Exchange Symmetry and Contracting Geometry of the Prime–Prime Exponentiation Lattice

This record contains the preprint and FAIR computational supplement for the study ``Exchange Symmetry and a Contracting Metric on the Prime--Prime Exponentiation Lattice: From the Classical Equation $x^y=y^x$ to Local Prime-Gap Geometry.'' The study develops a geometric framework for prime--prime exponentiation based on the exchange of base and exponent. For ordered prime pairs $(p_i,p_j)$, the logarithmic lattice is defined by \[L_{ij}=p_j\log p_i.\] The lattice is decomposed into symmetric and antisymmetric components. This decomposition provides exact coordinates for identity symmetry, reciprocal inversion, exchange orientation, and deterministic exchange correlation under the involution \[\mathcal{R}(p_i,p_j)=(p_j,p_i).\] The central scalar coordinate is \[\Phi(p)=\frac{\log p}{p}.\] It induces the exchange metric \[d_{\mathrm{ex}}(p,q)=\left|\Phi(p)-\Phi(q)\right|.\] Under this embedding, the arithmetically unbounded sequence of prime numbers becomes a bounded metric space whose tail contracts toward the missing coordinate zero. Its metric completion is obtained by adjoining one ideal endpoint, denoted by $\infty$, with \[\widehat{\Phi}(\infty)=0.\] The completed exchange space is \[\widehat{\mathbb{P}}_{\mathrm{ex}}=\mathbb{P}\cup\{\infty\}.\] It is compact and has a unique accumulation point. For every exchanged prime pair, the antisymmetric logarithmic coordinate is defined by \[A_{ij}=\frac{p_j\log p_i-p_i\log p_j}{2}.\] The bounded exchange-orientation and exchange-correlation coordinates are \[\Xi_{ij}=\tanh A_{ij},\qquad\Gamma_{ij}=\operatorname{sech}A_{ij}.\] They satisfy the exact identity \[\Xi_{ij}^{2}+\Gamma_{ij}^{2}=1.\] Under base--exponent exchange, the orientation coordinate changes sign, whereas the correlational coordinate remains invariant: \[(\Xi_{ij},\Gamma_{ij})\longmapsto(-\Xi_{ij},\Gamma_{ij}).\] The identity sector corresponds to \[(\Xi_{ii},\Gamma_{ii})=(0,1).\] The analysis also considers consecutive primes and local prime triples. The adjacent pair $(2,3)$ is shown to be the unique negative forward orientation, while the triple $(2,3,5)$ is the unique non-geodesic local fold in the arithmetically ordered prime chain. Beyond these initial exceptions, the exchange embedding is monotone. The normalized local metric balance asymptotically reproduces the relative contrast between consecutive prime gaps after removal of the common contracting scale. The computational supplement is organized into ten independent FAIR modules. It includes Wolfram Language scripts and Mathematica notebooks, machine-readable CSV tables, JSON audit metadata, publication-quality figures in PDF, PNG, and SVG formats, execution instructions, licensing information, and SHA-256 checksums for data verification. The calculations were performed in Wolfram Mathematica 14.3. Numerical audits generally used 50-digit working precision and a numerical zerotolerance of \[10^{-35}.\] The numerical experiments verify finite consequences of the analytical results, test the internal consistency of the implementation, identify exceptional configurations, and illustrate analytically established asymptotic behavior. They do not replace the mathematical proofs. The term ``exchange correlation'' is used in a deterministic algebraic sense and does not denote a statistical correlation coefficient. The construction does not propose a statistical model for the distribution of prime numbers, establish a new statistical law for prime gaps, or imply any result concerning the zeros of the Riemann zeta function or the Riemann hypothesis.