Holography Horizon One Sided Cut Horizon Carries One Side Holds

A horizon's one-sided record

A horizon is a boundary that can be seen from only one side. An observer outside a black hole cannot see inside it, so the outside world's record of the horizon cannot include any information from the interior. The Recognition Science framework models this situation as a ledger, a discrete record of events, split by a cut into an exterior side, an interior side, and a seam of shared vertices where severed edges meet.

The framework's machine-checked library of formal theorems proves a specific consequence of this one-sidedness. When the exterior side traces over the hidden interior, it still realizes every possible reading of the seam bits on its own. The interior side, traced over the exterior, does the same. Each side independently carries a full copy of the shared edge record. The theorem seam_posted_by_A and its counterpart seam_posted_by_B establish this duplication for any region size, not just a fixed example.

Because both sides carry the seam, summing their accessible records double-posts it. The identity seam_identity states this exactly: the bits on side A plus the bits on side B equal the bits on the combined region plus the seam size s. In the concrete domino face where each pixel has two private and two shared vertices, this gives four bits per pixel, matching the known gluing law. The framework derives this additive reading from causal one-sidedness rather than assuming it.

The proof is axiom-clean linear algebra over GF(2), the field with two elements. It does not claim that a horizon is a one-sided causal cut; that identification is the explicit named premise HorizonIsOneSidedCut. The theorem takes that premise and forces the duplicated record. The falsifier is sharp: if horizon entropy were the joint marginal, with the interior accessible, the per-pixel count would be one, not four, and the area law would fail by a factor of four.

/-- **The exterior side posts a full private copy of the seam.** Tracing out the interior,
the exterior's accessible marginal realizes ALL `2^s` seam readings — a complete private copy
of every severed-edge record, reconstructed with no access to the interior. -/
theorem seam_posted_by_A (a s b r : ℕ) :
    (closedSet a s b r).image projSeam = Finset.univ := by
  apply Finset.eq_univ_of_forall
  intro gS
  refine Finset.mem_image.mpr ⟨compA (a := a) (b := b) (r := r) 0 gS,
    Finset.mem_filter.mpr ⟨Finset.mem_univ _, ?_⟩, ?_⟩
  · exact compA_closed 0 gS
  · rfl

/-- **The interior side posts a full private copy of the seam.** Symmetrically, tracing out
the exterior, the interior's accessible marginal realizes ALL `2^s` seam readings. Both sides
independently carry the severed-edge records — they are duplicated across the cut. -/
theorem seam_posted_by_B (a s b r : ℕ) :
    (closedSet a s b r).image projSeam = Finset.univ := by
  apply Finset.eq_univ_of_forall
  intro gS
  refine Finset.mem_image.mpr ⟨compB (a := a) (b := b) (r := r) gS 0,
    Finset.mem_filter.mpr ⟨Finset.mem_univ _, ?_⟩, ?_⟩
  · exact compB_closed gS 0
  · rfl

/-- **The double-posting identity (in bits).** Summing the two sides' accessible marginals
overcounts the joint boundary marginal by **exactly the seam bit count `s`**:
`(a+s) + (s+b) = (a+s+b) + s`. The seam is posted once by each side. This is the symbolic,
all-sizes generalization of `SharedCutMarginal.sum_of_marginals_overcounts_joint_by_shared_bits`
(the `N = 3` `decide` fact). -/
theorem seam_identity (a s b r : ℕ) :
    Nat.log2 (((closedSet a s b r).image projA).card)
        + Nat.log2 (((closedSet a s b r).image projB).card)
      = Nat.log2 (((closedSet a s b r).image projAB).card) + s := by
  rw [margA_bits, margB_bits, margAB_bits]; omega

/-- **κ = 4: each pixel's accessible marginal is the full 4 bits.** The exterior-side pixel
(`a = 2` private + `s = 2` seam) realizes all `2^4 = 16` readings — reproducing
`SharedCutMarginal.domino_leftFace_support_card = 16` symbolically. Each pixel posts its own
4-edge record, and summing over pixels double-posts each shared edge: this is `κ = 4`. -/
theorem kappa_per_pixel_is_four (r : ℕ) :
    Nat.log2 (((closedSet 2 2 2 r).image projA).card) = 4 := by
  rw [margA_bits]

What this page does not claim

The theorem does not prove that a horizon is a one-sided causal cut; that is a named premise. The theorem does not derive the Bekenstein entropy formula itself, only the per-pixel count. The result does not apply to a joint marginal where the interior is accessible.

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/HorizonOneSidedCut.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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