structural_parallel_certificate
This result collects five properties of the J-cost function that parallel the functional equation satisfied by the Riemann zeta function. Number theorists working in Recognition Science would cite it when connecting multiplicative ledger conservation to zero-line confinement. The proof is a term-mode conjunction that directly invokes the symmetry, positivity, unit-zero, d'Alembert-satisfaction, and composition-identity lemmas already established in the Cost module.
claimThe J-cost function $J$ satisfies $J(x)=J(x^{-1})$ for all $x>0$, $J(1)=0$, $J(x)>0$ whenever $x>0$ and $x≠1$, the auxiliary map $H(t):=J(e^t)+1$ obeys the d'Alembert equation $H(t+u)+H(t-u)=2H(t)H(u)$ for all real $t,u$, and the composition identity $J(xy)+J(x/y)=2J(x)+2J(y)+2J(x)J(y)$ holds for all $x,y>0$.
background
In the PrimeLedgerStructure module natural numbers are interpreted as transactions on a discrete multiplicative ledger, with primes as the irreducible entries. The J-cost function, introduced in the Cost module, is given explicitly by $J(x)=(x-1)^2/(2x)$ for $x>0$ and measures deviation from the fixed point at unity. SatisfiesDAlembert is the predicate that a real function $H$ obeys the d'Alembert functional equation $H(t+u)+H(t-u)=2H(t)H(u)$ for all real $t,u$.
proof idea
The proof is a term-mode five-tuple. It applies Jcost_symm to obtain inversion symmetry, Jcost_unit0 to obtain the zero at unity, Jcost_pos_of_ne_one to obtain strict positivity off unity, rs_cost_satisfies_dalembert to obtain d'Alembert compliance for the shifted exponential, and dalembert_identity to obtain the composition law.
why it matters in Recognition Science
The declaration supplies the model-level identification between J-cost and the zeta functional equation that the module uses to motivate the Recognition Science prediction of the Riemann hypothesis. It directly supports the downstream theorem j_positive_off_fixed_point and sits inside the chain that links d'Alembert zero confinement to ledger conservation. The module records that d'Alembert structure is proved while the full Euler-product constraint on the completed zeta remains an open gap.
scope and limits
- Does not prove the Riemann hypothesis.
- Does not impose d'Alembert-type constraints on the Euler product.
- Does not locate zeros numerically or derive their explicit positions.
- Does not address higher-dimensional or non-multiplicative extensions.
formal statement (Lean)
227theorem structural_parallel_certificate :
228 -- J is symmetric under inversion (like ξ(s) = ξ(1-s))
229 (∀ (x : ℝ), 0 < x → Cost.Jcost x = Cost.Jcost x⁻¹) ∧
230 -- J has a unique zero at the fixed point (like RH's critical line)
231 (Cost.Jcost 1 = 0) ∧
232 (∀ (x : ℝ), 0 < x → x ≠ 1 → 0 < Cost.Jcost x) ∧
233 -- J satisfies d'Alembert (which forces zero-line confinement)
234 SatisfiesDAlembert (fun t => Cost.Jcost (Real.exp t) + 1) ∧
235 -- The d'Alembert identity governs cost composition
236 (∀ (x y : ℝ), 0 < x → 0 < y →
237 Cost.Jcost (x * y) + Cost.Jcost (x / y) =
238 2 * Cost.Jcost x + 2 * Cost.Jcost y +
239 2 * Cost.Jcost x * Cost.Jcost y) :=
proof body
Term-mode proof.
240 ⟨fun x hx => Cost.Jcost_symm hx,
241 Cost.Jcost_unit0,
242 fun x hx hne => Cost.Jcost_pos_of_ne_one x hx hne,
243 rs_cost_satisfies_dalembert,
244 fun x y hx hy => Cost.dalembert_identity hx hy⟩
245
246end
247
248end PrimeLedgerStructure
249end NumberTheory
250end IndisputableMonolith