Operator-Gram Sensing at Vanishing Visibility: Hölder Minimax Laws and Coherent Heterodyne Seclusion

Tao Lin

Published Sep 9, 2026• 0 citations

Abstract

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 $$D_x(t) = t^r A(t) x,$$ 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 $C^m$ operator germ we obtain the sharp fixed-dimensional minimax exponent: $$p_{m,r} = 2 + \frac{2r}{m}.$$ For noninteger Hölder smoothness $\beta = k+\alpha$, the exponent becomes: $$p_{k,\alpha,r} = 2 + \frac{2r}{\beta}.$$ 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: $$\dot{K}_0(s,t) = F(s)^* G(t) + G(s)^* F(t),$$ 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 $C^{k,\alpha}$ but not $C^{k+1}$ 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.

DOI: 10.5281/zenodo.22669896

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Operator-Gram Sensing at Vanishing Visibility: Hölder Minimax Laws and Coherent Heterodyne Seclusion

Tao Lin

Published Sep 9, 2026• 0 citations

Publisher

Abstract

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

D_x(t) = t^r A(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

C^m

operator germ we obtain the sharp fixed-dimensional minimax exponent:

p_{m,r} = 2 + \frac{2r}{m}.

For noninteger Hölder smoothness

\beta = k+\alpha

, the exponent becomes:

p_{k,\alpha,r} = 2 + \frac{2r}{\beta}.

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:

\dot{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

C^{k,\alpha}

but not

C^{k+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.

DOI: 10.5281/zenodo.22669896

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Hoang Duy Nguyen

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Record formation, classical definiteness, and Born probabilities are structural consequences of algebraic restriction under finite accessibility. **This paper is part of the Finite Accessibility series, which reconstructs gravitational geometry, quantum dynamics, gauge structure, cosmological perturbation theory, and the dark sector from a single operational primitive: a finite certification ceiling ω* governing the maximum rate at which physical relations can be maintained across extended domains.** **Introductory papers** - Geometric Limits to Coherent Timekeeping in Extended Reference Frames — https://doi.org/10.5281/zenodo.18824556 - Accessibility Selection Law (ASL) — https://doi.org/10.5281/zenodo.18824649 - Phase Boundary Gravity — https://doi.org/10.5281/zenodo.18824707 **Pillars (numbered I–IV)** - Paper I: Deriving Gravitational Geometry and Schrödinger Dynamics from Finite Accessibility — https://doi.org/10.5281/zenodo.18601600 - Paper II: Deriving Inertia, Motion, and Wave Propagation from Finite Accessibility — https://doi.org/10.5281/zenodo.18625174 - Paper III: Accessible State Spaces, Record Stability, and the Emergence of Quantum Statistics — https://doi.org/10.5281/zenodo.18625192 - Paper IV: Frame Dissolution, the Certification Ceiling, and Observable Deformations — https://doi.org/10.5281/zenodo.18654458 **Extension papers** - Dark Matter from Finite Accessibility — https://doi.org/10.5281/zenodo.18654433 - Dark Energy from Finite Accessibility — https://doi.org/10.5281/zenodo.18654523 - No-Go Theorems and the Admissibility Corridor — https://doi.org/10.5281/zenodo.18654493 - Gauge Structure from Finite Accessibility — https://doi.org/10.5281/zenodo.18928152 - Perturbation Theory from Finite Accessibility — https://doi.org/10.5281/zenodo.18928213 **Experimental paper** - Experimental Probes of Finite Accessibility: GW Dispersion and Vacuum Path Independence — https://doi.org/10.5281/zenodo.18931126 **Keywords:** Born rule derivation, measurement problem, quantum statistics, finite accessibility, operator subalgebra, record formation, definiteness emergence, wave function collapse, compositional consistency, accessible state spaces, action gap, irreversibility, sector restriction, quantum foundations, algebraic quantum mechanics, decoherencePapers I and II derived gravitational geometry and second-order dynamics from finite accessibility without invoking measurement, observation protocols, or collapse as primitive notions. This paper extends the operational program to the measurement sector, deriving the structure of record formation and probabilistic weights from the same finite accessibility and compositional constraints. Finite accessibility restricts admissible operations on a subsystem to a proper operator subalgebra. Because bounded certification excludes operators requiring nonlocal coordination or arbitrarily fine phase resolution, the accessible algebra is strictly smaller than the full operator algebra. When a system becomes correlated with orthogonal environment sectors, off-diagonal operators between those sectors fall outside the accessible algebra. The reduced state is therefore indistinguishable, under all admissible operations, from its diagonal ensemble. Definiteness emerges as an algebraic consequence of sector restriction, not as a collapse postulate. The weights of these sectors are fixed uniquely by compositional consistency. Phase invariance, multiplicativity under independent refinement, and stability under record coarse-graining constrain any admissible assignment of sector weights to satisfy a functional equation whose only continuous, non-contextual solution is quadratic in amplitude. The Born rule therefore arises as the unique invariant measure compatible with finite accessibility. Irreversibility follows from action scaling: reversal requires operators outside the admissible tensor algebra and incurs action cost proportional to the number of correlated subsystems. Since admissible operations are bounded by the action gap, macroscopic record reversal is excluded. Measurement introduces no new physical primitives beyond finite accessibility, locality, and compositional stability. Record formation, classical definiteness, and Born probabilities are structural consequences of algebraic restriction under finite accessibility. **This paper is part of the Finite Accessibility series, which reconstructs gravitational geometry, quantum dynamics, gauge structure, cosmological perturbation theory, and the dark sector from a single operational primitive: a finite certification ceiling ω* governing the maximum rate at which physical relations can be maintained across extended domains.** **Introductory papers** - Geometric Limits to Coherent Timekeeping in Extended Reference Frames — https://doi.org/10.5281/zenodo.18824556 - Accessibility Selection Law (ASL) — https://doi.org/10.5281/zenodo.18824649 - Phase Boundary Gravity — https://doi.org/10.5281/zenodo.18824707 **Pillars (numbered I–IV)** - Paper I: Deriving Gravitational Geometry and Schrödinger Dynamics from Finite Accessibility — https://doi.org/10.5281/zenodo.18601600 - Paper II: Deriving Inertia, Motion, and Wave Propagation from Finite Accessibility — https://doi.org/10.5281/zenodo.18625174 - Paper III: Accessible State Spaces, Record Stability, and the Emergence of Quantum Statistics — https://doi.org/10.5281/zenodo.18625192 - Paper IV: Frame Dissolution, the Certification Ceiling, and Observable Deformations — https://doi.org/10.5281/zenodo.18654458 **Extension papers** - Dark Matter from Finite Accessibility — https://doi.org/10.5281/zenodo.18654433 - Dark Energy from Finite Accessibility — https://doi.org/10.5281/zenodo.18654523 - No-Go Theorems and the Admissibility Corridor — https://doi.org/10.5281/zenodo.18654493 - Gauge Structure from Finite Accessibility — https://doi.org/10.5281/zenodo.18928152 - Perturbation Theory from Finite Accessibility — https://doi.org/10.5281/zenodo.18928213 **Experimental paper** - Experimental Probes of Finite Accessibility: GW Dispersion and Vacuum Path Independence — https://doi.org/10.5281/zenodo.18931126 **Keywords:** Born rule derivation, measurement problem, quantum statistics, finite accessibility, operator subalgebra, record formation, definiteness emergence, wave function collapse, compositional consistency, accessible state spaces, action gap, irreversibility, sector restriction, quantum foundations, algebraic quantum mechanics, decoherence

Feb 12, 20260 citations

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

D_x(t) = t^r A(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

C^m

operator germ we obtain the sharp fixed-dimensional minimax exponent:

p_{m,r} = 2 + \frac{2r}{m}.

For noninteger Hölder smoothness

\beta = k+\alpha

, the exponent becomes:

p_{k,\alpha,r} = 2 + \frac{2r}{\beta}.

\dot{K}_0(s,t) = F(s)^* G(t) + G(s)^* F(t),

C^{k,\alpha}

but not

C^{k+1}

hard pair, and give a structurally independent spherical self-calibration realization with

(r, m) = (2, k-2).

\le

admissible-pipeline infimum

\le

selected deployed risk. Across designs, deployed-pipeline rankings do not determine Blackwell order. Given a declared physical labeling and prior, finite-dose prior-averaged squared risk splits exactly into within-component conditioning, between-component ambiguity, and posterior regret. Label refinement transfers an exact amount from conditioning to ambiguity. On a common-displacement branch, Theorem 2.2 bounds selected deployed risk by an asymptotic surrogate with a

1/N

envelope of complete branch coefficients, dose-independent squared displacement, and decaying or persistent alias loss. The general fallback partition keeps

1/N

separate while coupling displacement and ambiguity in §5.1's report-to-estimand block envelope. Neither corresponds termwise to the exact Bayes split. Finite upper certification adds a statistical remainder and, for approximate-generator inputs, a physical-fidelity allowance. In a coherent-displacement model with nonzero target and nuisance sensitivities, a Gram-proportional readout's target-information fraction equals the squared sine of their signal-phase separation. Any measurement on a

K_\mathrm{mode}

-mode coherent-displacement family jointly identifies at most

2K_\mathrm{mode}

displacement parameters on an open neighborhood. For two jointly unknown parameters at acute signal-phase separation

\Delta\psi

, simultaneous quadrature has penalty

2/(1+\sin\Delta\psi\,H)

relative to the best single-local-oscillator homodyne sequence, with

H

the magnitude-symmetry factor. With nonsingular sensitivity Gram matrix, a local one-mode pure coherent-displacement model admits a homodyne allocation attaining the Holevo bound for every positive-definite quadratic weight. Under §0's detection laws, continuous rotational stabilizer generators of a rigid particle's detected response are rotationally Fisher-null. Independent fixed-registration fusion removes a null exactly when a positively allocated design supplies information in that direction. The mean fitted score cannot identify estimand displacement without an added relation, such as a calibrated regular-branch fidelity path or true-estimand identification from the full retained law. At a fixed endpoint with

0<F_1^\mathrm{eff}<\infty

, the regular ambiguity exit is the generalized last-exit boundary of an exact pair Bayes floor above the local comparator. Once defined for a fixed witness pair, endpoint, and branch, the endpoint ambiguity diagnostic and variance–displacement crossover delimit an informative window but do not certify feasibility. Under a fixed design, a target-moving true-map fiber or other class with one complete detected law at every dose has intrinsic minimax risk at least its estimand image's squared Chebyshev radius,

h^2/4

for a two-point gap

h

. A pair sharing one common law within every candidate design at every dose makes that floor design-set-wide. Under transfer calibration, pair-kernel identification, and signed kernel normalization, a single-exchange calculation for homogeneous, isotropic, dilute, far-zone ensembles of weak, lossless, single-signed scalar scatterers and their co-polarized dipole extension certifies both signs of the pair-order ensemble-mean exact-minus-multislice forward-channel response-phase difference outside an endpoint allowance. At full retention, bounded variation of the pair correlation beyond a fixed exclusion radius bounds that allowance independently of the scalar radial pair-visibility window. Full-response means require omitted-mean bounds, whereas estimand-level means require the resolved phase-to-estimand bridge.