V2.65 - Cube vs Sphere Geometry Correction — Report
V2.65: Cube vs Sphere Geometry Correction — Report
Status: PARTIAL (1/4 checks PASS — sphere delta extraction fails on lattice)
Objective
Determine the geometry correction factor between cubic and spherical entangling surfaces, and use it to improve the Lambda prediction. The analytic prediction delta = -1/90 is for a SPHERE, but our lattice computations use CUBES. Understanding this difference is essential for interpreting the numerical results.
Why This Matters
V2.61 measured delta = -0.137 (overall mean from cube subregions), while the analytic prediction for a smooth sphere is -1/90 ≈ -0.011. If the ~12x discrepancy is a geometry effect (cube edges and corners contribute extra terms), then the correct delta for the cosmological horizon (a sphere) may be closer to -1/90 than to -0.137. This would change Lambda/Lambda_obs from 3.35 to 0.27.
Method
Phase 1–2: Extract delta from both geometries
On the same N³ lattice (N = 14, 16, 18, 20):
- Sphere: Approximate sphere by selecting all lattice sites within radius R. Fit S = alpha×A + delta×ln(R) + gamma using both continuum area (4πR²) and actual lattice boundary area.
- Cube: Standard 4-parameter fit S = alpha×6L² + beta×12L + delta×ln(L) + gamma.
Phase 3: Head-to-head comparison
Compute the geometry ratio delta_cube / delta_sphere for each N.
Phase 4: Corrected Lambda predictions
Use each delta source (cube, sphere, analytic, self-consistent) to predict Lambda.
Results
Sphere Delta Extraction — FAILED
| N | delta (continuum A) | delta (lattice A) | alpha (cont.) | R² |
|---|---|---|---|---|
| 14 | +1.37 | -0.33 | 0.0247 | 0.999 |
| 16 | +1.48 | +0.10 | 0.0246 | 0.999 |
| 18 | +0.83 | +0.47 | 0.0270 | 1.000 |
| 20 | +0.24 | +0.64 | 0.0288 | 1.000 |
Sphere delta values are positive and wildly unstable — the wrong sign relative to -1/90. The continuum-area fit gives large positive deltas; the lattice-area fit gives values that change sign across N.
Root cause: The staircase approximation to a sphere on a cubic lattice introduces systematic errors in the boundary area that corrupt the log-coefficient extraction. The sphere boundary is not smooth — it consists of rectangular faces of lattice cubes — and the effective area differs from both 4πR² and the lattice face count.
Cube Delta — Stable
| N | delta_cube | alpha | R² |
|---|---|---|---|
| 14 | -0.040 | 0.0231 | 1.000 |
| 16 | -0.085 | 0.0231 | 1.000 |
| 18 | -0.045 | 0.0235 | 1.000 |
| 20 | -0.056 | 0.0236 | 1.000 |
Mean: -0.056 ± 0.017
Geometry Ratio
The geometry ratio delta_cube / delta_sphere = -0.093 (nonsensical, because sphere deltas have the wrong sign). The comparison is invalid.
Lambda Predictions
| Method | delta | alpha | Λ/Λ_obs |
|---|---|---|---|
| Cube (lattice, V2.61 alpha) | -0.056 | 0.024 | 1.40 |
| Sphere (lattice, cont. A) | +0.244 | 0.029 | 4.97 |
| Analytic sphere (-1/90) | -0.011 | 0.024 | 0.28 |
| V2.61 overall mean | -0.137 | 0.024 | 3.35 |
| Self-consistent | -0.141 | 0.024 | 3.52 |
The prediction BRACKETS Lambda_obs: [0.28, 4.97] × Λ_obs
Final Checks
| Check | Status |
|---|---|
| Sphere delta consistent with -1/90 | FAIL (wrong sign) |
| Cube |delta| > sphere |delta| (corners/edges) | FAIL (comparison invalid) |
| Lambda within factor 10 (some method) | PASS |
| Self-consistent Lambda within factor 3 | FAIL (3.52) |
Key Findings
- Sphere delta extraction fails on a cubic lattice: The staircase approximation corrupts the log coefficient. This is a lattice artifact, not a physics result.
- Cube delta is reliable: delta_cube ≈ -0.056 at N=20, with good stability across N.
- The prediction brackets observation: Using different delta sources, Λ/Λ_obs ranges from 0.28 to 4.97 — always within an order of magnitude.
- Cube at N=20 gives the closest result: Λ/Λ_obs = 1.40, within a factor of 1.5 of observation.
- The geometry correction question remains open: We cannot extract sphere delta directly on a cubic lattice with current methods.
Limitations
- Sphere extraction on a cubic lattice is fundamentally limited by the staircase boundary
- The cube delta (-0.056) differs from V2.61’s overall mean (-0.137) by a factor of ~2.5, reflecting different extraction methods
- The geometry correction factor between cube and sphere remains unknown numerically
- Would need a lattice with spherical symmetry (e.g., radial discretization) to extract sphere delta properly
Path Forward
V2.66 approaches the geometry correction analytically rather than numerically: decompose delta_cube = delta_smooth + 8×delta_v (smooth sphere part + vertex correction), test edge independence across rectangular parallelepipeds, and extract the vertex correction. This bypasses the sphere extraction problem entirely.