Holography Local Recognition Horizon Cut One Sided Horizon Record Ne Joint Margi

The one-sided record

The declaration oneSided_horizonRecord_ne_joint_marginal is a theorem in the framework's machine-checked library of formal theorems. It states a precise inequality about a horizon record, a discrete record of events kept at a boundary. In plain language: if a horizon record is one-sided, meaning it only stores data visible from the exterior, and if there is at least one seam bit, a boundary marker between exterior and interior regions, then the length of that record cannot equal the number of distinct joint boundary states. The record must be strictly larger.

The reason is a double posting. The theorem builds on a model where the one-sided cut forces the horizon record to double-post the seam, meaning the seam's contribution is counted twice in the record's length. The joint boundary marginal, by contrast, counts each boundary state once. With a nonempty seam, the double posting adds extra length, so the two quantities cannot match. The theorem's proof is direct: it assumes they are equal, derives that the seam size must be zero, and contradicts the premise that the seam is nonempty.

The declaration does not claim anything about the physical nature of the horizon. It does not assert that the horizon radiates, that it has a temperature, or that it obeys any continuum thermodynamics. It does not involve a stress tensor, a Ricci tensor, a focusing law, or an Einstein equation. The theorem is purely about the combinatorial structure of the record: given the one-sided model, the arithmetic of the double-posted seam forces the inequality. It is a statement about bookkeeping, not about spacetime geometry.

The practical consequence is a discrimination test. Any proposed horizon record that claims to be one-sided but has a length equal to the joint boundary marginal, with a nonempty seam, is internally inconsistent. The double posting is not an optional feature; it is load-bearing. This gives a formal criterion for what a one-sided record must look like, and it rules out a whole class of simpler candidates.

/--
Discrimination: when the seam is nonempty, the one-sided horizon record cannot
equal the joint boundary marginal. The seam double-post is load-bearing.
-/
theorem oneSided_horizonRecord_ne_joint_marginal {a s b r : ℕ} {kappa : ℝ}
    (H : LocalHorizonContext a s b r kappa)
    (hs : 0 < s) :
    H.horizonRecord ≠ Nat.log2 (((closedSet a s b r).image projAB).card) := by
  intro heq
  have h := horizonRecord_eq_joint_plus_seam H
  have hEq :
      Nat.log2 (((closedSet a s b r).image projAB).card) + s =
        Nat.log2 (((closedSet a s b r).image projAB).card) := by
    rw [← h, heq]
  have hs0 : s = 0 := Nat.add_eq_left.mp hEq
  exact (Nat.pos_iff_ne_zero.mp hs) hs0

/--
Given the one-sided-cut MODEL on the shared context, the horizon record equals
the joint boundary marginal bit count plus the seam bit count.
-/
theorem horizonRecord_eq_joint_plus_seam {a s b r : ℕ} {kappa : ℝ}
    (H : LocalHorizonContext a s b r kappa) :
    H.horizonRecord =
      Nat.log2 (((closedSet a s b r).image projAB).card) + s :=
  horizon_record_double_posts_seam a s b r H.horizonRecord H.oneSided

/--
Shared local-horizon context joining the one-sided-cut MODEL, the posted-record
carrier dimensions, and the near-horizon rate MODEL. No thermality or curvature
premise is included.
-/
structure LocalHorizonContext (a s b r : ℕ) (kappa : ℝ) where
  horizonRecord : ℕ
  oneSided : HorizonSumsPerSide a s b r horizonRecord
  rindler : NearHorizonRindlerForm kappa

What this page does not claim

The theorem does not claim that a horizon has a temperature or radiates. The theorem does not claim that the horizon record is physically realizable. The theorem does not claim that the one-sided model is the only possible model for a horizon record.

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

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