module module moderate

`IndisputableMonolith.Foundation.NineParities`

Organizes the nine Recognition Science parity indices by origin: spacetime, color, and generation. Defines the parity vector, vacuum parity, and the trichotomy that partitions the nine indices. Gap derivation imports the module. Content is definitional scaffolding plus counting and classification lemmas (nine-dimensional parity space, tick-reversal conjugation).

claimRecognition Science carries a nine-dimensional parity space: three spacetime parities, three color parities, and three generation parities. The vacuum has a distinguished parity vector. Tick reversal acts by conjugation and flips every parity sign. The indices admit a unique source trichotomy and a source decomposition of the parity vector.

background

Recognition Science forces spatial dimension $D = 3$ (DimensionForcing, chain step T8) and forces double-entry ledger structure from $J$-symmetry (LedgerForcing). Against that background this module isolates discrete $\mathbb{Z}/2$ labels that tag ledger events: parity indices.

The nine indices split by origin. Spacetime parities track orientation and causal structure in the forced $D = 3$ geometry. Color parities label the internal three-way structure tied to the recognition ledger. Generation parities mark the three-family copy structure on the $\varphi$-ladder. The vacuum parity is the reference vector against which flips are measured.

Constants enter only through the native tick $\tau_0 = 1$; the module does not re-derive $c$, $\hbar$, or $G$. Downstream gap work treats the parity count as fixed input when closing the coherence-energy exponent.

proof idea

Definition module with thin classification lemmas, not a long forcing proof. It introduces ParityIndex and ParityVector, the vacuum vector, and three predicates (spacetime / color / generation). Counting is a finite enumeration: the parity space has dimension nine. Trichotomy and source decomposition are case splits on those predicates. Tick-reversal conjugation is defined and shown to flip all nine signs. No deep analytic estimates; the work is bookkeeping once $D = 3$ and the ledger are assumed from upstream modules.

why it matters in Recognition Science

Supplies the discrete parity inventory that GapDerivation imports when it closes boundary item B-22: coherence energy exponent equals $D+2$, hence $E_{\mathrm{coh}} = \varphi^{-5}$ at $D = 3$. Without a fixed nine-fold parity count and source split, the configuration dimension of a recognition event is ambiguous.

Sits downstream of DimensionForcing (T8: $D = 3$) and LedgerForcing ($J$-symmetry to double-entry). The nine indices are the discrete skeleton on which later mass-ladder and generation structure hang; they are not themselves the mass formula, but they constrain how sources are labeled before rung arithmetic begins. Parent consumer is the GapDerivation module, not a single named theorem in the supplied edges.

scope and limits

Does not prove $D = 3$; that is assumed from DimensionForcing.
Does not derive the mass formula or $\varphi$-ladder rung arithmetic.
Does not compute $\alpha^{-1}$ or close the fine-structure band.
Does not force the eight-tick octave; only uses tick reversal as a $\mathbb{Z}/2$ action.
Does not claim experimental identification of each parity with a Standard Model quantum number.

used by (1)

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

IndisputableMonolith.Foundation.GapDerivation module_import

depends on (3)

Lean names referenced from this declaration's body.

IndisputableMonolith.Constants module_import
IndisputableMonolith.Foundation.DimensionForcing module_import
IndisputableMonolith.Foundation.LedgerForcing module_import

declarations in this module (28)

inductive ParityIndex L59
abbrev ParityVector L72
def vacuumParity L75
theorem parity_count_eq_nine L80
theorem parity_space_dimension L84
def isSpacetimeParity L90
def isColorParity L98
def isGenerationParity L105
theorem parity_trichotomy L111
theorem source_decomposition L118
def tickReversalConjugate L124
theorem parities_flip_under_tick_reversal L128
theorem tick_reversal_involutive L137
theorem vacuum_parities_vanish L152
theorem vacuum_is_zero_vector L158
theorem vacuum_not_fixed_by_tick_reversal L162
def basisVector L171
theorem basisVector_nonzero L175
theorem basisVectors_distinct L182
theorem parity_independence L191
theorem color_parity_count_from_D3 L200
theorem spacetime_parity_count L205
theorem generation_parity_count L210
def hammingWeight L215
theorem vacuum_hamming_weight L219
theorem tick_reversed_vacuum_hamming_weight L224
theorem total_parity_configs L230
theorem nine_parities_master L247