A Mathematically Guaranteed Multi-Axis Synergy in Causal-Set Order-Invariants, and Its Partial, Normalization-Specific Signature in Physically Mixed Populations
Year: 2026
Type: preprint
Abstract: Causal set theory defines spacetime as a locally finite partial order, encapsulated in Sorkin’s slogan “Order + Number =Geometry.” This structure maps naturally onto WCM-PL’s own two-layer architecture (PL-2): anoccupancy layer, builtfrom order-invariant point-set statistics, and arelational layer, sensitive to sequence and order. We ask a narrow, well-posed question suggested by this correspondence: can WCM-PL’s occupancy-geometry sensor (D5Sensor) detect genuinejoint (cross-axis, covariance-level) structure among a causal set’s order-invariant diagnostics — structure that is invisibleto any one of those diagnostics taken alone?We answer this in two parts. First, we construct a regime in which the answer is mathematically guaranteed to betestable without loopholes: we compare a population of real causal-set feature vectors against the same data with eachfeature independently permuted across instances, which provably forces every one-dimensional marginal to beexactlyidentical (Mann-Whitney AUC = 0.5000, p = 1.0, for all five axes, in every trial) while destroying only the cross-axiscorrelation structure. D5Sensor discriminates these two populations under all three of WCM-PL’s standardnormalizations, and a properly powered 20-draw meta-analysis confirms this cleanly (Stouffer combined z = 10.5‒15.1, signtest p < 10⁻⁵ for every normalization). This is, to our knowledge, the first demonstration that WCM’s occupancy geometrycarries genuine multivariate information beyond any single marginal statistic.Second, we ask whether this capability tracks anything in a physically generated (non-artificial) causal-set population.Three direct attempts to construct a physical pair of classes whose marginals overlap on all five axes failed, for aphysically informative reason: curvature and dimension are single scalar parameters that move all five order-invariantscoherently, leaving little room to vary joint structure independently of the marginals with the geometric knobs available tous. A fourth attempt — comparing a homogeneous single-curvature population against a genuine 50/50 population mixtureof two different curvatures, tuned so that all five marginalmeans coincide (Mann-Whitney AUC ≈ 0.50 throughout) —succeeds only partially. Diagnostic decorrelation shows that under minmax and rank normalization the apparentdiscrimination is entirely an artifact of the mixture’s larger marginal variance, not joint structure. Under robustnormalization alone, however, a genuine joint-structure-specific signal survives decorrelation, and a 15-draw replicationconfirms it is highly reliable (mean Δz = 7.37 ± 2.27, 15/15 draws positive, sign test p = 3.05 × 10⁻⁵) — while minmax andrank show no corresponding pattern ( minmax ’s delta is in fact slightly negative on average).We report both results honestly, including their limits: the artificial result is airtight but not yet physical; the physicalresult is real and well-replicated but confined to one of three normalizations, and does not (yet) demonstrate anyadvantage over the classical Myrheim-Meyer dimension estimator in the regime where that estimator is known to fail.WCM-PL has, as a result, moved a small but genuine step forward into a new domain: a decorrelation-based multi-axissynergy test now exists as a validated tool, and a first, narrow, physically-grounded lead — robust-normalization-specificsensitivity to cross-axis correlation in mixed causal-set populations — is available for further work. ※The lisence for codes is AGPL-3.0-or-later
Authors: Watabe, Masanori, Kurado, Claude Sonnet 5
Institution: Anthropic (United States)
Language: English
Subfield: Statistical and Nonlinear Physics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4302439716