Weyl curvature from the Hasse diagram: a parameter-free bridge formula for causal sets

Weyl curvature from the Hasse diagram: a parameter-free bridge formula for causal sets | Authorea try { document.documentElement.classList.add('js'); } catch (e) { } var _gaq = _gaq || []; _gaq.push(['_setAccount', 'G-8VDV14Y67G']); _gaq.push(['_trackPageview']); (function() { var ga = document.createElement('script'); ga.type = 'text/javascript'; ga.async = true; ga.src = ('https:' == document.location.protocol ? 'https://ssl' : 'http://www') + '.google-analytics.com/ga.js'; var s = document.getElementsByTagName('script')[0]; s.parentNode.insertBefore(ga, s); })(); Skip to main content Preprints Collections Wiley Open Research IET Open Research Ecological Society of Japan All Collections About About Authorea FAQs Contact Us Quick Search anywhere Search for preprint articles, keywords, etc. Search Search ADVANCED SEARCH SCROLL This is a preprint and has not been peer reviewed. Data may be preliminary. 2 April 2026 V2 Latest version Share on Weyl curvature from the Hasse diagram: a parameter-free bridge formula for causal sets Authors : David Alfyorov 0009-0003-6027-7837 [email protected] and Igor Shnyukov Authors Info & Affiliations https://doi.org/10.22541/au.177507192.22549325/v2 203 views 118 downloads Contents Abstract Supplementary Material Information & Authors Metrics & Citations View Options References Figures Tables Media Share Abstract We construct a statistical estimator on Poisson-sprinkled causal sets in four spacetime dimensions and derive, both analytically and numerically, a parameter-free relationship between this estimator and the electric part of the Weyl tensor. The estimator, denoted CJ, is a stratified covariance of path-count observables computed from the Hasse diagram. On a vacuum causal diamond of proper time T with N sprinkled elements, we find ⟨CJ⟩=4⋅83⋅19!⋅8π15⋅π24⋅N8/9EijEijT4⟨CJ⟩=4⋅38​⋅9!1​⋅158π​⋅24π​⋅N8/9Eij​EijT4 where each factor has a distinct geometric origin: 4 = 2² from the two-leg structure of the link score, 8/3 = c₄² from the squared Benincasa–Dowker normalisation, 1/9! from the nine-simplex beta overlap, 8π/15 from the angular average of the squared tidal deformation, and π/24 from the four-dimensional diamond volume. The combined numerical prefactor is C₀ = 32π²/(3·9!·45) ≈ 6.44 × 10⁻⁶. The rational coefficient contains no continuous free parameters; the factor 4 = 2² is derived from the two-leg structure of the link score in the bulk regime, and the exponent 8/9 is empirically established but not derived from first principles. The formula is verified against Monte Carlo data on exact pp-wave causal diamonds for N = 500–15,000: the ratio of measured to predicted CJ is 1.016 ± 0.015, with residual scatter below 4%. Eight independent diagnostic tests are reported, including a de Sitter null test (CJ = 0 exactly), Kottler cross-term, polarisation independence, a Lorentz boost test confirming CJ ∝ E² (blind to the magnetic Weyl component B², match < 1%), and Sorkin–Johnston entropy independence. The derivation rests on two explicitly stated conditions concerning the continuum limit of the stratified estimator. The N^{8/9} exponent is empirically established but not derived from first principles. All rational coefficient identities, including the general-d beta overlap (d!)²C(2d,d)(2d+1) = (2d+1)!, are formally verified in Lean 4 using the Mathlib library (105 sorry-free theorems). Five failed approaches and thirty closed spectral routes are documented. We construct a statistical estimator on Poisson-sprinkled causal sets in four spacetime dimensions and derive, both analytically and numerically, a parameterfree relationship between this estimator and the electric part of the Weyl tensor. The estimator, denoted CJ, is a stratified covariance of path-count observables computed from the Hasse diagram. On a vacuum causal diamond of proper time Supplementary Material File (sct_cj_bridge.pdf) Download 361.69 KB Information & Authors Information Version history V1 Version 1 01 April 2026 V2 Version 2 02 April 2026 Copyright This work is licensed under a Creative Commons Attribution 4.0 International License Keywords benincasa-dowker action causal diamond causal sets gravity hasse diagram mathematical model mathematical physics parameter-free formula path counting physics quantum gravity theoretical physics tidal deformation weyl curvature Authors Affiliations David Alfyorov 0009-0003-6027-7837 [email protected] View all articles by this author Igor Shnyukov View all articles by this author Metrics & Citations Metrics Article Usage 203 views 118 downloads .FvxKWukQNSOunydq8rnd { width: 100px; } Citations Download citation David Alfyorov, Igor Shnyukov. Weyl curvature from the Hasse diagram: a parameter-free bridge formula for causal sets. Authorea . 02 April 2026. DOI: https://doi.org/10.22541/au.177507192.22549325/v2 If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download. For more information or tips please see 'Downloading to a citation manager' in the Help menu . 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