Holography Cell Injection

The cell-injection test

Holography cell injection is a test about whether a three-dimensional object's interior is fully recorded on its surface. In Recognition Science, the object is a cube with one recognition bit, a binary value, on each of its 8 vertices. The boundary record is a list of six parities, one per face, each saying whether the four bits on that face sum to an even number. The question: does every change to the interior bits necessarily change the boundary record?

The answer, proved in the framework's machine-checked library of formal theorems, is a sharp dichotomy. Flipping any single vertex bit changes the boundary record from every one of the 256 possible configurations, and it changes exactly the three face records that meet at that vertex. No single interior distinction goes unrecorded. But the record map is not one-to-one: there are 16 distinct interior configurations, forming a group, that all produce the same boundary record. These blind moves are exactly the whole-face flips, including flipping an entire face's four bits, the global complement of all eight bits, and two inscribed tetrahedra.

The blindness is strictly global. Any two configurations with the same boundary record differ on at least 4 of the 8 vertices, half the cell. This bound is tight: flipping one whole face, a move of weight 4, is invisible from every base configuration. The record map has rank 4 and nullity 4, so the image and the kernel each have 16 elements, and 16 times 16 equals 256, the total number of configurations.

In Recognition Science, this test separates two futures for a principle called recognition complementarity, the idea that no interior distinction exists without a boundary-readable witness. The forced cell reduces that principle from a statement about all 256 bulk states to a statement about 16 explicit global parity moves. The module does not prove those 16 moves are gauge, meaning unobservable by fiat; it isolates and classifies them. The fork between whether complementarity stays an axiom or becomes a theorem is now a concrete choice about those 16 moves, computed rather than postulated.

/-- **THE CELL-INJECTION TEST (the panel's question, answered positively for local
moves).** Flipping any single vertex bit of the D=3 cell changes the boundary record,
from every configuration. No single bulk distinction is unrecorded. -/
theorem single_flip_posts :
    ∀ (c : CellCfg) (i : Fin 8), faceRecord (flipv c i) ≠ faceRecord c := by decide

/-- Sharper: a single vertex flip changes EXACTLY three of the six face records — the
three faces incident to that vertex. The posting is local and quantized. -/
theorem single_flip_posts_three :
    ∀ (c : CellCfg) (i : Fin 8),
      (List.zipWith (fun a b => a != b) (faceRecord (flipv c i)) (faceRecord c)).count
        true = 3 := by decide

/-- **The boundary record map is NOT injective on raw bulk states.** The countermodel
of the holography paper survives at cell level unless record-equal states are
physically identified. -/
theorem record_not_injective : ¬ Function.Injective faceRecord := fun h =>
  absurd (h complement_invisible) (by decide)

theorem recordKernel_card : recordKernel.card = 16 := by decide

/-- **THE DICHOTOMY (the module's headline).** Any two distinct bulk configurations
with the SAME boundary record differ on at least 4 of the 8 vertices, half the cell.
Together with `single_flip_posts`: the record misses nothing local; what it misses is
exactly a 4-bit group of cell-global parity moves. -/
theorem record_blind_only_global :
    ∀ c c' : CellCfg, faceRecord c = faceRecord c' → c ≠ c' →
      4 ≤ weight (xorCfg c c') := by decide

/-- Record rank of the whole cell: 4 posted bits (`log₂ 16`). -/
theorem record_rank_eq_four :
    Nat.log2 (Finset.univ.image faceRecord).card = 4 := by decide

/-- Record nullity of the whole cell: 4 free bits (`log₂ 16`). At whole-cell
granularity rank = nullity; the entropy fork splits only per face and under gluing. -/
theorem record_nullity_eq_four : Nat.log2 recordKernel.card = 4 := by decide

/-- First-isomorphism check of the actual cell record map:
`|image| · |kernel| = |domain|` (`16 · 16 = 256`). -/
theorem record_image_times_kernel :
    (Finset.univ.image faceRecord).card * recordKernel.card = 256 := by decide

What this page does not claim

The module does not prove the 16 blind moves are gauge, only that they are the complete set of unrecorded distinctions. This test does not establish recognition complementarity as a theorem; it reduces the question to a finite, classified choice. No claim is made about whether the boundary record fully determines the interior for cells larger than the 2³ cube.

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/CellInjection.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

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