The Classification Law as a Diagnostic for Mathematical Formalism Selection: When to Discretize and When to Use the Continuum

This paper extends the Classification Law (R₂ ∧ C ≠ C₀), originally formulated as a structural taxonomy for finite directed relational systems, into a pre-computational diagnostic for selecting between discrete and continuous mathematical formalisms.

Foundational paper. A Theory of Geometric Structure: The Classification Law (R₂ ∧ C ≠ C₀), Santiago 2026, DOI 10.5281/zenodo.18756471. The Diagnostic paper assumes the foundational law as established and tests a second function it appears to serve.

The diagnostic claim. Before any equations are written, inspect the relational graph of the system in question and apply the R₂ test. If R₂ = 1 (the graph is acyclic), the system possesses a natural directional ordering and discrete formulations will produce tractable solutions. If R₂ = 0 (the graph contains cycles), the system is symmetric or cyclic and continuous mathematics will be required. The R₂ test is mechanical, exact, and domain-independent.

Validation set. Eight independent systems plus one controlled comparison were analyzed under a pre-registered protocol that fixed the R₂ assignment before checking the historical outcome. The forward cases (predicted discrete) include Lattice QCD, the Lattice Boltzmann Method, Causal Dynamical Triangulations, the Arithmetic Black Hole Model and the hydrogen spectrum, Nyquist-Shannon digital signal processing, and AlphaFold-style graph-based protein folding. The reverse cases (predicted continuous) include the Ising model at criticality and the Traveling Salesman Problem. The controlled comparison is CDT versus EDT, where identical 4-simplex building blocks produce a four-dimensional de Sitter-like universe under causal ordering (R₂ = 1) and crumpled, unphysical geometry without it (R₂ = 0). The diagnostic predicts the historical outcome correctly in all cases.

Case selection methodology. Section 3 documents how the test cases were chosen, including pre-registration of the R₂ assignment and a counterexample search across four classes: systems with R₂ = 1 where naive discretization fails (chiral fermions on the lattice), symmetric systems where direct discrete simulation succeeds in trivial regimes (Monte Carlo on Ising away from criticality, small-n TSP enumeration), hybrid systems excluded from the test sample (plasma physics), and the canonical hybrid stress test (classical electromagnetism). No counterexample of the form "R₂ = 1 with discretization failing entirely" or "R₂ = 0 with discretization solving the hard regime" was found.

Structural insight. Section 9.1 develops the deeper principle behind the diagnostic. Wilson's renormalization group treatment of the Ising model is examined in detail: RG does not merely smooth a discrete substrate into a differentiable field. It manually constructs a directed acyclic flow on scale space (block-spin transformations always coarsen, never refine) that the physical interaction graph does not supply. The continuum is the substrate on which a missing directional ordering can be injected. The general claim is that every successful physics calculation requires R₂ = 1 somewhere in the formulation, supplied either by the physical system or constructed by the physicist on an auxiliary axis (scale, coupling-constant, relaxation, or path-ordered time).

Appendix A: numerical comparison. The ABHM and hydrogen spectrum case is supported by an explicit numerical comparison across ten transitions in the Lyman, Balmer, and Paschen series. ABHM produces the dimensionless transition energy form 1/k₁² minus 1/k₂² as an exact algebraic identity. Wavelengths derived from this form match measured hydrogen lines to better than 0.06% in all cases, with the residual error matching the Rydberg-versus-reduced-mass correction (R-infinity is the infinite-nuclear-mass limit; the actual hydrogen Rydberg differs by the electron-to-proton mass ratio). The form identity is exact; the proportionality constant is supplied by measurement, placing ABHM in the same epistemic position as the Bohr model.

Scope. The diagnostic is a structural correlation between R₂ status and historical formalism success in the tested sample. It does not claim that R₂ is the sole cause of tractability in all cases. The sample size is small relative to the total number of physical systems and a larger systematic survey is recommended as the next validation step. The paper's claims are bounded to formalism selection and do not propose new physical theories or derive physical constants.

Companion papers.

Foundational: A Theory of Geometric Structure: The Classification Law (10.5281/zenodo.18756471).
Technical depth: The Sieve Firewall (10.5281/zenodo.18854321), which applies the same structural framework to the Riemann Hypothesis and classifies its proof landscape as DARK_LOOP, reducing the open question to a single inequality (Theorem J).

Let's make pizza: AXONLang Labs Internal Research Paper (10.5281/zenodo.19651828)

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