pith. sign in
module module high

IndisputableMonolith.NumberTheory.PrimeCostSpectrum

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This module isolates the cost of each prime p as J(p) using the recognition cost function. Researchers extending the ledger to twisted L-series or prime phase distributions cite it as the basis for irreducible transaction costs. It is a definition module whose content consists of the primeCost definition together with immediate nonnegativity and monotonicity properties.

claimFor each prime $p$, the prime cost equals $J(p)$ where $J(x) = (x + x^{-1})/2 - 1$. The resulting values form the basis of the cost spectrum.

background

The module imports the J-cost definition from IndisputableMonolith.Cost and the ledger interpretation from IndisputableMonolith.NumberTheory.PrimeLedgerStructure. The latter states that primes are the irreducible transactions in the discrete multiplicative ledger and that the fundamental theorem of arithmetic supplies the balance sheet. J itself satisfies the Recognition Composition Law J(xy) + J(x/y) = 2J(x)J(y) + 2J(x) + 2J(y).

proof idea

This is a definition module, no proofs.

why it matters in Recognition Science

The module supplies the prime cost spectrum that CostTwistedLSeries generalizes to character-twisted arithmetic functions and that PrimePhaseDistribution and RecognitionTheta rely upon for the 8-tick character and phi-ladder weights. It directly implements the isolation of irreducible costs described in its own doc-comment.

scope and limits

used by (3)

From the project-wide theorem graph. These declarations reference this one in their body.

depends on (2)

Lean names referenced from this declaration's body.

declarations in this module (22)