This note unifies two major parts of a long research program: the vacuum structure of quantum field theory and the theory of distributed quantum records. It identifies a θ-condensate as a distributed displacement record whose relative displacement lies exactly along the field’s zero mode. The question of “which vacuum?” becomes a question of which branch information is accessible. The paper proves that the infrared invisibility results of vacuum sectors and the zero-fragment theorem of distributed records are the same underlying structure. In mode space, any subset excluding the zero mode carries exactly zero information about the θ-sector, while any subset containing the zero mode carries the complete evidence. In spatial regions, the behavior is opposite: the condensate becomes a broadcast record. Every region contains more than its proportional share of information, complementary regions contain overlapping evidence, and in the massless limit every region can access the condensate at full strength, producing maximal spatial redundancy. The paper also proves an absolute charge lock: all charge-neutral, shift-invariant observables see identical sectors, meaning that no neutral observer can determine the θ-sector regardless of the amount of spatial information available. The central result is that broadcasting and hiding are not contradictory behaviors. Space broadcasts the condensate, while charge locks it. These are two complementary aspects of one information structure connecting vacuum sectors, quantum records, and the emergence of objectivity. All results are pre-registered and verified with reproducibility tools. The work is presented as a structural unification in solvable lattice models, while interacting-field and fully dynamical-gravity extensions remain open.
This note unifies two major parts of a long research program: the vacuum structure of quantum field theory and the theory of distributed quantum records. It identifies a θ-condensate as a distributed displacement record whose relative displacement lies exactly along the field’s zero mode. The question of “which vacuum?” becomes a question of which branch information is accessible. The paper proves that the infrared invisibility results of vacuum sectors and the zero-fragment theorem of distributed records are the same underlying structure. In mode space, any subset excluding the zero mode carries exactly zero information about the θ-sector, while any subset containing the zero mode carries the complete evidence. In spatial regions, the behavior is opposite: the condensate becomes a broadcast record. Every region contains more than its proportional share of information, complementary regions contain overlapping evidence, and in the massless limit every region can access the condensate at full strength, producing maximal spatial redundancy. The paper also proves an absolute charge lock: all charge-neutral, shift-invariant observables see identical sectors, meaning that no neutral observer can determine the θ-sector regardless of the amount of spatial information available. The central result is that broadcasting and hiding are not contradictory behaviors. Space broadcasts the condensate, while charge locks it. These are two complementary aspects of one information structure connecting vacuum sectors, quantum records, and the emergence of objectivity. All results are pre-registered and verified with reproducibility tools. The work is presented as a structural unification in solvable lattice models, while interacting-field and fully dynamical-gravity extensions remain open.
This note removes the open-boundary constraint from the lattice Schwinger model and studies the periodic theory with its physical Wilson-line zero mode. The goal is to determine which features of the previous lattice results belong to the gauge theory itself and which were caused by the boundary. Using exact diagonalization of the periodic lattice model with fermions and flux degrees of freedom, the study reveals four main results. The transition is restored: the mass-gap collapse and the critical response reappear, with the mass-direction evidence developing an interior peak near the transition region. The θ-lock is recovered at zero fermion mass, with the sector energy sensitivity strongly reduced compared with the open-boundary case, while the mass-induced unlock follows the expected perturbative behavior and becomes critically enhanced near the transition. The transition-amplifier law also returns: sector evidence grows monotonically with the mass deformation, showing that the open-boundary inversion was a boundary artifact caused by pinning both sectors to a single vacuum. The central result is a record-migration law. In the massless theory, the θ-record is stored mainly in matter correlations and can be accessed through charged endpoints of spatial cuts. In the massive broken phase, the record moves into the topological flux sector: the competing vacua differ by a Wilson-loop flux, making the information invisible to purely matter-based local observers. This establishes how the location of quantum information changes across phases of an interacting gauge theory: symmetry breaking does not only create records, it determines who can access them. The results are obtained within the finite periodic lattice framework, with bounded system sizes, rotor truncation checks, and no continuum extrapolation.
This work tests whether the theory of quantum records survives the transition from finite lattice models to continuum quantum field theory. Using a free scalar field in 1+1 dimensions with two histories differing by a localized classical source, the record accessible to a spatial region is expressed exactly through the fidelity of equal-covariance Gaussian states. Four principal results are established. First, the continuum record functional is verified against direct truncated-Fock-space Uhlmann fidelity calculations with agreement at approximately 10^−10. Second, the lattice record converges monotonically toward the exact continuum mode-sum result under refinement from 256 to 2048 lattice sites, demonstrating that the framework is not a lattice artifact. Third, a continuum lock law is found: record evidence measured a distance L from a localized source decays as exp(−2mL), with the fitted decay rate agreeing with 2m to high precision. This identifies the finite-volume suppression observed in earlier discrete models as the lattice manifestation of a continuum correlation-length law. Fourth, the distinction between constraint records and arrangement records survives the continuum limit. As the screening length diverges, the ratio between net-charge and zero-charge rearrangement records grows approximately as 1/m², causing the constraint-class record to dominate without bound. An exploratory study of disjoint regions also finds uniformly subadditive record evidence, consistent with redundant access to a common source through continuum vacuum correlations. The results establish the first continuum realization of the record framework in an exactly controlled quantum field theory. The present analysis is restricted to a free scalar field in 1+1 dimensions and coherent mean-shift histories; interacting fields, gauge theories, massless limits, higher dimensions, and gravity remain open.
Quantum Darwinism explains classical objectivity through the redundant proliferation of records in an environment. This note studies that mechanism inside a gauge theory, where Gauss's law constrains which records can exist and where they can be stored. Using exact lattice Schwinger model calculations with both open and periodic boundaries, including the Wilson-line rotor, this work shows that record redundancy is determined by the relation between information and gauge constraints. Three fundamental record classes are identified. First, unscreenable charges produce perfectly redundant records: every accessible downstream region can read the same information because the constraint itself distributes the record without requiring copying dynamics. Second, screened charges produce bilocal records, with information divided between the screening cloud and the boundary-delivered image while intermediate regions remain blind. Third, symmetry-breaking order produces graded redundant records, while topological flux records represent the opposite extreme: globally distinguishable but invisible to local matter fragments. The study also reveals a constraint-induced discordant split. In the flux sector, mutual information can remain large while readable evidence is almost absent, showing that a gauge theory can contain correlations that no local fragment can convert into classical information. The central conclusion is that classical objectivity in gauge theories is not determined only by environmental interaction. It is controlled by the structure of the constraints: Gauss's law decides which records can be broadcast, localized, or hidden. All results are obtained through exact diagonalization with independent verification checks and documented numerical convergence tests.
This note studies how quantum information is distributed when an entangled environment decoheres a system. It brings together three previous results: information stored nonlocally in environmental correlations, the separation between coherence loss and locally readable leakage, and the partial-information curve describing how much evidence can be recovered from a fraction of the environment. The paper proves that these effects are different aspects of one exact structure: a duality law for distributed quantum evidence. The readable-evidence functions of any environment fraction and its complementary fraction are exact reciprocals, with the product of the two evidence exponents remaining constant across all partitions. The self-dual point occurs when half of the environment is available, where the accessible evidence reaches the vacuum-level value. The extreme limits reproduce the previously identified eavesdropper deficit, showing that hidden environmental correlations control how much information becomes locally accessible. The paper also computes exact mutual-information curves in a hybrid qubit–Gaussian model. It shows that mutual information and readable which-path evidence are not the same quantity. A fragment may contain correlations with the system while carrying zero locally readable information. The difference represents quantum correlations that cannot be converted into an observable record by local measurements. The results reveal a fundamental distinction between information existing in correlations and information becoming an objective classical fact. Broadcast environments create redundant public records, while entangled environments can store information in a hidden, nonlocal form. All results are derived in exactly solvable Gaussian models, verified analytically and numerically, and accompanied by a reproducibility package.
Topologically ordered matter carries exotic "anyonic" charges and stores information non-locally, shielded from local disturbance; the total topological charge of a region can be read from outside only by a measurement that encircles it. Quantum Darwinism, separately, says an observable is objective when the environment redundantly records it. This paper shows the two notions single out the same thing. For the toric code — a canonical solvable model — the paper proves, exactly and at arbitrary size, that a region's total topological charge is the objectively broadcast fact of its environment, while the local anyon content is not. Four results make this precise. The charge is recorded by a loop lying inside every fragment that encircles the region, so its redundancy grows with size. Each fragment holds a perfectly distinguishable copy, so the charge forms an exact broadcast structure. An individual anyon, by contrast, is invisible to any single fragment and cannot be copied. And the celebrated topological entanglement entropy is exactly the encoded, non-broadcast complement of the classical charge record — the region's entropy splits cleanly into the two. The same split gives an error-correction reading: the objective charge plays the role of the code's syndrome, and the hidden anyons its protected logical qubits. The abelian results are proved and verified by exact stabilizer computation at arbitrary size; the fully non-abelian broadcast is left open. In short: a topological phase publishes its total charge to the world while keeping its anyons private — and its celebrated hidden entanglement is exactly that private, non-broadcast complement.
The information content of quantum field theory states depends on which operator algebra is available to an observer. Casini, Huerta, Magán, and Pontello developed entropic order parameters based on relative entropy to quantify which algebras can access superselection-sector information and established entropic complementarity relations for complementary algebra choices. This work computes that framework exactly in the θ-vacuum manifold of the massless Schwinger model, providing a fully solvable quantum field theory realization of the entropic structure associated with sector information. The neutral-algebra restriction of a θ-state is shown exactly to be the uniform mixture of flux sectors and completely independent of θ. Consequently, the relative entropy between different θ-sectors on the neutral algebra vanishes: the neutral algebra carries no information about the vacuum angle. The entropic order parameter is computed in closed form. For equal sector weights, the relative entropy order parameter is exactly ln(N), while for general sector weights it becomes the Shannon entropy H(p) of the sector distribution. The conjugate pinching in the θ-basis provides the complementary disorder parameter, and the pair realizes the entropic complementarity relation, including saturation of the Maassen–Uffink bound at the extreme limits. Combining these results with the companion vacuum decomposition results gives an exact information-theoretic realization of the principle in this model: the θ-sector information is globally present and perfectly distinguishable, absent from the neutral algebra, absent from null-surface data within the stated universality framework, and carried by the charged/boundary sector while requiring no energy cost. This work provides a fully solvable benchmark for entropic superselection frameworks in quantum field theory. The declared limits are explicit: the calculation is performed for a finite sector structure in the 1+1 dimensional Schwinger model; continuum algebraic results are cited rather than re-derived; no universal theorem and no gravitational claim are made. All symbolic derivations and numerical checks are accompanied by reproducibility materials, verification scripts, audit procedures, and recorded outputs.