Holography Local Recognition Horizon Cut Horizon Record Eq Joint Plus Seam

The seam in the record

In the Recognition Science framework, a horizon is modeled as a boundary where information is recorded. The framework's library, a machine-checked collection of formal theorems, contains a result called horizonRecord_eq_joint_plus_seam. In plain terms, this theorem states that the total count of bits in the horizon record equals the number of bits needed to describe the joint state of the boundary's exterior and its seam, plus the number of bits in the seam itself. The seam is the shared edge between the exterior region and the rest of the universe.

The theorem's power comes from what it excludes. It does not claim that the horizon record is the same as the joint boundary marginal, which would be a simpler count. In fact, the framework proves the opposite: when the seam is nonempty, the one-sided horizon record cannot equal the joint boundary marginal. The seam must be double-posted, appearing once as part of the joint description and once as its own separate contribution. This double-counting is not an error but a structural necessity, and it is the load-bearing fact that makes the one-sided record distinct from a naive joint count.

The theorem operates within a specific model. It assumes a one-sided cut configuration, where data from the interior and the rest of the universe are not visible in the exterior record. It also assumes a near-horizon rate model with a positive surface gravity. Under these conditions, the theorem is proved as a formal consequence. The library's proof is axiom-clean, meaning it relies only on the standard axioms of the underlying type theory, with no additional assumptions specific to the framework.

What the theorem does not do is import any physics beyond the record bookkeeping. It does not involve a stress tensor, Ricci tensor, focusing law, curvature match, Unruh effect, or Einstein equation. The result is purely about the discrete structure of the record. It establishes that the ledger balances along every finite trajectory of closed local cuts, and that the exterior heat is exactly the change in the record's potential. This is a statement about the consistency of the ledger, not about the dynamics of spacetime.

In the broader context of the framework, this theorem provides a precise, formal answer to a question about information at a boundary. It shows that the one-sided record is not a simple projection but carries an extra seam contribution. This distinction is what allows the framework to discriminate between the one-sided record and the joint marginal, a difference that is essential for its later constructions. The theorem is a small but sharp piece of the framework's foundation, clarifying the exact structure of the horizon record.

/--
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

/--
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

/--
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

/--
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

What this page does not claim

The theorem does not claim that the horizon record is the same as the joint boundary marginal. The theorem does not claim any connection to Unruh radiation or continuum thermality. The theorem does not claim that the one-sided cut model is the unique or physically necessary description of a horizon.

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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