arXiv subjects

Jonathan Washburn

Publications and source records attributed to Jonathan Washburn.

At least 19 recordsLinked to original sources

arXiv2026

Molecular Transferability of a Noble-Gas Coordinate for Electronegativity Equalization

Charge-equilibration models predict how electrons redistribute over a molecule more cheaply than quantum chemistry, and each begins from a table of atomic electronegativities and hardnesses. A recent table replaces ionization energies and electron affinities with one geometric quantity: each main-group atom's fractional distance to the noble gas closing its row. Holding geometries, electrostatic damping and the molecule list fixed, we compare it with spectroscopic values, with its own kernel refitted, and with a quadratic control, over a primary set of fifty-two molecules and ions at the B3LYP/def2-TZVP level. Two structural consequences follow from the table construction rather than the equalization solver. Atoms at equal fractional distance receive identical electronegativities, so the dipole moments of chlorine monofluoride, iodine monobromide and sulfur dioxide vanish exactly, and the measured $0.72$ debye $\mathrm{HF}/\mathrm{HCl}$ gap is lost. Because the first period is excluded, hydrogen retains its spectroscopic electronegativity above every geometric value, reversing $\mathrm{O{-}H}$, $\mathrm{N{-}H}$ and hydrogen-halide polarity. Fitted alike, geometric and quadratic kernels agree to $0.0022$ electron per atom, so predictions are set by the fitted energy scales, not the kernel shape. Resolving the electronegativity scale period by period restores correct polarity in all five polyatomic heavy-atom-hydrogen tests.

physics.chem-ph↗

arXiv2026

Tilt control of coverage heterogeneity for hard spherocylinders locked on a sphere

We study hard spherocylinders on a sphere with axes rigidly locked to a tangential director field at fixed angle $\tilt$ to the meridian, necessarily singular at the poles. Three lengths set the problem: the rod length $\Lrod$, the diameter $\Drod$, and the host radius $\Rsph$. Monte Carlo simulations across fifteen geometries and four coverages give two main results for the polar marginal, the azimuthal average of rod-center density. First, the tilt is a continuous handle on the width of the depleted region that packing induces around each singularity. Under meridian locking it is set principally by the rod length; turning the director toward the latitude contracts it substantially, the contraction being spent well before latitude locking. Second, how uniform the polar marginal can be made is limited by geometry, not tilt. The smallest variance on the sampled grid follows a power law in $\Lrod^{2}/(\Rsph\Drod)$, which measures how far a straight rod's ends stand off the curved surface in rod diameters, with an effective exponent between $1.1$ and $1.3$. Long rods on small hosts cannot be made uniform at any sampled tilt; the remedy is geometric, not orientational. The polar marginal is equator-heavy almost everywhere, inverting only at high coverage and tilt; meridian locking is the least uniform choice, and the variance-minimizing tilt usually lies in a sampled band from $31.7^{\circ}$ to $55^{\circ}$. The golden-ratio tilt $\arctan(1/\phigold)$ is one of those angles: a benchmark, not one the model selects. Both results describe the infinitely locked athermal ensemble.

cond-mat.soft↗

arXiv2026

Fixed-order postselected CHSH reference acquisition for parity-constrained spatial-mode qubits on a commercial cloud photonic processor

We report a fixed-order Clauser-Horne-Shimony-Holt (CHSH) acquisition and reporting protocol for two encoded photonic qubits on Quandela's commercial, cloud-accessible Belenos processor, executed end-to-end by external users through the public cloud interface. Each logical qubit is a two-dimensional spatial-mode subspace in the seven-dimensional zero-sum sector of an eight-mode single-photon register, and a target postselected linear-optical controlled-$Z$ (ideal success probability $1/9$) couples the registers on $16$ of $24$ modes. The primary quantity is the operational CHSH score $S$ on accepted logical coincidences, with each complete four-setting pass as the experimental unit. Eight sequential same-day passes each gave a raw score above $2$; session means were $2.40$ and $2.58$ (sample standard deviations $0.15$ and $0.03$), with excess dispersion $Q/\nu=6.1$ ($\nu=7$). Count-pooled secondary descriptors are $S_{\mathrm{count}}=2.485\pm0.019$ and a fixed-ratio efficiency-reweighted model scenario $S^{\mathrm{rw}}_{\mathrm{count}}=2.380\pm0.021$, whose weakest reweighted pass ($2.040\pm0.062$) overlapped $2$ within $1\sigma$. The reweighting is an archived-metadata model scenario, not a corrected platform score; an ad hoc $\kappa\in[1.2,1.8]$ stress scan (not a calibrated uncertainty band) spans $2.341$-$2.441$. Setting order was fixed, the compiled mapping was not returned, and residual remote-setting marginals remain, so the data support an operational reference acquisition rather than an entanglement-witness or cross-platform benchmarking claim. The parity-check terminology labels the encoding subspace; no syndrome measurement was performed. Count records, job identifiers, circuit-construction code, and analysis are openly archived with content hashes for the submitted targets.

quant-ph↗

arXiv2026

Reciprocal Cost on the Positive Rationals

We study nonnegative solutions of the reciprocal cost law on the positive rationals and determine which of them admit regular extensions to the positive reals. Using the substitution $H=1+F$, we reduce the reciprocal composition law to d'Alembert's functional equation. We prove that every nonnegative solution on $\mathbb{Q}_{>0}$ is determined by one real weight $\alpha_p$ for each prime $p$, with only a global sign identification. Thus the rational solution space is infinite dimensional. We prove that every nonnegative rational solution has an algebraic extension to $\mathbb{R}_{>0}$, but a regular extension exists exactly when the prime weights satisfy $\alpha_p=\lambda\log p$ for some $\lambda\in\mathbb{R}$. In this case the extension is unique and belongs to the one-parameter family $F_\lambda(x)=\cosh(\lambda\log x)-1$. Otherwise the rational solution is unbounded on every nonempty open subset of $\mathbb{Q}_{>0}$, and the regular locus is closed and nowhere dense. We also extend the result to arbitrary nontrivial subgroups of $\mathbb{R}_{>0}$, where the alternative is governed by rational rank. Finally, we show that unit logarithmic curvature selects the canonical reciprocal cost $J(x)=(x+x^{-1})/2-1$, while, for a carrier $G$ whose logarithm is dense in $\mathbb{R}$, the single asymptotic condition $F(e^s)\sim s^2/2$ implies both regularity and calibration.

math.GM↗

arXiv2026

The $\delta$-calculus: from distinction to arithmetic

Let $\delta$ denote the primitive act of distinction, formally realized as the one-step extension $r\mapsto Sr$ of a finite record. We prove that the $\delta$-orbit is initial among $\delta$-algebras: every $\delta$-algebra admits a unique structure-preserving map from the orbit. The map is injective when the successor operation is injective and the base point is not a successor. If the $\delta$-algebra also satisfies induction for all predicates, the map is bijective and gives the unique isomorphism with the generated $\delta$-orbit. The $\delta$-calculus is an intuitionistic first-order proof system over the signature $\{0,S,+,\cdot\}$. Each derivation carries a ledger recording the use of the law of excluded middle, the limited principle of omniscience, Markov's principle, and induction on quantified formulas. Every closed formula derivable in the forced fragment is true in the standard model. Starting from $\delta$, we construct the choice-free number tower $\delta \leadsto \mathbb{N}_\delta \hookrightarrow \mathbb{Z}_\delta \hookrightarrow \mathbb{Q}_\delta$. The metatheoretic systems $\mathbb{N}$, $\mathbb{Z}$, and $\mathbb{Q}$ each admit an explicit injection into $\mathbb{N}_\delta$. Under the law of excluded middle, every recognizer is either injective or has kernel congruence $\equiv_{i,p}$ for a unique pair $i\geq0$, $p\geq1$. In the noninjective case, the quotient is a finite monogenic monoid $M(i,p)$. A decidable congruence together with an explicit pair of distinct related elements implies the classification without additional nonconstructive principles. For a decidable congruence different from equality, Markov's principle is needed. For an arbitrary congruence, the dichotomy requires the law of excluded middle. The reverse implications show that the last two prices are exact. They are distinct from the syntactic ledger of derivations in the $\delta$-calculus.

math.LO↗

arXiv2026

Finite-Resolution Identifiability and Measurement Design for Molecular Conformer Spectroscopy

Conformer assignment is meaningful only when the measurement can distinguish the candidate structures. We formulate conformer spectroscopy as a measurement-model-dependent identifiability problem in which each modality defines an observation law with explicit experimental and theoretical uncertainty. Equality of observation laws defines exact classes that refine as modalities are added, whereas finite-resolution ambiguity is non-transitive and is represented by a Bayes-error graph. We also give rank and constrained-Fisher criteria for population recovery from unnormalized additive spectra. The framework is applied to audited B3LYP-D3(BJ)/def2-TZVP ensembles of 1,2-difluoroethane, ethylene glycol, and n-pentane. Under the declared shared-covariance working model, IR separates all non-mirror pairs, leaving only mirror pairs that are exactly degenerate under the achiral observation maps. This within-method result does not imply general IR sufficiency: in a six-case PBE0-to-B3LYP diagnostic, fixed calibration recovers one intended representative, whereas scale-and-shift profiling with a conservative cross-method covariance recovers five representatives and all six achiral classes. Under a combined stress-test covariance, n-pentane develops one non-mirror quotient ambiguity. Three individual Raman windows remove it, while no tested IR window does. Mirror-collapsed class populations remain identifiable and well conditioned under the calibrated-scale working model, whereas separate mirror-partner populations are exactly unidentifiable from achiral additive spectra.

physics.chem-ph↗

arXiv2026

Auditing Haldane Consistency in Reversible Enzyme Kinetics: A Curated Two-Sided Backbone and a Labeled Fold-Error Benchmark

Reversible enzyme kinetic constants can be audited through the Haldane relation: the apparent equilibrium constant implied by the rate law should match biochemical thermodynamics under matched conditions. We use the reciprocal cost $C_{\mathrm{Haldane}}=J(K'_{\mathrm{eq,kin}}/K'_{\mathrm{eq,thermo}})$, with $J(x)=\tfrac12(x+x^{-1})-1=\cosh(\ln x)-1$, as a calibrated, direction-symmetric reporting scale. The score is zero at agreement, penalizes reciprocal over- and underestimates equally, encodes the free-energy discrepancy in $RT$ units, and ranks records identically to $|\Delta\Delta G|$; the contribution is therefore biochemical curation, a reproducible workflow, and fold-error calibration rather than a new ordering. We apply the score to a curated demonstration set and, under prespecified inclusion criteria, assemble a two-sided backbone of twenty-one audited single-study records. Eight genuinely independent tests pair kinetics fit without a thermodynamic prior against separately measured equilibria; all eight fall within twofold (maximum $C_{\mathrm{Haldane}}=0.069$), although this remains a feasibility demonstration. Across the full backbone, eighteen records fall within twofold and three are flagged. The backbone is concentrated in carbohydrate isomerases and epimerases, so these results are within-family observations. Because real records carry no ground-truth labels, a semi-synthetic benchmark (twenty-nine within-twofold seeds, $1{,}885$ injected known-error cases) quantifies detectability: AUC $0.784$ ($95\%$ bootstrap CI $0.725$--$0.838$), conditional on the injected error taxonomy and invariant under monotone rescaling of $|\ln x|$. All data, code, protocol, and benchmark generator are archived for exact reproduction.

physics.chem-ph↗

arXiv2026

Curvature-induced smectic-C order of tangentially anchored hard spherocylinders on a sphere with a rigidly locked director field

We study the strict locked-orientation limit of hard spherocylinders on a sphere, in which the rod axes are rigidly locked to a prescribed tangential director field and cannot reorient. Because the bulk hard-rod phase diagram contains no smectic-C phase, any coherent tilt isolates a geometric curvature mechanism rather than a finite-stiffness equilibrium effect. A ratio-symmetric recognition cost fixes the layer spacing at the bulk close-contact value and yields a hierarchy of geometric statements: the lower edge of the smectic-area window at $45^\circ$ follows from reciprocal symmetry; the upper edge at $58.3^\circ$ is a falsifiable channel-saturation hypothesis; the smectic-A to smectic-C boundary is a closed-form prediction; and the rod tilt angle is set by the rod-to-radius ratio, modulated by a chirality envelope peaking near $24^\circ$. Locked-orientation Monte Carlo across fifteen geometries confirms these predictions with no fitted elastic constants: the smectic area peaks at $55^\circ$, and a coherent smectic-C window is detected.

cond-mat.soft↗

arXiv2026

Characterization of nested Walsh parity-check filters in a single-photon eight-mode register on a cloud photonic processor

We characterize two nested Walsh parity-check filters implemented on Quandela's Belenos cloud photonic processor in a single-photon eight-mode spatial register. The modes are indexed by the vertices of the cube $Q_3$. The filters realize the classical $[8,7,2]$ single-parity-check code, the zero-sum neutral subspace $\mathcal{N}$ and the $[8,4,4]$ extended Hamming code, the parity-checked subspace $\mathcal{S}\subset\mathcal{N}$ with one DC and three face-parity syndrome channels. These are first-quantized path/mode encodings of classical codes: the experiment verifies leakage suppression and syndrome routing, not error correction or protection against photon loss, and all probabilities are conditional on postselected single-photon detections. Across more than 340,000 detections, neutral inputs show residual DC-port leakage of $0.02\%$-$1.1\%$ (mean $0.6\%$), corresponding to $\approx21\times$ suppression relative to the ideal $0.125$ DC-capture baseline and $31.6\times$ relative to the measured non-neutral control. Injected DC contamination gives a monotonic soft error signal, and the three face-parity syndrome channels route to their predicted ports with $94$-$99\%$ selectivity. A sector-preserving unitary core keeps leakage far below non-neutral controls over one to three applications, with differences dominated by calibration and compilation systematics rather than gate-cycle physics. We quantify these limits, including fixed-pattern separator bias, $\pm 0.02$ calibration offsets, and compilation scatter near the $10^{-3}$ leakage level, and report a Hong-Ou-Mandel degradation episode in which suppression vanished and recovered after recalibration.

quant-ph↗

arXiv2026

d'Alembert's Functional Equation and a Globally Convex Free-Action Principle on Positive Paths

We study the kinetic action that d'Alembert's functional equation induces on positive paths in $\Rplus$, and prove it strongly convex. Calibrated d'Alembert forces the cosh cost $\Jcost(x)=\tfrac12(x+x^{-1})-1$, i.e.\ $\Jlog(\xi)=\cosh\xi-1$ in the log coordinate $\xi=\log x$. Evaluating this log-cost at the log-\emph{velocity} $\dot\xi$ rather than the log-position -- a single postulate (Postulate~\ref{post:step}) -- yields $\actionA[\gamma]=\int_a^b(\cosh\dot\xi-1)\,dt$, strongly convex under geometric (log-space) interpolation. This convexity has three consequences, none requiring an Euler--Lagrange equation, a Fr\'echet derivative, or a second variation. First, a one-sided chord condition characterizes global minimality. Second, the unique fixed-endpoint minimizer is the uniform-log-velocity path. Third, the action gap obeys an exact Bregman / Pythagorean identity $\actionA[\gamma]-\actionA[\gamma_*]=\int D_\Kkin(\dot\xi\,\|\,\dot\xi_*)\,dt$, sharpened by a quantitative Friedrichs--Poincar\'e bound on $\log(\gamma/\gamma_*)$. It has a dually-flat / Hessian-manifold reading in the additive coordinate $\xi$. \\ This theorem is purely mathematical, and we delimit it. The bridge to Newtonian and rapidity mechanics is \emph{conditional}, requiring structure beyond Postulate~\ref{post:step}: a kinematic embedding, a mass coupling, a time calibration, and a Hamiltonian-primary Legendre structure. Granted these, the cosh action recovers the Newtonian small-step limit and the rapidity profile $\Kkin_m(\phi)=m(\gamma_L-1)$; yet the cosh-dual Hamiltonian is \emph{not} the special-relativistic free-particle Hamiltonian (Proposition~\ref{prop:not-SR}), the agreement being one of profile, not an identity of Hamiltonians. Global minimality is a free-sector phenomenon: once a non-affine strictly convex potential is added, joint convexity is lost and the classical stationary-action picture returns.

math.OC↗

arXiv2026

A Finite-Lattice Model from a Reciprocal Cost Action: Spectral and Reflection-Positivity Properties

We study the finite-lattice statistical-mechanical model whose nearest-neighbor bond potential is the reciprocal cost $J(e^\varepsilon)=\cosh\varepsilon-1$, selected by the d'Alembert functional equation under the stated regularity and calibration assumptions. The structural inputs are stated explicitly; once they are fixed, the analysis is rigorous mathematics about the bond action $V(\Delta\phi)=\cosh(\Delta\phi)-1$ on finite boxes in $\mathbb Z^3\times\mathbb Z/8\mathbb Z$. Our main result pairs a negative and a positive statement about reflection positivity. For the continuous noncompact model the natural temporal kernel $K(u)=\exp[-(\cosh u-1)]$ fails the Bochner positive-definiteness test: an interval-certified quadrature gives $\widetilde K(3)<0$. Thus the standard Bochner route to Osterwalder-Schrader reflection positivity is obstructed. For a finite-alphabet variant, with field values restricted to a finite symmetric set $\Phi=v_0\{-N,\ldots,N\}$, reflection positivity holds whenever the finite crossing-bond Toeplitz matrix $(K_{\Phi(v_0,N)})_{a,b}:=K(b-a), a,b\in\Phi$, is positive semidefinite. For $v_0\in\{1.2,1.5,2.5\}$, this is discharged by a rigorous diagonal-dominance certificate uniform in $N$, and the associated one-step transfer operator is then positive and self-adjoint in an explicit reflection-positivity inner product. These finite-volume results do not provide a continuum Wightman theory, Osterwalder-Schrader reconstruction, LSZ scattering, or a continuum mass gap.

cond-mat.stat-mech↗

arXiv2026

Golden and Metallic Structures on Hessian Manifolds

We consider the reciprocal cost function $ J(x)=\frac12(x+x^{-1})-1 $ and its $n$-dimensional extension $J(x_1,\ldots,x_n) = \frac12(R+R^{-1})-1, R=\prod\limits_{i=1}^n x_i^{\alpha_i}, \alpha=(\alpha_1,\ldots,\alpha_n)\in\mathbb{R}^n\setminus\{0\}.$ In logarithmic coordinates $t_i=\log x_i$, the Hessian of $J$ has a rank of one at every point. The associated Hessian geometry is degenerate and does not define a Riemannian metric. To obtain a nondegenerate geometric structure, we introduce a family of Hessian metrics $h_\lambda$. Combining the rank-one tensor with the Hessian metric $h_\lambda$, we construct a $(1,1)$-tensor field $A_\lambda$. Its trace normalization defines a projector $P_\lambda$, which induces an almost product structure and the corresponding golden and metallic structures. We study several properties of the projector $P_\lambda$ and the induced structures, including eigendistributions, parallelism, integrability, and curvature. The construction is given in an arbitrary dimension, and explicit formulas are obtained in the two-dimensional case. In particular, we show that the projector $P_\lambda$ is generally not parallel with respect to either the canonical flat affine connection or the Levi-Civita connection $\nabla^\lambda$ of the Hessian metric $h_\lambda$.

math.DG↗

arXiv2026

A Finite-State Gibbs Construction from a Recognition Cost

On a finite outcome space, the canonical Gibbs distribution is usually obtained by maximizing Shannon entropy at fixed mean of an externally supplied energy functional. This paper studies the finite-state consequences of a ratio-cost construction instead: after adopting the normalized d'Alembert degree-two closure called the Recognition Composition Law (RCL), with unit log-curvature calibration at the reference ratio, the continuous nontrivial positive branch is $J(x)=\tfrac12(x+x^{-1})-1=\cosh(\log x)-1$. Given the induced cost vector $X_\omega=J(r_\omega)$, multinomial counting and convex duality recover the finite-state Gibbs weights and the identity $F_{\mathrm{R}}(q)-F_{\mathrm{R}}(p)=T_{\mathrm{R}}\,D_{\mathrm{KL}}(q\Vert p)$; the entropy-maximization steps are classical once the cost is fixed. New technical content includes a non-asymptotic Stirling bound and soft-shell constrained-type theorems for real-valued costs. A three-state example compares the Gibbs law to squared-log, affinity-as-energy, and Tsallis alternatives at the same cost vector and mean-cost constraint, with sample-size power calculations at fixed RCL ground truth. The framework is conditional on axioms (A1)--(A3) and restricted to finite outcome spaces with strictly positive weights; it does not derive the composition law from a more primitive principle.

cond-mat.stat-mech↗

arXiv2026

A Noble-Gas-Centered Coordinate for Within-Period Atomic Property Trends

We introduce a single dimensionless landscape function $J_{\rm chem}(\rho) = \cosh(\rho \ln \varphi) - 1$, $\varphi = (1+\sqrt{5})/2$, on the noble-gas-centred coordinate $\rho = d/L_p \in [0,1)$, and show that it organizes four central atomic observables: first ionization energy \IE$_1$, electron affinity EA, Mulliken electronegativity $\chi_M$, and Pearson chemical hardness $\eta$, on one periodic-table axis. The outward step $\Delta J_{\rm chem}^{+}$ delivers IE$_1$; the inward gap $\Delta J_{\rm chem}^{-} = J_{\rm chem}(1) - J_{\rm chem}(\rho)$ delivers EA and $\eta$; $\chi_M$ follows by Mulliken's identity. Three results establish the empirical content. (i) The within-period IE$_1$ envelope reproduces the full noble-gas-to-alkali ordering across periods 2--6: of 34 atoms compiled across periods 2-4, 26 lie on the predicted monotone descent and the 8 upward deviations occur exactly at the textbook anomaly sites $\{p^3, d^5, f^7, s^2, d^{10}\}$. (ii) Two golden-ratio identities, ${\rm IE}_1(G_p)/{\rm IE}_1(G_{p+1}) \approx \varphi^{1/4}$ on three heavy noble-gas pairs and ${\rm IE}_1(\text{halogen})/{\rm IE}_1(\text{alkali}) \approx \varphi^2$ on four within-period pairs, agree with NIST data to MAD $\approx 1\%$ and $\approx 5\%$, respectively. (iii) The shared kernel $\Delta J_{\rm chem}^{-}$ provides single-parameter analytical fits to EA across periods 4--6 (MAE $0.3$--$0.4$~eV), to Pearson hardness $\eta$ across periods 2--4 (MAE $\sim 1$~eV on noble-gas maxima up to $10.8$~eV), and to Mulliken $\chi_M$ across a 15-atom four-class benchmark ($R^2 = 0.73$). \edit{At the period-averaged scale level, the shared-kernel relation ${\rm EA}/\eta \approx C^{(p)}_{\mathrm{EA}}/C^{(p)}_{\eta}$ is supported on period-4 NIST data: the empirical nine-atom mean $\overline{{\rm EA}/\eta} = 0.180$ agrees with the predicted constant $0.182$ to better than $1\%$, although individual-atom scatter ($\sigma \approx 0.13$) is much larger.

physics.chem-ph↗

arXiv2026

The Cactus Criterion: When Nonlinear Hodge Theory Reduces to Linear on Graphs

Let $G$ be a finite connected simple graph with a chosen orientation of its edges. For the edge potential $\psi(t)=\cosh t-1,$ we minimize $\sum_{e\in E^\to}\psi(z_e)$ over each affine class $\omega+dC^0(G)\subset C^1(G)$. The minimizer is the unique representative satisfying the nonlinear coclosed equation $\delta\sinh z=0,$ and hence defines a nonlinear selector $\Picc:C^1(G)\to C^1(G).$ We show that $\Picc$ is real analytic, identify its image as $\imop \Picc=\Mcc=\operatorname{arsinh}(\ker\delta),$ and compute its differential as a weighted Hodge projector. In particular, $\Picc$ agrees with the ordinary Hodge projector $\PiH$ to first order at the origin, and the first nonlinear correction is cubic. Our main global theorem is a graph-theoretic criterion: for every admissible edge potential -- even, $C^2$, strictly convex, and non-quadratic -- the associated nonlinear selector coincides with $\PiH$ on all of $C^1(G)$ if and only if $G$ is a cactus graph. Finally, we work out the two-triangle graph, the smallest connected simple obstruction, and record a self-concordant Newton method for computing $\Picc$.

math.CO↗

arXiv2026

Multidimensional cost geometry

In this paper, we study the geometric structure induced by the canonical reciprocal cost function and its natural $n$-dimensional extension. In logarithmic coordinates, the potential depends only on the linear combination $S=\alpha\cdot t$, and the associated Hessian metric has rank one at every point. The geometry is intrinsically degenerate and effectively one-dimensional, with an $(n-1)$-dimensional null distribution. On the other hand, when the same function is expressed in the original $x$-coordinates, the corresponding Hessian is generically nondegenerate and defines a pseudo-Riemannian metric away from explicit singular hypersurfaces. We further analyze affine and Levi-Civita geodesics and compare their behavior. In particular, affine geodesics in logarithmic coordinates are globally defined, while in $x$-coordinates their behavior is restricted by the domain and the singular set. Finally, we relate the construction to symmetrized Itakura-Saito and Bregman divergences, and give a Fisher-Rao realization of the logarithmic Hessian metric.

math.DG↗

arXiv2026

The d'Alembert Inevitability Theorem

We study functions satisfying the composition law $F(xy)+F(x/y)=P(F(x),F(y))$ with a symmetric polynomial combiner $P$. We prove that symmetry together with a quadratic degree bound on $P$ forces a composition law of d'Alembert type. We establish a degree mismatch exclusion criterion showing that symmetric polynomial combiners with $\mbox{deg} P(u,v) \ge 3$ do not admit nonconstant continuous solutions, provided the leading term does not cancel (Theorem 3.1.). For continuous nonconstant functions $F:\mathbb{R}_{>0}\to\mathbb{R}$ with $F(1)=0$ satisfying the composition law with a symmetric polynomial $P$ of degree at most two, the combiner is necessarily of the form $P(u,v)=2u+2v+c\,uv$, $c\in\mathbb{R}$ (Theorem 3.3.). The equation reduces in logarithmic coordinates to the classical d'Alembert functional equation. For $c\neq 0$, one obtains hyperbolic or trigonometric branches, while $c=0$ yields the squared-logarithm family. Under the cost-function assumptions $F\ge 0$ and convexity, only the hyperbolic branch with $c>0$ remains. A unit log-curvature calibration selects the canonical value $c=2$, which yields the canonical reciprocal cost $F(x)=\tfrac12(x+x^{-1})-1$. For $c\neq0$, the result extends to $\mathbb{R}_{>0}^n$: every solution depends only on a single linear combination of coordinate logarithms; for $c=0$, the solution is a general quadratic form $\sum_{i,j}a_{ij}\ln x_i\ln x_j$. In either case, nontrivial coordinate-wise separable costs are excluded.

math.CA↗

arXiv2026

Matching Rules as Cocycle Conditions: Discrete Potentials on Penrose and Canonical Projection Tilings

Aperiodic tilings support two classically studied but hitherto separately presented structures: matching rules, which enforce global order via local constraints, and height functions, which encode global geometry through integer-valued potentials. Their precise relationship has remained implicit in the literature. This paper bridges them via a cochain-first framework, establishing a four-way equivalence -- between matching rules, Ammann bar continuity, cycle closure of the associated $1$-cochains, and height-function existence -- proved for candidate tilings without presupposing any of the four conditions. The proof proceeds via a half-edge/gluing construction: for each Ammann bar family, we assign to every directed edge a signed bar-crossing count, yielding an antisymmetric $1$-cochain. A tile-side crossing function and a global cochain are built in two stages; the global cochain exists precisely when adjacent tiles agree on shared edges. Gluing implies cycle closure; the discrete Poincar\'{e} lemma then produces a scalar potential coinciding with the classical Ammann height function. The framework extends uniformly to canonical projection tilings (CPTs) from $\mathbb{Z}^N$: lattice-coordinate cochains reconstruct vertex positions via $v = \sum_{k=1}^N x_k(v)\,\mathbf{e}_k^*$, and (for CPTs with generic window) form a $\mathbb{Z}$-basis for $\check{H}^1 \cong \mathbb{Z}^N$ (Forrest--Hunton--Kellendonk), yielding a conservation-forced structure with recognition gap $\mathcal{R}(\mathcal{T}) \cong \mathbb{Z}^N$. The framework is verified for the Fibonacci chain, Penrose P2, Ammann--Beenker, and the icosahedral Ammann tiling; whether conservation forcing characterises exactly the Pisot substitution CPTs is left as an open conjecture.

math.CO↗