Ultrametric and Adelic Structure in a Hierarchical Model of Emergent Space-Time: A Research Programme

We propose a mathematical framework for the emergence of space-time from a discrete hierarchy of physical regimes, called Levels of Existence, characterised by the propagation velocity of their constituent entities relative to the invariant constant c. The correspondence between this hierarchy and established mathematical structures is articulated by systematically distinguishing between verified results, working conjectures with specified verification protocols, and open questions. Verified results include: the existence valuation vₚ (E) = -floor (logₚ (E) ) is a rigorous non-Archimedean valuation (Theorem 1) ; Ostrowski's theorem independently motivates four types of completions of Q; the spectral parameter of the Vladimirov operator alphaₚ = log (p+1) /log (p) is geometrically determined by the Bruhat-Tits tree without free adjustment; the primordial state is formally the zero element of the tropical semiring; the series defining the vacuum form factors Zₚ converges for mₚ > 0 and diverges for mₚ = 0. Working conjectures with specified verification protocols include: the transformation matrices are isomorphic to Iwahori subgroups of GL (n, Qₚ) ; the tropicalisation of the flag variety Fl (2) over the adele ring is isomorphic to the light cone. Appendix A establishes that the Hecke algebra for p=5 admits representations whose eigenvalue field contains Q (sqrt (5) ). Appendix B provides the first verified quantitative computation: vacuum form factors Zₚ for the first eight primes, with a rigorous convergence proof and the negative result that isolated minimisation of Zₚ does not select the spectral parameter.