IndisputableMonolith.Mathematics.GoldbachFromRS
The GoldbachFromRS module models primes as recognition equilibria under the J-cost function to derive even-number decompositions into prime pairs. Researchers connecting number theory to the Recognition Science framework would cite it for its structural definitions. The module organizes the argument via gap categories, symmetric pair costs, and explicit certificates without containing any proofs.
claimPrimes $p$ satisfy the equilibrium condition $J(p) = 0$ where $J(x) = (x + x^{-1})/2 - 1$, with unit cost at $p=1$. Even integers are certified as sums of such prime pairs via symmetric cost-zero representations.
background
The module imports the Cost module, which supplies the J-cost function measuring deviation from recognition equilibrium. In this setting J(1) = 0 defines unit cost, and primes are equilibria where J(p) = 0. The module introduces PrimeGapCategory to classify gaps, primeGapCategoryCount to enumerate them, prime_unit_cost and prime_pair_symmetric to assign costs to pairs, and GoldbachRSCert together with goldbachRSCert to certify decompositions.
proof idea
This is a definition module, no proofs. It declares the six sibling objects that structure the Goldbach derivation: gap categories, their counts, unit costs for primes, symmetric pair predicates, and the certificate type with its constructor.
why it matters in Recognition Science
The module supplies the definitions needed to embed the classical Goldbach statement inside Recognition Science equilibria. It feeds downstream results that treat Goldbach representations as consequences of the forcing chain and RCL. The GoldbachRSCert object directly encodes the zero-cost prime-pair condition that links to the phi-ladder mass formula and eight-tick octave.
scope and limits
- Does not prove Goldbach independently of RS axioms.
- Does not enumerate explicit pairs for specific even integers.
- Does not address the odd Goldbach conjecture.
- Does not derive prime distribution asymptotics.