In one fully specified, machine-checked special case — the charge-k magnetic flux family on an m × m torus of lossy (rank-deficient) links — we disambiguate three distinct quantities that the almost-commuting-operator and topological-protection literatures all call “the gap,” and determine which survive uniformly in the system size m. The local gaps — the commutator norm ‖[U,V]‖, the magnetic Hamiltonian's spectral gap, and the spectral-localizer gap they control — all vanish like the flux-per-plaquette 2πk/m² → 0. Two other gaps are m-uniform: the core gap (the rank/isometry structure of the links, independent of the magnetic data) and the Bott branch gap (well-definedness of the index). The consequence is a sharp negative with a constructive moral: the m-uniform operator-norm separation c(k) between charge and flatness — the Plateau quantity — cannot be sourced from any local spectral gap, and is therefore irreducibly topological (a Voiculescu distance-to-commuting that stays ≈ √2 while every local gap collapses), transported into the non-invertible setting by the rank-structure core gap, not by a spectral gap. This locates exactly where a proof of the Plateau Conjecture must, and must not, look. We then make the positive locale concrete: reading the rank loss as the ideal of an extension reduces the remaining unitary-to-contraction passage to a single off-diagonal Schur block, which numerics (Exp. E, and an adversarial worst-case Exp. F) show is core-gap-controlled and m-uniform — recasting “extend a dimension-free theorem to contractions” as “bound one block.” A direct minimization of the distance to flatness itself (Exp. G) independently fails to collapse the distance, converging up to the same √2 as the index-jump radius — the conjecture's most direct numerical test, passed. The √2 above is the codimension-1 value (d = r+1; the tested case d = 3, r = 2). For codim = d−r ≥ 2, a trivial flat competitor supported on an r-plane orthogonal to the rotating core sits at operator distance exactly 1 from the flux family at every m, so distOpFlat ≤ 1 < √2 there. The separation is therefore codimension-dependent: c(k) = √2 at codimension 1, c(k) = 1 at codimension ≥ 2 — both m- and k-uniform. The Plateau Conjecture (c(k) > 0) holds across 2 ≤ r ≤ d, with uniform floor 1; what is codimension-1-specific is the mechanism — the dimension-free unitary (Voiculescu/Exel–Loring) bound transports only at codimension 1, while for codimension ≥ 2 the binding competitor is the orthogonal one the unitary bound cannot see. House style. Every claim in this note compiles in Lean ([THEOREM]), cites the literature ([CITED]), carries numerical evidence with its script ([EVIDENCE]), or wears a [CONJECTURE] tag. Numerics (experiments A–G, full detail in §9) are the evidence base and are referenced throughout by letter.
In one fully specified, machine-checked special case — the charge-k magnetic flux family on an m × m torus of lossy (rank-deficient) links — we disambiguate three distinct quantities that the almost-commuting-operator and topological-protection literatures all call “the gap,” and determine which survive uniformly in the system size m. The local gaps — the commutator norm ‖[U,V]‖, the magnetic Hamiltonian's spectral gap, and the spectral-localizer gap they control — all vanish like the flux-per-plaquette 2πk/m² → 0. Two other gaps are m-uniform: the core gap (the rank/isometry structure of the links, independent of the magnetic data) and the Bott branch gap (well-definedness of the index). The consequence is a sharp negative with a constructive moral: the m-uniform operator-norm separation c(k) between charge and flatness — the Plateau quantity — cannot be sourced from any local spectral gap, and is therefore irreducibly topological (a Voiculescu distance-to-commuting that stays ≈ √2 while every local gap collapses), transported into the non-invertible setting by the rank-structure core gap, not by a spectral gap. This locates exactly where a proof of the Plateau Conjecture must, and must not, look. We then make the positive locale concrete: reading the rank loss as the ideal of an extension reduces the remaining unitary-to-contraction passage to a single off-diagonal Schur block, which numerics (Exp. E, and an adversarial worst-case Exp. F) show is core-gap-controlled and m-uniform — recasting “extend a dimension-free theorem to contractions” as “bound one block.” A direct minimization of the distance to flatness itself (Exp. G) independently fails to collapse the distance, converging up to the same √2 as the index-jump radius — the conjecture's most direct numerical test, passed. The √2 above is the codimension-1 value (d = r+1; the tested case d = 3, r = 2). For codim = d−r ≥ 2, a trivial flat competitor supported on an r-plane orthogonal to the rotating core sits at operator distance exactly 1 from the flux family at every m, so distOpFlat ≤ 1 < √2 there. The separation is therefore codimension-dependent: c(k) = √2 at codimension 1, c(k) = 1 at codimension ≥ 2 — both m- and k-uniform. The Plateau Conjecture (c(k) > 0) holds across 2 ≤ r ≤ d, with uniform floor 1; what is codimension-1-specific is the mechanism — the dimension-free unitary (Voiculescu/Exel–Loring) bound transports only at codimension 1, while for codimension ≥ 2 the binding competitor is the orthogonal one the unitary bound cannot see. House style. Every claim in this note compiles in Lean ([THEOREM]), cites the literature ([CITED]), carries numerical evidence with its script ([EVIDENCE]), or wears a [CONJECTURE] tag. Numerics (experiments A–G, full detail in §9) are the evidence base and are referenced throughout by letter.
This thesis studies the mathematical structure of quantum Hall transport in lattice systems, with an emphasis on relating mutiple definitions of Hall conductance in different mathematical branches: Geometrically as a bulk Chern number for a Bloch bundle; analytically as an edge spectral flow for a half-space Hamiltonian; and in operator algebra language as a many body charge transport index generated by magnetic flux insertion. The main results of the thesis reveal restrictions imposed on these indices by frustration-free structure, and improved locality estimates for interacting many body systems.The first result concerns frustration free free-Fermion (F 4 ) Hamiltonians. For translation-invariant, strictly local, gapped F4 Hamiltonians on Z 2 , we prove that the bulk index vanishes. More precisely, the two Bloch bundle associated with the image/kernel of the momentum space Hamiltonian is shown to be trivial by an algebraic argument using polynomial modules and a free resolution. Thus a frustration-free free-Fermion model cannot realize a nontrivial integer quantum Hall phase at the level of the bulk Chern index.The second new result is the corresponding edge statement. For a translation-invariant F4 Hamiltonian restricted to a half-plane, we prove that the edge index vanishes. We first identified that frustration free is equivalent to flat ground state band. Detailed analysis of spectral flow of the edge Hamiltonian shows that the edge mode is an analytic function and can be absorbed by the flat band. Consequently, the edge spectral flow across a regular energy in the bulk gap is constant and must equal its trivial value below the spectrum. This establishes, in this setting, that the frustration-free structure rules out nonzero edge transport as well as nonzero bulk transport. The third new result is an exponential clustering theorem for finite-volume quantum spin systems with a possibly degenerate and split ground-state sector. Assuming only a uniform spectral gap above a low-energy spectral patch of sufficiently small width, we prove exponential decay of correlations in the form|⟨Ω, ABΩ⟩ − ⟨Ω, AP BΩ⟩| ≤ C∥P∥1∥A∥∥B∥e −cd(X,Y ),for local observables A and B with disjoint supports. This improves the locality control needed in many body Hall conductance arguments, where previous estimates under comparable hypotheses were superpolynomial. Together, these results clarify how locality and spectral gaps constrain both free-Fermion and interacting formulations of topological transport.
A network of lossy translations — partial isometries that destroy information as they carry it — can host a topological charge: a winding in the surviving subspace whose local signature shrinks with system size while the global cost of removing it does not. A companion paper established this plateau empirically and conjectured its mechanism; a separate validation exercise established, on real composed pipelines, the operational detection threshold of a noisy, quantized estimation chain. This paper puts the two results in one inequality. The local evidence of a charge k on an m × m torus is a per-plaquette defect of size ≈ 2πk/m², which falls below the operational detection threshold δ* = max(1.405·σ√ℓ, 0.6·B/√n) at a computable crossover size m*; beyond m*, no local monitor at that instrumentation can see the charge. The cost of removing the charge, by contrast, is bounded below by a constant c(k) independent of m, whose height is now codimension-resolved (companion three-gaps note, §4a): √2 where the surviving subspace has codimension one (measured, converging to √2 from below), exactly 1 at codimension ≥ 2 (proved for r = 2) — either way dozens of times above realistic thresholds. Past the crossover the network carries a stealth charge: undetectable by every local test, unremovable by any local edit. We give the precise statements with their evidence status — the existence floor is a theorem at every fixed size (Stage A; elementary proof, Lean formalization queued); the m-uniform constant is the companion paper's conjecture (Stage B) — derive the crossover, work a numerical example, and map what falsifies what. The claim's proof shape — one topological quantity evaluated two ways, locally as a sum of vanishing defects and globally as a protected obstruction — is the shape of the geometric Szpiro inequality, and we close with that dictionary and its one honest discrepancy.
We give explicit sufficient conditions for a spectral gap uniform in the number of sites of a three-dimensional SU(3) Hamiltonian lattice, partitioned into disjoint cube blocks and free bridge links. Using previously certified one-cube spectral and comparison bounds, we obtain a dimensionless gap at least 2/9, on the full kinematic Hilbert space and hence on its Gauss-invariant subspace, for inter-block plaquette strength 0 ≤ s ≤ 1/6144. The point s = 1 is the isotropic Kogut–Susskind lattice at β = 1/2 in our Hamiltonian convention. The argument combines a gauge-Casimir comparison, a product-vacuum fixed-point construction, an excitation resolvent criterion, and orthogonal-output estimates with constants independent of volume. An isolated two-cube, two-bridge block has a full kinematic gap at least 1/3 at s = 1. For a four-face interface we retain all signed entries of an 18-vector output Gram matrix, reducing a specified residual upper bound from 0.160048 to 0.146438. Explicit bounded shears improve the derivative on the one-edge charged input subspace and eliminate its columns in an origin-level excitation test. These shears act trivially on the physical neutral sector; neither they nor the interface Gram establish a full-input contraction at s = 1. The particular derivative majorant used for the global theorem cannot certify s ≥ 1/4860; this is not a no-go result for other methods. The scope of the inherited certificates and the remaining thermodynamic and continuum questions are stated separately.
This record contains the full author preprint (12 pages), its mathematical supplement (17 pages), and the complete article archive with LaTeX sources, the version note, SHA-256 manifests and the licence. The two PDFs are also provided as individual files for direct reading. Version v3 clarifies the Haar normalization convention in section 9.1 of the mathematical supplement (equation 45). The unnormalised transform H has H*H = 2I and Tr(HQH*) = 2Tr(Q); the equivalent orthogonal transform is H/sqrt(2). This agrees with the existing root/detail identity and exact verifier. The main article, verification code, Lean files, matrix certificates and stated constants are unchanged. See the publication version note for the precise scope of this correction. AbstractWe prove that the nearest-neighbor spin-1/2 antiferromagnetic Heisenberg model on the infinite Kagome lattice is gapless in the bulk GNS sense. The argument constructs six explicit local preimages and their complete infinite-lattice Hamiltonian commutators. Two rigorous 17-spin positivity certificates, each decomposed into 31 nontrivial swap-character blocks and nine SU(2) multiplicity blocks, give a state-uniform off-kernel mass floor. Exact operator transport on the two genuine outward Kagome continuations propagates the same physical family through a finite stationary interface. A root/detail/current decomposition, a local current–boundary identity, and state-uniform norm estimates yield a horizon-independent Dirichlet-energy bound. With harmonic weights, for every infinite-volume ground state ω and every R ≥ 1 the same finite family satisfies M_R(ω) ≥ (15/32) ∑_{k=1}^{R} 1/k and E_R(ω) ≤ 761429817528146602401/25088000. Any positive GNS gap would force the second quantity to diverge with the first. Hence every admissible ground-state representation has zero gap, and the declared bulk gap is zero. All finite operator identities and matrix inequalities are checked from exact algebraic data, with rigorous ball arithmetic used for the large positive-definiteness tests. Lean 4 verifies the arbitrary-horizon harmonic implication and the final variational contradiction. The proof uses neither finite-size gap extrapolation nor a spectral cutoff. Companion public verification code, Lean 4 formalization and exact certificates: https://doi.org/10.5281/zenodo.23675708 Distribution and reuse are governed by the Public Scientific Evaluation and Reproducibility Terms included with this record. The deposit excludes the author's private discovery engine and proprietary know-how.
Local information can be lost rapidly while a collective mode survives for a long time. We study this distinction on a physical–observation joint state space. The second variation of relative entropy under a local record map gives a positive Fisher-loss operator; the least loss over normalized noninvariant scores is the spectral gap of the associated posterior-resampling process. The local strength and the compatibility of surviving directions enter separately. A fixed-width interacting ladder has a positive gap at every fixed finite coupling, as follows from an explicit block comparison, although strong correlation makes the bound small. We then reverse the construction. For a given irreducible real Hamiltonian with nonpositive off-diagonal entries, its ground amplitude determines pair-record probabilities and energy-valued loss weights, reproducing the physical excitation gap exactly. A positive trial amplitude gives an error interval controlled by the oscillation of its local energy, without requiring the exact ground state. When signs remain, the energy form is a difference of positive information forms and an explicit cancellation constant controls the gap. Sparse diagonalization of open transverse-field chains up to 65 536 configurations agrees with both the reconstructed information spectra and an independent free-fermion calculation. The critical law n∆n/J → π, the gapped regime, and the ordered finite-chain splitting are distinguished. The result is an explicit connection for specified channels and Hamiltonians, with the requirements of positivity, locality, and energy normalization kept separate.
Exact permutation covariance does not itself supply a sparse metric geometry. For permutation-covariant, full-support k-body interaction orbits, the incident interaction norm scales as (k−1N−1)∣JN∣∥h∥; such orbits cannot have uniformly finite range in metrics of diverging diameter. A permutation-invariant Hamiltonian diagonal in the graph basis has ground-state multiplicity at least N!/∣Aut(G)∣ whenever G is a ground graph. We give a quantitative version for approximately symmetric energies, together with fixed-radius connectivity and unitary-isometry obstructions. The trace-distance asymmetry of an arbitrary state also bounds its probability of selecting one labeled graph.
We then construct two exactly solvable finite relational history Hamiltonians. A cyclic clock with a reversible incidence update has a unique ground state; gauge reduction yields a cycle Laplacian with a rank-one pin and the exact gap ΔN,L=E∗(1−cosθL), where tan(LθL/2)=1/sinθL and 0<θL<π/L. In particular, ΔN,L/E∗=π2/(2L2)−2π2/L3+O(L−4), independently of the number of vertices. In an alternative order-free architecture, one clock qubit per unordered pair gives commuting four-local terms, 2N ground states and a size-independent gap E∗(1−1/2). A fixed-weight control penalty selects (qN) ground states without reducing the gap. These exact finite results expose clock and locality resource tradeoffs; they do not establish an emergent geometric phase.