cert
plain-language theorem explainer
Packages three elementary facts about the D=3 domain cost into a single certificate: diagonal vanishing, non-negativity for positive arguments, and positivity of the canonical threshold. Cited by anyone needing an inhabited D=3 combinatorics certificate in the structural forcing chain. Proof is a structure constructor wiring three sibling lemmas.
Claim. There is a D=3 combinatorics certificate asserting: (i) the domain cost of any nonzero real against itself is zero; (ii) the domain cost of any two positive reals is nonnegative; (iii) the canonical threshold is strictly positive.
background
The module develops D=3 combinatorics in Recognition Science: three spatial dimensions, three quark colors, three neutrino flavors, three families, and related ternary structure, all forced from configDim = D = 3. Status is a structural theorem (zero sorry, zero axiom).
The certificate structure bundles three properties of the domain cost functional used in this combinatorics layer: vanishing on the diagonal away from zero, non-negativity when both mass-like and energy-like arguments are positive, and positivity of a fixed canonical threshold. Upstream, non-negativity of recognition cost is already known from ObserverForcing: every recognition event has cost at least zero via the J-cost minimum.
Sibling lemmas domainCost_at_eq, domainCost_nonneg, and canonicalThreshold_pos discharge the three fields; this definition merely packages them.
proof idea
One-line structure constructor. The three fields of D3CombinatoricsV2Cert are filled by the sibling results domainCost_at_eq (diagonal vanishing), domainCost_nonneg (non-negativity for positive arguments), and canonicalThreshold_pos (threshold positivity). No additional reasoning; the definition is pure packaging.
why it matters
Gives an inhabited certificate for the D=3 combinatorics layer that the module advertises as a structural theorem. In the Recognition forcing chain this sits under T8 (D = 3 spatial dimensions) and the broader claim that ternary structure in particle physics and logic is forced by configDim = 3 rather than postulated. No downstream users are recorded yet; the natural consumer is any theorem that needs a single named witness that domain-cost hygiene and threshold positivity hold in the D=3 setting. Closes the certificate interface so later combinatorics arguments can assume the bundle rather than re-prove the three facts.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.