generationSlotCount
plain-language theorem explainer
The number of charged-generation slots is the face-pair count of the spatial cube: for forced dimension $D=3$ this is exactly three. Mass and torsion bridge proofs cite it as the geometric slot budget. It is a one-line abbreviation of the ParticleGenerations face-pair map at the forced dimension.
Claim. Define the generation slot count as the number of opposite face-pairs on the $D$-cube, evaluated at the forced spatial dimension $D=3$. Equivalently, $\mathrm{generationSlotCount} := \mathrm{face\_pairs}(D)$ with $D=3$, hence three slots.
background
The Generation Torsion Bridge module derives the charged-generation torsion schedule ${0,11,17}$ from $Q_3$ cube combinatorics rather than raw numerals. Spatial dimension $D$ is fixed at $3$ by the forcing chain (T8/T9): linking and recognition geometry force three spatial directions.
On a $D$-cube, opposite faces come in pairs; the count of such pairs is exactly $D$. ParticleGenerations records this as $\mathrm{face_pairs}(D):=D$, and at $D=3$ there are three pairs. Those pairs are the geometric carriers of independent generation slots in the bridge.
The same integer later unifies with the independent loop count on $Q_3$ and with the CW-filtration story (ground / passive-edge / face+edge modes) that builds torsion increments $0$, then $11$, then $17$.
proof idea
Pure definitional abbreviation: unfold to $\mathrm{face_pairs},D$ with the global constant $D=3$. No tactics or lemmas; downstream equalities (generationSlotCount_eq_three, generationSlotCount_eq_loopCount) discharge the arithmetic by simpa against face_pairs_at_D3 and the loop/face-pair identification.
why it matters
This is the named slot budget inside the cube-geometry source of truth for generation torsion. CubeGenerationFiltration packages it as the first field (slot_count : generationSlotCount = 3), together with loop/face-pair unification and incremental torsion steps. Theorems generationSlotCount_eq_three and generationSlotCount_eq_loopCount make the count equal both $3$ and the independent $Q_3$ loop count; minimalLoopExcitation_matches_generation_slots ties minimal loop-excitation profiles to the same integer.
In the RS landmarks this sits under T8 ($D=3$) and the eight-tick/$Q_3$ combinatorics that organize fermion generations. It does not yet derive the ground/edge/face coupling rule from the Recognition Composition Law; the module still treats that assignment as an explicit structural premise with uniqueness, upgrading only the "no hardcoded numerals" side of the gap.
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