In label-free microscopy, the experimenter's recurring question is not “Which named modality is best?” but “What operator should I use to estimate this physical quantity in this sample?” Named modalities are preset design families spanning illumination, sample interaction, post-sample optics, detector channels, readout, and dose allocation. Within a design, intrinsic minimax risk \(\le\) 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.
In label-free microscopy, the experimenter's recurring question is not “Which named modality is best?” but “What operator should I use to estimate this physical quantity in this sample?” Named modalities are preset design families spanning illumination, sample interaction, post-sample optics, detector channels, readout, and dose allocation. Within a design, intrinsic minimax risk (\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.
Exact identification of a target instrument need not imply a linear stability bound for nearby observations. We study fixed finite-dimensional complete instruments with rank-one target Choi blocks, all outcomes retained, a fixed diagnostic contamination path with strictly positive branch noise, and an all-device bound on Choi mass outside the target rays. Under these hypotheses, linear prescribed-target certification is equivalent to injectivity of the joint observation and completeness map on the physical target-tangent space. An invisible nonzero tangent yields an exactly feasible positive instrument pair with square-root separation; a separate target-ray condition distinguishes sharp square-root behavior from constant-order ambiguity. For two prescribed three-basis readouts on a four-dimensional incidence-record benchmark, we obtain linear and square-root compatible-instrument radii, respectively. We then distinguish arbitrary probability residuals from simultaneous physical preparation and readout errors. Positivity and normalization give a support-amplitude estimate under which both protocols have linear calibration-only sensitivity, although their responses to contamination and unrestricted probability error differ. A fixed-sample application uses additional exactly trusted references to acquire operator-error bounds and obtain a simultaneous full-instrument confidence statement. A separate named-source power calculation reaches the prescribed error threshold with a conservative sufficient allocation of approximately 6.12 × 10^21 trials. The results concern a fixed laboratory-frame target and supplied common-device and sampling assumptions; they establish neither practical efficiency nor device-independent calibration. Written proofs are supported by exact finite certificates and a returned kernel-checked Lean subset; the full argument is not formalized end to end.
Information quantique basée sur la mesure avec des peignes de fréquences optiques Ce manuscrit porte sur l’étude théorique de techniques expérimentales récemment développées pour réaliser des protocoles d’information quantique en variables continues. Les états Gaussiens multi-modes produits par conversion paramétrique de peignes de fréquences optiques jouent un rôle centrale dans ce travail. Ce phénomène permet de générer de façon déterministe un grand nombre d’états Gaussiens de la lumière. L’état de sortie peut ensuite être de-Gaussifié par soustraction ou addition d’un photon dans une superposition cohérente de modes du champ, puis mesuré par détection homodyne. La thèse est organisée en trois projets principaux. Le premier concerne l’optimisation du spectre du laser de pompe pour manipuler l’état de sortie de la conversion paramétrique. Nous avons développé les outils mathématiques pour traiter des profils spectraux avec amplitude et phase spectrales arbitraires. On a ensuite utilisé un algorithme d’optimisation pour trouver les spectres maximisant des différentes propriétés de l’état de sortie. Une importance particulière est donnée à la production d’"états cluster" en variables continues. Les optimisations ont été développées pour prendre en compte les limitations expérimentales pour assurer la faisabilité des forme spectrales dans les expériences. Dans le deuxième projet nous avons étudié comment les états non-Gaussiens obtenus par soustraction d’un photon d’un état comprimé peuvent être utilisés pour le calcul quantique. Nous proposons un protocole inspiré par le paradigme de "calcul quantique basé sur la mesure" qui combine l’état de-Gaussifié et la mesure homodyne pour approximer des opérateurs unitaires non-Gaussiens. On montre que les mêmes résultats peuvent être obtenus avec des mesure projectives sur des états de photon unique. Finalement, le troisième projet porte sur le partage de secret quantique ("quantum secret sharing"). Dans les protocoles de partage de secret quantique un donneur veut distribuer de l’information codée dans un système quantique à plusieurs joueurs d’une façon qui oblige des sous-ensembles de joueurs à collaborer s’ils veulent retrouver l’information originale. Nous avons développé un protocole qui peut être transféré aux expériences de notre groupe et nous avons participé à la formulation d’une preuve de concept expérimentale. À partir de cela, nous avons dérivé des résultats généraux sur le partage et la reconstruction d’états arbitraires de la lumière en utilisant des ressources Gaussiennes.
Cette thèse aborde plusieurs problèmes de statistiques quantiques, où il faut partir de l'objet et non des données après mesure. Nous utilisons des méthodes de sélection de modèles en tomographie quantique homodyne, et appliquons nos résultats à la calibration d'un compteur de photons. Nous étudions la discrimination optimale minimax d'états quantiques ou de canaux de Pauli. Nous mettons au point une méthode d'estimation d'opération unitaire de vitesse de convergence 1/n. Nous donnons un critère suffisant pour qu'une mesure soit propre, au sens de Buscemi et al. , et nous en servons pour caractériser les mesures propres sur les qubits. Nous démontrons qu'il ne peut pas y avoir cinq sous-algèbres complémentaires isomorphes à M2(C) dans M4(C). Le thème principal reste la normalité asymptotique locale quantique forte. Nous prouvons que les expériences i.i.d. sont asymptotiquement équivalentes à des expériences de décalage gaussien quantiques. En d'autres termes, de nombreuses copies d'un système de dimension finie correspondent d'un point de vue statistique à une copie d'un état gaussien d'une algèbre CCR de bonne dimension, dont le paramètre inconnu est la moyenne, au sens où il existe des canaux transformant l'un en l'autre, sans connaître l'état précis. Nous montrons comment un couplage atome-champ usuel permet de réaliser le canal pour des qubits. Ainsi, tous les problèmes résolus pour les expériences de décalage gaussien quantiques le sont asymptotiquement pour les expériences i.i.d. En particulier, nous donnons explicitement une méthode d'estimation optimale pour toute «bonne» fonction de perte, dans les cadres minimax ou bayésien uniforme.
An unknown optical phase can hide quantum non-Gaussianity in every individual mode while leaving it certifiable in the joint light. This paper shows when Gaussian measurements can exclude the complete Gaussian convex hull without knowing the source's common phase, and gives explicit finite-sample tests. The results guide whether to preserve phase relations between emissions and how to process the resulting measurement records. 1. Certification against the full Gaussian model. Let Gn be the trace-norm-closed convex hull of all physical n-mode Gaussian states. For a fixed known ρ∈/G1, form the common-phase state Ωn=∫02π(UϕρUϕ†)⊗n2πdϕ,Uϕ=e−iϕN. Finite Gaussian measurements with a laboratory phase reference and bounded classical scores achieve arbitrarily small false-positive and missed-detection probabilities. The null includes arbitrary Gaussian entanglement and classical correlations, with unrestricted displacement, squeezing and energy. In particular, infσn∈Gn∥Ωn−σn∥1⟶2. The norm is the full, unhalved trace norm. An unavailable common phase therefore differs operationally from independently randomising the phase of each emission. 2. An exact coherence boundary and a joint-only resource. For the displaced two-Gaussian-packet family, centred by its actual mean before pure loss, the Gaussian-hull boundary is ρη,ν∈G1⟺ν≤νc,νc=exp[−2(1−η)(v−1/2)2v2R]. Here v>1/2 is the packet quadrature variance, R the momentum displacement, 0<τ,η<1 the packet weight and transmissivity, and 0≤ν≤1 the coherence visibility. The same membership boundary holds after common-phase averaging for every n≥3. At (v,R,τ,ν,η)=(20,2/9,9/50,1,1/2), each phase-averaged single mode is a Gaussian mixture, while the joint common-phase state lies outside Gn for n≥3. Independent phase averaging instead gives a Gaussian mixture at every block size. These states are separable and Wigner-positive: their difference is carried by phase relations across emissions. 3. The same two-packet source has an explicit finite certificate. Heterodyne phase training, a finite homodyne score and an eight-direction bounded tail test provide a complete certificate for the preceding parameter choice. With ideal Gaussian readout, the sufficient raw acquisition budget and certified errors are nraw=1015+2(4×1015)=9×1015,PFP<3.86×10−22,PFN<0.002068. This conservative budget establishes finite-sample certification of the joint-only resource; it is not an optimised experimental acquisition requirement. The PDF contains the measurement coefficients and complete proof. 4. A complementary two-million-trial implementation. For a displaced lossy single-photon source with unknown common phase, the target conditional state is ωφ=D(eiφ)2∣0⟩⟨0∣+∣1⟩⟨1∣D†(eiφ). A bounded homodyne score uses the full-Gaussian one-photon benchmark 33/(4e), 128 phase candidates and decision threshold 0.490. Including the specified source and readout tolerances, n=2×106,PFP<1.86×10−4,PFN<7.908×10−3. All vacuum emissions, loss events and measurement outcomes count towards the raw budget. This is a separate source family from the two-packet example. Together, the results distinguish unknown common phase from destructive phase randomisation and connect full-model exclusion to explicit Gaussian-measurement scores, error bounds and acquisition costs. The 15-page PDF includes the manuscript, two figures and complete supplementary proofs.
En optique quantique, la reconstruction de l'état quantique (fonction de Wigner ou matrice de densité infini-dimensionnelle) d'un faisceau de lumière correspond en statistique à un problème inverse trés mal posé. Premièrement, nous proposons des estimateurs de la matrice de densité basés sur les fonctions \textit{pattern} et des estimateurs à noyau de la fonction de Wigner. Nous faisons l'hypothèse que la matrice de densité inconnue appartient à une classe non paramétrique définie en accord avec les exemples étudiés par les physiciens. Nous en déduisons pour la fonction de Wigner associée à cette matrice des propriétés de décroissance rapide et de régularité. Deuxièmement, nous estimons une fonctionnelle quadratique de la fonction de Wigner par une U-statistique d'ordre deux sur une classe plus large. Cette fonctionnelle peut être vue comme une indication sur la pureté de l'état quantique considéré. Nous en déduisons un estimateur adaptatif aux paramètres de régularité de la fonction de Wigner. La dernière partie de ce manuscrit est consacrée au problème de test d'adéquation à la matrice de densité. Cette procédure est construite à partir d'un estimateur de type projection sur les fonctions \textit{pattern}. Nous étudions les bornes supérieures de type minimax de toutes ces procédures. Les procédures d'estimation de la matrice de densité et de test d'adéquation à une matrice de densité sont implémentées et leurs performances numériques sont étudiées.
This paper develops a framework-relative ontology of determinacy. Its core question is not what the world is made of, but what must minimally be accounted for if a stable deter- minate identity is internally self-representing. The primitive and functional core of Version 3.8 is retained: Being, the possibility of distinction, and domain-relative realizability define the sparse primitive basis; coherent self-representation then requires three non-equivalent func- tional questions—Support, Internal Encoding, and Correspondence. Explicit finite witness pairs establish uniform non-determination of each role from the other two in the unrestricted baseline class, while stronger canonical or equivariant subclasses remain open. Version 4.1.1 is a corrective consolidation of Version 4.1 rather than a new upstream on- tology step. It retains the Version 4.0 bridge revision and the Version 4.1 constructional layer. That layer adds the reverse evaluative direction Physics ⇒ Mathematics ⇒ Ontology and records where downstream structures constrain, but do not retroactively constitute, the ontol- ogy. Several previous bridge claims are weakened or rejected. Symmetry classification, tensor factorization, entanglement of a state, and path-dependent dynamics are kept as four distinct mathematical layers. A conditional expectation is treated as a localization map rather than as actualization; a branch instrument defines possible conditional events but does not select one realized branch; and a projection onto admissible spectral sectors is not called conservative dynamics without an independently derived dilation. The causal and propagation bridge is sharpened in two directions. First, a previous can- didate definition of a null relation by vanishing accumulated interval volume is proved degen- erate: every comparable pair admits a chain of links, so that criterion vanishes throughout the causal order. Link ensembles remain useful near-null approximants, but the null sector is therefore a limiting or measure-relative object rather than a pair-local primitive in the de- clared causal-set model. Second, the long-range free-chain calculation at 𝛼 = 2 is separated into two non-commuting limits. The infinite-range spectral symbol has maximal group veloc- ity 𝜋𝐽𝑎, whereas the envelope of finite-cutoff maxima tends to 2𝐽𝑎 Si(𝜋). These are spectral group-velocity statements; a causal signal cone still requires separate Lieb–Robinson-type hy- potheses. Isotropy and a unique physical Lorentz cone remain open. The defect bridge is updated with the completed 𝐾 = 512 data layer of the frozen ratio campaign. All 903 prescribed values of 𝜃 are present with eight branches each; 897 aggregate points are admissible and six lie in a short rejected interval. Neither √ 2 nor 𝜙 is a local or global minimum. The original preregistered primary ratio test nevertheless remains non- evaluable because its 𝐾 = 64 training gate is universally degenerate. Thus 𝐾 = 512 completes a characterization block; it does not retrospectively repair the primary estimator. Provenance registration and cross-platform verification remain pending. Version 4.0 turned Test 28A into an explicit reconstruction audit, and Version 4.1 retains that audit unchanged. For the standard three one-body extractions on ℋ = (C 3 ) ⊗3 , the combined linear map has exact rank 25 and kernel dimension 704. The hidden space splits as 192 two-body plus 512 fully traceless three-body directions; even the full 𝑆3-invariant class retains an exact 156-dimensional hidden sector. The preregistered continuation identifies an exact energy-selected fixed point, the totally antisymmetric state |𝜀⟩, while holonomy and variance objectives yield set-valued argmin correspondences. The historical non-triviality gate is therefore preserved but shown not to be a universal criterion, and no retroactive 𝑉28.4 verdict is declared. A prospectively frozen two-local QPH family provides an exact computer-assisted existence witness: its cycle-phase differential has rank 704/704 on the marginal kernel for the primary family and two preregistered replications, so marginals plus relational cycle phases are locally in- formationally complete in the fixed regularity stratum. Canonical modular-mismatch selectors are then audited rather than assumed. Identical additive qutrit references and every single pure mismatch under exact 𝑆3 protection obey strict capacity/rank no-go bounds. On the protected 156-dimensional domain a physically generated mixed mismatch has sufficient raw capacity but standard phase-only QPH remains incomplete. Exact cumulant analysis explains why order three is the natural next layer: the 𝑆3-hidden sector decomposes as 156 = 36 + 120, and the third connected cumulant is a complete affine coordinate on fibres of fixed proper marginals. A historical preregistered cubic-moment completion is retained only as a delicate T-NUM wit- ness, not as a theorem premise. This is the declared stopping point of the arity/cubic line: selecting one specific cubic observable would require a new independent physical law. Test 28B remains suspended. Version 4.0 added one further macro-level correction, retained in Version 4.1. Causal and conformal structure, transport geometry, stable matter sectors, and quantum correlations are treated as parallel downstream representations of one substrate model, not as a serial matter– space–time chain. A proposed identification of the matter admissibility projector Πadm with spacetime volume is rejected in its literal form: the projector is matter-branch specific, need not be nested under generative updates, and has no canonical finite trace in a type-III con- tinuum. In its place a conditional causal–transport conformal closure theorem is proved. If a causal reconstruction supplies a Lorentzian conformal class and an independent transport limit supplies a positive spatial metric on the same spacelike distribution, the conformal factor is fixed pointwise exactly when the two induced spatial metrics are conformally compatible. A scale-free generalized-eigenvalue defect provides a direct falsifier. The corresponding clock– cone–matter gate is also corrected by conformal weight: frequencies and dimensionful spectral scales have lapse weight one under a pure lapse deformation, whereas spectral ratios, localiza- tion fractions, and dimensionless stability-window coordinates have weight zero. A Tolman– thermal-time comparison is retained only as a relative lapse-profile gate; absolute time units remain open. Version 4.1 introduced a controlled common-substrate realization. Version 4.1.1 corrects its scope: parity alone excludes only an odd or linear mass/source identification with the same signed coordinate; an even law such as 𝑀 ∝ 𝑑 2 is not excluded by parity. On the declared graph Klein–Gordon sector, charge conjugation reverses Noether/gauge-current orientation while leaving mode energy and localization intensity unchanged. A grounded-graph Green- kernel theorem fixes the spatial uniformity of the cross-term sign, while the Coulomb sign itself is conditional on realizing a non-dynamical 𝐴0-type constraint sector; the same kernel in a dynamical scalar sector gives the opposite sign. A separate no-go proves that matter Dirichlet energy alone cannot select a nontrivial interior geometry. A state-selected positive geometry Hessian yields the conditional linear response 𝑦 = 𝐾 −1 𝐸 𝜏 , so weak 𝐶-even stress sources add while 𝐶-odd currents can cancel. The principal new bridge is a conditional stationary common causal–transport–matter con- struction. If one protected center supplies a common base, the transport branch supplies a positive spatial metric ℎ, and several modular/physical clocks agree on one relative lapse 𝑁, then 𝑔CTM = −𝑐 2 *𝑁2𝑑𝑇2 + ℎ has one timelike sign, predicts the same null front for all canonically normalized matter sectors, and fixes its stationary volume element without an additional scalar measure. Different masses and opposite 𝑈(1) representation weights remain lower-order data and do not alter the characteristic cone. The independently reconstructed HSMI conformal class remains a held-out test rather than an input to this construction. A second independent Version 4.1 result supplies a conditional physical law selecting com- plex rank two. A nonzero-Chern line cannot be closed under conjugation at rank one on 𝑆 2 , whereas 𝐿 ⊕ 𝐿¯ is the minimal conjugation-closed carrier; independently, ClC 2 ≃ 𝑀2(C) has minimal irreducible complex module dimension two. The existing odd-Chern defect-link con struction realizes this pair. If its Clifford algebra is physically admissible, the local carrier has 𝑀2(C) qubit kinematics. This is not an assertion that opposite electric charges form a coherent qubit, nor that every particle is a qubit. The resulting Version 4.1.1 is a constrained foundational proposal with a frozen primitive core, stronger positive and negative bridge results, a first common stationary geometry can- didate, an independently motivated minimal rank-two carrier, and a reproducibility-oriented ledger for every step from ontological determinacy toward physics. It also makes the external spin-sign, soft-graviton, equivalence-principle, and mass-budget inputs explicit without treat- ing them as evidence for the ontology.