This paper constructs a finite deterministic hybrid pilot medium whose ordinary configuration histories converge to the minimal Bell jump process on a fixed finite graph and physical-time horizon. Canonical bond currents drive conservative packet export. An independently prepared spatial gas generates an approximation to marked Poisson contact histories, finite opposite-packet recombination suppresses surplus traffic, and carrier-population tracking produces the residence-weight denominator. The combined comparison controls complete tagged paths, including nodes, current reversals and null intervals. For a fixed compiled clock and quantitatively calibrated initialization, explicit resource scales yield an O(N^(-1/70)) error bound. An autonomous material Hamiltonian includes source, fuel, pending and loss states, physical copies, reset receivers, an inaccessible reference and noncommuting continuation. Monomial copy boundaries are crossed exactly once on the clock’s first pass, establishing faithful records of actual sampled configurations. The result is a controlled effective realization under the declared P1–P4 interaction and preparation assumptions. Exact finite-resource Bell dynamics and a smooth Hamiltonian realization of the entire hybrid source are not established. This Version 2 preprint presents the proofs and appendices as a self-contained companion to the monograph.
This paper constructs a finite deterministic hybrid pilot medium whose ordinary configuration histories converge to the minimal Bell jump process on a fixed finite graph and physical-time horizon. Canonical bond currents drive conservative packet export. An independently prepared spatial gas generates an approximation to marked Poisson contact histories, finite opposite-packet recombination suppresses surplus traffic, and carrier-population tracking produces the residence-weight denominator. The combined comparison controls complete tagged paths, including nodes, current reversals and null intervals. For a fixed compiled clock and quantitatively calibrated initialization, explicit resource scales yield an O(N^(-1/70)) error bound. An autonomous material Hamiltonian includes source, fuel, pending and loss states, physical copies, reset receivers, an inaccessible reference and noncommuting continuation. Monomial copy boundaries are crossed exactly once on the clock’s first pass, establishing faithful records of actual sampled configurations. The result is a controlled effective realization under the declared P1–P4 interaction and preparation assumptions. Exact finite-resource Bell dynamics and a smooth Hamiltonian realization of the entire hybrid source are not established. This Version 2 preprint presents the proofs and appendices as a self-contained companion to the monograph.
How can deterministic microscopic dynamics reproduce quantum jump histories, including their timing, while creating records faithful to actual earlier events? This paper constructs a conditional realization of the minimal Bell jump process using a deterministic hybrid model on a fixed finite configuration graph. The model combines a canonical complex field, conservative packet production, carrier–packet reactions, finite-rate recombination, and a finite spatial gas. Under explicit initial calibration and gas-independence assumptions, the ordinary-configuration path law converges in total variation to the Bell jump process without rescaling physical time. Quantitative estimates control the gas contact history, recombination error, and population tracking, including wavefunction nodes, current reversals, and zero-current intervals. An autonomous quantum clock carries a finite measurement circuit that retains fuel, loss channels, copied archives, reset receivers, feedback, and an inaccessible reference. At specified copying gates, the Bell path crosses the gate exactly once during the first pass. The archive therefore records its own actual earlier configuration and preserves that record under subsequent label-preserving operations. An explicit resource hierarchy provides an input-uniform polynomial convergence bound for a fixed engineered circuit. The theorem depends on the stated interaction catalogue and statistical preparation premises. It does not derive those premises, establish laboratory realization, or embed the complete hybrid dynamics in a single smooth material Hamiltonian. A supplementary smooth construction covers isolated contact modules.
This monograph develops the Shadow Theory quantum measurement programme through two distinct physical constitutions, connecting source dynamics to actual events, Born-rule measurement statistics, material records and subsequent measurements. The pilot-medium construction combines canonical current production, conservative packet export, independently prepared deterministic gas contacts, finite recombination and carrier-population tracking. Under its stated assumptions, the complete ordinary tagged path converges in total variation to the minimal Bell process in unchanged physical time. An autonomous finite-clock construction supplies faithful first-pass records, retained resources and noncommuting continuation. The separate massive-configuration construction adopts Bohmian guidance and complete initial quantum equilibrium. It supplies explicit material detectors, archives, reset operations and autonomous control, with separate bounds for retained outputs and historical record fidelity. The volume integrates proofs, counterexamples and the programme’s detector, preparation and protection results. It presents conditional internal completions within declared premises, rather than a derivation of those premises from source/readout incompleteness. The pilot result is a controlled effective limit; the massive result is a separate operational completion. This preprint is offered for independent scrutiny.
This paper develops a semibounded material realization of a finite quantum measurement chain within a continuous-configuration theory. Its physical premises are spinor Schrödinger dynamics, Bohmian kinetic-momentum guidance and complete initial quantum equilibrium with a finite prepared resource stock. Smooth forced harmonic traps provide explicit pointer trajectories and crossing-time distributions. The construction retains pending excitation, loss products, fuel, reset receivers and inaccessible references. Physical archives, transported-trap reset and noncommuting continuation are incorporated into the same material inventory. A retained massive controller implements the programme autonomously. Weighted derivative estimates bound the complete retained-wave error, while absolute surface-current estimates separately control historical archive corruption. The closure theorem combines output, archive and pre-reset transfer errors. Explicit rival dynamics demonstrate that reliable records and Born-compatible endpoint statistics do not uniquely determine the underlying motion law. This is a separate constitutive completion, not a derivation of discrete Bell jumps. Guidance and initial equilibrium remain postulates, and the conclusions concern fixed finite programmes within the stated domain. This Version 2 preprint is a self-contained companion to the monograph and pilot-medium paper.
A measurement apparatus must do more than produce the correct final outcome frequencies: its records must faithfully preserve its own earlier declarations. This paper constructs autonomous quantum measurement chains that address both requirements within a specified nonrelativistic Schrödinger–Bohmian model. Assuming Bohmian guidance, complete initial quantum equilibrium, and a finite apparatus stock prepared independently of the unknown input, the manuscript gives a Hamiltonian bounded below that implements each fixed finite measurement programme to any prescribed positive tolerance. The complete model retains the source, pointers, internal clock, coherent resource products, copied archives, reset receivers, and an inaccessible quantum reference. Forced harmonic traps supply exact pointer motion. Retained orthogonal keys permit stationary record storage, while explicit copying and transfer preserve earlier records before working registers are reset. A finite-mass quantum clock replaces external time-dependent control. Weighted derivative estimates quantify its approximation error and bound the absolute probability current through archive decision surfaces. The resulting theorem separates final quantum-output accuracy from historical-record fidelity, including preservation throughout designated holding intervals. It retains coherent null outcomes and provides reproducible finite examples. Every positive tolerance uses finite resource choices for the fixed programme and observation horizon, although Gaussian packets retain unbounded tails. This is a conditional implementation and error-control result. It does not derive the initial Born law, uniquely determine Bohmian guidance, establish closeness of complete microscopic trajectories, or demonstrate laboratory realization of the admitted interactions.
Can an autonomous quantum apparatus produce approximately Born-distributed outcomes and faithfully preserve its own earlier measurement label without assuming complete initial quantum equilibrium? This paper presents two finite autonomous Schrödinger–Bohmian constructions with explicit error bounds under restricted nonequilibrium initial laws. Each construction addresses three separate requirements: outcome calibration, copying the actual earlier configuration label, and retaining that label throughout a specified holding interval. The first model combines periodic focusing, a finite recoiling reference rotor, and a delayed compensated oscillator. Its record-distribution error is at most 0.004434840087, while failure to copy and retain its actual earlier label is at most 0.002327254798. The second model begins with a small radial Gaussian, includes autonomous preparation and activation, and incorporates the record writer from the original entrance. The joint law of its earlier label and complete symbolic holding record differs from an ideal constant Born-labelled record by less than 0.006086770113 in total variation. Its actual-label copy-and-hold failure probability is below 0.000552421956. The proofs combine bounded-variation calibration, exact recoil reduction, conditional-rank current estimates, finite-clock derivative bounds, and absolute probability-current estimates over the entire holding interval. The specialized derivations and coefficient calculations are included in the manuscript’s appendices. The two models retain distinct Hamiltonians and statistical assumptions. Neither establishes universal relaxation from arbitrary initial laws, supplies physical sources for every prescribed interaction, or demonstrates laboratory realization. External specialist verification remains open.
We construct an operational quantum model using the free overlap Hamiltonian on a complete nonsingular aperiodic D₆/H₃ carrier and a specified Fibonacci clock. From the quantitative resolvent estimate we derive O(h) continuum error for actual H⁴ cell-average preparations without a fixed spectral cutoff; variable-step midpoint has O(h+δmax²) error. Compatible mixed states and finite measurement protocols converge in trace norm and total variation. Postselected states converge when the limiting success probability is positive, with an explicit denominator bound. The carrier supports an exact two-aperture interference identity and a controlled continuum, paraxial and far-field construction of two-slit fringes. Uncertainty is stated through quadratic forms for finite-variance states, and marker complementarity includes mixed, unequal-prior and nonideal readouts on consistently weighted ensembles. An explicit number-conserving quartic interaction, calibrated with pulse strength O(h⁻¹), prepares a mode singlet during one actual Fibonacci lapse with O(h) state error for fixed smooth source modes while free dynamics remains active. Transport from a common source and quantitative truncation produce complete binary readouts in disjoint laboratories at finite spacing. A preparation family with momentum k=h⁻¹ᐟ⁵ satisfies |Sh+2√2|≤Ch¹ᐟ⁵, retaining all events. The specified local algebras obey the Tsirelson bound and operational no-signalling exactly. Operator-norm calibration bounds also control imperfect readouts and combine with state and finite-sample errors. These are realization and approximation theorems under stated Hilbert-space, probability, composition, preparation and instrument assumptions. Established quantum phenomena are implemented with explicit regulator errors; quantum probability and apparatus dynamics are not inferred from static geometry alone.