law, and its Rindler rate is inherited unchanged.

The theorem also shows that on a fixed context, the varying-context heat reduces to the already proved exterior step heat, so the moving-cut result is a strict generalization. The certificate bundles these identities into a single structure, and the theorem movingRecognitionHorizonCutCert constructs that structure from the component proofs.

What the certificate does not claim is as important as what it proves. It is a discrete moving-boundary theorem. It does not supply a smooth interpolation, null covectors, Raychaudhuri, Ricci curvature, or a sourced field equation. Those belong to continuum physics, and the framework does not pretend this discrete step reaches them.

theorem movingRecognitionHorizonCutCert :
    MovingRecognitionHorizonCutCert where
  capacity_step := extendCut_apertureCount
  record_step := extendContext_horizonRecord
  heat_step := movingStepHeat_extendCut
  zero_decoy := posted_zero_extension_decoy

/-- 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

@[simp] theorem extendContext_horizonRecord
    {a s b r : ℕ} {kappa : ℝ}
    (H : LocalHorizonContext a s b r kappa) :
    (extendContext H).horizonRecord = H.horizonRecord + 1 :=
  rfl

/-- 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 certificate does not provide a smooth interpolation between discrete cuts. It does not derive null covectors, Raychaudhuri, Ricci curvature, or a sourced field equation. It does not claim the moving-cut identities hold for arbitrary context changes beyond the single append move.

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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