Six gravity numbers from one golden ratio.
The strong-theory companion calculator, revised 2026-07-02 after the Beltracchi audit (the audit corrected the log coefficient and the dark-energy block; the fixes are in the docstring, not hidden). Six self-contained blocks: the SI bridge from the native constants ℏ = φ⁻⁵ and G = φ⁵/π, the Regge residual scaling (now measured by the live triangulation lab), the black-hole log coefficient cRS = −(log φ)/2, the Page time of a solar-mass hole, the vacuum rung φ⁻⁵⁸⁸ against the observed dark-energy density, and the equation-of-state drift w(z). Five internal integration checks all pass, and each block carries its own tier.
ℏ = φ⁻⁵, G = φ⁵/π
Native constants map to an SI substrate length (2.86×10⁻³⁵ m), time (9.56×10⁻⁴⁴ s), and mass with no free fit. Planck-ratio consistency checks all pass.
Strictly between LQG and string
cRS = −(log φ)/2 = −0.2406. Lean proves cRS ≠ −1/2 (LQG) and cRS ≠ −3/2 (string), zero sorry. An earlier version of this script claimed −3/2; the audit caught it. Current bounds cannot discriminate the three; LISA-era ringdowns can.
Magnitude and shape
The rung n = 588 fixes today's magnitude; the J-cost kernel fixes the shape. Together they make a falsifiable (w₀, wa) pair. The exponent matches Lean (particleHorizonRungCount = 294, exponent −588 proved); the printed ratio 1.144 is this script's arithmetic on a hardcoded particle-horizon length, not a Lean number.
The dark-energy drift
The kernel predicts a thawing equation of state: w(0) = −0.9828 today, relaxing toward −1 in the past. That sign of drift (w₀ > −1 with wa < 0) is the same quadrant DESI-era fits prefer, and the pair is testable at Euclid-class precision. A confirmed freezing signature (w₀ < −1, wa > 0) kills the formulation.
Headline numbers and their tiers
| Quantity | Value | Tier |
|---|---|---|
| ℏRS | φ⁻⁵ = 0.0901699 | Definitional identity, exponent forced in Lean (GapDerivation); the SI anchor step is MODEL |
| GRS | φ⁵/π = 3.5301107 | THEOREM in RS-native units (G_eq_phi_fifth_over_pi); the SI anchor step is MODEL |
| cRS | −(log φ)/2 = −0.240606 | THEOREM (algebra, Lean) · HYPOTHESIS (as BH coefficient) |
| tPage(M☉) | 1.355×10⁶⁷ years | DERIVED (standard Hawking/Page algebra; the Lean module's t_Page_SI label differs and is flagged for repair) |
| ρΛ,RS / ρΛ,obs | 1.144 (rung n = 588) | HYPOTHESIS |
| (w₀, wa) | (−0.9828, −0.0172) | HYPOTHESIS (Euclid/DESI falsifiable) |
Run it yourself
$ python3 qg_six_derivations.py [PASS] All consistency checks passed. No fit anywhere. RESULT: c_RS = -(log phi)/2 = -0.240606 RESULT: rho_Lambda,RS / rho_Lambda,obs = 1.144 RESULT: (w0, wa) = (-0.9828, -0.0172) ALL INTEGRATION CHECKS PASSED
Mixed tiers by block; the table above carries them, and any combined claim takes the
weakest. D1 splits: G = φ⁵/π is a THEOREM in RS-native units
(G_eq_phi_fifth_over_pi) and ℏ = φ⁻⁵ is a definitional identity whose
exponent is forced in GapDerivation; only the SI anchor step is MODEL, and none of it is a prediction. D2's
h⁴ scaling is MEASURED on the
Regge lab; this script only reprints the power-counting companion. D3's algebraic
value is THEOREM in Lean; identifying it with the physical black-hole log coefficient is HYPOTHESIS, and no
current observation discriminates it from LQG or string values. D5's exponent −588 matches Lean, but the
ratio 1.144 is this script's own arithmetic on a hardcoded particle-horizon length, not a Lean number (the
Lean docstring's particle-horizon variant lands near 1.00); the residual is reported, not absorbed. Lean also
carries a parallel THEOREM-grade static track, ΩΛ = 11/16 − α/π
≈ 0.684 (inside Planck 2σ), which this lab does not run. D6 names its falsifier: a freezing
(w₀ < −1, wa > 0) signature at 5σ.