Holography Pixel Glued Plaquette

The glued domino count

In Recognition Science, the framework that derives physical structure from a forced cost of recognition, a ledger is a discrete record of events. The module PixelGluedPlaquette studies a tiny ledger: a 2 by 1 domino made of two square faces sharing an edge. Each of the domino's six vertices carries one recognition bit, and a configuration is closed when both faces post a balanced loop, meaning each face's four vertex bits XOR to zero.

The question is how the number of recognition sectors, the distinct closed configurations up to symmetry, behaves when area doubles. A single face carries exactly 4 sectors, a theorem proved elsewhere. If the count were an area density, two faces glued along an edge should carry 8 sectors. The machine-checked theorem glued_sector_count proves the actual number is 9, not 8. The count is super-additive: 9 equals 2 times 4 plus 1, where the extra 1 is the boundary correction from the shared edge. The stabilizer here is the Klein four-group of the rectangle, not the larger symmetry of a square, and sectors are counted by canonical orbit representatives.

This result is evidence against the count-to-area ansatz, the idea that the per-pixel integer 4 is a multiplicative area coefficient. The integer 4 per face remains a theorem, but identifying it as an area density is a genuine modeling step, not a consequence of the enumeration. In plain language, the module shows that recognition sectors do not scale extensively with area; the boundary between two regions contributes its own sector. This is a negative result for a specific hypothesis, not a refutation of the framework itself.

The finding matters because it closes a live bet in the framework's development. It does not refute the framework, but it shows that the count-to-area ansatz cannot route through the count being the area. The count is not extensive. This is a concrete, machine-checked boundary correction that any future area model must accommodate.

/-- **THEOREM (axiom-clean, by `decide`). The glued 2×1 domino carries exactly 9
recognition sectors** — NOT `2 · 4 = 8`. -/
theorem glued_sector_count : admissibleSectors.card = 9 := by decide

/-- **The sector count is super-additive (the LB1 finding).** Two faces glued along an
edge give `9 = 2·4 + 1`, where `4` is the single-face count
(`PixelLocal.recognition_sector_count`) and the `+1` is the shared-edge correction. The
count is therefore NOT an extensive area density: the per-pixel integer `4` is not the
area coefficient in the additive sense. -/
theorem glued_super_additive :
    admissibleSectors.card = 2 * PixelLocal.admissibleSectors.card + 1 := by decide

What this page does not claim

This module does not refute the framework or the theorem that a single face carries 4 sectors. The result does not establish any positive area law; it only rules out the simple extensive ansatz. The boundary correction of +1 is not yet connected to any physical constant or measured quantity.

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/PixelGluedPlaquette.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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