V2.293 - Golden Ratio Universality Test
V2.293: Golden Ratio Universality Test
Motivation
V2.292 found α_B/α_s = φ (golden ratio) to 0.033% at n=20. This experiment tests:
- Does the match improve at larger n (up to n=40)?
- Is φ universal across dimensions (D=3 vs D=4)?
- How does the ratio depend on angular cutoff C?
Key Results
1. D=4: The Ratio Passes Through φ at n ≈ 32
| n | α_B/α_target | Dev from φ |
|---|---|---|
| 8 | 1.625157 | +0.440% |
| 12 | 1.621216 | +0.197% |
| 16 | 1.619714 | +0.104% |
| 20 | 1.618965 | +0.058% |
| 25 | 1.618450 | +0.026% |
| 30 | 1.618148 | +0.007% |
| 35 | 1.617952 | −0.005% |
| 40 | 1.617817 | −0.013% |
The ratio crosses φ = 1.618034 between n=30 and n=35. At n=30, the match is 0.007% — one part in 14,000. This is the closest approach to φ seen in any lattice computation in this program.
Convergence fits (R(n) = R_∞ + c/n^p):
| Model | R_∞ | Dev from φ | R² |
|---|---|---|---|
| 1/n | 1.61560 | −0.150% | 0.976 |
| 1/n² | 1.61769 | −0.021% | 0.997 |
| 1/n^p (p=1.72) | 1.61734 | −0.043% | 0.99998 |
The best fit (R² = 0.99998) gives R_∞ = 1.61734, which is 0.043% below φ. This is within the systematic uncertainty from C-convergence.
2. D=3: The Ratio is NOT φ — Golden Ratio is NOT Universal
| n | D=3 ratio α_B/α_W |
|---|---|
| 8 | 1.427 |
| 16 | 1.442 |
| 25 | 1.445 |
| 30 | 1.446 |
D=3 extrapolated ratio: ~1.447, which is 10.5% below φ. The golden ratio does NOT appear in D=3.
This means: if φ appears in D=4, it is dimension-specific, not a universal property of tridiagonal matrices or entanglement entropy.
3. C-Dependence
At fixed n=16, the ratio α_B/α_W is nearly C-independent:
| C | α_B/α_W |
|---|---|
| 4 | 1.590 |
| 6 | 1.597 |
| 8 | 1.600 |
| 10 | 1.601 |
The B/W ratio varies by only 0.7% across C=4–10, making it a robust intrinsic property of the lattice.
However, the ratio α_B/α_target depends on C because α_target = 1/(24√π) is the double limit (n→∞, C→∞):
| C | α_B/α_target (n=12) |
|---|---|
| 8 | 1.621 |
| 10 | 1.648 |
| 12 | 1.663 |
| C→∞ | ~1.696 |
This suggests the true C-converged ratio might be HIGHER than φ. But since α_W also increases with C (approaching the true α_s from below), the double limit (C→∞, n→∞) of α_B/α_target requires simultaneous extrapolation in both variables.
4. Assessment of the φ Conjecture
Evidence FOR:
- At C=8: ratio passes through φ at n≈32 with 0.007% precision
- The crossing is smooth and monotonic (not an oscillation)
- Best 1/n² fit gives 0.021% from φ
- The B/W ratio is C-stable, suggesting intrinsic structure
Evidence AGAINST:
- Best unconstrained fit (1/n^1.72) gives 0.043% below φ — slight tension
- C-convergence analysis suggests the true ratio might be higher than φ at C→∞
- The ratio CROSSES φ (from above to below), meaning R_∞ < φ unless there are higher-order corrections
- D=3 gives 1.447, not φ — the golden ratio is NOT universal
Verdict: The D=4 data at C=8 is tantalizingly close to φ (within 0.04% at best), but the evidence does not CONFIRM φ. The crossing at n≈32 means the n→∞ limit at C=8 is slightly below φ. Whether the full double limit (n→∞, C→∞) equals φ remains open.
Physical Interpretation
D=4 vs D=3: Why Different?
In D=4, the entangling surface is S² with angular decomposition using spherical harmonics Y_lm (degeneracy 2l+1). In D=3, the surface is S¹ with Fourier modes e^{imθ} (degeneracy 2 for m≠0). The correction factor depends on the angular structure:
- D=4: weighted by l(2l+1) (quadratic growth) → ratio ≈ 1.617
- D=3: weighted by m (linear growth) → ratio ≈ 1.447
The golden ratio, if it appears in D=4, would be related to the specific structure of spherical harmonic sums. The factor 24 in α_s = 1/(24√π) already connects to the D=4 angular structure (V2.232: k_4 = 24.0).
Implications for the α_s Proof
Even if the ratio is not EXACTLY φ, the decomposition α_s ≈ α_boundary/R with R ≈ 1.617–1.618 is still useful:
- α_boundary is the simpler quantity (boundary correlator integral, no Williamson decomposition needed)
- R is a universal constant (nearly C-independent, depending only on n and dimension)
- The proof could target R directly via the structure of the Williamson decomposition for the Srednicki chain
The fact that R differs between D=3 and D=4 means R encodes dimensional information. In D=4: R ≈ φ (connected to S² angular structure). In D=3: R ≈ 1.447 (connected to S¹ angular structure).
Honest Assessment
Achieved:
- Pushed D=4 ratio to n=40 with 0.007% match at n=30
- Definitively showed φ is NOT universal — D=3 gives 1.447
- Demonstrated C-stability of B/W ratio (0.7% variation over C=4–10)
- Best convergence fit: R² = 0.99998 with R_∞ = 1.6173
Not achieved:
- Cannot confirm or reject R = φ at current precision (0.04% ambiguity)
- C-convergence introduces systematic uncertainty that isn’t fully resolved
- No analytical understanding of WHY R ≈ 1.617 in D=4
For the overall science:
The golden ratio hypothesis from V2.292 is neither confirmed nor definitively rejected. The D=4 data shows a striking near-miss at 0.04%, which could be φ with higher-order corrections or a nearby but distinct constant. The key new result is that the ratio is dimension-specific: D=3 gives a completely different value, ruling out a universal mechanism. If the ratio IS φ in D=4, it must arise from the specific angular structure of spherical harmonics, not from general properties of tridiagonal matrices.
Open question: Does the double limit (n→∞, C→∞) of α_B/α_s converge to φ? This requires C ≥ 12 with n ≥ 40, which is computationally intensive but feasible.