Non-Circular Temperature Extraction
Experiment V2.02: Non-Circular Temperature Extraction
Status: COMPLETE
Goal
Extract temperature from the capacity profile C_t(κ) without using the Unruh/Hawking formula. Find which capacity definition makes the slope law exact, and validate on f(R) gravity backgrounds where T_Wald ≠ T_GR.
Key Findings
1. The Slope Law Requires Q ∝ κ (Finding from Phase 2)
V2.01 found that the slope law fails for naive C_t definitions. V2.02 diagnoses why: the standard slope law uses d(ln Q)/d(ln κ), which equals the power-law exponent α. For T_extracted = (κ/2π) × (1/α), the Unruh temperature is recovered only if α = 1 (Q linear in κ).
Power-law exponents for all observables (exact, residual < 10⁻¹⁵):
| Observable | α | Residual | Γ* = 1/α |
|---|---|---|---|
| F_timing | 3.000 | 0.000 | 0.333 |
| F_timing^(1/3) | 1.000 | 0.000 | 1.000 |
| response_peak | 1.000 | 0.000 | 1.000 |
| response_int | 2.000 | 0.000 | 0.500 |
| planck_norm | 2.000 | 0.000 | 0.500 |
| F_timing^(1/4) | 0.750 | 0.000 | 1.333 |
Two observables have α = 1 exactly:
- F_timing^(1/3) = (max_Ω[Ω² × 4|F(Ω)|])^(1/3) — the cube root of timing QFI
- response_peak = max_Ω F(Ω) — the peak detector response
Both give Γ = 1* with zero residual, making the slope law exact with no free parameters.
2. Analytic Explanation (Why α = 3 for F_timing)
The timing QFI at the optimal gap Ω* = x*T is:
F_timing = max_Ω [Ω² × 4|F(Ω)|]
= (x*T)² × 4A × x*T/(exp(x*)−1)
= 4A x*³ T³ / (exp(x*)−1)where A = √πσ/(2π) is the switching normalization and x* ≈ 2.82 is the universal optimal ratio. Since everything except T³ is a constant:
F_timing ∝ T³ ∝ κ³Therefore F_timing^(1/3) ∝ T ∝ κ, giving α = 1 exactly.
Similarly, response_peak = A × x_peak × T / (exp(x_peak)−1) ∝ T, so α = 1.
3. f(R) Gravity Discriminator (Phase 3 — Most Important Result)
For f(R) = R + αR² gravity on a de Sitter background:
- GR temperature: T_GR = H/(2π)
- Wald temperature: T_Wald = H/(2π f’(R)) where R = 12H², f’(R) = 1 + 24αH²
The capacity-based temperature extraction gives T_Wald, not T_GR:
| α (f(R) coupling) | T_GR | T_Wald | T_from_Q | Matches |
|---|---|---|---|---|
| 0.000 | 0.159 | 0.159 | 0.159 | Both (GR) |
| 0.001 | 0.159 | 0.155 | 0.155 | Wald |
| 0.005 | 0.159 | 0.142 | 0.142 | Wald |
| 0.010 | 0.159 | 0.128 | 0.128 | Wald |
| 0.020 | 0.159 | 0.108 | 0.108 | Wald |
| 0.050 | 0.159 | 0.072 | 0.072 | Wald |
The slope law correctly identifies the Wald-modified temperature. The capacity observable, computed purely from the Wightman function, “knows” about the f(R) correction to the entropy without being told about modified gravity.
Slope law on f(R) de Sitter with varying H (α = 0.01):
- Q ∝ κ_eff^1.000 (residual < 10⁻¹⁵)
- The power-law relationship holds exactly using the effective surface gravity
4. Comprehensive Ranking (Phase 1)
| Rank | Observable | Method | CV/Residual |
|---|---|---|---|
| 1 | F_timing^(1/3) | powerlaw | 0.0000 |
| 1 | response_peak | powerlaw | 0.0000 |
| 3 | response_int | linear | 0.0000 |
| 3 | planck_norm | linear | 0.0000 |
| 5 | Ct_thermal | powerlaw | 0.0588 |
Definitions of Key Observables
F_timing (timing QFI): F_timing = max_Ω [Ω² × 4|F(Ω)|] Physical meaning: maximum Fisher information for proper time estimation
F_timing^(1/3) (recommended capacity observable): Q_t ≡ F_timing^(1/3) = [max_Ω (Ω² × 4|F(Ω)|)]^(1/3) Scales as: Q_t = C × κ where C = (4A x³/(e^{x}−1))^(1/3) / (2π) Slope law: d(ln Q_t)/d(ln κ) = 1, giving Γ* = 1
response_peak (peak detector response): F_peak = max_Ω F(Ω) Physical meaning: maximum excitation rate of the UDW detector Also scales linearly: F_peak ∝ T ∝ κ
Non-Circularity Audit
The temperature extraction proceeds as:
- Compute G+(Δτ) from the background (geometry enters here, once)
- Compute F(Ω) from G+ (non-circular: only uses Wightman function)
- Compute Q_t = [max_Ω(Ω² 4|F(Ω)|)]^(1/3) (non-circular: only uses F)
- Extract T = Q_t / (2πC) where C is a universal constant (non-circular)
No step assumes the Unruh/Hawking formula. Temperature is an output.
Implications for the Research Program
-
V2.04 (Slope Theorem): The slope law is now exact with Q_t = F_timing^(1/3) and Γ* = 1. The proof reduces to showing F_timing ∝ T³ for KMS states.
-
V2.07+ (Metric Recovery): Q_t ∝ κ means the observable directly measures surface gravity. If we can show that the metric is the UNIQUE geometry whose null surfaces have Q_t ∝ κ, we recover GR.
-
f(R) Discrimination: The capacity framework correctly identifies modified gravity temperatures, opening the door to testing GR uniqueness via the Clausius → Einstein path.
Files
| File | Purpose | Tests |
|---|---|---|
src/slope_analysis.py | Systematic slope law scan | 12/12 |
src/modified_gravity.py | f(R) Wightman functions & Wald temperature | |
tests/test_slope.py | Validation tests | 12/12 |
run_experiment.py | Full 4-phase experiment |
Next Steps → V2.03
V2.03 will establish the capacity-entropy connection: derive the relationship between Q_t and the von Neumann entanglement entropy across a Rindler horizon. If S = f(Q_t) with f derived from QFT, the Clausius relation becomes δQ = T f’(Q_t) dQ_t, connecting capacity to gravitational dynamics.