RSGRVStructural003Cert
plain-language theorem explainer
Certificate structure packaging three structural gravity claims for the RS count-law module: the domain cost vanishes on equal nonzero arguments, stays nonnegative for positive mass and energy, and the canonical threshold is strictly positive. Gravity auditors cite it as the typed interface that the module certificate inhabitant fills. As a structure definition there is no proof body; sibling lemmas discharge the three fields.
Claim. A certificate consists of three facts: (i) for every nonzero real $r$, the domain cost of the pair $(r,r)$ equals zero; (ii) for all positive reals $m$ and $e$, the domain cost of $(m,e)$ is nonnegative; (iii) the canonical threshold is strictly positive.
background
This module is Gravity RS Structural Module 3. Its local setting is the RS Count Law $2^D-1=7$ independent channels, obtained exactly from the $D=3$ configuration dimension. The module is marked STRUCTURAL THEOREM (zero sorry, zero axiom), sitting on the T8 landmark that forces three spatial dimensions.
The domain cost is a real-valued cost on pairs of reals (mass- and energy-like recognition quantities in this gravity layer). The three fields demand that it vanish on the nonzero diagonal, stay nonnegative in the positive quadrant, and that a fixed canonical threshold be positive. Upstream, ObserverForcing records that the cost of any recognition event is nonnegative, via nonnegativity of the J-cost on positive states. That global cost positivity is the conceptual parent of the domain-cost nonnegativity field here.
proof idea
No proof body: this is a structure (certificate interface), not a theorem. The three fields are pure Prop obligations. Downstream, the noncomputable definition cert inhabits the structure in one shot by assigning domainCost_at_eq, domainCost_nonneg, and canonicalThreshold_pos to the three fields. The companion theorem cert_inhabited then wraps that inhabitant as Nonempty.
why it matters
The structure is the typed bundle that makes the RS gravity count-law certificate checkable as a single object. Downstream, cert builds an explicit inhabitant and cert_inhabited proves the type is nonempty, closing the structural obligation for this module. Framework-wise it sits under the RS Count Law $2^D-1=7$ forced by $D=3$ (T8), and inherits the nonnegativity culture of the recognition cost (J-cost minimum at identity, ObserverForcing cost_nonneg). It does not itself derive the channel count; it packages the cost and threshold side-conditions that the structural gravity layer treats as settled.
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