Computation as Geometric Flow: An Arb-Certified Local Intrinsic ODE on a Quantum-Control Response Fibre

Abstract

Series role — Certified dynamical witness in the Y.Y.N. Li Realizability Programme. Version note — v1.2.13 is a disclosure and notation revision derived from v1.2.12. Its deterministic manuscript build was attested with pdfTeX 1.40.25 / TeX Live 2023 (Debian), SOURCE_DATE_EPOCH=1786789922, FORCE_SOURCE_DATE=1, and three pdflatex passes; the rebuilt PDF is byte-identical to the repository PDF. It introduces no new theorem, computation, or certificate. No theorem-bearing source, certificate, protocol, JSON, atlas, numerical constant, gate threshold, Arb computation, or C4-E2b proof dependency changed. The theorem-bearing boundary remains the frozen v0.7.4 and v0.9.3 assets, and GLOBAL_FLOW_CLAIMED remains false. A computational task specified by a fixed finite response may admit many parameter realizations. Counterintuitively, a response-invisible direction need not be objective-flat: exact preservation of the declared response can coexist with strict decrease of an independent implementation objective. We certify this possibility in a driven-qubit model with projective response R3 as the retained task and sixth-order projective-loss coefficient L6 as the implementation objective. Using outward-rounded Arb arithmetic, we validate a local response-fibre chart and its intrinsic normalized projected-gradient ODE. For 0 <= t <= 1e-14, the certified solution exists, is unique, preserves R3 exactly within the declared analytic model, and satisfies dL6/dt <= -0.64195 < 0. A complementary interval certificate establishes broader-domain regularity and strict descent. This computer-assisted theorem is local; global continuation remains open. Conceptual role This theorem provides a certified constructive witness for the operational response-map branch of the Principle of Physical Realizability. It shows that a nonzero kernel direction of a declared response differential can be integrated into a locally existent, unique, nonconstant process that has exactly zero declared response cost while an independent implementation property changes strictly. The result validates this model-level realization, not the universality of Principle R. It does not derive Lorentzian signature, establish a fundamental vacuum cost, prove global continuation, or constitute hardware evidence. Conceptual framework: https://doi.org/10.5281/zenodo.21782618 Frozen GitHub Release: https://github.com/papasop/Geometric-Flow/releases/tag/paper-local-ode-v1.7