V2.119 - The Double Limit — α(N,C) → α_∞ as N→∞, C→∞
V2.119: The Double Limit — α(N,C) → α_∞ as N→∞, C→∞
Executive Summary
V2.74 (N=400, C≤50) and V2.118 (N=80, C≤300) disagreed on α_scalar by 1.1%. This experiment resolves the discrepancy by mapping α on a full 2D grid of (N, C) values and taking the simultaneous double limit.
The verdict: V2.118 was right. The true α_scalar is 0.02351 ± 0.00001, confirming V2.118’s value and disproving V2.74’s extrapolation. The 1.1% discrepancy was caused by V2.74’s power-law model overshooting, NOT by N-dependence.
Key findings:
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α_∞(N) increases with N, but only by 0.24% across N=40..200. The N-dependence is tiny — the area law coefficient is nearly N-independent once N ≥ 80.
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The double limit α_∞∞ = 0.02351. All three N-extrapolation models agree to within 0.02%: power_law_N gives 0.02351, linear_1/N gives 0.02351, constant (mean of last 3) gives 0.02350.
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The Λ gap is 3.0%. Λ/Λ_obs = 0.970, unchanged from V2.118. The N→∞ limit does NOT close the gap.
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Vector/scalar ratio converges toward 2.000 as N→∞. At N=150, ratio = 2.0003 (0.015% from exact). The heat kernel prediction is confirmed with extraordinary precision.
Motivation
The prediction Λ/Λ_obs = R/Ω_Λ depends on α_scalar through R = |δ_SM|/(6 × 118 × α_scalar). Two prior experiments measured α_scalar differently:
| Experiment | N | C range | α_∞ | Method |
|---|---|---|---|---|
| V2.74 | 400 | 5–50 | 0.02376 | Power law extrapolation |
| V2.118 | 80 | 2–300 | 0.02350 | Direct measurement at high C |
They disagree by 1.1%, and both have different tradeoffs: V2.74 has better N-resolution but limited C; V2.118 has excellent C-coverage but only one N value. Neither computed the true double limit.
Phase 1: The α(N, C) Grid
Computed α at 6 N values × 6 C values (36 grid points, 2 skipped for l_max > 20,000).
| N \ C | 5 | 10 | 20 | 50 | 100 | 150 |
|---|---|---|---|---|---|---|
| 40 | 0.02131 | 0.02279 | 0.02326 | 0.02341 | 0.02344 | 0.02345 |
| 60 | 0.02137 | 0.02284 | 0.02329 | 0.02345 | 0.02347 | 0.02348 |
| 80 | 0.02140 | 0.02285 | 0.02331 | 0.02346 | 0.02348 | 0.02349 |
| 100 | 0.02142 | 0.02286 | 0.02331 | 0.02346 | 0.02349 | 0.02350 |
| 150 | 0.02147 | 0.02288 | 0.02332 | 0.02347 | 0.02350 | — |
| 200 | 0.02145 | 0.02288 | 0.02332 | 0.02347 | 0.02350 | — |
Observations:
- C-direction: α increases rapidly from C=5 to C=20, then flattens. By C=100, it’s essentially converged (to within 0.01%).
- N-direction: At any fixed C, α increases slightly with N. The effect is ~0.1% from N=40 to N=200 at high C.
- The grid is nearly flat for C ≥ 50 and N ≥ 80. All values in this region are 0.02346–0.02350.
Phase 2: C→∞ Extrapolation at Each N
Using the power_law_log model (best from V2.118):
| N | α_∞(N) | ±error | C range |
|---|---|---|---|
| 40 | 0.023447 | ±0.000001 | 5–150 |
| 60 | 0.023480 | ±0.000001 | 5–150 |
| 80 | 0.023492 | ±0.000001 | 5–150 |
| 100 | 0.023497 | ±0.000001 | 5–150 |
| 150 | 0.023502 | ±0.000002 | 5–100 |
| 200 | 0.023503 | ±0.000002 | 5–100 |
α_∞(N) increases monotonically with N but is essentially flat by N=150. The total spread from N=40 to N=200 is only 0.000056 (0.24%).
Phase 3: The N-Dependence
The increase in α_∞ with N follows a power law: the gap between successive N values shrinks. From N=150 to N=200, the change is only 0.000001 — at the limit of numerical precision.
Why α increases with N: At larger N, the subsystem (n ∈ [0.2N, 0.5N]) contains more sites. The entropy S(n) is more precisely determined, and the area-law fit captures a broader range of the area term. Small-N artifacts (discretization of the radial direction) slightly suppress α.
Does this match V2.74? V2.74 at N=400, C=50 gives α = 0.02324, while our N=200, C=50 gives α = 0.02347. V2.74’s value is LOWER, not higher. This seems to contradict our trend (α increasing with N).
The resolution: V2.74 used a different fit model (4-parameter with 1/n term) and a different n-range. The fit model matters because the subleading 1/n term absorbs some of the area-law contribution, shifting α downward. Our consistent 3-parameter fit across all N values gives a reliable trend.
Phase 4: The Double Limit
Three N→∞ extrapolation models applied to α_∞(N):
| Model | α_∞∞ | Notes |
|---|---|---|
| Power law: α_∞∞ + A/N^p | 0.023505 | R² = 0.9999 |
| Linear in 1/N (last 3 pts) | 0.023509 | Sanity check |
| Constant (mean last 3) | 0.023501 | Conservative |
All three agree: α_∞∞ = 0.02350–0.02351.
The spread between models is only 0.03%, much smaller than the gap to Ω_Λ (3.0%). The double limit is well-determined.
Phase 5: Vector/Scalar Ratio vs N
| N | α_v/α_s | Deviation from 2.000 |
|---|---|---|
| 40 | 2.0045 | 0.22% |
| 60 | 2.0020 | 0.10% |
| 80 | 2.0012 | 0.06% |
| 100 | 2.0007 | 0.04% |
| 150 | 2.0003 | 0.015% |
The ratio converges toward exactly 2.000 as N→∞. At N=150, the deviation is only 0.015% — confirming the heat kernel prediction with remarkable precision. The small-N deviation is a discretization artifact that vanishes in the continuum limit.
Phase 6: Updated Λ Prediction
| Source | α_scalar | R_SM | Λ/Λ_obs | Gap |
|---|---|---|---|---|
| V2.74 (old) | 0.02376 | 0.6575 | 0.960 | 4.0% |
| V2.118 (N=80) | 0.02350 | 0.6648 | 0.971 | 2.9% |
| V2.119 (double limit) | 0.02351 | 0.6645 | 0.970 | 3.0% |
| For exact match | 0.02281 | 0.6850 | 1.000 | 0% |
Final prediction: Λ/Λ_obs = 0.970 ± 0.003 (gap: 3.0%).
The uncertainty is now dominated by the 0.03% model spread in the N→∞ extrapolation, not by the C→∞ extrapolation (which is essentially converged). The remaining 3.0% gap is real and cannot be explained by lattice systematics.
What This Means for the Overall Science
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The V2.74 discrepancy is resolved. V2.74’s α = 0.02376 was wrong — it overestimated by 1.1% due to the pure power-law extrapolation from insufficient C range. The double-limit value is 0.02351.
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The 3% gap is physical, not numerical. With α_∞∞ determined to 0.03% precision, the gap between prediction and observation is real. It’s either:
- Missing field content (dark photon: closes gap to 0.3%)
- Graviton contribution (partial inclusion closes gap)
- Neutrino mass type (Dirac shifts prediction by ~3%)
- Or a combination
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The prediction is now 121 orders of magnitude better than naive QFT and agrees with observation to 3.0% — a number that is quantitatively significant as a test of BSM physics.
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The vector/scalar ratio = 2.000 in the double limit. The heat kernel prediction is confirmed to 0.015% at N=150, the best precision achieved in this program.
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The systematic error budget is closed:
- C→∞ extrapolation: ±0.01% (converged at C=150)
- N→∞ extrapolation: ±0.03% (models agree)
- Mass effects (V2.117): ±0.03% (physical masses negligible)
- Interaction corrections (V2.117): ±0.01%
- Total lattice systematic: ±0.05%
- Remaining gap: 3.0% — 60x larger than systematics
What Did NOT Work
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The N-dependence does NOT close the gap. α increases with N, which makes R = |δ|/(6α) smaller, which WIDENS the gap. The N→∞ limit goes in the wrong direction.
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The double limit gives essentially the same answer as V2.118. The elaborate 2D grid and N-extrapolation only shifted α by 0.005% from V2.118’s value. The N-dependence was a red herring.
Implications
The 3.0% gap, with all lattice systematics bounded at 0.05%, means the gap must come from one of:
- Physics beyond the Standard Model (most exciting)
- The self-consistency derivation (factor f = 6 having corrections)
- The heat kernel assumption α_Weyl = 2α_scalar (never verified on lattice for fermions)
These are the only remaining sources of uncertainty larger than 0.05%.
Technical Notes
- Lattice: Lohmayer radial chain, N=40..200, C=5..150
- Fit: 3-parameter (S = α × 4πn² + βn + γ), consistent fractional range n/N ∈ [0.2, 0.5]
- C-extrapolation: power_law_log model α(C) = α_∞ + (A + B ln C)/C^p
- N-extrapolation: power_law α_∞(N) = α_∞∞ + A/N^p
- All 11 tests pass
- Runtime: 485 seconds (dominated by N=150,200 entries)