module module high

`IndisputableMonolith.Foundation.SpectralEmergence`

SpectralEmergence defines the D-dimensional binary hypercube graph, with vertex set V of cardinality 2^D, edges E, faces F, face pairs, automorphism order, and the concrete 3-cube Q3 objects including Euler characteristic. Researchers deriving spectral properties from the Recognition Science forcing chain cite it to ground the combinatorial base for dimension emergence. The module consists solely of definitions, importing the time quantum from Constants and cost structures from Cost to parameterize the graph.

claimThe D-dimensional hypercube has vertex set $V$ with $|V|=2^D$, edge set $E$, face set $F$, face pairs, and automorphism group order. For the 3-cube these specialize to $Q3$ vertices, edges, faces, face pairs, automorphism order, and Euler characteristic.

background

This module establishes the graph structure underlying spectral emergence in Recognition Science. It defines the binary cube in D dimensions, where vertices correspond to binary strings of length D, yielding |V|=2^D as stated in the module documentation. Key objects include the edge set E, faces F, face pairs, and the automorphism order, with explicit Q3 instances for the 3-cube and its Euler characteristic.

proof idea

This is a definition module, no proofs.

why it matters in Recognition Science

The module supplies the combinatorial primitives that feed parent theorems on spectral emergence in the Recognition Science framework. It fills the graph-theoretic step supporting the forcing chain T0-T8 where D=3 emerges from the eight-tick octave, providing Q3 objects for downstream spectral and physical derivations including the alpha band and phi-ladder mass formula.

scope and limits

Does not derive spectral eigenvalues or emergence laws.
Does not prove forcing chain steps such as D=3.
Does not incorporate the J-function or Recognition Composition Law.
Does not extend beyond the 3-cube Q3 specialization.

depends on (2)

Lean names referenced from this declaration's body.

IndisputableMonolith.Constants module_import
IndisputableMonolith.Cost module_import

declarations in this module (57)

structure of L12
def V L81
def E L85
def F L89
def face_pairs L93
def aut_order L97
theorem Q3_vertices L101
theorem Q3_edges L102
theorem Q3_faces L103
theorem Q3_face_pairs L104
theorem Q3_aut_order L105
theorem Q3_euler_characteristic L109
theorem Q3_self_dual_vertex_count L114
inductive SpectralSector L131
theorem sector_dim_sum L165
theorem gauge_sector_dim L174
theorem conjugate_dim_forced L182
theorem gauge_generators_eq_edges L191
theorem three_generations L204
abbrev Generation L207
theorem generations_eq_dimension L210
def fermion_flavors L219
theorem fermion_count_24 L228
theorem fermions_eq_D_times_V L233
theorem fermions_eq_half_aut L238
def total_fermion_states L242
theorem fermion_states_eq_aut L246
theorem quark_lepton_ratio L250
def J_phi L263
theorem J_phi_pos L266
theorem J_one_zero L275
theorem J_phi_sq_identity L279
def mass_rung L284
theorem mass_rung_step L288
theorem mass_rung_pos L296
theorem rung_ratio L301
theorem rung_separation L309
structure Q3State L331
def consciousness_ground L348
theorem consciousness_zero_cost L353
theorem consciousness_is_zero_defect L358
theorem consciousness_or_particle L363
theorem zero_defect_unique L371
theorem any_deviation_costs L378
structure SpectralViability L396
theorem D3_viable L402
theorem D1_fails_sync L408
theorem D2_fails_sync L411
theorem D4_fails_sync L414
theorem D5_fails_sync L417
theorem gap_sync_unique L420
theorem D3_unique_viable L427
structure SpectralEmergenceCert L449
theorem spectral_emergence L495
def SelfConsistent L532
theorem framework_self_consistent L543
theorem numerological_summary L559