Unified Formula — Relativity + Quantum Mechanics¶
Contents¶
- the complete equation — below on this page
scripts/dvpt/dvpt_unified_formula_test.py— numerical verification
The formula in three lines¶
ds² = -f_k(r) dt² + dr²/f_k(r) + r² dΩ²
f_k(r) = 1 - 2GMr / (r² + G/(π·α(k,β)))
α(k,β) = W(E·e^(E+c_k)) / E
where: E(β) = -ζ'(β)/ζ(β) + (k+1)·ζ'((k+1)β)/ζ((k+1)β) c_k = 1 - ln ζ(k+1) W = Lambert W function
How to test it¶
Run python scripts/dvpt/dvpt_unified_formula_test.py (or paste it into Colab). Each test asserts its tolerance, so the final
"ALL TESTS PASSED" line is printed only if every check holds.
Expected output: - TEST 1: sech as ground state (μ=1, ε=0), corr > 0.9999 - TEST 2: E(β), c_k, S_k consistent - TEST 3: Lambert W = brentq (10^-16) - TEST 4: commutation S_J - S_k = 0 (10^-15); α is defined by this condition, so this is a consistency check of the Lambert-W solution - TEST 5: metric f_k(r) differs for each k - TEST 6: α_∞(k) = e/ζ(k+1) (10^-15)
What it means¶
- Relativity: the metric f_k(r) is modified Schwarzschild
- Quantum: H(μ,ε) is the Schrödinger operator
- Bridge: α(k,β) connects gas and metric via Lambert W
- Price: the metric depends on k, it is not universal
References¶
- Jusufi & Anand (2026) — entropy-geometry metric
- Gentile (1940) — intermediate statistics
- Bakas & Bowick (1991) — arithmetic gases
- Corless et al. (1996) — Lambert W function
License¶
UNIFIED FORMULA — Relativity + Quantum Mechanics¶
THE EQUATION¶
┌─────────────────────────────────────────────────────────┐
│ │
│ ds² = -f_k(r) dt² + dr²/f_k(r) + r² dΩ² │
│ │
│ f_k(r) = 1 - 2GMr / (r² + G/(π·α(k,β))) │
│ │
│ H(μ,ε) ψ = E₀ ψ │
│ H = -d²/dr² + ε·cos(2πr/Φ) - 2μ·sech²(r) │
│ │
│ α(k,β) = W(E·e^(E+c_k)) / E │
│ E(β) = -ζ'(β)/ζ(β) + (k+1)·ζ'((k+1)β)/ζ((k+1)β) │
│ c_k = 1 - ln ζ(k+1) │
│ │
│ MATCHING: r² = α(k,β)·E(β)/π │
│ │
└─────────────────────────────────────────────────────────┘
THE 5 PIECES¶
1. Relativistic side (metric)¶
f_k(r) = 1 - 2GMr / (r² + ℓ_ζ²)
ℓ_ζ² = G / (π·α(k,β))
2. Quantum side (Schrödinger)¶
H(μ,ε) = -d²/dr² + ε·cos(2πr/Φ) - 2μ·sech²(r)
Limits:
- μ=1, ε=0 → Pöschl-Teller, ψ₀ = sech(r), E₀ = -1
- μ=0 → pure Aubry-André
3. The Gentile gas¶
Z_k(β) = ζ(β) / ζ((k+1)β)
E(β) = -ζ'/ζ + (k+1)·ζ'((k+1)β)/ζ((k+1)β)
S_k(E) = E + ln E + c_k, c_k = 1 - ln ζ(k+1)
4. The map Φ (Lambert W)¶
α(k,β) = W(E·e^(E+c_k)) / E
Asymptotic: α(k, β→∞) = e^(c_k) = e/ζ(k+1)
Values:
k=1 → α_∞ = e/ζ(2) = 1.6525171940
k=2 → α_∞ = e/ζ(3) = 2.2613586938
k=∞ → α_∞ = e = 2.7182818285
5. Commutation¶
S_J(A) = A/(4G) + ln(A/(4G))
Condition: S_J(αE) = S_k(E)
→ uniquely determines α(k,β)
Note on α: here α(k,β) is the energy→area map, A/(4G) = α·E. It is not the α of Jusufi & Anand (arXiv:2608.23528), which is the coefficient of ln(A/4G) in the entropy (fixed to 1 by the pole of ζ at β = 1) and enters their metric as ℓ² = α·G/π. With their map E = A/(4G) the metric is the same for every k (DVPT Gentile supplement); the k-dependence below comes from choosing the map S_J(αE) = S_k(E), which absorbs c_k.
VERIFICATION¶
α_∞(k) = e/ζ(k+1) verified to 10^-15 for k = 1,2,3,5,10,20,50,100
Commutation verified to 10^-15 (consistency check: α is defined by it)
Lambert W = brentq verified to 10^-16
Schrödinger sech verified to 10^-4 (μ=1, ε=0)
THE PRICE¶
You cannot have simultaneously: - a universal metric (independent of k) - an exactly commuting diagram
The obstruction is c_k = 1 - ln ζ(k+1).
ℓ²(k) = G·ζ(k+1) / (π·e)
k=1 → ℓ² = 0.1926 G
k→∞ → ℓ² = 0.1171 G
CONSTANTS¶
α_G = 5.905e-39 (gravitational scale ratio)
1/α_G = 1.69e38 (number of fractures)
E_crack = α_G · E_spike