The Information-Scaling Manifold and the Arithmetic Structure of Quantum Gravity Geometry, Coherence, and the Timeless Ground

We present a unified treatment of two complementary frameworks within the Spectral Computing Programme: the Information-Scaling Manifold (ISM), which constructs the arithmetic geometry of the Riemann zeta function from the ground up via prime-power curvature sources, and the Wheeler-DeWitt/ISM Connection, which descends from quantum gravity to identify the ISM constraint surface with the WdW superspace. The central thesis of this amalgamated paper is that these two frameworks, developed independently, describe the same mathematical object from opposite directions, meeting at the critical line σ=1/2. The ISM provides eight proved results: the self-metrising property gₛs = κI, the curvature dipole law Λ_ρ·Λ₁-⏠ = e^2π, the three-constant unification through the χ₄ character, the geodesic theorem identifying σ=1/2 as a geodesic of the completed metric, and the coherence-decoherence symmetry argument for the critical line. The WdW paper provides four structural theorems connecting the WdW constraint surface to the ISM manifold, the WKB semiclassical branch to the Reintegration Cycle, and zero simplicity to clock non-degeneracy. An original analysis using three mathematical frameworks — Heisenberg uncertainty, Poincaré recurrence and topology, and the Renormalisation Group — reveals nine hidden structures in the amalgamated framework. Key new results include: the critical line as the minimum-uncertainty ground state of the ISM Heisenberg algebra; zero simplicity as a necessary consequence of the Page-Wootters clock commutation condition; the critical line as a Poincaré concentration set with infinite invariant measure and zero mean return time; the Wheeler-DeWitt equation identified as the RG fixed-point equation of the ISM effective action; σ=1/2 as the unique marginal fixed point of arithmetic RG flow; and the three constants e^{2π, e^π, π/4} identified as the scaling dimensions of operators at levels 2, 1, 0 of the ISM RG tower.