U0 Special Monograph: The Universe as a Closed Electrical Circuit — Composition Theorem, Complex Power Circulation, and Carrier-Dual Memory

U0 Special Monograph: The Universe as a Closed Electrical Circuit develops a precise mathematical and systems-theoretic answer to the following question: under what explicit conditions can a modeled total system be represented as a closed, nested, multidomain electrical circuit?

The monograph does not assume that every physical effort is literally a voltage or that every physical flow is an electric current. Instead, each physical domain retains its own variables, units, constitutive laws, validity conditions, storage mechanisms, dissipation channels, boundaries, and oriented ports. Electrical circuit theory supplies a common interconnection grammar based on relative potentials, effort–flow pairings, transferred power, storage, conversion, phase, reference, and return.

The construction follows a continuous chain:

conservation → circulation → phase → double cycle → carrier-dual structure → closure

The first part establishes the distinction between a genuine multidomain electrical representation and a merely verbal analogy. It defines physical objects as typed subcircuits, formulates domain circuits and effort–flow pairs, and constructs a universal interconnection graph without introducing a fictitious external ground, a universal voltage, or a unique current of the Universe.

The second part derives energy closure from local conservation laws. It treats charge, matter, momentum, probability, electromagnetic exchange, Poynting transport, field–matter coupling, dissipation, heat, entropy production, and relative exergy. Losses are not deleted: they must be assigned to explicit internal conversions or reservoirs.

The third part develops the finite composition of physical domain circuits. Port-Hamiltonian aggregation, passive and active elements, distributed systems, ideal and nonideal transducers, measurement chains, inverse problems, model reduction, nested circuits, and parent–child interfaces are integrated into a composition theorem. The monograph also identifies false-closure mechanisms produced by omission, averaging, projection, implicit sources, incomplete boundaries, illegal reduction, or ontological relabeling.

The fourth part separates phase, delay, latency, and memory. It analyzes active, reactive, and apparent power; defines complex-power circulation; distinguishes the physical phase cycle from the five-function causal cycle; and introduces projective return only for observables proven to remain invariant under the declared quotient.

The fifth part constructs a finite carrier-dual memory layer, called Coho, using a sixteen-state level-four system, Weyl operators, finite Fourier analysis on (G_4), carrier and dual variables, mixed classes, and scalar spectral energy. It proves that identical carrier and dual marginals can retain different mixed coupling information. This establishes a finite non-reducibility result: the mixed invariant cannot, in general, be reconstructed from either marginal alone.

The central result is the U0 Composition-Closure Theorem. A finite rooted hierarchy satisfying the declared local laws and interface contracts receives three independent certificates:

1. a root-boundary certificate detecting unmatched external ports;
2. an extended-energy certificate testing boundary power, storage drift, and declared internal reservoirs;
3. a mixed-memory certificate detecting carrier-dual information lost through projection or reduction.

When these exact certificates vanish, the representation is closed relative to the declared finite model. Numerical conformance is treated separately and requires normalized diagnostics to remain below preregistered tolerances.

A reproducible electromechanical–thermal reference model demonstrates that the admissible class is nonempty. Independent controls test open ports, unattributed dissipation, hidden active sources, incompatible phase conventions, incomplete bridges, removed dual information, eliminated memory, illegal projective return, and other false-closure mechanisms. The release includes the scientific monograph, editable and source formats, figures, quality-assurance records, numerical results, executable source code, registries, provenance records, and integrity checksums.

The result is architectural and relative to the declared model. It does not establish a universal cosmic voltage, a unique current of the Universe, a general scalar energy for arbitrary spacetime geometry, an automatic identity between active–reactive power and Coho modes, a universal conservation law for the mixed invariant, or the ontological identity of the physical Universe with an electrical circuit. The continuous physical carrier-dual bridge remains an open research problem.

This publication is the complete scientific English edition aligned with the French Cahier spécial U0 and coordinated with the foundational architecture of the Treatise on the Electrical Universe.