j_zero_at_fixed_point
The J-cost function vanishes at the multiplicative identity. Researchers working on the Recognition Science ledger and its link to prime factorization would cite this when establishing the zero locus of the cost function. The proof is a direct one-line reference to the unit-zero lemma for the explicit J-cost definition.
claimThe cost function satisfies $J(1)=0$, where $J(x)=((x-1)^2)/(2x)$.
background
In the Recognition Science framework the natural numbers form a discrete multiplicative ledger in which primes are the irreducible transactions. The J-cost function is defined by the squared-ratio expression $J(x)=((x-1)^2)/(2x)$ and measures deviation from the fixed point at unity. The module situates this zero result inside the broader analysis of the d'Alembert equation on the ledger and the structural parallel between J and the zeta function.
proof idea
This is a one-line wrapper that applies the Jcost_unit0 lemma from the Cost module. The lemma itself follows by direct simplification from the definition of Jcost.
why it matters in Recognition Science
The declaration supplies the base zero of J inside the PrimeLedgerStructure module, which the module doc-comment lists as part of the proved d'Alembert zero-structure theorem. It supports the T5 J-uniqueness step of the forcing chain and the model identification J ↔ ζ. The module treats the result as a prerequisite for the open question whether the Euler product imposes d'Alembert-type constraints on the completed zeta function.
scope and limits
- Does not prove uniqueness of the zero without the explicit J-cost formula.
- Does not derive the full functional equation satisfied by J.
- Does not address the Riemann Hypothesis prediction stated in the module.
- Does not extend the zero result to complex arguments or other number fields.
formal statement (Lean)
173theorem j_zero_at_fixed_point : Cost.Jcost 1 = 0 := Cost.Jcost_unit0
proof body
Term-mode proof.
174
175/-- J is strictly positive away from the fixed point. -/