CompletedZetaLedgerCert
plain-language theorem explainer
CompletedZetaLedgerCert defines a certificate that packages the completed Riemann zeta function as a balanced arithmetic ledger whose reciprocal symmetry forces the critical line as the fixed locus of the map s to 1-s. Researchers decomposing the physical thesis data in the Recognition Science framework cite this structure when recovering the arithmetic ledger from the classical zeta. The definition records the balanced property and the uniqueness statement without internal proof steps.
Claim. A certificate for the completed Riemann zeta function $F$ asserting that $F(1-s)=F(s)$ for all $s$ in the complex plane and that every fixed point of the involution $s=1-s$ satisfies $s.re=1/2$.
background
The Completed Zeta as a Balanced Ledger module treats Mathlib's completed zeta functional equation as the reciprocal balance law for an arithmetic ledger. BalancedArithmeticLedger F is the structure requiring reciprocal symmetry $F(1-s)=F(s)$ together with the fixed-locus condition that $s=1-s$ implies $s.re=1/2$. The upstream balanced predicate from LedgerForcing states that a ledger is balanced precisely when its event list is balanced. The zeta abbreviation supplies the standard arithmetic zeta function that equals 1 on positive integers.
proof idea
This is a structure definition whose two fields are supplied externally. The downstream completedZetaLedgerCert populates the balanced field via the completedZeta_balanced lemma and the critical_line_unique field via the reciprocal_fixed_re_eq_half uniqueness result.
why it matters
The certificate supplies the completedLedger field required by both RSPhysicalThesisDataLogic and RSPhysicalThesisData. These structures decompose the physical thesis data while retaining the classical analytic zeta stack and recording compatibility with the recovered prime ledger. Within the Recognition framework the declaration certifies that the zeta functional equation realizes the reciprocal balance law, thereby linking the arithmetic ledger to the forcing chain and the critical-line uniqueness property.
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