13 July 2026. A gravity/cosmology companion, alongside the particle-sector docs.
TFT’s gravity is emergent geometry sourced by the phase field’s energy (the 1/r shape and universal sign are derived; the toy solar system and Mercury’s 42.9″ follow). Two further derived facts seed a black hole: √(2GM/r) is the inflow rate of space toward a mass (reaching c at the Schwarzschild radius), and time is the turning of the phase (dτ = ℏ dθ/E). Reading those literally gives a complete, and in places distinctive, black-hole picture — each piece a consequence of the same substrate.
Labels: [derived] · [computed] · [consistency] (matches GR) ·
[structural] · [floor] (an absolute value the U(1) field does not fix).
Pre-registrations: BH0_, BHB0_, BHE0_.
bh_study.py)The horizon is where space’s inflow overtakes light [derived route]. The derived inflow rate v = √(2GM/r) equals c exactly at r_s = 2GM/c². Inside, space falls in faster than phase can propagate out, so outgoing light is dragged inward — a horizon, obtained from TFT’s own inflow rate rather than imposed (the river / Gullstrand–Painlevé picture). The location r_s matches GR [consistency].
Time freezes at the horizon, literally [structural]. Since time is phase cycling, the rate of time ∝ √g₀₀ = √(1−r_s/r) → 0 at r_s. An outside observer watches infalling phase freeze at the horizon — the old “frozen star” made exact: the phase-cycling that is time stops there.
bh_study.py)The distinctive departure from GR. The phase field is bounded: the phase gradient cannot exceed ~one turn per coherence length (|∇θ| ≲ 1/ℓ₀) and the amplitude is finite, so the energy density cannot diverge — it caps at ~Planck density. The mass therefore sits in a regular core, not a point singularity: for a solar mass, r_core ≈ 4.5×10⁻²³ m — about 3×10¹² Planck lengths across (a real, extended object, not a Planck point), buried ~10⁻²⁶ of the way in from the horizon. A TFT black hole is a horizon wrapped around a Planck-density, phase-frozen core — in the regular-black-hole / Planck-star / gravastar family, but here the singularity resolution comes specifically from the phase field’s boundedness. [TFT-native prediction]
bh_bounce.py)That core is not static — it bounces. Taking the framework’s own stable particle (the φ⁶ Q-ball), squeezing it out of equilibrium, and evolving the full field equations, the core density peaks and re-expands, oscillating and staying finite — it breathes rather than collapsing [computed]. The mechanism is the same boundedness: the potential is repulsive at high density (dV/d(ρ²) climbs from +0.17 to +1.13 as it compresses) — a field “degeneracy pressure” that forbids collapse to a point [derived]. No ad-hoc quantum gravity: the same boundedness that removes the singularity drives the bounce.
An observable follows. In the core’s proper time the bounce is fast (~t_Planck), but the horizon’s extreme time dilation stretches it into an external delay ~(M/m_P)² t_P (the Rovelli–Vidotto Planck-star scaling, here motivated by the TFT bounce). Working it through, a primordial black hole of ~6×10²² kg would be completing its bounce now — arriving as a short high-energy burst. The exact mass is scaling-dependent (~10¹¹–10²⁴ kg by the assumed bounce law), so this is a candidate signal, not a hard number — but the structure is real: TFT black holes are delayed rebounders, not eternal sinks. [derived scaling; observable model-dependent]
bh_entropy.py)The deepest piece, split honestly.
The area law is derived and TFT-native [computed]. The puzzle of black-hole entropy is that it scales with the horizon area, not volume. The reason, computed here: the entropy is the entanglement entropy of the phase field’s massless Goldstone across the horizon (Srednicki’s mechanism, applied to TFT’s own field). A radial-lattice calculation — build the Goldstone’s vacuum, trace out the interior, sum the entanglement over angular momenta — gives S ∝ R^1.9, the area law, definitively not the volume law (R³). The field’s vacuum correlations are short-ranged, so only boundary-hugging modes contribute → entropy tracks area. The holographic surprise falls out of the phase field’s ground-state entanglement, parameter-free.
The coefficient ¼ is located, but not conjured [structural / floor]. In induced gravity (Sakharov, Jacobson, Susskind–Uglum — TFT’s lineage), the same field fluctuations that give S_ent = C·A/ε² also induce 1/G = C′/ε²; the Planck-scale cutoff divides out of the ratio, so S_ent = A/4G. The ¼ is inherited and tied to G by the one field — not a free fit — but pinning it to the last digit needs the phase field’s exact induced-G spectrum. That is the strain point, the direct analog of the Immirzi parameter in loop quantum gravity or the microstate count in string theory: a constrained floor. The whole coefficient reduces to one number — entropy per Planck cell = ¼ nat — which reproduces S ~ 10⁷⁷ for a solar mass and S ∝ M² exactly [consistency].
a0_de_study.py)TFT’s dark energy is the same ultralight phase field, with the sine-Gordon cosine potential — a pseudo-Nambu-Goldstone thawing quintessence, an ordinary scalar, so w ≥ −1 always: it cannot cross into phantom (w < −1) [derived]. The field mass that gives a₀ ∝ cH₀ (~H₀) makes it just-thawing now; matched to the observed w₀ = −0.88 it predicts wₐ ≈ −0.20 (updated 15 Aug 2026; supersedes −0.24) with the field mass ~0.72 H₀ — one field for both a₀ and the equation of state [computed]. DESI’s w₀wₐCDM fit prefers a phantom crossing (w < −1 in the past), which a thawing scalar cannot produce. The falsifier is sharp — and as of DESI DR2 (2025) the tension runs against the prediction: the evolving-DE preference firmed from 2.6σ (DR1, DESI+CMB) to 3.1σ (DR2), 2.8–4.2σ with supernovae — refreshed 7 Aug 2026 to 2.7σ (DESI+CMB) / 3.2σ (+SNe) by DR2 Results IV with the full-shape Lyman-α forest (arXiv:2607.27410), a softening that does not change the direction (every parameterisation still has w₀+wₐ < −1, i.e. a crossing) — and the DR2 best fit (w₀ = −0.838 ± 0.055, wₐ = −0.62) crosses w = −1 in the past, with DESI reporting non-phantom models disfavoured. TFT and DESI agree that dark energy evolves (both disfavour ΛCDM); they part on the magnitude of wₐ. The prediction w ≥ −1 stands and the current data do not meet it; if the crossing hardens with DR3 / Euclid, TFT’s dark-energy sector is excluded.
| statement | label |
|---|---|
| horizon at r_s from TFT’s own inflow rate reaching c | derived route (r_s = consistency) |
| time (= phase cycling) freezes at the horizon | structural |
| no singularity — a regular Planck-density core (bounded phase field) | derived (TFT-native) |
| the core bounces (φ⁶ degeneracy pressure); Planck-star burst | computed (observable model-dependent) |
| black-hole entropy ∝ area, from the Goldstone’s entanglement | computed |
| S = A/4G — the ¼ is the induced-gravity coefficient, tied to G | structural (constrained) |
| dark energy cannot go phantom (w ≥ −1) — falsifier vs DESI | derived (falsifiable) |
| exact ¼; absolute scales (ℓ₀=ℓ_P?, Λ_cc, H₀); Kerr; full GR bounce | floor / open |
TFT reproduces GR where GR is tested (r_s, the thermodynamic scales) and adds distinctive, in-principle-testable structure where GR breaks down — no singularity, a bouncing core, and a first-principles area law for the entropy. What it does not do is conjure the absolute coefficients (the exact ¼, the Planck scale, Λ_cc) from nothing — those are the framework’s recurring floors, open here as everywhere. Deriving the area law while honestly leaving the ¼ as a constrained floor is the state of the art; no framework does better without a tunable input.
pip install numpy
mkdir -p outputs
python3 bh_study.py # horizon (river model), frozen time, singularity-free core
python3 bh_bounce.py # the core bounces (Planck-star burst scaling)
python3 bh_entropy.py # entropy area law from the Goldstone's entanglement
python3 a0_de_study.py # dark energy cannot go phantom (DESI falsifier)
Inputs: physical constants and the repo field only. Each script prints its
pre-registered gate and verdict; JSON lands in outputs/.