IndisputableMonolith.Foundation.NineParities
Catalogues nine independent parity indices forced by the recognition ledger and D=3, grouped by spacetime, color, and generation origin. Supplies the discrete parity space used when deriving the coherence gap and related selection rules. The module is mostly definitional: it names the indices, proves there are exactly nine, and records how they transform under tick reversal.
claimThere is a nine-dimensional parity space whose basis splits into spacetime, color, and generation parities. The vacuum carries a distinguished parity vector; under tick reversal each parity flips by a fixed conjugation. The count identity $|\mathrm{ParityIndex}|=9$ and the trichotomy of origins are part of the package.
background
Recognition Science forces spatial dimension $D=3$ (DimensionForcing) and a double-entry ledger from $J$-symmetry (LedgerForcing). Once the ledger and the eight-tick octave are in place, discrete sign data on ledger edges become parity indices.
This module organizes those indices by origin: spacetime parities (linked to the three spatial directions and time orientation), color parities, and generation parities. The vacuum parity vector is the reference configuration; a tick-reversal conjugation acts on the whole space.
Constants supplies the RS time quantum $\tau_0=1$ tick against which reversal is defined. The nine-count and the trichotomy are the structural facts the rest of the foundation imports.
proof idea
Definition-heavy module with a short cluster of counting and classification lemmas. ParityIndex and ParityVector are introduced; vacuumParity is fixed; parity_count_eq_nine and parity_space_dimension establish cardinality nine. Predicates isSpacetimeParity, isColorParity, isGenerationParity plus parity_trichotomy and source_decomposition partition the basis by origin. tickReversalConjugate and parities_flip_under_tick_reversal record the $Z_2$ action. No deep analytic argument; the content is naming, enumeration, and the reversal law.
why it matters in Recognition Science
GapDerivation imports this module to close boundary item B-22: the coherence energy exponent equals $D+2$, yielding $E_{\mathrm{coh}}=\varphi^{-5}$ at $D=3$. The nine parities supply the discrete labels that sit on recognition events when the configuration dimension is counted.
In the broader forcing chain the module sits after T8 ($D=3$) and ledger forcing, and before gap and mass-ladder work. Anyone citing the nine-fold parity structure or tick-reversal conjugation on the ledger will land here.
scope and limits
- Does not derive the Standard Model gauge group or fermion spectrum from the nine parities alone.
- Does not prove dynamical stability of the vacuum parity vector.
- Does not compute numerical coupling constants or mass ratios.
- Does not address continuous symmetries beyond the discrete tick-reversal action.
used by (1)
depends on (3)
declarations in this module (28)
-
inductive
ParityIndex -
abbrev
ParityVector -
def
vacuumParity -
theorem
parity_count_eq_nine -
theorem
parity_space_dimension -
def
isSpacetimeParity -
def
isColorParity -
def
isGenerationParity -
theorem
parity_trichotomy -
theorem
source_decomposition -
def
tickReversalConjugate -
theorem
parities_flip_under_tick_reversal -
theorem
tick_reversal_involutive -
theorem
vacuum_parities_vanish -
theorem
vacuum_is_zero_vector -
theorem
vacuum_not_fixed_by_tick_reversal -
def
basisVector -
theorem
basisVector_nonzero -
theorem
basisVectors_distinct -
theorem
parity_independence -
theorem
color_parity_count_from_D3 -
theorem
spacetime_parity_count -
theorem
generation_parity_count -
def
hammingWeight -
theorem
vacuum_hamming_weight -
theorem
tick_reversed_vacuum_hamming_weight -
theorem
total_parity_configs -
theorem
nine_parities_master