arXiv subjects

Jonathan Washburn

Publications and source records attributed to Jonathan Washburn.

24 records · Page 2Linked to original sources

arXiv2026

A Complete Derivation of the Fermion Spectrum from the Recognition Composition Law

We present a first-principles derivation of the masses of all twelve known fermions -- three charged leptons, six quarks, and three neutrinos -- and the fine-structure constant $α^{-1}$, from a single discrete functional equation, the Recognition Composition Law (RCL), with \textbf{zero continuously adjustable parameters}. The mass spectrum follows from the RCL supplemented by four regularity conditions and eight structural theorems (T1--T8): the golden ratio $φ=(1+\sqrt{5})/2$ emerges as the unique hierarchy base (T6); an 8-step period is fixed by the 3-cube Hamiltonian cycle (T7); three spatial dimensions are selected by a unique combinatorial identity (T8). All integers entering the mass formula are the six combinatorial invariants of the 3-cube $Q_3$; none is fitted. The sole empirical input is the electron mass, which fixes an irreducible unit-conversion constant~$τ_0$. Predictions are confronted with PDG measurements. Charged-lepton masses are reproduced at sub-ppm accuracy for the muon and $\sim\!10^{-4}$ for the tau (Table~\ref{tab:lepton_validation}). All six quark masses are predicted at integer level; first-generation quarks agree to better than $1\%$, while second/third-generation residuals of $2$--$16\%$ are expected integer-precision effects (Table~\ref{tab:quark_validation}). Neutrino mass-squared splittings agree with NuFIT~5.3 within $1$--$2σ$, normal ordering is predicted, and $Σm_ν\approx 0.063$~eV satisfies cosmological bounds. All structural claims are machine-verified in Lean~4 (179 files, 0~\texttt{sorry}; \texttt{github.com/\allowbreak jonwashburn/\allowbreak recognition-science}).

physics.gen-ph↗

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

Uniqueness of the Canonical Reciprocal Cost

We study a rigidity problem for functions $F:\R_{>0}\to\R_{\ge 0}$ that penalize deviation of a positive ratio from equilibrium $x=1$. Assuming (i) a d'Alembert-type composition law on $\R_{>0}$, and (ii) a single quadratic calibration at the identity (in logarithmic coordinates), we prove that $F$ is uniquely determined. The composition law implies the normalization $F(1)=0.$ The unique solution is called the canonical reciprocal cost, namely the difference between the arithmetic and geometric means of $x$ and its reciprocal. Our proof uses the logarithmic coordinates $H(t)=F(e^t)+1$, where the composition law becomes d'Alembert's functional equation on $\R$. The calibration provides the minimal regularity needed to invoke the classical classification of continuous solutions and fixes the remaining scaling freedom, selecting the hyperbolic-cosine branch. We also establish necessity of each assumption: without calibration the composition law admits a continuous one-parameter family, without the composition law the calibration does not determine the global form, and without regularity the composition law admits pathological non-measurable solutions. Finally, we establish a stability estimate for approximate solutions under bounded defect and characterize some properties of the canonical cost.

math.CA↗

arXiv2026

The Coercive Projection Theorem for Canonical Reciprocal Costs

We develop a finite-data framework for certifying \emph{zero-defect} (neutral) configurations of positive vectors under the canonical separable reciprocal cost. We show that this scalar cost is characterized among non-constant continuous costs by the Recognition Composition Law together with a local quadratic calibration at balance; in particular, reciprocity symmetry and the normalization at the neutral point follow from the composition law. Under a conservation constraint and short-window observations of a rational (finite--state) signal class, we construct a canonical decision procedure that is \emph{locally maximal on the identifiability locus} among all sound procedures: any sound rule that resolves a datum must agree with the canonical output, and cannot resolve strictly more cases. The method is organized as $Φ^\ast=A\circ B\circ P$: the projection/coercivity core is forced by the canonical-cost axioms, while the aggregation/reconstruction step is specified on a non-degenerate identifiability locus (e.g.\ a Hankel invertibility condition).

math.OC↗

arXiv2026

Reciprocal Convex Costs for Ratio Matching: Functional-Equation Characterization and Decision Geometry

We study ratio-induced mismatch costs of the form $c(s,o)=J(ι_S(s)/ι_O(o))$, built from positive scale maps $ι_S:S\to(0,\infty)$ and $ι_O:O\to(0,\infty)$ and a penalty $J:(0,\infty)\to[0,\infty)$. Assuming inversion symmetry, strict convexity, normalization $J(1)=0$, and a multiplicative d'Alembert identity, we show that $f(u):=1+J(e^u)$ satisfies the additive d'Alembert equation and hence $J(x)=\cosh(a\log x)-1=\tfrac12(x^a+x^{-a})-1$ for some $a>0$. We then analyze the associated argmin mapping over feasible scale sets: existence under explicit subspace-closedness hypotheses, geometric-mean decision boundaries for finite dictionaries with stability away from boundaries, exact compositionality for product models, and an optimal sequential mediation principle given by a geometric mean (or its log-space projection when infeasible). The paper is purely mathematical; any semantic interpretation is optional and external to the theorems proved here.

math.OC↗

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 $δ\mathbb{Z}$) 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 Poincaré 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↗