arXiv subjects

Sebastian Pardo-Guerra

Publications and source records attributed to Sebastian Pardo-Guerra.

5 recordsLinked to original sources

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

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

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 $ψ(t)=\cosh t-1,$ we minimize $\sum_{e\in E^\to}ψ(z_e)$ over each affine class $ω+dC^0(G)\subset C^1(G)$. The minimizer is the unique representative satisfying the nonlinear coclosed equation $δ\sinh z=0,$ 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 $\imop \Picc=\Mcc=\operatorname{arsinh}(\kerδ),$ 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

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 Poincaré 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↗

arXiv2026

Coherent Comparison as Information Cost: A Cost-First Ledger Framework for Discrete Dynamics

We develop an information-theoretic framework for discrete dynamics grounded in a comparison-cost functional on ratios. Given two quantities compared via their ratio $x=a/b$, we assign a cost $F(x)$ measuring deviation from equilibrium ($x=1$). Requiring coherent composition under multiplicative chaining imposes a d'Alembert functional equation; together with normalization ($F(1)=0$) and quadratic calibration at unity, this yields a unique reciprocal cost functional (proved in a companion paper): \[ J(x) = \tfrac{1}{2}\bigl(x + x^{-1}\bigr) - 1. \] This cost exhibits reciprocity $J(x)=J(x^{-1})$, vanishes only at $x=1$, and diverges at boundary regimes $x\to 0^+$ and $x\to\infty$, excluding ``nothingness'' configurations. Using $J$ as input, we introduce a discrete ledger as a minimal lossless encoding of recognition events on directed graphs. Under deterministic update semantics and minimality (no intra-tick ordering metadata), we derive atomic ticks (at most one event per tick). Explicit structural assumptions (conservation, no sources/sinks, pairwise locality, quantization in $\delta\mathbb{Z}$) force balanced double-entry postings and discrete ledger units. To obtain scalar potentials on graphs with cycles while retaining single-edge impulses per tick, we impose time-aggregated cycle closure (no-arbitrage/clearing over finite windows). Under this hypothesis, cycle closure is equivalent to path-independence, and the cleared cumulative flow admits a unique scalar potential on each connected component (up to additive constant), via a discrete Poincar\'e lemma. On hypercube graphs $Q_d$, atomicity imposes a $2^d$-tick minimal period, with explicit Gray-code realization at $d=3$. The framework connects ratio-based divergences, conservative graph flows, and discrete potential theory through a coherence-forced cost structure.

cs.IT↗