RSSTDStructural001Cert
plain-language theorem explainer
Certificate bundle for the first Standard Model structural module: diagonal domain cost vanishes, domain cost is nonnegative on the positive quadrant, and the canonical threshold is strictly positive. Cited by anyone assembling or discharging the RS_STD_Structural_001 inhabitation proof. The structure itself is pure data; the concrete witness is assembled elsewhere from three sibling lemmas.
Claim. A certificate consists of three facts: (i) for every nonzero real $r$, the domain cost of the pair $(r,r)$ is zero; (ii) for all positive reals $m,e$, the domain cost of $(m,e)$ is nonnegative; (iii) the canonical threshold is strictly positive.
background
This module packages the first structural Standard Model prediction in Recognition Science: the J-cost $J(x)=\frac12(x+1/x)-1$, the golden ratio $\varphi$ as self-similar fixed point, and $D=3$ forced (T5--T8 of the forcing chain). Status is structural theorem with no sorry and no axioms.
Domain cost is the local cost functional on pairs of positive reals used in this Standard Model layer; it inherits nonnegativity from the global recognition cost. Upstream, ObserverForcing records that every recognition event has nonnegative cost via $J$-cost nonnegativity on positive states, and the identity event sits at the $J$-minimum $x=1$. The canonical threshold is the positive cutoff against which structural comparisons in this module are made.
The certificate is the Prop-bundle that later definitions must inhabit: zero on the diagonal, nonnegative off it (for positive arguments), and a positive threshold.
proof idea
No proof body: this is a structure declaration. The three fields are pure propositions. Inhabitation is deferred to the sibling definition cert, which fills the fields by domainCost_at_eq, domainCost_nonneg, and canonicalThreshold_pos respectively. Nonemptiness is then a one-line constructor application.
why it matters
Gives a single named interface for the three structural side-conditions of RS_STD_Structural_001. Downstream, cert builds a concrete inhabitant and cert_inhabited records Nonempty of the certificate type, closing the module's structural claim.
In the broader framework this sits under the Standard Model structural layer that advertises J-uniqueness (T5), $\varphi$ forced (T6), and $D=3$ (T8). The certificate does not itself prove those forcing steps; it only packages the cost and threshold hygiene needed so later SM structural theorems can cite one object rather than three scattered lemmas.
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