The substrate condensate: formation, the dark gauge gap, and time activation for matter

This deposit contains one paper and the two deterministic scriptsreproducing every computed statement in it. It merges two earliernotes — on substrate condensation and the dark gauge gap, and on darkenergy and time activation — into a single line of argument: onecondensate, followed through its formation and two of itsconsequences.The paper derives the condensate from a single explicitly fixed energyfunctional. The symmetric phase is proved variationally to be a saddlerather than a metastable state, so condensation requires no nucleationand no barrier: arbitrarily small fluctuations suffice. The condensateprofile is given in Thomas-Fermi form and by relaxation of the fullEuler-Lagrange equation, with the validity range of the former stated.The interface tension saturates the Bogomolny bound. Two dynamics aredistinguished, correcting a dimensional inconsistency of an earliernote: the front is luminal in the relativistic case (v* -> c, confirmedto 0.3% in a 1+1d simulation with compact-support seed) andv* = 2 sqrt(D Gamma a) in the dissipative case (0.5%). On an expandingbackground Hubble friction slows the spinodal instability but can neverquench it, so completion is local, requires no communication acrosshorizons, and obeys a derived local-completion law confirmed in a 1+1dFRW simulation.It then gives the composite dark gauge sector its mass. Given thecondensate and the framework’s compositeness and minimal-couplingassumptions — both marked as assumptions — the gauge field inherits thecondensate profile, producing the Cornwall / Aguilar-Binosi-Papavassiliou structure of a dynamically generated mass that vanishesin the ultraviolet and saturates in the infrared. Strict positivity ofthe gap is proved, and a 1/omega^2-tail criterion proves that the gapis a genuine bound state strictly below the continuum edge; thecriterion is confirmed numerically as a phase boundary in the spectralkinetic coefficient. The converged value is lambda_0 = 0.96416 g^2 v^2,m_gap = 0.982 g v, at the light end of the dark scale.Finally it identifies the condensate with dark energy — one object, not two— and derives a second function at model level: because the emergenttime direction is the condensate-activated phase fiber, the temporalvielbein component of matter inherits the condensate amplitude, somatter evolves in time, and gauge bosons act on matter, only where thecondensate is present. The activation profile vanishes identically inthe ultraviolet region below the core radius, a region without mattertime. After completion the condensate is homogeneous, so the mechanismpredicts w = -1 exactly and no varying constants, consistent with theframework’s DESI DR2 analysis. Finally, the condition for thecondensate’s own causal completion is not an anthropic selection but isforced by the see-saw relation between the substrate scale, the Plancklength and the de Sitter radius, giving H/sigma of order 1e-31: thehabitable zone is guaranteed rather than selected.A closing remark ties the parts together: the amplitude of thecondensate sources Lambda, gaps the composite gauge sector and switcheson matter time, while its phase carries the winding that counts thefermion generations in the companion note — the zeros of the amplitudebeing simultaneously the region without matter time and the vortexcores that host the families.Claims are graded [proved], [derived], [computed], [imported],[estimated] or [assumed]. The gauge-gap and dark-energy results areexplicitly model-level: the former is conditional on the compositenessand minimal-coupling assumptions, and the central open step of thelatter is the full vielbein derivation from the substrate action. Fiveopen points are listed. Three propositions, each with a proof.The paper is written to be self-contained: Section 2 lists all sixframework inputs it uses, each with the formula in which it enters, sothat a reader unfamiliar with the parent framework can treat them asthe axioms of a model problem and follow everything else as derivedhere. The origin of the inputs is referenced rather than assumed.