Holography Moving Recognition Horizon Cut Posted One Extension Bundle

A moving boundary, one bit at a time

In the Recognition Science framework, a recognition event, a discrete act of recording, happens across a cut that separates an exterior, what is being observed, from an interior, the observer's record. A ledger, a discrete record of events, tracks these cuts. The theorem posted_one_extension_bundle describes what happens when that cut moves: it exposes one new exterior aperture, a channel for information, and appends one new bit to the record. The theorem proves three exact, simultaneous facts about this single step: the number of exterior apertures increases by one, the one-sided horizon record count increases by one, and the active heat across the moving cut equals the value of the newly exposed bit, which is 1 for a posted one and 0 for a posted zero.

This is a discrete moving-boundary theorem. It is not a statement about smooth spacetime. It does not supply a continuous interpolation of the cut's motion, nor does it introduce null covectors, Raychaudhuri's equation, Ricci curvature, or a sourced field equation. The theorem is a precise, combinatorial identity about what happens when a boundary steps outward by one unit in a discrete model. The extension is not postulated independently; its one-sided record law is derived from the old context's law, and its Rindler rate, a measure of acceleration, is inherited unchanged. The theorem shows that the new context is forced by the old one, not added by hand.

The theorem is part of a machine-checked library of formal theorems, meaning the proof is verified by a computer. The library contains the full development: snocReading appends a new coordinate to an exterior reading, extendContext builds the new context, extendCut extends the cut itself, and movingStepHeat defines the heat as the change in active posted bits. The bundle theorem assembles these pieces into a single statement. A companion theorem, posted_zero_extension_decoy, shows that exposing a zero still grows the area capacity but posts no active heat, a useful contrast that highlights the role of the bit's value.

In Recognition Science, this result is a certificate for the first varying-context recognition-cut move. It establishes that a moving boundary in this discrete setting behaves like a simple information-processing step: one new bit of information is exposed, and the heat it carries is exactly that bit's value. This is a foundational result for the framework's holographic ideas, but it is a discrete combinatorial theorem, not a claim about the smooth geometry of general relativity.

/-- 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]⟩

/--
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

Not a claim about smooth spacetime geometry, null covectors, Raychaudhuri's equation, Ricci curvature, or a sourced field equation. Not a derivation of the fine-structure constant or any other physical constant from this theorem alone. Not a statement about the physical mechanism that realizes a recognition event in the world.

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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