IndisputableMonolith.Mathematics.PrimeCostSpectrumFromJCost
PrimeCostSpectrumFromJCost derives the recognition cost spectrum for primes from the J-cost function, establishing zero cost precisely when a prime sits at ratio 1 on the lattice. Number theorists and Recognition Science researchers cite it to ground prime distributions in the underlying cost structure. The module consists of definitions and supporting lemmas that import the Cost module and build the spectrum directly from J.
claimThe prime cost spectrum is the map sending each prime $p$ to its J-cost evaluated at the ratio $r(p)$, with the property that the cost vanishes when $r(p)=1$ (exact lattice alignment).
background
The module imports the Cost module, which supplies the J-cost function $J(x) = (x + x^{-1})/2 - 1$. It introduces the prime cost spectrum as the restriction of this function to primes, using the convention that ratio 1 corresponds to lattice placement. The local setting is the mathematics of deriving prime distributions from recognition costs, with notation and conventions carried forward from the Cost definitions.
proof idea
this is a definition module, no proofs
why it matters in Recognition Science
The module supplies the cost spectrum used by downstream siblings including PrimeDistributionRegime and primeDistributionCount. It fills the step that connects J-cost to prime lattices inside the Recognition Science mathematics layer, supporting the broader derivation of distributions from the forcing chain.
scope and limits
- Does not compute explicit numerical costs for individual primes.
- Does not treat composite integers or their cost spectra.
- Does not prove uniqueness of zero-cost lattice points.
- Does not link the spectrum to physical constants such as alpha or G.