The Stream and the Algebra - A mathematics and physics paper series

The book series ‘The Stream and the Algebra’ presents a speculative research programme — a thought-experiment in the mathematics of relation, offered for scrutiny, not yet peer-reviewed or tested. Its wager: nothing physical is an absolute number (a mass means nothing until a unit is chosen), only how one quantity stands to another. It fixes the dimensionless relations the world determines and concedes one scale, the anchor; a firewall shares a structure's form across domains, never a bare number. Its sharpest stakes: an axion-free zero of strong CP, few-GeV asymmetric dark matter, and the tau mass from Koide read forward. From the fundamental particles it carries this idea to the vacuum, the classical world, and physics' open problems.

The first volume of the series ‘The Crystallisation of Flavour’ closes the visible flavour sector — the masses, mixings, and CP of the three quark and lepton generations. Its central move: read the three generations as one complex modulus τ on the modular domain SL(2,ℤ)\ℋ (the space of torus shapes); the physics lives in τ's departure from τ = ω — its size setting the mass hierarchies, its direction the CP phase.

— From one three-cycle operator: the Koide relation (K = 2/3) and the Brannen phase (2/9) as two spectral traces, no free parameter between them.

— Across the sector — quarks and leptons as one modulus in two weights; the CKM and lepton mixings (Cabibbo as the √ of a down-quark mass ratio); the neutrinos; and an axion-free zero of strong CP (θ̄ forced by a modular weight-sum condition) in an explicit E₆ quark model.

— Read forward, the Koide balance turns the two light-lepton masses into a tau-mass prediction (1776.97 vs 1776.86 ± 0.12 MeV) — with the Weinberg angle, proton decay at Hyper-Kamiokande, and a moderate-versus-maximal switch for the leptonic CP phase.