We investigate the information retained after eliminating hidden degrees of freedom from a finite-dimensional positive-definite Hermitian operator. A single energy-dependent elimination signature connects orthogonal residual geometry, linear innovation, Feshbach–Schur reduction, resolvent response, and dynamical memory. Its value at zero determines the minimum energy after elimination, while its negative first derivative is the Gram matrix of the full-state reconstruction. The main result solves an extremal problem over all positive realizations sharing the same signature value, first derivative, and known lower spectral threshold of the hidden block. We construct a minimal canonical realization that saturates a matrix-valued lower bound throughout the subthreshold energy interval. We determine the exact minima and suprema of the first indexed eigenvalues, allowing arbitrary finite hidden dimensions. All lower minima are attained simultaneously by one realization, while a single family approaches all upper suprema. Operator equality with the extremal bound at one energy enforces the entire extremal signature. The deviation from this bound controls the high-energy component of the static reconstruction and satisfies optimal comparisons across energies. The full signature identifies the active hidden realization up to unitary equivalence, whereas first-order data determine only its minimum required dimension. Noncommuting and three-mode examples demonstrate the benefit of retaining complete matrix information and distinguish reduced visible-state weight from destructive interference in a fixed spectral readout.
We investigate the information retained after eliminating hidden degrees of freedom from a finite-dimensional positive-definite Hermitian operator. A single energy-dependent elimination signature connects orthogonal residual geometry, linear innovation, Feshbach–Schur reduction, resolvent response, and dynamical memory. Its value at zero determines the minimum energy after elimination, while its negative first derivative is the Gram matrix of the full-state reconstruction. The main result solves an extremal problem over all positive realizations sharing the same signature value, first derivative, and known lower spectral threshold of the hidden block. We construct a minimal canonical realization that saturates a matrix-valued lower bound throughout the subthreshold energy interval. We determine the exact minima and suprema of the first indexed eigenvalues, allowing arbitrary finite hidden dimensions. All lower minima are attained simultaneously by one realization, while a single family approaches all upper suprema. Operator equality with the extremal bound at one energy enforces the entire extremal signature. The deviation from this bound controls the high-energy component of the static reconstruction and satisfies optimal comparisons across energies. The full signature identifies the active hidden realization up to unitary equivalence, whereas first-order data determine only its minimum required dimension. Noncommuting and three-mode examples demonstrate the benefit of retaining complete matrix information and distinguish reduced visible-state weight from destructive interference in a fixed spectral readout.
Eliminating hidden degrees of freedom is often treated as a straightforward substitution problem. In infinite-dimensional systems, however, an effective response may remain well defined even when no hidden state realizes the corresponding extremal value, and several notions that coincide in finite dimensions can separate. This work develops an operator-theoretic framework for response reduction in Hilbert spaces that keeps these distinctions explicit. It studies hidden-sector elimination, singular limits, complex response correlations, and the relation between energetic and oriented reductions without assuming that conventional inverse or pseudoinverse formulas remain available. The framework also addresses approximation and regularization. Finite-dimensional truncations, singular hidden sectors, and successive reductions are compared at the operator level in order to determine which parts of the effective response remain stable and which require additional geometric information. The resulting picture separates effective value, attainability, solvability, orientation, and correlation structure. In infinite dimensions these are genuinely different layers of information, and a reduced description that preserves only one of them can miss features that survive in the full response.
How much internal quantum memory is needed to reproduce a stationary stochastic process, and how does reducing that memory change the environmental resources required when output records are retained? This paper establishes exact quantum-source constructions and resource bounds under explicitly specified readout and retained-output conditions. A general completion theorem identifies the minimum Hilbert-space dimension for stationary triple data in the orthogonal-current-symbol source class: it equals the sum of the complex positive-semidefinite ranks of the middle-symbol slices. An attaining construction reproduces the entire canonical stationary order-two process. For the delayed trine process, the minimum classical hidden Markov model requires nine states at every non-independent parameter value. Quantum instruments reproduce the same complete output law using four-dimensional memory throughout the parameter range, with four dimensions proved necessary in a certified low-noise regime and three dimensions both sufficient and necessary on an explicit high-noise interval. The paper then separates reusable memory dimension from environmental dilation cost. On the certified high-noise interval, the exact one-step Pareto frontier consists of a three-dimensional source with recorded-channel Choi rank nine and a four-dimensional source with rank three. Here, Choi rank measures the minimum initially pure environmental remainder dimension for the channel retaining both the emitted classical symbol and final quantum memory. A separate mixed-support qutrit construction at noise parameter 0.99 improves complete-history costs and gives upper bounds close to universal lower bounds. These results show why optimizing a single step need not optimize a retained multi-step history. The analysis also distinguishes complete channels, fixed-input states and archive-only outputs, and demonstrates how access to a retained reference can distinguish generators with identical passive output statistics. Full-word proofs, algebraic complete-positivity certificates, a technical companion and reproducibility materials accompany the results. The exact intermediate memory boundary and complete-history optimum remain open. All results use standard quantum probabilities and apply to the stated operational resource models; they do not establish total hardware, energy or thermodynamic minima.
This record contains the fifteenth technical paper in the Boundary Compensation research programme. This paper develops a finite-dimensional normal-form theory for Schur/Feshbach response classes. Previous modules of the programme established that effective response data generally do not determine a unique hidden-sector lift. Instead of treating this non-uniqueness as a defect to be eliminated by additional speculative assumptions, the present work organizes it as a controlled quotient structure. The central object is a positive effective response operator obtained from a finite-dimensional Schur/Feshbach reduction. The paper shows that, after whitening the hidden metric, the lift problem becomes a Gram-factorization problem. It then identifies the natural redundancies of this representation: hidden-frame changes, positive scale gauges and decoupled hidden enlargements. The main result is that, after removing decoupled hidden directions and quotienting by hidden-frame freedom, every finite-dimensional positive response class admits a canonical intrinsic representative. This representative is support-valued: its target is the support of the effective response operator itself, rather than a fixed external model space. This distinction is essential, because choosing a fixed framed model space would require an additional convention and cannot be made globally canonical over rank strata. The paper also proves that the minimal hidden support dimension is equal to the rank of the effective response operator, and that minimal whitened factorizations of the same response are unique up to a unitary hidden-frame transformation. On rank-stable parameter intervals, the canonical square-root representative varies smoothly, while rank-changing boundaries are identified as the natural singular walls to be studied in the next module. Boundary Compensation XV therefore converts hidden-lift non-identifiability into a precise mathematical language of quotient classes, gauge redundancies, minimal supports and canonical normal forms. The work is deliberately finite-dimensional and structural. It does not claim reconstruction of a physical hidden Hamiltonian, derivation of particle masses, a Standard Model derivation, a cosmological-constant calculation or an empirical prediction. Its purpose is to complete the regular quotient geometry needed before studying singular rank-changing walls in subsequent Boundary Compensation modules. Keywords:
We develop a finite operational response theory for controlled quantum systems in which observer closure, hidden microscopic structure, and higher-order response are treated within a single algebraic framework. The central distinction is between generic higher-order observability and Closure–Residue (CR)-constrained generative response. Standard finite-dimensional quantum theory does not, by itself, impose the specific CR accessibility condition introduced here; physically realizable controls are ordinarily determined by the microscopic Hamiltonian, hardware connectivity, conservation laws, and environmental couplings of the system under study. We therefore formulate First-Residue Accessibility as an additional finite selection principle for the coherent-control sector studied here: the infinitesimal deformation of the observer closure algebra generated by any admissible tangent in the declared target control manifold must lie in the first-residue derivation space inherited from an underlying microscopic CR sector. The algebraic consequences are rigid. Within this coherent-control sector, the maximal abelianness of the population algebra implies that the infinitesimal action of a control on the closure algebra determines its anti-Hermitian generator uniquely modulo a diagonal population gauge. First-Residue Accessibility therefore forces the declared coherent control class, up to this gauge, to lie in a canonical graph-internal subtype extracted from the microscopic CR sector. For diagonal prepared states, the visible first-order population response is silent, while the second variation factorizes through a graph-divergence map of a microscopic bilinear edge flow. A three-site single-excitation benchmark realizes the factorization exactly and yields an even response law of the form Sigma(epsilon X) = epsilon^2 Q_CR(X,X) + O(epsilon^4). We also separate exact discrete support transport from continuous-time quasilocality. A hard one-step support cone has a singular continuous-time limit whenever the limiting generator genuinely spreads local observables; finite propagation in the continuous limit must therefore be formulated by soft tails rather than exact support. We introduce a normalized-trace visible-leakage profile and derive a soft causal readout bound. In the coherent single-excitation sector this profile is uniformly equivalent, up to fixed observer-volume constants, to intrinsic operator-norm quasilocality. For finite-range exchange with simultaneous site dephasing we prove an N-uniform normalized-trace quasilocality estimate without using rank preservation. This genuinely nonunitary extension establishes a common finite-time exponential locality class with standard open-system quasilocality, while the stronger two-sided intrinsic equivalence is claimed only for the coherent unitary subclass. We further show that divergence-image membership is non-discriminating for connected full-population readout, construct an explicit standard CPTP control family that is first-order silent but violates the CR component constraint on a disconnected graph, and isolate experimentally measurable null tests. We distinguish a declared target-control manifold from the larger independently calibrated manifold of physically available coherent controls, thereby separating a conditional declared-manifold accessibility hypothesis from the stronger Physical First-Residue Accessibility statement. Under star-compatible oriented provenance we also derive an observable cross-prediction bound that links independently fixed static CR data to a measurable quadratic response coefficient. A framework-discriminating test requires both that the CR structure be fixed without using the target Hessian data and that the resulting constraint be stronger than what follows from the same calibrated microscopic information in a standard quantum model. We further exhibit a connected three-site collective-mode candidate in which CR Accessibility predicts an exact left–right quadratic null, while a same-support standard comparator constrained only by independently fixed aggregate tangent norms permits a nonzero response envelope. This yields a strict prediction-set separation under an explicit Dataset-A information condition, while also making clear that the separation disappears if standard microscopic calibration independently fixes the same collective tangent relation. We map this strict comparator onto a single-electron linear triple quantum dot, construct a four-point mixed-Hessian estimator, and propagate swap-odd instrumental asymmetry, finite-difference bias, and shot noise into a preregistered experimental envelope. The deeper derivation of the accessibility principle from the full Generative Map is left to the subsequent axiomatic stage of the program.
We formulate a conditional geometric framework for a phase bypass of a zero sector. The calibrated order parameter is a smooth section of a Hermitian complex line bundle, not a complete normalized quantum state. At a transverse zero, its normalized section has no canonical continuous extension to a full neighbourhood in the original space, whereas its pullback extends uniquely and smoothly to the exceptional boundary of the real oriented blow-up. The boundary value is determined by the vertical derivative and records the incoming normal direction; it does not select an outgoing path. On a controlled scattering component, a Hamiltonian flow supplies the missing dynamical datum under explicit transversality, two-sided non-trapping, completeness, and uniqueness assumptions. The resulting first-exit map is a symplectomorphism and is C1 persistent under small C2 perturbations of shifted Hamiltonians with uniform C3 control and quantitative regularity, flight-time, boundary-separation, and intersection-isolation margins. A Koopman–cocycle operator transports an internal fibre over this map but remains a classical construction unless a separate quantum realization is supplied. For a symmetry-constrained cubic exceptional-point slice, analytic continuation gives exact C3 monodromy, and a companion generator admits a formal GKSL-compatible no-jump embedding after a uniform loss shift. An engineered retain/erase benchmark provides programmed positive control and independently calibrated erase and lifted-EP3 nulls. The register-free bridge forbids deliberate injection of the orientation label, defines a natural carrier as minimal process memory across a certified causal break, and separates Hilbert-space spectral subspaces from Liouville-space operator modes. Exact memory identification is stated as a reversible full-algebra sufficiency criterion. A separate approximate theorem converts finite recovery and predictive-factorization errors into quantitative information and output-distribution bounds. A unified analytic benchmark derives the section, first-exit dynamics, resolved-to-control map, spectral family, and Kato transport from one calibrated joint model. These results define a falsifiable certification programme; they do not claim a completed hardware realization or a universal law of phase bypass.
A coarse description of a conservative amplitude dynamics is not determined by an energy compression alone. We develop a conditional account of refinement, unresolved contrasts, and effective preparation for finite weighted graphs and a supplied one-dimensional local chain. Norm conservation, covariance, and additive refinement permit different site measures. Repeated-value refinement closes exactly only when the measure-normalized external rates agree within each coarse block. When they do not, eliminating the complementary coordinates yields coherent memory and an initial-data forcing term, not stochastic noise or irreversible decoherence. A three-site example separates return to the coarse subspace at one time from autonomous coarse evolution. On a uniform pair-refined cycle, a contrast of vanishing norm retains finite energy feedback; a corrected nearest-neighbor generator has a fixed-band, finite-time second-order compressed estimate. Static elimination induces both stiffness and a generally nondiagonal metric. For uniformly positive nonuniform couplings, the normalized static graph has fourth-order low-energy eigenvalue accuracy, but its general prepared-state estimate is separately only first order. A contracting inverse series controls finite-range approximation, with additional costs for normalization and evolution. First-resolvent dressing improves an isolated low mode's state and phase ceiling to fourth order under quantitative spectral separation. The contribution is the source-bound combination of these conditional statements and obstructions, not a new general Schur, projection-memory, or subspace-perturbation mechanism. Geometry, measure, couplings, preparation, clock, action scale, and any empirical readout interpretation remain supplied.