We formulate quotient observation as a nonlinear map on the cone of finite-dimensional positive definite precision operators and show that it is governed by a minimum-energy variational principle. From this we obtain well-posedness, positive homogeneity, monotonicity, operator concavity, and composition along nested observation towers. For positive upstairs corrections, the finite-observation law is the global tangent ceiling for the visible quotient correction. We also separate compression from quotient by a positive semidefinite Schur gap and identify, on $H_0$-adapted sectors, the first genuinely quotient-specific term as a sign-definite quartic hidden-mediation contribution. We then analyse the latent geometry of quotient observation. Finite hidden sectors preserve tangent support, and the visible laws beneath a fixed tangent ceiling are completely classified by a unique positive hidden-load operator, which also determines the minimal hidden latent dimension and the associated contraction volume. This shows that the class closed by marginal elimination is the quotient Gaussian precision class itself, while the finite-state Donsker--Varadhan Hessian survives as the tangent ceiling and microscopic skew-square cone inside that larger attenuation class. For hidden-local precision graphs we prove exponential suppression of long-range visible couplings by hidden graph distance and conditioning, identify corridor and flat-bundle extremal architectures, and confirm the predicted signatures in synthetic tests and in open EIT and passive-network examples. Full nonlinear finite-state Markov Donsker--Varadhan re-embedding remains open.
We formulate quotient observation as a nonlinear map on the cone of finite-dimensional positive definite precision operators and show that it is governed by a minimum-energy variational principle. From this we obtain well-posedness, positive homogeneity, monotonicity, operator concavity, and composition along nested observation towers. For positive upstairs corrections, the finite-observation law is the global tangent ceiling for the visible quotient correction. We also separate compression from quotient by a positive semidefinite Schur gap and identify, on H0-adapted sectors, the first genuinely quotient-specific term as a sign-definite quartic hidden-mediation contribution. We then analyse the latent geometry of quotient observation. Finite hidden sectors preserve tangent support, and the visible laws beneath a fixed tangent ceiling are completely classified by a unique positive hidden-load operator, which also determines the minimal hidden latent dimension and the associated contraction volume. This shows that the class closed by marginal elimination is the quotient Gaussian precision class itself, while the finite-state Donsker--Varadhan Hessian survives as the tangent ceiling and microscopic skew-square cone inside that larger attenuation class. For hidden-local precision graphs we prove exponential suppression of long-range visible couplings by hidden graph distance and conditioning, identify corridor and flat-bundle extremal architectures, and confirm the predicted signatures in synthetic tests and in open EIT and passive-network examples. Full nonlinear finite-state Markov Donsker--Varadhan re-embedding remains open.
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 study the local identifiability of continuous parameters describing intermediate history interventions in a graph-local quantum system from a single final-time measurement architecture in the presence of nuisance parameters. For a fixed-axis commuting Hamiltonian, we prove that back-propagated single-site observables obey an exact one-hop support closure and derive a topology-dependent ceiling on the rank of the history Jacobian. Under A3, which provides one single-site scalar readout per site, a chain geometry confines a five-dimensional history space to at most four history-sensitive observation channels, yielding exact local non-identifiability. We then introduce weak noncommuting perturbations. Although the deficient raw history direction opens at first order in the perturbation strength ε, under the fixed A4 row partition and the baseline exact structural zeros of A5, graph-local support structure and nuisance-space placement force that first-order correction to remain inside the baseline joint history–nuisance observation space, J_h,1v₀ ∈ col 𝓙₀. Hence no new independent nuisance-quotient history direction can open at first order. Combining rigorously interval-certified nonzero second-order coefficients with real-analytic genericity, we obtain, outside a proper analytic exceptional set within a fixed right-analytic structural component, σ_raw=Θ(ε) and σ_quot=Θ(ε²). Unknown-input observability, nuisance projection, graph distance–relative degree, graph-local perturbative suppression, second-order identification, and singular-value perturbation are established prior results. To our knowledge, within the prior art we examined, the narrow contribution not directly anticipated is the sufficient-condition chain by which graph-local support placement structurally traps an already first-order-visible raw history tangent inside the baseline joint history–nuisance observation space, delaying only the first new quotient-identifiable direction to second order, together with the resulting strict raw/quotient order split.
A finite Hilbert space supports many inequivalent tensor-product decompositions, so any Hilbert-space-first account of locality must explain more than why a single candidate performs well in a discrete search; it must also determine whether that preference governs nearby continuous refactorizations. Earlier work in this area established three key ingredients: that decomposition space is empirically non-flat under simple support-based selectors in explicit few-qubit studies; that the physically meaningful object in exact finite searches is a quotient class rather than a uniquely labeled factorization, defined modulo local gauge transformations and subsystem permutations; and that, for exactly graph-local Hamiltonians, the associated quadratic form has a kernel precisely given by the local gauge Lie algebra under explicit visibility and genericity conditions. Building on these results, the present paper proves a local theorem showing that, in a neighborhood of the graph-local class, the static selector is quantitatively equivalent to the square of the quotient distance. As a consequence, the preferred decomposition class is a strictly isolated local minimum: small selector values enforce closeness in the quotient sense, and no sufficiently small nongauge refactorization can replicate the preferred structure. The result is intentionally local in scope. It does not assert global uniqueness, a global spectral gap, or a full reconstruction of geometry, but instead isolates the strongest quantitative statement currently supported by the available analytic framework.
We study active coherent experiments in which a hidden vector- or operator-valued response is observed only through scalar powers. The local response has the form Dx(t)=trA(t)x, where the field-amplitude visibility vanishes to order r, and the observable information factors through the hidden-codomain gauge-invariant right-Gram geometry. For a Cm operator germ we obtain the sharp fixed-dimensional minimax exponent: pm,r=2+m2r. For noninteger Hölder smoothness β=k+α, the exponent becomes: pk,α,r=2+β2r. The lower bound is not stable under an arbitrary coherent query algebra: a remote nonzero same-mode field can act as a heterodyne reference and restore linear sensitivity. We therefore introduce the observable Gram-tangent kernel: K˙0(s,t)=F(s)∗G(t)+G(s)∗F(t), and prove a continuous query-operator criterion separating three regimes: Exact quadratic seclusion The sharp quadratic regime Heterodyne-dominated shortcutting We then realize the Hölder law in an actual finite-difference coherent dark-port experiment, including a genuinely Ck,α but not Ck+1 hard pair, and give a structurally independent spherical self-calibration realization with (r,m)=(2,k−2). Polarization, phase retrieval, classical quadratic-functional minimax theory, nulling interferometry, and curvature reconstruction are treated as prior-art inputs. The contribution claimed here is the coupling of: vanishing visibility, right-Gram quotient sensing, the sharp Hölder minimax law, and the global Gram-tangent heterodyne-seclusion criterion, rather than novelty of any one imported component.
This paper proves a finite-dimensional structural theorem for recursive holographic descent and source recognition. The theorem begins with a declared finite source carrier whose states are passed through a specified chain of completely positive trace-preserving channels and compared through the distinctions accessible at a terminal level. It determines which source distinctions survive composed descent, which become terminal-invisible, and when terminal data recognize source-level structure. The main theorem constructs composed accessibility quotients, survivor spaces, terminal-invisible source kernels, and a recursive compression invariant. Accessibility quotients form a quotient tower, survivor spaces decrease under further descent, terminal-invisible source kernels increase, and compression is monotone. Cumulative compression is not a sum of unrelated one-step losses; it is the codimension of the composed survivor range, with increments given by successive losses of already surviving source distinctions. The source-recognition theorem gives the factorization criterion. A source observable is recognizable at a terminal level exactly when it lies in the survivor space. A general source invariant on a state class is recognizable exactly when it factors through the composed accessibility quotient. A terminal value identifies a declared source identity class only when the recognized invariant is source-complete for that identity relation. Thus terminal data acquire source significance only when they survive the composed channel chain, factor through the composed accessibility quotient, and are complete for the declared source identity relation. The theorem belongs to a broader sequence of structural results: upstream carrier and admissibility results supply finite source carriers and lawful descendant structures from fixed carrier, completion, comparison, and realization data, while the companion one-step boundary-compression theorem identifies what survives a single boundary channel. Here the zero-compression case is full source separation, equivalently informational completeness of the composed terminal law; physical completely positive trace-preserving recovery requires additional recovery data. The theorem therefore provides a finite-dimensional criterion for recursive source recognition and a carrier-level audit condition for boundary, measurement, selection, holographic, and source-identification claims. License note: Distributed under CC BY-NC-ND 4.0.
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.