Collective charge response is not determined by the few-particle sectors

Abstract

Preprint and reproducibility archive; not peer reviewed. WHAT IS THE QUESTION? Many-body physics is often inferred from simpler information: single-particle motion, two-particle interactions, few-particle spectroscopy, few-particle Hamiltonian tomography, or an accurately known many-body ground state. A natural question is whether sufficiently complete few-particle information must eventually determine a collective many-body response. This work asks a deliberately strong version of that question: Suppose two interacting systems have the same complete few-particle Hamiltonians, including their full dependence on every static Peierls gauge phase, and suppose they also have the same finite-density zero-twist many-body ground-state ray. Must they then have the same collective static charge response? The answer is no. The paper constructs explicit finite-range positive Hamiltonians, called response twins, that are indistinguishable by the specified complete few-particle gauge data and share the same zero-twist many-body ground ray, yet have different many-body static charge curvatures. MAIN EXACT RESULT: RESPONSE TWINS For any prescribed finite blind depth p, the construction gives two positive finite-range parent Hamiltonians whose complete static Peierls-gauge Hamiltonians coincide at every static link phase in every particle-number sector N <= p, and first differ in sector N = p + 1. At zero twist, the two parents have the same unique completely symmetric Dicke ground ray in every fixed-number sector. Despite this exact agreement, their collective static charge curvatures can differ extensively at finite density. For the basic spectator-weighted hard-core-pair family, the response is obtained in closed form. Along spatial direction mu, kappa_(mu;N,V) = [N(V-N)/(V-1)] [1 + 2(N-1) Lambda_mu/(V-2)], so at fixed density N/V -> nu, rho_mu(nu) = nu(1-nu) [1 + 2 Lambda_mu nu]. The equality of the shallow sectors is exact at the operator level for arbitrary static link phases, while the many-particle response difference remains finite per site. The response difference therefore cannot be attributed to uncertainty in a fitted one-particle Hamiltonian or to a different zero-twist many-body wavefunction. INFORMATION-THEORETIC INTERPRETATION The construction is an exact non-identifiability witness for a specified information map. Any estimator whose complete input is restricted to the identical few-particle Peierls-sector Hamiltonians receives exactly the same input on the two response twins. Yet the target collective static curvature is different. Therefore the specified shallow information map is not sufficient to identify that collective response on the stated Hamiltonian class. This is an information-contract statement. It is not a blanket theorem that every named many-body method, Hamiltonian-learning method, density-functional method, tensor-network method, variational method, or electronic-structure method must fail. A theorem about a particular named method would additionally require showing that the complete input of that method factors through the information map that is identical on the response twins. WHERE DOES THE MISSING INFORMATION FIRST APPEAR? The paper identifies the blind-depth boundary exactly within the controlled hard-core class. A spectator polynomial of body order D can remain invisible through sector N = D and first become observable at N = D + 1. An explicit bridge construction saturates this bound. In this controlled class, blind depth therefore equals spectator body order, while the first sector capable of detecting the hidden structure lies one particle deeper. This result does not mean that arbitrary information depth can be achieved at fixed microscopic complexity. Unbounded blind depth requires increasing local body order. On a fixed-dimensional bounded-coordination lattice, it also requires growing support because a bounded-radius neighbourhood contains only finitely many independent spectator sites. The arbitrary-depth construction is therefore not a theorem of arbitrary depth at uniformly fixed body order and uniformly fixed geometric range. A COMPLEMENTARY SUFFICIENCY RESULT The paper also identifies a restricted class in which shallow information is sufficient. For an exact-pair response tower satisfying the stated triple-closure, zero-twist parent-intertwining, and one-body twist-insertion hypotheses, the compressed second-order response has pair-slot order at most two. Within that structural class, connected one- and two-pair information is sufficient, and depth two is minimal. This theorem is intentionally restricted. It is not claimed to hold for arbitrary interacting Hamiltonians. The coexistence of the negative response-twin theorem and the positive depth-two closure theorem is central to the interpretation: whether few-particle information is sufficient depends on the Hamiltonian class and on the physical task being predicted. THE HIDDEN RESPONSE CAN BE PROGRAMMED The ambiguity is not restricted to an overall scalar change in stiffness. By using independent displacement families, the difference of the symmetric response tensors can be programmed locally at interior filling. The d(d+1)/2 dyads generated by the displacement set {e_i} union {e_i + e_j}_{i<j} span the full vector space of symmetric d-dimensional response tensors. This gives the exact number of independent scalar controls required for local surjectivity onto response-tensor differences around a common positive baseline. A particularly transparent two-dimensional construction gives response twins whose diagonal charge curvatures are exactly equal while the off-diagonal shear response differs. Thus the two systems can agree along the coordinate axes and separate only through shear, providing an internal null control. “Programmable” here refers to exact mathematical programmability of the response-tensor difference inside the model class. It is not a claim that the complete Hamiltonian has already been microscopically programmed in existing hardware. DIRECT FINITE-SIZE DISCRIMINATOR The hard-core construction is also mapped to qubits. For a compact 4 x 4, 16-qubit example at half filling, the x-direction response density changes exactly from 4/15 to 8/15, while the y-direction remains 4/15 and acts as a null control. A compressed equal-time measurement protocol uses seven settings: - four compatible Bell-measurement colour classes; - one global X setting; - one global Y setting; - one global Z setting. The same measurement schedule can additionally certify the intended common Dicke ground state. This provides an operator-evaluation and state-certification blueprint. It is not a claim that the complete controlled-projector Hamiltonian has already been implemented on a quantum processor. Hardware-specific controlled-projector integration and calibration remain implementation requirements. STATISTICAL CLARIFICATION Any archived wording referring to a “5-sigma” or “five-standard-deviation” shot budget should be interpreted as a signal-to-standard-error planning criterion under the stated readout-noise model. It should not be interpreted as a distribution-free Gaussian-tail 5-sigma significance certificate. A ratio of signal to standard error equal to five is not, by itself, equivalent to a rigorously certified 5-sigma tail probability, especially at very small sample counts. Sampling uncertainty, concentration bounds, SPAM errors, calibration drift, and other device systematics are separate quantities from the exact model-level response contrast. WHY THIS MATTERS The broader question addressed by the paper is: What information is actually sufficient to determine a collective many-body response? The response twins show that “knowing a system perfectly in every shallow few-particle sector” and “having enough information to determine a collective many-body task” are not equivalent statements. An interaction may be exactly inactive in all calibrated sectors and become relevant only when enough particles are present. Conversely, the positive closure theorem shows that additional information is not always required indefinitely. For certain structured Hamiltonian classes, shallow information can be proved sufficient. The relevant scientific question is therefore not simply how much information is available, but whether that information is sufficient for the particular observable and model class under consideration. This distinction connects the work to broader problems in: many-body response; Hamiltonian learning; few-body tomography; quantum simulation; effective theories; interaction hierarchy; response reconstruction; quantum geometry; and the limits of prediction from incomplete calibration data. SCOPE AND NONCLAIMS The observable throughout this work is the selected-branch static second-order Kato curvature under a U(1) Peierls twist. The paper does not claim equality with finite-frequency conductivity, a universal dynamical superfluid weight, a transition temperature, or a material-specific response. It does not prove a theorem for every interacting paired system. It does not establish a blanket failure of DFT, DMRG, VMC, tensor-network methods, Hamiltonian learning, downfolding, or any other named computational framework. A method-specific no-go statement would require a separate proof that the complete information used by that method factors through the shallow information map that is identical on the response twins. The finite-size robustness argument establishes an open neighbourhood of the exact c