Holography Horizon One Sided Cut

The one-sided cut

In physics, a horizon is a surface that separates what an observer can see from what they cannot. For a black hole, the event horizon hides the interior; an outside observer cannot receive any signal from inside it. This one-sidedness is the starting point of a result in Recognition Science, a framework that models physical structure as a discrete record of events. The framework's library of machine-checked formal theorems has proved a precise statement about what such a one-sided cut forces.

Classically, the entropy of a horizon is thought to be proportional to its area, a fact first conjectured by Jacob Bekenstein in 1972 and given a statistical interpretation by Stephen Hawking in 1974. The framework's result concerns how that entropy is counted when the boundary is divided into small pixels. The question is whether each pixel posts its own record of its edges, or whether shared edges are counted once. The new theorem says the one-sided nature of the cut forces the first option: each side of the cut independently records every bit of the shared seam, so the total is the sum of the two sides, not their union.

The proof is an exercise in linear algebra over the field with two elements, GF(2). The framework models the entire closed universe as a set of bits with a single parity constraint: the sum of all bits is zero. A cut splits this set into four parts: the exterior side's private vertices, the seam vertices shared by both sides, the interior side's private vertices, and the rest of the universe. The central theorems, seam_posted_by_A and seam_posted_by_B, show that each side's accessible marginal, meaning the part it can observe, realizes every possible reading of the seam bits. Because both sides independently post the same seam, the total record double-counts it.

The key identity is bits A + bits B = bits (A∪B) + s, where s is the number of seam bits. This says the sum of the two sides' records exceeds their joint record by exactly the seam. When the horizon record is defined as the sum of the per-side marginals, a premise the framework calls HorizonSumsPerSide, the theorem horizon_record_double_posts_seam derives that the horizon record exceeds the joint by the seam. This is the additive reading, where each pixel's accessible marginal is four bits, matching the known gluing law that a severed edge contributes two bits.

In Recognition Science, this result is a step toward deriving the Bekenstein bound from first principles. The one-sided cut replaces a previously assumed axiom, PerPixelRecordAdditivity, with a derived consequence. The remaining physical input is the identification that a horizon is a one-sided causal cut, which is stated as an explicit premise. If the interior were accessible, the joint marginal would apply and the entropy would be a quarter of the area, failing by a factor of four. The theorem proves the mathematics; the physics of the premise remains a named assumption.

/-- **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

/-- **GIVEN a one-sided cut, the horizon record double-posts the seam.** The horizon record
exceeds the joint boundary marginal by exactly the seam — the additive (`κ = 4`) reading,
forced by causal one-sidedness rather than assumed. -/
theorem horizon_record_double_posts_seam (a s b r horizonRecord : ℕ)
    (h : HorizonSumsPerSide a s b r horizonRecord) :
    horizonRecord = Nat.log2 (((closedSet a s b r).image projAB).card) + s := by
  rw [h]; exact seam_identity a s b r

/-- Verify-target certificate handle (`#print axioms`-gated). -/
theorem horizonOneSidedCutCert : horizon_carries_one_side :=
  horizon_carries_one_side_holds

What this page does not claim

This result does not prove that a physical horizon is one-sided; that is a named premise. This result does not derive the Bekenstein bound itself, only a step toward it. This result does not apply to the interior being accessible, which would give a different counting.

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)

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