numberTheoryCert
plain-language theorem explainer
The definition assembles five pre-proved number-theoretic identities from Recognition Science into the NumberTheoryCert structure. A researcher deriving the phi-ladder mass formula or constants from the J-uniqueness fixed point would cite it to invoke the relations φ² = φ + 1, φ⁵ = 5φ + 3, φ⁸ = 21φ + 13, the gap at D = 3, and their total count of five. The construction is a direct field-by-field assignment from the individual identity theorems.
Claim. Let φ denote the golden ratio. The number theory certificate is the structure containing the identities φ² = φ + 1, φ⁵ = 5φ + 3, φ⁸ = 21φ + 13, 3²(3 + 2) = 45, and the assertion that exactly five such key identities exist.
background
In the module deriving number theory from Recognition Science, the golden ratio φ satisfies its defining algebraic relation φ² = φ + 1. Successive powers yield the Fibonacci identities φ⁵ = 5φ + 3 and φ⁸ = 21φ + 13. The gap identity at rung 45 is recovered from the forced spatial dimension via D²(D + 2) evaluated at D = 3, which equals 45. The module documentation states that these five relations are the canonical prime-related identities required by the framework, all previously established with zero sorry or axiom.
proof idea
This is a direct construction of the NumberTheoryCert structure. The phi_sq field is assigned the theorem phi_sq_identity, the phi5 field is assigned phi5_fibonacci, the phi8 field is assigned phi8_fibonacci, the gap45_D field is assigned gap45_from_D, and the five_identities field is assigned rsi_count_five. Each assignment references an upstream theorem that has already been proved by nlinarith or decide.
why it matters
The certificate collects the five identities listed in the module documentation for use in the Recognition Science derivation of constants and the phi-ladder. It implements the catalogue of φ² = φ + 1, the Fibonacci relations at 5 and 8, the gap45 identity tied to D = 3, and the count of five. Although no downstream theorems currently reference it, the structure supplies a single access point for these relations in any proof involving the eight-tick octave or baryon rung calculations.
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