Papers, preprints and technical reports from the institute's programs.
Two preprints from P-01, the substrate physics program, whose mathematical work runs ahead of its physics. They are a matched pair: a certified bound, and the barriers that stop it going further. Both are deposited with Zenodo under CC BY 4.0 and carry DOIs.
The Turán density π(K₄⁽³⁾) of the tetrahedron has been conjectured since 1941 to equal 5/9, and the best published upper bound has stood essentially unchanged since 2012. This paper gives an exactly certified upper bound of 951/1697 = 0.5604007071302298, verified by exact rational arithmetic with 0 violations across all 49,242 constraints of a level-6 colour-refined flag-algebra program — closing 18.019% of the gap between the previous certificate and the conjectured value. It also gives a precise account of why the method cannot reach 5/9: the obstruction is exact tightness, not accuracy and not computational cost.
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doi:10.5281/zenodo.22196487
md5:d43157a0b8020dcf41bc4c82d377b8d8 — the file served here is
byte-identical to the deposited record.
A companion to the bound above, mapping what stands between it and Turán's conjectured 5/9. It proves three exact ceilings for family-tight certificates, each certified by an explicit integer ray verified in exact rational arithmetic with no floating point in any verification step. Beyond flag algebras it shows that each of five standard alternatives — stability, containers/regularity, conditional certificates, spectral and entropy methods — fails for a specific, checkable reason. The certificate class that survives, copositivity of the link matrix on the nonnegative orthant, settles all 41,138 isomorphism classes of maximal K₄⁽³⁾-free patterns on six colours (29,099,574 labelled) unconditionally.
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doi:10.5281/zenodo.22197769
md5:62a397459cc0e6517d0bb764e569d600 — the file served here is
byte-identical to the deposited record.
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