Holography Pixel Local Sector Count Eq Two Pow

The pixel sector count

The theorem sector_count_eq_two_pow establishes a precise combinatorial fact. In three dimensions, the framework's forced eight-tick cell is a cube, which has eight vertices and six faces. Consider one square face of that cube. The framework places one recognition bit, a single binary value, on each of the face's four vertices, giving sixteen possible configurations. The theorem states that, after applying two natural conditions, exactly four of those configurations remain distinct.

The first condition is ledger closure, a discrete record of events that must balance to zero. For a closed loop of recognition around the face, this means the four vertex bits must sum to zero under XOR, also known as even parity. This rule admits eight of the sixteen configurations. The second condition identifies configurations that differ only by a symmetry of the square: a rotation or a reflection. The square's symmetry group, called the dihedral group D4, has eight elements. Quotienting the eight admissible configurations by these eight symmetries leaves four orbits, represented by the bit patterns 0000, 0011, 0101, and 1111.

These four orbits are the theorem's content. They correspond to the empty loop, an adjacent-edge loop, a diagonal loop, and the full loop. The theorem proves that this count equals 2^(3-1), or 2 squared, which is 4. The proof is a direct computation, checked by a machine-checked library of formal theorems. The result is a pure counting statement about a square's vertices under a parity rule and a symmetry group. It involves no entropy, no area, and no length scale.

In Recognition Science, this count is the substrate-native realization of the integer 4 that appears in the holographic pixel area formula a_pix = 4 · H · ℓ_P². The framework separates that formula into three parts: the integer 4, the per-event entropy H, and the area scale ℓ_P². This theorem supplies the integer leg of that split. It does not, however, connect the sector count to the area coefficient in the formula, nor does it derive the length scale ℓ_P². Both links remain open targets, deliberately not proved here.

The honest scope is narrow. The theorem proves the count is 4, nothing more. It does not claim a derived Bekenstein-Hawking entropy of one quarter. The sector count is a combinatorial fact about a cube face, while the area coefficient in the holographic formula requires a separate argument. The length scale is provably unreachable from the current dimensionless data, as a separate no-go result shows. This result closes the integer leg and leaves the other two legs for future work.

/-- The geometric exponent realized: `2^(D-1)` at `D = 3` is the sector count. -/
theorem sector_count_eq_two_pow : admissibleSectors.card = 2 ^ (3 - 1) := by decide

/-- The four sectors are exactly the orbit minima `{0000, 0011, 0101, 1111}`
(empty loop, an adjacent-edge loop, a diagonal loop, the full loop). -/
theorem admissibleSectors_eq :
    admissibleSectors = ({0, 3, 5, 15} : Finset FaceCfg) := by decide

/-- The forced 8-tick cell in D=3 is the cube: `2³ = 8` vertices. -/
theorem cube_vertices : Fintype.card (Fin 3 → Bool) = 8 := by decide

/-- The cube has 6 faces (an axis together with a side). -/
theorem cube_faces : Fintype.card (Fin 3 × Bool) = 6 := by decide

What this page does not claim

This theorem does not derive the area coefficient in the holographic pixel area formula. This theorem does not derive the length scale ℓ_P². This theorem does not claim a derived Bekenstein-Hawking entropy of one quarter.

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/PixelLocal.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.

Derived articles

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: