Holography Pixel Local Recognition Sector Count

The sector count

In the Recognition Science framework, a recognition event is a discrete act of noting, and the framework keeps a ledger, a record of which events balance out. The declaration recognition_sector_count is a theorem about a simple geometric setup: take one square face of a cube, put a single bit (a 0 or 1) at each of its four corners, and ask how many distinct patterns can occur under two rules. The first rule is ledger closure: the four bits must have even parity, meaning their XOR sum is 0, which represents a balanced loop of recognition events around the face. The second rule identifies patterns that differ only by a symmetry of the square, such as a rotation or reflection, as the same physical sector.

The theorem, proved by exhaustive computation in the framework's machine-checked library of formal theorems, states that the number of admissible sectors is exactly 4. The four sectors are the empty loop (0000), a loop along two adjacent edges, a loop along a diagonal, and the full loop around all four edges (1111). This count matches the formula 2^(D-1) for D=3 spatial dimensions, giving 2² = 4. The proof is pure combinatorics on the cube's face: it uses no notions of entropy, area, or any physical scale. The framework's library also confirms the cube has 8 vertices and 6 faces, consistent with the 2³ structure.

In Recognition Science, this result is one leg of a larger effort to derive the area of a recognition pixel, written as a_pix = 4 · H · ℓ_P². The number 4 here is the sector count just proved. The other two factors, the per-event entropy H and the area scale ℓ_P², are separate quantities that this theorem does not address. The framework explicitly leaves open whether this sector count is the same as the geometric coefficient 4 in the area formula, and it notes that the length scale ℓ_P² is provably unreachable from the current dimensionless data, requiring a new normalization that carries a length.

What the theorem does not do is derive the Bekenstein-Hawking entropy formula or any area law. It establishes a finite combinatorial count on a discrete lattice, nothing more. The honest scope is narrow: the forced integer 4, realized concretely as four orbit classes under the square's symmetry group. This is a building block, not the finished building.

/-- **THEOREM (axiom-clean, by `decide`). The D=3 forced lattice yields exactly 4
recognition sectors per boundary face.**

`card(AdmissibleBoundarySectors / FaceStabilizer) = 4 = 2^(D-1)`. No `H`, no `ℓ_P`,
no area input enters any definition; the count is pure ledger-closed plaquette
combinatorics on the forced 8 = 2³ substrate. -/
theorem recognition_sector_count : admissibleSectors.card = 4 := 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 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

/-- **THEOREM (axiom-clean, by `decide`). The D=3 forced lattice yields exactly 4
recognition sectors per boundary face.**

`card(AdmissibleBoundarySectors / FaceStabilizer) = 4 = 2^(D-1)`. No `H`, no `ℓ_P`,
no area input enters any definition; the count is pure ledger-closed plaquette
combinatorics on the forced 8 = 2³ substrate. -/
theorem recognition_sector_count : admissibleSectors.card = 4 := by decide

What this page does not claim

The theorem does not derive the Bekenstein-Hawking entropy formula or any area law. The sector count is not shown to be the same as the geometric coefficient 4 in the area formula. The length scale ℓ_P² is not determined by this result.

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)

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