Holography Pixel Glued Plaquette Act By

The domino action

The declaration actBy is a precise rule for rearranging a domino-shaped configuration. The domino is a 2 by 1 rectangle made of two square faces sharing an edge, and each of its six corners carries a single bit, either 0 or 1. The rule takes one of the rectangle's four symmetries, such as a horizontal mirror or a 180 degree rotation, and applies it to the six bits, moving each bit to the corner where its counterpart lands. This is the formal machinery behind counting recognition sectors, the distinct ways a ledger-closed configuration can look up to symmetry.

The classical setup here is elementary combinatorics. A ledger, a discrete record of events, is closed when every face has an even number of 1 bits on its four corners. For a single square face, the count of such closed patterns, modulo its symmetries, is exactly 4. The natural guess, called the count to area ansatz, is that this number acts as a density: twice the area should give twice the count. The domino, with twice the area of one face, would then carry 8 sectors.

The machine-checked result contradicts that guess. The glued domino carries exactly 9 sectors, not 8. The extra one comes from the shared edge, a boundary correction that the simple multiplicative picture misses. The theorem states this as 9 = 2 times 4 plus 1, and the full list of the nine canonical representatives is verified by direct computation. This is a theorem in the framework's machine-checked library of formal theorems, proved by deciding each case, with no extra axioms beyond the standard three.

In Recognition Science, this settles a specific open question, labeled LB1, about whether the per-pixel integer 4 is an area coefficient. The answer is no: the count is super-additive, not extensive. The framework does not treat this as a failure. The single-face count of 4 remains a theorem, but identifying it as a multiplicative area density is now shown to be a modeling choice, not a consequence of the enumeration. The path from counts to physical area cannot run through simple multiplication.

What actBy does not claim is just as important. It does not establish any physical area value, any connection to the Planck scale, or any statement about real holographic screens. It only governs a finite combinatorial object, a six-bit domino, and its symmetry group. The result is a negative finding about a specific ansatz, not a positive construction of geometry from counts. The question of how recognition sectors actually relate to area remains open.

/-- **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

/-- The explicit nine orbit minima of the domino. -/
theorem admissibleSectors_eq :
    admissibleSectors = ({0, 7, 9, 14, 18, 21, 27, 45, 63} : Finset DominoCfg) := by decide

What this page does not claim

No physical area value or Planck-scale coefficient is derived. No statement about real holographic screens or quantum gravity follows from this finite combinatorial result. The single-face count of 4 is not refuted; only its role as an area density is rejected.

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)

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: