matter_phi45_complementarity

This module records kernel-checked arithmetic identities between imported Standard Model degree-of-freedom counts and combinatorial quantities at $D=3$. Per the module status note (re-scoped after external review), it does not derive the SM spectrum: matter representations, the minimal-neutrino convention $g_f=90$, and the Fermi-Dirac $7/8$ weight are imported physics.

The dimension gap is $\Delta(d)=(\text{parity count})\times(\text{coherence exponent})=d^2(d+2)$. At the T8-forced value $D=3$ one gets $9\times 5=45$. Total fermionic DOF is the sum of quark, charged-lepton, and neutrino contributions over three generations, fixed at 90 under that standard tally.

Upstream, $D=3$ is the spatial dimension forced by DimensionForcing/T8; the fermionic count is the imported SM total from the cosmology $g_*$ derivation.

Both sides are closed natural numbers once definitions unfold: left side is $90/2$, right side is $\Delta(3)=3^2(3+2)=45$. The proof is a one-line native_decide that discharges the equality by kernel integer evaluation. No intermediate lemmas beyond the definitions of fermionic DOF, dimension gap, and $D$.

Feeds the parent certificate fermion_dof_gap_certificate, which packages five exact identities: $\Delta(3)=45$, $g_f=2\times\Delta(3)$, the $D=3$ weight $7/8$, assembled $g_*=28+(7/8)\times 90=106.75$, and this half-count form $g_f/2=\Delta(D)$.

In the Recognition framework it is Unification bookkeeping that rewrites the imported SM fermionic tally against the T8 gap $D^2(D+2)$. The doc-comment is explicit: a re-expression of one integer, recorded exactly, not independent evidence for the $\eta_B$ rung. The honest split keeps RS-derived landmarks ($D=3$, eight-tick $2^D=8$, generation count 3) separate from imported representation data. Closing a true RS derivation of $g_*=106.75$ would still require deriving gauge reps, Higgs content, chiral neutrinos, and the thermal integral from RS premises; none of that is claimed here.