module module high

`IndisputableMonolith.Holography.CellInjection`

Defines the forced D=3 recognition cell: eight vertices carrying bits, packed as configurations in Fin 256. Introduces face records, single-bit flips, XOR of configs, and weight. Supplies the discrete bulk state space used by the entropy-fork holography chain. Downstream RecordMonotonicity imports it to prove weak complementarity from no free erasure.

claimA cell configuration is an element of $\mathrm{Fin}\,256 \cong \{0,1\}^8$, one recognition bit on each vertex of the unit 3-cube. Vertex $v\in\mathrm{Fin}\,8$ has coordinate bits $(v\bmod 2,\lfloor v/2\rfloor\bmod 2,\lfloor v/4\rfloor)$. Face records, single flips, XOR, and Hamming weight are the elementary operations on this space.

background

Recognition Science forces $D=3$ spatial dimensions (T8) and an eight-tick octave of period $2^3$ (T7). The natural discrete bulk unit is therefore the 3-cube with eight vertices. This module packages that geometry: each vertex carries one recognition bit, so a full-cell state is a bitstring of length 8, identified with $\mathrm{Fin},256$.

Coordinate bits of vertex $v$ are read off by masks $(v\mathbin{&}1,v\mathbin{&}2,v\mathbin{&}4)$. Face records extract the four bits on a chosen face; single flips and face flips generate local moves; XOR and weight give the linear and counting structure needed for injection and monotonicity arguments.

The module sits at the start of the entropy-fork development (panel holo_unconditional): cell injection is step 1, before the Clausius selector and record monotonicity.

proof idea

Definition module. It introduces CellCfg as Fin 256, bit accessors, closed-on predicates, face records, flip and XOR operations, the zero cell, complement, and elementary lemmas such as single_flip_posts and single_flip_posts_three. No deep theorems; the content is the discrete state space and its elementary algebra for later injection and monotonicity proofs.

why it matters in Recognition Science

Feeds RecordMonotonicity (module import), step 3 of the entropy-fork chain. That downstream module proves weak complementarity (injection from physical bulk states into boundary records) from no free erasure, on the forced cell defined here. The holography manuscript isolates weak complementarity as the minimal sufficient form of recognition complementarity; this module supplies the concrete bulk configuration type on which that injection is stated. Anchors T7 (eight-tick) and T8 (D=3) in the discrete geometry used by the holography development.

scope and limits

Does not prove injectivity of any bulk-to-boundary map; only defines the cell state space.
Does not address continuous geometry or continuum limits of the 3-cube.
Does not establish weak complementarity or Clausius-type inequalities.
Does not fix dynamics or a Hamiltonian on configurations; only static bit configurations and flips.

used by (1)

From the project-wide theorem graph. These declarations reference this one in their body.

IndisputableMonolith.Holography.RecordMonotonicity module_import

declarations in this module (28)

abbrev CellCfg L77
def vbit L80
def closedOn L84
def faceRecord L90
def flipv L99
def xorCfg L103
def weight L107
def cell0 L110
def cellComplement L113
def faceFlip L116
theorem single_flip_posts L123
theorem single_flip_posts_three L128
theorem complement_invisible L137
theorem record_not_injective L142
def recordKernel L147
theorem recordKernel_card L150
theorem recordKernel_eq L155
theorem record_image_card L162
theorem record_rank_eq_four L165
theorem record_nullity_eq_four L170
theorem record_image_times_kernel L174
theorem record_blind_only_global L183
theorem face_flip_invisible_everywhere L189
theorem faceFlip_weight L192
theorem invisible_iff_kernel L197
def target_cell_injection L207
theorem target_cell_injection_holds L218
theorem cellInjectionCert L223