The Substrate Condensate: Formation, the Dark Gauge Gap, and Time Activation

Abstract

One object does three things. This note derives the substrate condensate of Spectral Quantum Gravity from a single, explicitly fixed energy functional and follows the same condensate through two of its consequences. What is confirmed.By consistency: one condensate carries the dark energy, the gauge gap and the phase clock, without conflict between the three roles, and it meets the scale relation between substrate, Planck length and de Sitter radius.Against data: a homogeneous dark energy with w = −1 and no varying constants is compatible with present observations. The never-phantom bound w ≥ −1 sits at the edge of current DESI fits with supernovae and will be decided by the coming data releases. Results. Formation [proved]. The symmetric phase is proved variationally to be a saddle, not a metastable state, so condensation needs no nucleation and no barrier. It completes as a front: luminal in the relativistic dynamics (v* → c, to 0.2 %) and v* = 2√(DΓa) in the dissipative case (to 0.5 %). Hubble friction slows the instability but can never quench it, so completion is local and needs no communication across horizons. The dark gauge gap [proved, given the stated assumptions]. Given the condensate and the framework's compositeness and minimal-coupling assumptions, the composite gauge sector inherits a Cornwall-type mass profile. The gap is proved to be a genuine bound state strictly below the continuum edge, by a 1/ω²-tail criterion, whenever g²v²ω_c² > κ/4, with m_gap = 0.982 gv. Dark energy and time activation. The same condensate is the dark energy, one object rather than two. Because the emergent time direction is the condensate-activated phase fiber, the condensate sets the stiffness of the phase clock: a clock runs only where the condensate is present. After completion the condensate is homogeneous, so the mechanism predicts w = −1 exactly and no varying constants. Scale. With ℓ_s = 39 µm the gap is Δ ≃ ħc/ℓ_s ≈ 5 meV, inside the framework's derived condensation band, and the amplitude is v ≈ 40–50 meV. A guaranteed habitable zone. The condition for the condensate's own causal completion, H ≪ σ, is not an anthropic selection. It is forced by the see-saw relation ℓ_s = √(L_Pl L_dS), which gives H/σ ≃ 2 × 10⁻³¹. The gauge-gap and dark-energy sections are model-level and say so. Claims are graded [proved], [derived], [computed], [imported], [estimated], [argued] or [assumed]. Derivations, verification code and drafting were carried out in collaboration with Claude (Anthropic). Version notes (Feld „Additional notes“) Scale placement corrected: Δ ≃ ℓ_s⁻¹ (≈ 5 meV), with v ≈ 40–50 meV from the matching lemma and the band, replacing 16–34 meV. The power-law tail of the condensate profile below the Thomas–Fermi edge is recorded. The geometric fiber and the zero-mode operator of matter are condensate-independent; the matter fiber term is no longer weighted by |ρ_c|/v, which would have destroyed the family count of the generations companion. Status section opens with Confirmed (the three roles of one condensate are consistent; w = −1 compatible with present data; w ≥ −1 at the edge). Family-hosting statements aligned with the generations companion (embedding and four-dimensional family reading open). Duplicate labels removed; grades completed.