Systems with N Degreesof Freedom

Rongzhen Dai

A persistent internal state can retain preparation information without improving prediction of achosen output. We construct a finite-time comparison, under matched dynamics, between theconditional risk of a policy that reads the record once and the worst-preparation risk of everypredictor restricted to one common output law. Separately controlled reference and policy-riskbudgets turn elementary metric separation into a sufficient certificate of predictive advantage.For a stochastic Schlögl network, a coupling stopped at record-band exit combines write failure,escape and the accumulated variation of reporter rates. An analytic boundary-crossing boundtransfers these budgets from a finite-capacity chain to the unbounded count process. The selectedchemical point gives certified upper and lower risks of 0.04423 and 0.13845. An engineeredbound-pair model gives corresponding trace-distance bounds of 0.06683 and 0.28242. For thesame stored channel distinction, a fixed position measurement retains a positive certificate, whilemomentum measurement has exactly zero separation. These results distinguish persistence ofa record, access through a reading policy and predictive effect on a selected observable. Forthe charge-field programme, which separates inherited identity from active internal degreesof freedom, the comparison supplies a calculable test of finite-time record-based constraint.Chemical reporter feedback is neglected; quantum preparation and detection are specified modelinputs. The capacity bound controls state-space truncation separately from numerical integrationerror.

И. Д. Феранчук, A. A. Ivanov, Van-Hoang Le +1 more

Walter E Thirring

Yoshida, Kazuyuki

We propose a unified operational perspective on two seemingly disparate mysteries of modernphysics: quantum nonlocality and the cosmological dark sector. The common structural feature,we argue, is a projection from a nontrivial “virtual” geometry—an unobservable context field thatorganizes which observables can in principle become actual—to the “real” axis of spacetime-localmeasurement outcomes. On the cosmological side, we show that any ΛCDM background compatiblewith observations can be equivalently described by Einstein’s equations with an effective stress–energy tensor Tµν[N] associated to an in-principle unobservable context field N(x). The usualdark-matter and dark-energy components are then reparameterizations of the same context field,whose influence is visible only through its gravitational shadow. Redshift phenomenology, includingthe evidence for accelerated expansion, can be expressed as a functional Z[N; γ] of the context fieldalong null geodesics. On the quantum side, we show that the apparent “superluminal” characterof entanglement correlations arises precisely when the internal context geometry is suppressed andone insists on describing experiments solely in terms of real, local spacetime events. In both cases,paradoxical features emerge only after a dimension-reducing projection from virtual context degreesof freedom to real outcomes. We suggest, at an interpretational level, that the unobservability of thecontext field places it in the same structural role as the “self” or “awareness” in phenomenologicalaccounts: a condition for observation that cannot itself be observed as an object. The proposal isempirically equivalent to standard cosmology and quantum mechanics, but relocates what is meantby “dark” from unknown substances to the background context that makes any observation possible.

Yan Zhuang

Within the framework of ZTD (Marker Ontology), this paper provides a downgradederivation of the Schrödinger equation from two ontological axioms. The paper consists oftwo parts.Part I (Physics): From the finitude axiom, we rigorously derive that the change ofprimitive states requires time, excluding zeroth-order dynamics. Second-order dynamicsis chosen as the minimal nontrivial dynamics (ontological parsimony assumption). Fromground-state stability, elastic restoring force is derived; from nearest-neighbor coupling, thecoupling term is derived, yielding the coupled oscillator equation. Key clarification: spatialprimitives are the minimal distinguishable units of free fluctuation, not small balls with fixedsize. The refresh of primitives is asynchronous—synchronous refresh would requireinfinite information to coordinate, violating Axiom 2. Asynchronous refresh allows collectivefluctuation modes to persist. The offset ϕn is a measure of the collective fluctuationmode, not the size deviation of a single primitive. The oscillator equation naturally yieldstwo real degrees of freedom—displacement and normalized velocity. These two real degreesof freedom differ by 90◦ in phase in simple harmonic motion, naturally encoding a complexamplitude, from which the imaginary unit i emerges. Under the condition that the randomtessellation continuum limit holds, the oscillator equation converges to the Klein-Gordonequation, and then through a rigorous algebraic transformation to the Schrödinger equation.The old diffusion equation is shown to be a special case of the oscillator equation in theoverdamped limit.Part II (Mathematics): A conditional rigorous proof is given for the continuum limitof the Poisson-Voronoi coupled oscillator network. Rigorously proved are: energy conservation,uniform boundedness, H1loc boundedness of the piecewise linear interpolation, weakcompactness, the discrete edge sum identity, and the complete weak-form limit derivationfrom the unique core assumption (H’) to the Klein-Gordon equation. The unique core openproblem (H’) is precisely stated: does the vector graph Laplacian of the Poisson-Voronoiadjacency graph weakly converge to the continuous gradient operator with scaling O(l20)?This paper is a conditional rigorous proof: every step from (H’) to the conclusion is rigorous;the only remaining (H’) is an independent open mathematical problem.Keywords: Schrödinger equation; downgrade; coupled oscillator model; asynchronousrefresh; Poisson-Voronoi tessellation; graph Laplacian; spectral convergence; Klein-Gordonequation; ZTD theory

Walter E Thirring