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plain-language theorem explainer
Records that matching Standard Model fermionic degrees of freedom to the D=3 dimension-gap count is exact arithmetic, while whether that match is structural or accidental stays open. Cosmology and unification readers cite it for the honest split between imported SM bookkeeping and RS-forced D=3/8-tick facts. No proof body: it is documentary framing, not a derived claim.
Claim. In the fermion degree-of-freedom / dimension-gap bridge, the identities $90 = 2\times\mathrm{dimensionGap}(3)$, $\tfrac{7}{8}=(2^3-1)/2^3$, and $28+(\tfrac{7}{8})\times 90=106.75$ are exact on imported counts. Whether this match is structural (forced by Recognition Science) or accidental remains an open question; the module poses the arithmetic precisely and does not answer it.
background
The module sits in Unification and, after external review, is explicitly re-scoped to arithmetic identities only. It does not derive the Standard Model spectrum. Imported inputs include SM matter representations, the minimal-neutrino convention $g_f=90$, the Fermi–Dirac thermal weight $7/8$, and the high-$T$ assembly $g_\star=106.75$ (see Standard Model relativistic DOF bookkeeping and temperature-dependent $g_\star(T)$ thresholds).
RS-derived upstream facts cited, not re-proved: spatial dimension $D=3$ (T8 / dimension forcing), the eight-tick period $2^D=8$, and three generations. The spin-statistics exchange sign is upstream; the $7/8$ thermal weight itself is the standard Fermi/Bose integral ratio and is imported.
Local notation freezes $D:=3$. Sibling quantities include the eight-tick cadence, $\mathrm{dimensionGap}$, degrees of freedom per generation, generation count, and total fermionic DOF. The auxiliary $g_{\star,D}(d)$ varies only the thermal weight $(2^d-1)/2^d$; bosonic 28 and fermionic 90 stay fixed at $D=3$ SM values.
proof idea
No proof obligations. Claim status is definitional framing (structure/documentation), not a theorem. The surrounding module discharges only kernel-checked arithmetic on already-known counts: expand $\mathrm{dimensionGap}(3)$, compare to 90, evaluate $(2^3-1)/2^3=7/8$, and assemble $28+(7/8)\cdot 90=106.75$. Interpretive glosses (identity-tick reading of $7/8$, matter/$\varphi^{45}$ duality) are marked hypothesis and are not forced here.
why it matters
Keeps the Recognition unification layer honest about $g_\star$. Framework landmarks in play are T8 ($D=3$) and the eight-tick octave $2^3=8$; the identities rewrite SM tallies in that $D$-flavored notation without claiming an RS derivation of gauge reps, Higgs content, or chiral neutrinos.
Downstream edges attached to this name are largely incidental name collisions (path interpolation, $J$-cost reciprocal automorphisms, crystal-system tables) rather than true consumers of a fermion-gap theorem. The real parent use is local: precise posing of the structure-versus-accident question so later work cannot silently upgrade imported SM bookkeeping into an RS prediction.
Open question left explicit: a genuine $g_\star(D)$ theory would need matter content derived at each $D$; that is unavailable. $D=2$ and $D=4$ rows are counterfactual arithmetic only, not condensed-matter confirmations.
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