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.
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.
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.
The Parity-Entanglement Bound (PEB) fixes the clock-system mutual information of a ℤ₂ constrained bipartite pure state at exactly two bits. We show that this invariant functions as the zero point of a metric geometry built from information. Six results are given. (i) The deviation Π = 1 - 1/2 I is the unique local-unitary (LU) invariant of the minimal state (pure, ℤ₂, bipartite); relaxing each hypothesis adds a definite number of further axes, which we count numerically (mixedness +1, ℤn+(n-2), tripartite +2; a continuous gauge group opens a continuous axis), (ii) The correlation quantity d = -ln κ is non-negative, symmetric, and vanishes in the coincidence limit; in the gapped regime, at separations above a floor r*(m), it obeys the triangle inequality and is a metric. (iii) The validity window is mapped: the floor r*(m) grows with the gap and the window closes entirely once the correlation length falls below the lattice scale (m ≥ 1.7 in 1D, ≥ 2 in 2D, lattice units); in the opposite, gapless limit the inequality survives with growing slack (0 violations atm m = 0.02) and an asymptotic argument establishes it analytically in the logarithmic regime. Non-Gaussianity per se does not damage the inequality: on a φ⁴ chain the observed violations are reproduced quantitatively by a free field of matched effective gap. (iv) Local noise acts on d as an exactly additive per-region dilation, d → d + δA + δB with δ ≥ 0; pure dephasing leaves d exactly invariant, and local noise can only increase the triangle slack, so the metric is noise-robust by construction. (v) The PEB value equals a combinatorial crossing number of stabiliser generators, hence is independent of system size, clock size, and clock position; it is formally analogous to a topological entanglement entropy. (vi) Following the Dirac-equation construction [6], the two-dimensionality dim ℋC = 2 underlies a distance/time split. The construction is verified by exact and Monte-Carlo simulation (geodesic locking 60/60 across five seeds; zero triangle violations above the floor in more than 1.6 × 10⁴ triples spanning gapped, near-critical, interacting, and noisy regimes) and, as a proof of concept, by a state-preparation study on quantum hardware (IBM FEZ). Formal correspondences to gravitational behaviour geodesic bending in a non-uniform gap and a volume-to-area transition in the crossing count are recorded in Sec. no gravitational derivation is claimed. Statements are labelled [P] proved, [K] known-with-citation, [N] numerical, or [0] open.
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.
Large-scale Quantum Key Distribution (QKD) networks rely on trusted repeaters, making the security properties of the selected communication path an essential part of end-to-end assurance. At the same time, network operators may be unwilling to disclose their internal topology. We present a topology-certification mechanism that lets a provider prove in zero knowledge that policy-compliant routes between two endpoints exist, without disclosing the routes in an individual presentation. Our main idea is to certify nodes and edges independently using multi-message signatures, rather than signing the complete graph as one object. For a route chosen in advance, proof size and cryptographic proving and verification work depend on its positions and attributes, independently of the overall network size, whereas route discovery and certification have separate graph-dependent costs. The construction hides the actual route length up to a public bound and sketches extensions to node-disjoint routes and monotonic additions within a graph epoch. We give a formal security model and conditional proofs of unforgeability and graph hiding for a generic construction. For a BBS-based construction, presentation-size estimates remain below 300\,KiB at ℓ=m=n=50, and measured generation and verification times remain below 400\,ms at ℓ=64 and m=n=8.
We investigate whether an intrinsic duality can define a label-free distinction between two local sectors, without assuming that distinction from the outset. Given a system with an exact, order-2 duality automorphism D pairing two families of local operators, we show the geome-try/matter split can be defined purely by whether D is locally preserved or broken. We develop this construction on the {4, 4} (square) cellulation of T 2. This choice is not an incidental convenience: we derive an Euler-characteristic obstruction showing that no triangulated (simplicial) cellulation of non-trivial topology can support the construction, sharpening an earlier self-duality + non-trivial-topology + locality argument. Our main results are: (1) an exact duality auto- morphism of the minimal L = 2 toric code, with defects localizing exactly where D is broken, together with a continuum Bloch-vector analogue; (2) a generator-operator axiomatization and a Trichotomy Theorem classifying every local extension as defect-creating (a non-empty, even-cardinality defect set), free, or purely topological, verified by exact GF(2) computation on tori and a genus-2 surface, together with an exact discrete defect-pairing theorem and a numerically- supported continuum charge-conservation law; (3) a structural result excluding local sourcing between the two dual sectors by construction under the generator axioms, replaced by an exact topological braiding phase realizing Z2 mutual statistics; and (4) a proof that two natural lattice discretizations of a Gauss-map degree are each, separately, honest topological degrees under an explicit coherence bound, for odd lattice size, with their equality reduced to a single precisely-stated conjecture that we believe but do not prove and for which we identify the specific geometric obstruction a direct proof must overcome — even lattice size breaks the argument at an earlier, purely combinatorial step, which we leave open. All claims are either exact verified computations, proved propositions, or explicitly flagged as conjectural or open. Section 2 situates the construction relative to the pre-existing Kramers–Wannier/duality-defect and loop-quantum-gravity literature.
A recurring structural pattern appears across several recent preprints in quantum information, quantum sensing, and open-system dynamics: the act of *extracting* classical information from a quantum system is not a passive readout but an active, resource-consuming operation whose cost and fidelity depend jointly on measurement complexity, channel structure, physical noise geometry, and the topological character of the state being probed. This paper offers a heuristic reading — not a formal derivation — of five specific findings as instances of a single organizing question: *under what conditions does the quantum-to-classical interface become a bottleneck that determines what is knowable about a system, rather than merely a technical inconvenience?* Drawing from preprints in quant-ph and cond-mat.stat-mech, we identify four candidate mechanisms: (1) a sharp complexity-driven hidden-to-visible transition in quantum Fisher information accessibility [corpus:arxiv:2606.09765]; (2) the unification of cryptographic security and metrological sensitivity through channel symmetrization [corpus:arxiv:2606.10700]; (3) the reinterpretation of topological state discrimination as a weak-value measurement via Kirkwood-Dirac quasiprobabilities [corpus:arxiv:2606.11002]; and (4) the superposition of noise channels as a resource for recovering quantum capacity that individual channels cannot support [corpus:arxiv:2606.10744]. A fifth finding — that bosonic cyclic codes trade error-detection range for logical gate accessibility [corpus:arxiv:2606.11010] — is treated as a weakly connected addendum, connected by analogy rather than by shared mechanism. The falsification path is concrete: the complexity-threshold prediction of [corpus:arxiv:2606.09765] can be tested by constructing circuits of depth exactly at the critical threshold and measuring the Fisher information ratio as a function of system size. --- Authorship: Saluca Agentic AI Research Team (Saluca LLC). AI-drafted from arXiv preprint corpus on the date in the filename. Cited arXiv preprints: 2606.07339, 2606.09678, 2606.09734, 2606.09765, 2606.09805, 2606.10700, 2606.10744, 2606.10865, 2606.11002, 2606.11010