A charge-sector structure and the gauge dynamics built on it need not be controlled by the same microscopic information. We show this for a charge on a graph G = (V, E) with compact link variables, held near one site by a penalty Delta, with epsilon = t/Delta. The Gauss-sector combinatorics is fixed by the vertex count and the first Betti number b1 = |E| - |V| + 1, while the magnetic stiffness kappa_B generated by transport around closed cycles appears only at the order set by the girth g, as kappa_B = c_g Delta epsilon^g. The two are independent: a five-cycle and a four-cycle with a pendant edge agree in |V|, |E|, b1, record-site degree z, sector data and leading record leakage, yet their stiffnesses differ by a whole power of epsilon. With 1 - F = z epsilon^2 to leading order for the retained record weight F, this gives the compatibility relation kappa_B = c_g Delta [(1-F)/z]^{g/2} in the same regime. We also prove, from an exact symmetry of the transfer graph rather than order by order, that a shift-type link forces the within-sector electric stiffness to vanish at every coupling; a second electric channel then suffices to complete the dynamics within the static class studied. A finite record lifetime sets a further validity condition with no counterpart when a Gauss constraint is imposed exactly. The low-energy theory is standard compact gauge dynamics; the contribution is the constraint that localizing the charge places on which completions are accessible.
A charge-sector structure and the gauge dynamics built on it need not be controlled by the same microscopic information. We show this for a charge on a graph G = (V, E) with compact link variables, held near one site by a penalty Delta, with epsilon = t/Delta. The Gauss-sector combinatorics is fixed by the vertex count and the first Betti number b1 = |E| - |V| + 1, while the magnetic stiffness kappa_B generated by transport around closed cycles appears only at the order set by the girth g, as kappa_B = c_g Delta epsilon^g. The two are independent: a five-cycle and a four-cycle with a pendant edge agree in |V|, |E|, b1, record-site degree z, sector data and leading record leakage, yet their stiffnesses differ by a whole power of epsilon. With 1 - F = z epsilon^2 to leading order for the retained record weight F, this gives the compatibility relation kappa_B = c_g Delta [(1-F)/z]^{g/2} in the same regime. We also prove, from an exact symmetry of the transfer graph rather than order by order, that a shift-type link forces the within-sector electric stiffness to vanish at every coupling; a second electric channel then suffices to complete the dynamics within the static class studied. A finite record lifetime sets a further validity condition with no counterpart when a Gauss constraint is imposed exactly. The low-energy theory is standard compact gauge dynamics; the contribution is the constraint that localizing the charge places on which completions are accessible.
Preprint and reproducibility archive; not peer reviewed. WHAT IS THE QUESTION? Quantum phases are often characterized through properties of their many-body states: order parameters, correlations, reduced density matrices, entanglement spectra, and local-unitary equivalence classes. Transport, however, is a response to a specified physical probe. This paper asks a sharp version of that distinction: Can two exact condensates be indistinguishable by every spatial reduced-density spectrum, yet exhibit different charge stiffness when the same physical gauge field is applied to both? The answer is yes. The paper constructs an exactly solvable finite-range flat-band pairing model in which two number-projected condensates are related by onsite product unitaries and therefore have identical spatial reduced-density spectra for every bipartition, while their response to the same fixed physical U(1) Peierls twist is qualitatively different. MAIN EXACT RESULT The model contains an exactly degenerate manifold of number-projected condensates. Two branches in this manifold are related by onsite product unitaries. For every spatial region R, their reduced density matrices are therefore unitarily conjugate and have exactly identical spectra. Consequently, all spatial entanglement spectra and all Rényi entropies coincide between the two branches. Nevertheless, under the same fixed physical U(1) twist, one branch is exactly soft while the other has positive static charge curvature. The difference is exact rather than a small perturbative correction. The soft tower has zero curvature, while the rigid branch satisfies an exact all-filling response law K_N^r(epsilon) = [N(M-N)/(M-1)] K_1(epsilon), with M = 2L, and retains a nonzero response density in the fixed-filling density limit evaluated in the paper. An arbitrarily weak symmetry-allowed selector epsilon > 0 uniquely chooses the rigid texture while the odd-charge gap remains open. The odd-charge gap is obtained exactly: Delta_odd(epsilon) = 3/4 + epsilon/2. Thus the switch between zero and finite static curvature is not produced by closing the protected odd-charge gap. WHAT INFORMATION IS MISSING? The two condensates are equivalent as static states under onsite local unitary transformations. However, the external gauge coupling is held physically fixed rather than being transformed together with the state. The paper formulates this distinction as an equivariance problem for the twist-defect map. Static onsite-local-unitary orbit invariants determine the charge curvature only if the physical twist-defect map transforms equivariantly along the orbit. In the constructed model, it does not. The conceptual conclusion is therefore more precise than the familiar observation that condensate fraction and superfluid density are different quantities. Here the two states agree on an exceptionally strong collection of static state information: every spatial reduced-density spectrum. They nevertheless respond differently to the same physical probe. The missing datum is the relative orientation between the state orbit and the physical gauge embedding. In other words: state equivalence does not automatically imply response equivalence when the probe is specified independently of the equivalence transformation. The result does not deny that a full microscopic wavefunction written in a fixed physical basis, together with the complete physical coupling to the external field, determines the response. Rather, it identifies exactly which information is lost when one reduces the system to probe-independent state-orbit data. TWIST-INDUCED UNSELECTION AND NONCOMMUTING LIMITS The work also studies competition between the infinitesimal selector and a finite gauge twist. In the joint scaling limit epsilon -> 0+ with A_x^2/epsilon fixed, the leading texture energies cross at lim_(epsilon -> 0+) A_(x,c)^2(epsilon) / epsilon = 24/7. The twist can therefore drive the selected rigid texture back toward the soft texture while the states remain within the same static onsite-local-unitary orbit. This provides an explicit example of noncommuting limits: selecting the state with epsilon > 0 and then taking epsilon -> 0+ does not reproduce every feature of setting epsilon = 0 from the outset. WHY THIS MATTERS For readers outside flat-band superconductivity, the broader issue is: What information about a quantum state is sufficient to determine its response to an external probe? Entanglement spectra, reduced-density spectra, and local-unitary orbit data are powerful state descriptors. This construction shows that even the complete collection of spatial reduced-density spectra does not, by itself, determine static charge response when the physical gauge coupling supplies additional orientation information. The distinction is conceptually relevant to: - quantum many-body response; - quantum geometry; - transport inferred from wavefunction information; - exact paired states; - entanglement-based classification; - local-unitary invariants; - gauge coupling and probe dependence. The value of the construction is that this separation is realized exactly, without disorder, uncontrolled effective theory, or approximate numerical diagonalization. SCOPE AND NONCLAIMS Throughout this work, “stiffness” means finite-volume static charge curvature K_ij^(N) = [d^2 E_(0,N)(A) / dA_i dA_j]_(A=0) and the explicitly evaluated selected density limit. The result is not identified with an unspecified dynamical order of limits, finite-frequency optical conductivity, or a universal dynamical definition of superfluid weight. The thermodynamic sequence used in the manuscript is an L x 1 torus, periodic in both directions. It is not claimed to be a fully two-dimensional thermodynamic family in which both linear dimensions diverge. Neutral collective excitations can soften with increasing size, so finite-twist uniformity and dynamical response limits remain separate problems. VERSION CLARIFICATION The current release should be quoted using the asymptotic scaling statement A_(x,c)^2(epsilon) / epsilon -> 24/7 as epsilon -> 0+. Earlier wording could be read as claiming the stronger finite-epsilon identity A_x^2 = (24/7) epsilon. That stronger interpretation is not supported. The current manuscript explicitly shows finite-epsilon crossing values displaced from 24/7 and states the scaling limit correctly. This clarification does not alter the paper’s main exact result: identical spatial reduced-density spectra can coexist with zero versus finite static charge curvature under the same physical twist. It also does not alter the exact odd-charge-gap formula. REPRODUCIBILITY The deposit contains the Letter, Supplemental Material, exact and numerical certificates, figure generators, manuscript-build scripts, SHA-256 manifests, and a claim ledger. The archive retains refuted or quarantined claims instead of silently removing them and separates exact identities from finite-precision numerical stress tests. The assumptions behind the selected-state, curvature, gap, and scaling-limit statements are explicitly recorded. SEARCH AND INDEXING CONTEXT Relevant concepts include: flat-band superconductivity; static charge stiffness; Kohn curvature; Peierls twist; exact condensates; eta pairing; quantum geometric nesting; reduced density matrix; entanglement spectrum; Rényi entropy; local-unitary orbit; onsite product unitary; response from wavefunction data; gauge embedding; physical probe orientation; twist-defect map; transport switching; noncommuting limits; quantum many-body response.
This work extends the theory of quantum records from free continuum field theory to an interacting gauge theory using the exactly solvable massless Schwinger model. The study asks whether the continuum record law survives when the relevant correlation length is not imposed by a bare mass but generated dynamically through the chiral anomaly. Using the open-boundary staggered lattice Schwinger model at fixed physical volume, external charge configurations are introduced through Gauss-law strings and records are quantified by the fidelity distinguishability of reduced edge states relative to the vacuum. Four main results are obtained. First, the gauge-theory record shows systematic convergence under lattice refinement. Although the available lattice sizes do not yet establish a high-precision continuum value, successive differences decrease and independent discretization schemes approach one another. Second, the record obeys an exponential lock law. Its distance dependence is extremely close to E ∼ exp(−2MD), where M = g/√π is the anomaly-generated Schwinger boson mass. Independent lattice sizes and source separations reproduce the predicted decay rate to within a few percent. Third, the record functional detects bosonization. The same physical record is calculated in the bosonized description as a Gaussian record of a free massive scalar field. The fermionic-to-bosonic record ratio decreases monotonically toward an order-one value under refinement, providing a direct information-theoretic bridge between the two descriptions. Fourth, a net-charge configuration produces a stable constraint record under refinement when its Gauss-law flux necessarily crosses the observer's region. The results show that the continuum record framework survives its first interacting gauge-theory test and that the exponential information-locking scale is controlled by a dynamically generated physical mass. The present study is restricted to the 1+1-dimensional massless Schwinger model, finite lattice sizes, integer external charges, and open boundaries. Higher-dimensional gauge theories and the full continuum extrapolation remain open.
Preprint and reproducibility archive; not peer reviewed. WHAT IS THE QUESTION? In an ordinary dispersive band, charge stiffness is closely related to kinetic energy. In a perfectly flat band, ordinary band velocity vanishes, so superconducting or superfluid stiffness can instead depend on quantum geometry and interactions. In the quantum-geometric-nesting (QGN) class of frustration-free flat-band models, previous work established a rigorous bound relating the many-body superfluid stiffness to the inverse mass of a Cooper pair and reported numerical saturation at approximately the 0.1% level. This raises a natural question: Is perfect quantum geometric nesting by itself sufficient to make the pair-mass bound an exact equality? This work does not assume that the answer is yes and does not attempt to prove a universal saturation theorem. Instead, it calculates the finite-size static charge-twist response exactly and identifies the structural obstruction that decides whether saturation occurs. MAIN RESULT: TWO EXACT RESPONSE CHANNELS For the isolated QGN condensates studied here, the finite-size second derivative of the ground-state energy under a charge twist has the exact structure E_0^(N)''(0) = alpha C_3(N) + gamma C_2(N), where C_3(N) and C_2(N) are fixed hypergeometric filling factors. The amplitudes alpha and gamma are independent of pair number N and can be determined entirely within the one-pair sector. The second amplitude satisfies gamma >= 0, so its contribution has a fixed sign. This converts the question of saturation from a numerical observation into a structural condition: The familiar one-channel pair-mass form is saturated exactly when the second channel vanishes. The paper then determines how this second channel can arise. The spin trace first decides whether a generator contributes at all. Traceless spin structures make the relevant contribution vanish identically. For the contributing charge-type sector of the on-site construction, the remaining finite-size term is an aliasing contribution associated with nonzero lattice Fourier or Poisson modes rather than the continuum mode. EXPLICIT FINITE-SIZE COUNTEREXAMPLE The paper constructs an explicit admissible QGN model with a numerically certified unique gapped ground state for which gamma / alpha = 0.846 at N_k = 4. Relative to the one-channel saturating form, the finite-size departure is approximately 42% at half filling and rises to approximately 169% at N = 3, where the reference one-channel contribution is itself small. These percentages are finite-size departures from the one-channel form. They are not claimed to be thermodynamic violations of the rigorous pair-mass bound. THERMODYNAMIC RECOVERY The same analysis also explains why near-perfect saturation may appear in larger regular systems. Under the stated joint analyticity and regularity assumptions, uniform normalization assumptions, and a uniform charge-gap condition, the finite-size aliasing contribution decays and the conjectured one-channel relation is recovered in the thermodynamic limit. The work therefore separates two logically different statements: 1. perfect QGN alone does not force exact finite-size saturation; 2. with additional regularity and gap assumptions, the finite-size obstruction disappears in the thermodynamic limit. This distinction is important because an exact identity at finite system size and a thermodynamic definition of stiffness are not interchangeable statements. ADDITIONAL STRUCTURAL RESULTS The paper derives further constraints on the pairing form factor. When the generators that actually enter the Hamiltonian act irreducibly on each flat-band fibre and the momentum graph generated by them is connected, the pairing form factor has a degenerate, momentum-independent singular-value spectrum and the condensate is unique up to an overall scale. Counterexamples to converses are given, so these properties are not claimed to follow from QGN alone. The admissible band-space bilinears are also organized by a Z2-graded real structure. In an appropriate real basis, the even sector is real symmetric and the odd sector is i times a real antisymmetric matrix, with dimensions m(m+1)/2 and m(m-1)/2, respectively. This grading clarifies which algebraic sectors can carry the finite-size obstruction. The work additionally checks that, in the translation-invariant setting considered, optimization over the relevant all-spin matrix-valued antisymmetrized-geminal-power variational family reproduces the exact finite-size curvature to second order. The additional response channel is therefore not merely an artifact of evaluating a single fixed trial vector. WHY THIS MATTERS The scientific point is not simply that one numerical conjecture has a finite-size exception. The stronger result is that the apparent equality between a collective many-body charge response and a two-particle pair-mass quantity has an identifiable structural condition. The exact two-channel decomposition explains both: - why many QGN examples may appear almost perfectly saturated, and - how a valid QGN system can depart strongly from the one-channel form at finite size. For a broader reader, the work illustrates an important problem in many-body physics: When can information obtained from a low-particle sector determine a collective many-body response? In this QGN class, the one-pair sector carries unexpectedly strong information: it determines both response amplitudes. However, an additional algebraic and finite-size aliasing condition decides whether the response collapses to the commonly assumed pair-mass form. The lesson is therefore not that few-particle information is generally insufficient, but that its sufficiency must be established structurally rather than inferred from numerical agreement. SCOPE AND NONCLAIMS The observable treated here is the static charge-twist curvature in the conventions stated in the manuscript. The work does not claim that the 42% or 169% finite-size departures survive as thermodynamic violations. The thermodynamic recovery theorem requires the regularity, normalization, and gap hypotheses stated in the paper. Finite-range or infinite-volume situations outside those hypotheses are separate questions. The surjectivity theorem in release v1.1 explicitly assumes a momentum-independent, invertible singlet nesting matrix. That hypothesis is essential and should not be omitted when quoting the theorem. The relation between the graded algebraic sectors and the complete QGN interaction class also requires the stated orbital-coverage conditions; the manuscript gives cases in which those conditions fail. VERSION CLARIFICATION Release v1.1 tightens the formal scope of the v1.0 statements. Most importantly, it explicitly adds the momentum-independence hypothesis required by the surjectivity argument and separates analytically proved statements from numerically certified properties of the explicit counterexample. The finite-size two-channel response result and the thermodynamic scope should be quoted in the v1.1 form given above. REPRODUCIBILITY The deposit includes the manuscript and a fail-closed reproducibility package containing exact or symbolic checks, numerical certificates, figure generators, deterministic reference outputs, and a frozen manifest. The explicit finite-size counterexample, response formulas, convention factors, gap diagnostics, and auxiliary algebraic claims are mapped to executable verification routines. SEARCH AND INDEXING CONTEXT Relevant concepts include: flat-band superconductivity; quantum geometric nesting; QGN; superfluid stiffness; static charge curvature; Kohn twist; Cooper-pair mass; pair-mass bound; quantum metric; quantum geometry; exact pairing; antisymmetrized geminal power; AGP; finite-size aliasing; Poisson summation; thermodynamic limit; frustration-free Hamiltonian; Z2 grading.
This note extends the record-classification framework into a two-dimensional lattice gauge theory by studying pure Z2 gauge theory on finite tori. Using exact diagonalization in winding sectors, the work tests how quantum information is stored, distributed, and destroyed across different gauge phases. The classification survives and becomes sharper. The constraint record becomes a topological redundancy: any enclosing loop can recover the enclosed charge parity exactly at every coupling and in every phase, as a direct consequence of Gauss's law. String records appear only where confinement localizes flux into physical strings. The toric-code sector qubit forms a matter-blind record class, preserved in the topological phase but destroyed by confinement. Across the phase diagram, different records migrate or disappear, but the constraint record remains invariant. The study also identifies an important boundary case: antipodally placed charges do not produce a missing record, but an undefined local record because degenerate winding routes prevent a unique assignment. The results show that quantum records in gauge theories are determined by the underlying constraint structure and topology. Some information becomes redundantly accessible through global constraints, some becomes localized through confinement, and some survives only as nonlocal topological information. All results are obtained through exact finite-lattice calculations with documented verification checks and reproducibility data.
In a compression-based account of charge, five distinct objects share the single name “charge”: a global conserved charge , an einselected reduced record , a -odd Yang–Lee ordering operator , a decaying Gaussian occupation , and a trajectory charge-transfer count . We separate them inside an instrumented, charge-conserving open quantum process and ask which relations are invariants of the full process and which are only diagnostics of a monitoring scheme or of the no-jump conditional slice. In minimal completely-positive models we establish: (i) [THM] if every jump commutes with and , charge-diagonal monitoring makes the reduced record an exact fixed point, and [CALC] its survivor -algebra is the two-dimensional diagonal (charge-sector) algebra; a charge-violating perturbation makes it a finite-lifetime metastable record whose lifetime rises with the monitoring rate in the strong-monitoring (Zeno) branch. (ii) [THM] -covariance does not imply charge conservation — amplitude damping is covariant yet lowers the occupation. (iii) [CALC] the asymptotic () stationary fixed-point -algebra obeys a three-way classification: for a bare record; with unchanged charge centre when a non-abelian seed survives (intra-sector protected structure); and a refined centre when a second abelian label survives. Charge labels all asymptotic survivors iff the stationary and there are no nonzero-frequency peripheral modes; the weaker excludes neither intra-sector survivors nor rotating (nonzero-frequency) peripheral modes, which are counted separately from . (iv) [CALC] a genuine finite-time trajectory Lee–Yang zero of a two-site transport counting statistics exists (complex Newton root, residual , winding number one); we do not call it an edge. The two strongest results are negative: [NEG] the no-jump effective Hamiltonian is not a representation invariant of the physical generator — at an exceptional point the eigenvalue splitting opens from to under a redundant Lindblad representation that leaves fixed to , so the exceptional point itself is not a process invariant; and [NEG] in the stated model, at a no-jump exceptional point no non-analyticity of the physical scaled cumulant generating function is detected in the scanned range. One minimal model simultaneously instantiates three charge objects () and two derived dynamical objects (); and remain comparison entries supplied by other minimal models / companions. In that model the conserved record and the zero-generating transferred charge are different objects: the ledger and language are unified, but a direct, representation-invariant bridge from the record to the dynamical Lee–Yang zeros is not established. We do not derive , a dynamic exponent , or any universality class.
Compact lattice gauge theories are formulated in terms of angular variables and integer electric fluxes, while bosonic quantum hardware provides oscillator modes with continuous, unbounded quadratures. We bridge this gap with a one-to-one encoding. After Gauss's law is solved, each remaining gauge degree of freedom is carried by a single oscillator mode, with its interactions built from trigonometric gates, and a Gottesman--Kitaev--Preskill (GKP)-type stabilizer provides the compactness that the hardware does not. The encoding becomes exact in the limit of infinite squeezing, and at finite squeezing, the leading imperfections act as small, computable shifts of physical observables rather than uncontrolled leakage. We apply the construction to compact QED3 and derive the error budget at finite squeezing, characterizing the leading errors in closed form, and showing that they can be corrected, subtracted, or extrapolated away. We construct syndrome-extraction protocols that detect and remove the displacement component of photon loss, delimit the noise it does not reach, compare two choices of dynamical variables, and collect the scaling of mode count, gate count, and measurement cost. A one-plaquette example reproduces the exact compact-rotor dynamics, and real-time spectroscopy with controlled extrapolations recovers the exponentially small energy splitting between charge sectors, the seed of the monopole physics of the theory, at the percent level against its exact value.