Papers

We prove that every finite point set of size at least $10^{11055931}$ has four collinear points or six points that pairwise see each other. This resolves the first open case of the big-line-big-clique conjecture of Kára, Pór, and Wood.

The likelihood of a finite simple undirected graph $G$ on $n$ vertices is the probability that the uniform sequential attachment process, which at each step joins a new vertex to a uniformly random subset of uniformly random size of the vertices already present, outputs a graph isomorphic to $G$. Dervovic, Mocherla and Severini conjectured that the likelihood is minimised by the balanced complete bipartite graph. We prove that, among complete bipartite graphs of a given order, the balanced one uniquely minimises the likelihood. Exact computation shows that it also minimises over all graphs for every order from $6$ through $14$. At $n=15$, the exact global minimisers are $C_5[\bar K_3]$ and its complement, giving the first counterexample. Completed exact searches show that the global minimisers at order $16$ are $K_{8,8}$ and its complement. At order $17$, the unique minimiser is a self-complementary mixed blow-up of $C_5$. At order $18$, the minimisers are $K_{9,9}$ and its complement. We show that the failure is not sporadic by proving that the likelihood of the balanced complete bipartite graph is $2^{-(1/2-1/(8\ln 2)+o(1))n^2}$, whereas the minimum over all graphs of order $n$ is $2^{-(1/2+o(1))n^2}$, so the conjectured minimiser exceeds the minimum by a factor exponential in $n^2$. We also determine the Shannon entropy of the process to leading order, namely $n^2/(4\ln 2)$ bits, which shows that the conjectured minimiser is in fact more likely than a typical output of the process. The proofs rest on a vertex deletion recurrence which evaluates the likelihood in time $O(2^n n)$ and which closes on the blow-ups of any fixed base graph.

The twisted Foulkes character polynomial is an algebraically defined polynomial attached to an integer partition. We determine precisely how much combinatorial information this polynomial encodes by completely classifying all pairs of partitions that give rise to the same polynomial. Our main result shows that equality of twisted Foulkes character polynomials admits a purely combinatorial characterization in terms of two explicit local operations on partitions.

At fixed pulse energy, which drive waveform extracts the most $n$th-order nonlinear conversion from a resonator ($n=2$ for second harmonic)? A short pulse couples poorly to a narrow resonance, a long one dilutes its energy, and no linear rule fixes the compromise. We solve the problem exactly for a single mode of amplitude decay rate $\kappa$. Eliminating the drive turns fixed incident energy into a constraint on the stored field alone, and the optimization becomes a sharp Gagliardo–Nirenberg inequality whose extremal is the ground-state soliton of the nonlinear Schrödinger equation. The optimal stored field is $\mathrm{sech}^{1/(n-1)}[(n-1)\kappa t]$, sustained by an asymmetric input that rises as $e^{\kappa t}$ and falls as $e^{-(2n-1)\kappa t}$; the largest converted energy follows in closed form. A rising exponential, the time-reversal recipe, retains at most $79.0\%$ of the bound at $n=2$ and $2/e$ at large $n$; a two-rate pulse retains above $97\%$. Critical coupling generalizes to $n$-fold overcoupling, with optimal input coupling $n$ times the intrinsic loss rate. The bound applies from microrings to superconducting circuits and caps the per-pulse brightness of broadband photon-pair sources.

We provide a numerical criterion to find the curves on a surface of general type that are contracted to its minimal model. We then apply this approach to product-quotient surfaces, where the criterion determines whether all such contracted curves are contained in reducible fibres of the two natural fibrations.

Partial entanglement may provide enough connections in the universe to satisfy the definition of a small world. To investigate this possibility, we define a network of particles on a given space-like hyper-surface with a long-range link between any two particles that are connected by a chain of exchanged particles involving a limited number of interactions. Considering the mean free paths of particles in different regions of space and the resulting probability distributions of entanglement connections vs. distance, we find evidence of small-world or random network structure on all but the smallest scales - corresponding to stars and planets - on which other types of connections would need to be added to complete the small world picture.

We prove level-2 large deviation upper bounds for potentials on countable Markov shifts and for suspension semi-flows over countable Markov shifts. For strongly positive recurrent potentials, we establish equidistribution of weighted empirical measures toward the corresponding equilibrium state. We then apply these results to interval maps, obtaining, in particular, equidistribution and large-deviation estimates for measures supported on boundary points. For the Gauss map, this yields equidistribution results on rational numbers, including a theorem of David and Shapira.

Henning and Yeo conjectured an upper bound on the identifying vertex cover number of a graph in terms of its order, size, and maximum degree. A two-parameter family $H_{t,r}$ of connected diameter-two graphs disproves the bound for every maximum degree at least four; after denominators are cleared, its margin is exactly $-(t-1)(r-1)$. The complement relation $\tau_D=n-\rho$ exposes the mechanism: diameter-two fibres admit at most one packing vertex, while degree deficit accumulates under tree gluing with controlled port loads. Writing $A_\Delta$ for the supremal additive gap at maximum degree exactly $\Delta$, an exact transfer formula gives $A_\Delta=+\infty$ for every $\Delta\ge 4$, using Petersen fibres in degrees four and five and the original $H_{t,r}$ blocks in higher degrees. If $c_\Delta$ denotes the corresponding supremal gap per vertex, rooted rook-graph fibres match a universal square-graph packing bound to first order. Consequently, $c_\Delta\sim 1/\Delta$, equivalently $\Delta c_\Delta\to 1$ as $\Delta\to\infty$.

Let $E\subset\mathbf R^n$ be a set of dimension $2$ with an isolated singular point $0$, such that $E-\{0\}$ is an orientable $\mathcal C^2$ submanifold of $\mathbf R^n$ of dimension $2$, and hence a Riemann surface with the conformal structure induced by the ambient metric. We study the extension of its conformal structure across $0$, under hypotheses that are metric rather than analytic. Denoting by $S(0,r)$ the sphere of radius $r$ centred at $0$, we call the singularity $0$ $\mathit{M\text{-}regular}$ if (i) the pair $(E-\{0\},\{0\})$ satisfies a Whitney condition (the radial direction becomes asymptotically tangent to $E-\{0\}$ at $0$) and (ii) the length of the spherical slice $S(0,r)\cap E$ is bounded above by a linear function of $r$ for small $r$ ($\mathit{metric\ decay}$). The main result (1989) is that an $M$-regular singularity is $\mathit{conformally\ point\text{-}like}$, that is, a punctured neighbourhood of $0$ in $E$ is conformal to the punctured unit disc $\Delta^*$, so that the conformal structure extends across $0$. The proof uses the modulus of rings (extremal length), which generalizes the classical planar technique to $2$-dimensional submanifolds of $\mathbb{R}^n$. Two classes of surfaces are shown to be conformally point-like: all surfaces of revolution generated by a single $C^1$ curve, and orientable subanalytic sets of dimension $2$ with isolated singularities, for which $M$-regularity is proven directly from Hironaka's structure theory. In this case the metric decay condition follows from a direct estimate of the volume of the slices $S(0,r)\cap E$ (Lemma 7.5), based on Hironaka's Whitney theorem and Stasica's decomposition into $L$-analytic pieces. A further original result is a $C^\infty$ counterexample showing that a strict Whitney condition alone does $\mathit{not}$ imply the linear length bound: the two hypotheses defining $M$-regularity are independent.

In this work, we establish a strong factorization for Hardy-Orlicz spaces on the upper half-plane. We show that the product of two functions belonging respectively to Hardy-Orlicz spaces $H^{\Phi_{1}}$ and $H^{\Phi_{2}}$ lies in a third space $H^{\Phi_{3}}$, and every holomorphic function in $H^{\Phi_{3}}$ admits such a decomposition. We then provide an atomic decomposition for certain Hardy-Orlicz spaces, which allows us to describe the topological dual of these spaces when the associated function is concave. Finally, these results are applied to the study of the continuity of the Hankel operators in this setting.

We consider the task of simulating the real-time dynamics of weakly interacting fermionic systems. In particular, we focus on computing the expectation value of a local observable $A$ at time $t$. By analyzing the convergence of the perturbative expansion in the interaction strength $\lambda$ for the Heisenberg-picture observable, we propose a polynomial-time algorithm for estimating this expectation value in the weakly interacting regime $\lambda |t|^{2D+1}=\mathcal{O}(1)$, when the Hamiltonian is geometrically local on a $D$-dimensional lattice. Importantly, this condition is independent of the system size. If the goal is instead to approximate the time-evolved observable in normalized Frobenius norm, we extend the convergence regime to $\lambda |t|=\mathcal{O}(1)$ with quasi-polynomial runtime. When the non-interacting part exhibits Anderson localization, our polynomial-time algorithm can be extended up to $\lambda |t|=\mathcal{O}(1)$, modulo polylogarithmic factors. Our algorithm brings together ideas from continuous-time QMC, diagrammatic QMC, and Majorana Propagation, but with a new Heisenberg-picture operator-growth analysis that makes the sampling complexity rigorously controllable. This leads to provably efficient classical algorithms in regimes where the interaction is weak enough that the sampling variance remains bounded independently of system size. Together, these results identify broad regimes in which weak interactions, locality, and localization can be leveraged to make real-time fermionic dynamics classically tractable.

The pearl necklace number, $N_P$, of a knot is the smallest number of unit spheres required to construct a knot if each sphere is tangent to two neighbors and no sphere overlaps with another. It has been speculated that the pearl necklace number is equal to the minimum number of lattice sites required to embed a knot on a face centered cubic (FCC) lattice $N_L$, implying that a trefoil knot cannot be constructed from fewer than 15 spheres. A large language model (LLM) was prompted to find configurations of knot for which $N_P<N_L$, and this manuscript describes its findings, attempts at validating them, and their implications. No 14-vertex example was found for the trefoil knot, but examples with $N_P=N_L-1$ were found for all knots from 5 to 7 crossings as well as $8_{19}$, and $10_{124}$. One example, $8_1$, could be constructed with $N_P=N_L-2$.

In this study, a two-dimensional second-order macroscopic traffic flow model is proposed to capture the complex dynamics of disordered traffic by explicitly incorporating coupled longitudinal and lateral interactions. The framework extends the classical continuity equation to two spatial dimensions and introduces acceleration equations consisting of self-driven, interaction-induced, and road-boundary-related components. The macroscopic longitudinal acceleration is formulated based on the Full Velocity Difference Model (FVDM), while lateral dynamics are governed by interaction principles consistent with the Optimal Velocity Model (OVM). The explicit inclusion of road-boundary effects enables the representation of vehicle confinement and space-sharing behavior that are essential in lane-free and weakly lane-disciplined traffic systems. The model is examined through a series of numerical experiments in which longitudinal and lateral dynamics are analysed individually as well as simultaneously within a coupled two-dimensional setting under a range of initial conditions, including transitions between free-flow, medium congestion, and heavy congestion, and different lateral density configurations. The simulations demonstrate the model's ability to reproduce key traffic features such as shockwave propagation, lateral dispersion, vehicle rearrangement, and stable density evolution across the road width. The numerical scheme, based on upwind and Lax-Friedrichs discretisations, ensures stable solutions of the coupled partial differential equations. Overall, the proposed framework provides a robust macroscopic description of disordered traffic and offers a consistent basis for analysing two-dimensional vehicular flow dynamics.

We present a new impossibility region for the $5\times5$ symmetric nonnegative inverse eigenvalue problem. The region lies in the low-trace regime and, to the best of our knowledge, has not been identified previously. The proof uses a suitable diagonal shift to transform the problem to a critical high-trace boundary and then reduces a complementary commuting matrix to a weighted five-cycle. The characteristic polynomial and spectral identities of this five-cycle provide the main tools for deriving the resulting contradiction.

We study the transport of polynomial differential operators under the two-parameter commuting evolution \[ P_{t,s} = \exp\left( \frac{t}{2}D^2-\frac{s}{3}D^3 \right), \qquad D=\frac{d}{dx}. \] Conjugation of the position operator gives \[ P_{t,s}xP_{t,s}^{-1} = x+tD-sD^2, \] which leads to a recursive normal-ordering expansion and to a family of Heat–Airy polynomials arising as its derivative-free coefficients. Our main purpose is to study the interaction of this transport with moving absorbing boundaries. For functions satisfying \[ v_t=\frac12v_{xx}, \qquad v_s=-\frac13v_{xxx}, \qquad v(t,s,f(t,s))=0, \] the boundary condition generates a hierarchy of relations among the spatial jets of \(v\). Combined with the restrictions of transported differential equations and their spatial derivatives, these identities produce compatibility equations for \(f\). We prove a general restriction principle showing that, at every spatial-jet level, the compatibility equations of the classical Heat problem are obtained by restricting the corresponding Heat–Airy hierarchy to the slice \(s=0\). For second-order operators, this framework yields a finite characteristic system and nonlinear boundary equations, together with explicit solvable examples. For a third-order linear-potential operator, the additional Airy-flow identity permits a stronger sequential recovery: the first Heat–Airy compatibility equation determines \(f_s(t,0)\), and the next spatial-jet equation then reproduces the nonlinear compatibility condition obtained in the pure Heat theory from a \(4\times4\) characteristic determinant. Thus the additional commuting flow reorganizes the moving-boundary compatibility problem into a hierarchy while retaining the classical Heat theory as a distinguished restriction.

Designing compact, high-efficiency thermal architectures requires resolving the competing demands of solid conduction, fluid convection, and flow resistance within highly constrained physical envelopes. Density-based topology optimization provides a systematic framework for synthesizing these coupled layouts, yet the reproducibility and numerical stability of the resulting designs depend critically on the underlying discrete solvers and adjoint sensitivity mechanics. In this work, we present a transparent, self-contained two-dimensional finite-volume formulation on a staggered Marker-and-Cell grid for conjugate heat transfer governed by design-dependent energy transport coupled to Stokes–Brinkman or Darcy flow at fixed solid volume. To prevent spurious artificial thermal sources in porous, weakly compressible Brinkman domains, the discrete advection operator is constructed to satisfy the identity $\mathbf{u}\cdot\nabla T=\nabla\cdot(\mathbf{u}T)-T(\nabla\cdot\mathbf{u})$ cellwise, ensuring that uniform temperature fields remain exact discrete nullspaces even under inexact continuity satisfaction. Reverse-mode derivatives are evaluated via the implicit function theorem rather than unrolled iterative loops, yielding exact discrete adjoints with bounded memory requirements. The discrete operators are systematically validated through the method of manufactured solutions and directional Taylor remainder tests. Four representative thermofluid design benchmarks are optimized using a projected-gradient scheme with $\beta$-continuation, wherein candidate iterates are accepted and published only upon satisfying rigorous, predeclared gates on residual convergence, mass conservation, volume feasibility, and numerical finiteness. The resulting formulation provides an inspectable, deterministic reference stack for verifiable conjugate thermofluidic topology optimization.

The titular object of this paper is an analogue of the Mandelbrot set over the 3-dimensional real algebra whose multiplication is the bilinear extension of the rock-paper-scissors operation. In order to study the dynamics of the mappings $x\mapsto x^2+c$ in this setting, a notion of holomorphy for functions on general finite-dimensional real algebras is introduced. Under a mild assumption it is shown that for such algebras holomorphy is always equivalent to solving a finite system of linear first-order PDEs generalizing the Cauchy-Riemann equations. Code is provided for generating animations of these fractals and for exploring them in a video game format.

Covariate adjustment improves estimation efficiency in randomized experiments, but standard calibration and augmentation methods, when applied to distribution or survival functions, do not preserve monotonicity—a fundamental property of the estimand. We propose using empirical likelihood with covariate-balancing constraints to construct a covariate-adjusted empirical measure for each treatment arm. Estimators of a broad class of distributional functionals, including cumulative distribution functions, survival functions, quantiles, and restricted mean survival times, are then derived as plug-in functionals of this measure, automatically inheriting proper shape constraints. We establish asymptotic normality with an explicit, guaranteed efficiency gain over unadjusted estimators. The asymptotic distributions are invariant to the randomization scheme, providing a unified inference procedure under simple randomization and all commonly used covariate-adaptive designs satisfying a mild balancing condition. This unified construction, adjusting the empirical measure once and deriving all estimators from it, offers a principled reconciliation of covariate adjustment with shape preservation. Simulations and an application to the SURPASS-4 trial confirm the theoretical gains.

First formulated by Amarilli, Monet, and Suciu (arXiv:2401.16210, 2024), the Non-Cancelling Intersections (NCI) conjecture is an open problem in combinatorics stating that any set union can be constructively built from its algebraically non-cancelling intersections using only disjoint unions and subset complements. In the same paper, two orthogonal possible strengthenings are proposed: using only left-linear trees, and using non-trivial intersections only positively or only negatively depending on the sign of their Möbius value. Here we show that using only left-linear trees, the conjecture is false (independent of the other strengthening). Our argument is non-constructive. We prove the existence of a counterexample, though it is of immense size.

Work of Toda identifies groups of metastable vector bundles on even-dimensional spheres with stable homotopy groups of certain stunted projective spectra. Using Weiss' unitary calculus, the second-named author generalized this identification to show that metastable, stably trivial vector bundles on even cell complexes can naturally be identified with stable homotopy classes of maps into a shifted stunted projective spectrum. Thus, certain classical questions about vector bundles (or homotopy of unitary groups) can be rephrased as stable computations. In this note, we show that certain generalized cohomology theories arising in chromatic and equivariant homotopy theory can be used to deduce the existence of non-trivial, stably trivial vector bundles on spheres and complex projective spaces.

In this paper, we examine the structure of strongly regular graphs with parameters $\lambda = 1$ and $\mu = 2$. In particular, we provide a complete classification of induced subgraphs of order seven and determine their relative frequencies. These findings contribute to a finer understanding of the local structure of such graphs and may be useful in related combinatorial and algebraic investigations.

We introduce a holonomy groupoid for Lie algebroids. This construction generalizes both the holonomy groupoid of a foliation and the adjoint representation of a Lie algebra. We prove that the resulting groupoid is longitudinally smooth and compute its Lie algebroid. When the algebroid of the holonomy groupoid coincides with the given Lie algebroid, the holonomy groupoid is the canonical terminal integration: every source-connected integration admits a unique morphism into it inducing the identity infinitesimally. To construct the groupoid structure on the holonomy groupoid, we utilize the notion of the "flow product" of time-dependent sections of a Lie algebroid. This provides an alternative way to define the groupoid structure for the Weinstein groupoid.

We study a penalisation problem for two-dimensional Brownian motion. Starting from the Wiener measure, we consider a family of probability measures obtained by weighting paths by a nonnegative functional $F_t$ depending on $t \geq 0$, $F_t$ being measurable with respect to the $\sigma$-algebra generated by the path up to time $t$. Under suitable assumptions on the penalisation process, we establish the weak convergence of these measures when $t \rightarrow \infty$. The limiting law is identified explicitly in terms of a $\sigma-$finite measure $\mathbf{W}^{(2)}$, which admits a path decomposition involving the last hitting time of a circle. This decomposition plays a central role in the analysis and yields a martingale representation of the limiting measure. where ordering and local time techniques are no longer available. The proofs rely on Laplace transform methods and Tauberian theorems, which replace excursion-theoretic tools and allow a precise identification of the limiting measure and its structural properties.

We introduce Deep-MKV-TS, a path-dependent McKean-Vlasov framework for financial scenario generation. The stochastic dynamics are chosen by matching selected path and volatility features of generated scenarios to those observed in the data. Starting from an interpretable reference model, Deep-MKV-TS preserves the reference drift and adjusts its volatility, while a regularization penalty limits unnecessary departures from the calibrated dynamics. We solve the resulting control problem using a neural, sample-based implementation of the stochastic maximum principle. We validate the method against an exactly computable oracle. On Heston and Heston-mixture models, Deep-MKV-TS substantially reduces path-dependent and volatility-related deficiencies of the reference model. In delayed-volatility experiments, the correction remains effective as the forecasting horizon increases, while direct training becomes less reliable. On held-out intraday equity-index futures, the corrected model improves conditional forecasts relative to the reference and reaches a level of performance comparable to flexible generative and historical baselines. The resulting scenarios also support greater exposure than the reference under a fixed drawdown-risk target. These results show that path-dependent McKean-Vlasov control can enrich an interpretable reference model without replacing it.

We introduce outer contact billiards as an odd dimensional counterpart to outer symplectic billiards. Until now, outer billiards have only been considered in even dimensional symplectic vector spaces. By projectivizing outer symplectic billiards, we obtain outer contact billiards, where the affine midpoint condition descends to its projective analog, namely harmonic conjugation. We prove that outer contact billiards generate contactomorphisms. For quadratic surfaces in $\mathbb{RP}^3$, we show that the correspondence is completely integrable: its domain is foliated by invariant quadrics and, on each leaf, the dynamics is determined by the iteration of an explicit linear transformation. We also establish two rigidity results for periodic trajectories: outer contact billiards admit no 3-periodic orbits and, among quadratic tables, only one admits 4-periodic orbits.

In a parabolic contact geometry a choice of contact form determines a Weyl form consisting of three components: a soldering form, a Weyl connection, and a Rho tensor. An explicit description of the soldering form is well-known. We express the Rho tensor in terms of the Weyl connection in the setting of torsion-free normal parabolic contact geometries. Parabolic contact geometries with a specified contact form have a second canonical connection, the Tanaka-Webster connection. We express the Weyl connection in terms of the Tanaka-Webster connection in the setting of normal parabolic contact geometries. As an application of some of these results we derive formulas for the harmonic curvature components in integrable Legendrian contact geometry, recovering a result of S. Ivanov, D. Vassilev, and S. Zamkovoy.

We present a high-order radial basis function-generated finite difference (RBF-FD) method for partial differential equations on moving manifolds $\mathcal M(t)\subset\mathbb R^3$ of co-dimension one. Our method builds on the tangent-plane formulation of surface RBF-FD and combines Lagrangian and Eulerian treatments: the manifold and material derivative are evolved in a Lagrangian fashion, while the remaining surface differential operators are reconstructed on the instantaneous point cloud and stabilized, when necessary, by a quasi-analytical hyperviscosity formulation. Lagrangian marker drift is handled by adaptive rearrangement using a compact global parametric model of the moving surface, which also supplies accurate normals and geometry-based quadrature. After rearrangement, we reconstruct the multistep history by backward semi-Lagrangian tracing and interpolation before resuming the Lagrangian time discretization. We exploit temporal coherence through three update strategies: defect correction for the local RBF-FD weights, a curvature-based update of the hyperviscosity coefficients between spectral recomputations, and a global defect-correction iteration that reuses an incomplete LU (ILU) factorization before preconditioned generalized minimal residual (GMRES) iterations. Finally, we enforce the prescribed global mass balance through a scalar projection based on the evolving surface quadrature. Numerical experiments demonstrate high-order convergence, conservation to roundoff in source-free problems, stable long-time integration, and substantial savings from the proposed update strategies.

We present here a generalized construction of a finite solvable group whose prime character degree graph has the shape and structure of the bowtie graph. As with the original bowtie, the graphs obtained by this generalized construction, under certain restrictions, cannot be realized by the usual method of taking direct products of smaller graphs. Within the condition of $n=1$, we show how this recovers the original bowtie graph, which has five vertices. We also provide examples and explicit choices of primes which generate graphs with more vertices.

These notes develop four interconnected ways of reading a quantum circuit. A circuit for us begins as an operational composition of gates; then, it becomes a diagram whose local equalities may be used as calculations; next, it becomes a protected process once errors, syndromes, and logical degrees of freedom are separated; and finally, it becomes geometric when its connectivity, topology, and boundary data are treated as physical design parameters. The development begins at the level of bits and qubits before appealing to Deutsch's and Grover's algorithms as basic examples of quantum circuits. With the basics in hand, we interpret quantum circuits diagramatically, leading us to compact closed string diagrams and the ZX-calculus. After that, we consider how to correct quantum circuits by introducing the Knill–Laflamme condition, homological surface codes, and related concepts with a view towards thinking of these as operations on diagrams. The lectures eventually arrive at the properties of hyperbolic quantum codes and the prospect of physical superconducting circuits emulating the negatively-curved lattices needed to support those codes. These mathematical ideas and physical experiments, taken together, represent one way to impart a geometric layer onto quantum circuits. By the very end, we bring the ideas nearly full circle by assessing the extent to which these device physics experiments operationalize the basic ZX diagrams encountered much earlier in the story. While the later material reports on original research, and while the discussion becomes increasingly mathematical as the sections progress, no prior knowledge of quantum information, quantum computing, or quantum error correction is actually assumed.

Statistical watermarks for language models live in the freedom of the signifier: they choose among tokens that are nearly equivalent in meaning, and they are therefore eroded by exactly those transformations which move the form of a text while leaving its content in place. The literature measures such transformations by their endpoint, through the semantic similarity between the original and the rewritten text. We show that the endpoint is the wrong statistic. Adapting the formalism of linguistic loops, we prove that the invariant of a chain of meaning-preserving transformations factorises canonically into an endpoint part and a holonomy in the stabiliser of the initial state, the second of which the semantic deficit cannot see; the loop rotation is parallel transport on the unit sphere of the embedding space, so that the analogy with the Wilson loop becomes a theorem rather than a figure of speech. On the side of the detector we prove an exact identity: the residual statistic is proportional to the number of positions whose seeding window survived intact, from which the decay law $\rho^{h+1}$ follows as the independent-edit corollary. The identity has a disconcerting consequence, which we confirm to three decimal places: at one and the same retention rate the surviving signal may be one half of the original, one quarter of it, or exactly nothing, according only to where the edits fall.