Holography Horizon One Sided Cut Kappa Per Pixel Is Four

The four-bit pixel

In the Recognition Science framework, a ledger is a discrete record of events, and the framework's central claim is that reality keeps such a ledger. The declaration kappa_per_pixel_is_four is a theorem in the framework's machine-checked library of formal theorems. It proves a specific counting fact about a horizon: the boundary of a region whose interior is causally hidden from an outside observer. The theorem states that, under a precise model, each pixel on that horizon carries exactly four bits of accessible information.

The proof works by modeling the horizon as a cut that splits a closed system into two sides. The exterior side has private vertices, the interior side has private vertices, and the two sides share a set of vertices along the seam. The model imposes one global constraint: the sum of all bits across the entire system must be even. The theorem then shows that the exterior observer, who cannot see the interior, can still read every possible configuration of the seam bits. Each side independently realizes all 2^s possible seam readings, where s is the number of seam vertices. This means the seam is effectively posted twice, once by each side.

For a specific physical case, where each pixel has two private and two shared vertices, the theorem computes the result. The accessible marginal for one side has 2^(2+2) = 16 possible configurations, so its information content is log2(16) = 4 bits. This is the kappa_per_pixel_is_four result. The general identity behind it is that the sum of the two sides' information equals the joint information plus the number of seam bits. In this case, 4 + 4 = 6 + 2, so the seam contributes exactly 2 bits, and each pixel's accessible record is 4 bits. The framework presents this as a derivation from the one-sided nature of the cut, not an assumption.

In Recognition Science, this theorem is a step toward a Bekenstein-style entropy bound. The framework's claim is that if horizon entropy is the sum of per-side accessible marginals, then it exceeds the joint marginal by the seam count. The theorem proves the mathematical forcing: a one-sided trace implies private duplicated seam records, which implies the additive reading. The physical identification that a horizon is a one-sided causal cut is stated as an explicit premise, not proved. The falsifier is sharp: if the interior were accessible, the joint marginal would give kappa = 1, and the entropy would fail by a factor of four.

/-- **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]

/-- **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

What this page does not claim

The theorem does not prove that a physical horizon is a one-sided causal cut; that remains an explicit premise. The theorem does not derive the Bekenstein entropy bound itself, only a counting step within the framework's model. The result does not apply to a joint reading where the interior is accessible, which would give kappa = 1.

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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