SpinQuantumNumCert
plain-language theorem explainer
Certificate packing three analytic side-conditions on the domain cost and the canonical threshold used when deriving the electron spin quantum number from configuration dimension. Anyone citing the D=2 spinor/lattice route to s=1/2 needs this bundle. It is a pure structure definition: no proof body, only field types.
Claim. A certificate consists of three facts: (i) the domain cost vanishes on the diagonal, $\mathrm{cost}(r,r)=0$ for all $r\neq 0$; (ii) the domain cost is nonnegative for positive mass and energy arguments; (iii) the canonical threshold is strictly positive.
background
The module derives the electron spin quantum number from configuration dimension: $s=1/2=1/D$ with $D=2$ for the spinor representation (Clifford algebra supporting SU(2)). In Recognition Science this is read as spin emerging from a D=2 binary recognition lattice, so $s=1/2^1=1/2$.
The domain cost is the local cost functional on mass/energy (or scale) pairs that the spin argument uses; the canonical threshold is the positive cutoff against which that cost is compared. Upstream, recognition-event cost is already known to be nonnegative via the J-cost minimum at the identity ratio $x=1$ (ObserverForcing).
This structure does not itself compute $s$; it packages the three inequalities and identities the later certificate instance must discharge.
proof idea
No proof: the declaration is a structure (interface) with three Prop-valued fields. Downstream, cert fills the fields by the sibling lemmas that the domain cost vanishes on the diagonal, is nonnegative for positive arguments, and that the canonical threshold is positive; cert_inhabited then wraps that instance as Nonempty.
why it matters
Gives a named, reusable hypothesis bundle for the structural theorem that electron spin is $s=1/2$ from config dimension $D=2$, matching the module status (0 sorry, 0 axiom). Downstream cert inhabits the structure and cert_inhabited records nonemptiness, so later physics lemmas can assume one object rather than three separate side-conditions.
In the broader RS chain this sits under the dimensional forcing that yields $D=3$ spatial dimensions and the eight-tick octave: here the spinor sector is the $D=2$ binary lattice, so the quantum number is the reciprocal half. It does not reopen T5–T8; it only certifies the cost/threshold hygiene needed for the spin readout.
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