Experiment V2.92: Self-Consistency Factor Analysis

Status: COMPLETE

Summary

The self-consistency ratio R = |delta|/(f * alpha) should equal 1 for the Lambda prediction to close. But what is f? V2.64 derives f = 6 from the Cai-Kim horizon first law. V2.76 uses f = 12 without derivation. This experiment traces the origin of f, checks whether f = 12 is justified, and discovers that the “factor of 3 gap” may actually be a “factor of 1.5 gap.”

The most striking numerical finding: |delta_SM|/alpha_SM = 3.943, which is only 1.4% from the integer 4. With f = 4, R_SM = 0.986 — near-perfect self-consistency.

Key Results

The Self-Consistency Factor f

Factor f	Source	R_SM	Gap	Lambda/Lambda_obs
4	Empirical (best fit)	0.986	1.01x	0.986
6	V2.64 derivation	0.657	1.52x	0.657
12	V2.76 convention	0.329	3.04x	0.329

V2.64’s Derivation Chain

1. Entropy: S = alpha * A + delta * ln(A)
2. Clausius at apparent horizon: T * dS = -dE, with T = H/(2pi)
3. Standard Friedmann: H^2 = 8piG*rho/3 + Lambda_bare/3, G = 1/(4alpha)
4. Log correction: Delta(H^2) = -delta/(6alphaL_H^2)
5. Lambda_bare = 0: Lambda = |delta|/(2alphaL_H^2)
6. De Sitter self-consistency: LambdaL_H^2 = 3 => **|delta| = 6alpha**

The factor 6 = 2 x 3, where 2 comes from the Lambda formula coefficient and 3 = (d-1)(d-2)/2 from the 4D Friedmann equation.

Factor of 2 Discrepancy: V2.64 (f=6) vs V2.76 (f=12)

Possible source	Factor	Verdict
ln(A) vs ln(r) convention	1	No difference
Definition change V2.64 -> V2.76	2	Unaccounted
Newton constant convention	1	G = 1/(4alpha) is standard
Clausius correction coefficient	1	V2.64 derivation is consistent
Friedmann equation factor	1	Lambda*L_H^2 = 3 is standard

No physical source for the factor of 2 was found. V2.76’s f = 12 appears to be unjustified relative to V2.64’s derived f = 6.

The Critical Ratio |delta_SM|/alpha_SM

Quantity	Value
delta_SM	-1991/180 = -11.0611
alpha_SM	118 * 0.02376 = 2.8051
	delta_SM
Nearest integer	4
Deviation from 4	1.42%
alpha_scalar for ratio = 4 exactly	0.02344
alpha_scalar measured (V2.74)	0.02376
Shift needed	1.37%

D-Dimensional Generalization

d	Friedmann factor (d-1)(d-2)/2	f_sc	R_SM
3	1	2	1.97
4	3	6	0.657
5	6	12	0.329
6	10	20	0.197

Note: f = 4 does not correspond to any integer spacetime dimension d. Interestingly, f = 12 (the V2.76 convention) corresponds to d = 5.

Per-Species R Values (f = 6)

| Species | |delta|/alpha | R(f=6) | |---------|-------------|--------| | Scalar | 0.468 | 0.078 | | Weyl fermion | 1.286 | 0.214 | | Vector (photon) | 14.460 | 2.410 | | Full SM | 3.943 | 0.657 |

The vector field massively dominates the delta/alpha ratio.

Species Sensitivity

| Variation | |delta|/alpha | R(f=6) | |-----------|-------------|--------| | SM (baseline) | 3.94 | 0.657 | | SM without Higgs | 4.07 | 0.678 | | SM + graviton (HK) | 4.72 | 0.786 | | Photon only | 14.46 | 2.410 | | SM + 3 RH neutrinos | 3.81 | 0.636 |

Adding the graviton moves R_f6 from 0.657 to 0.786 (closer to 1).

Discussion

The “Gap” Is Likely 1.5x, Not 3x

The central finding: V2.64’s derivation gives f = 6 as the self-consistency factor. V2.76 uses f = 12 without providing a derivation. No physical source for the additional factor of 2 was identified. If f = 6 is correct:

• R_SM = 0.657 (not 0.329)
• The gap is 1.52x (not 3.04x)
• Lambda_pred/Lambda_obs = 0.66 (within a factor of 1.5!)

This substantially improves the status of the prediction.

The Near-Integer |delta_SM|/alpha_SM ~ 4

The ratio 3.943 being within 1.4% of 4 is intriguing. If alpha_scalar were 1.37% smaller (0.02344 instead of 0.02376), the ratio would be exactly 4 and R(f=4) = 1 exactly. However:

1. The 1.4% discrepancy is within the uncertainty of V2.74’s C->infinity extrapolation (V2.74 reports ~2% convergence uncertainty)
2. There is no known derivation that gives f = 4 as the self-consistency condition
3. The near-integer property is specific to the SM spectrum and may be coincidental

What Could Close the Remaining 1.5x Gap (f = 6)?

1. Graviton contribution: Adding the graviton moves R from 0.657 to 0.786 (20% improvement). But this requires quantizing gravity at the entangling surface.

2. Edge modes: Gauge field entanglement entropy has contact term corrections (Donnelly-Wall). If these reduce alpha_vector, R increases.

3. Higher-order entropy corrections: Terms beyond S = alphaA + deltaln(A) could modify the self-consistency condition.

4. Interaction corrections: All computations use free-field values. QCD corrections to gluon alpha/delta at the Hubble scale are unknown.

5. The c_d coefficient: V2.64 derives c_d = 2 (in Lambda = |delta|/(c_d * alpha * L_H^2)). If c_d receives radiative corrections, this changes f.

Comparison with V2.89’s R(mu) Analysis

V2.89 found R(mu) crosses 1 at mu* ~ 0.02 eV (using f = 12), but the Lambda prediction failed by 10^63. With f = 6:

• R(mu_Hubble) = 2 * 1.205 = 2.41 (photon only, overshoots)
• R(mu_SM) = 2 * 0.329 = 0.657 (full SM, undershoots)
• The R = 1 crossover still exists but at a different scale

The fundamental R(mu) picture is unchanged — there is always a crossover between the UV (all species, R < 1) and IR (photon only, R > 1). But the gap at both ends is reduced by a factor of 2 relative to V2.76’s analysis.

Next Steps

1. Verify the V2.64 derivation independently: Re-derive Delta(H^2) from the Clausius relation from scratch, ideally using sympy for symbolic verification. Check if the coefficient is -delta/(6alphaL_H^2) or -delta/(3alphaL_H^2).

2. Find or disprove f = 4: Check if there is ANY derivation path that gives Lambda = 3|delta|/(4alphaL_H^2). This could involve modified Clausius relations, different temperature definitions, or higher-order corrections.

3. Edge modes: Compute the Donnelly-Wall edge mode correction to alpha_vector and check if it modifies R significantly.

4. Improve alpha_scalar: Run V2.74’s convergence study at higher N (N=800+) to determine if alpha_scalar = 0.02344 (the value giving |delta|/alpha = 4 exactly) is within the true uncertainty.

5. Check V2.76’s source: Review the git history of V2.76 to understand why f = 12 was chosen. Was it a deliberate factor of 2 correction, or an oversight?

Thoughts

What worked: The factor analysis cleanly identifies the discrepancy between V2.64 (derived f=6) and V2.76 (used f=12). This is a purely analytical result that doesn’t require new lattice computations.

What’s uncertain: Whether f = 6 is truly correct. V2.64’s derivation makes assumptions (first-order correction, Lambda_bare = 0, pure de Sitter background) that could each contribute factors. The “correct” f might be somewhere between 4 and 6.

Honest assessment: The finding that the gap is 1.5x (not 3x) is important but not yet conclusive. A skeptic would say:

• “Show me the derivation of f = 4” (we can’t, yet)
• “V2.64’s derivation could have its own factor errors” (true)
• “|delta|/alpha ~ 4 could be coincidence” (possible)

The strongest statement we can make: The self-consistency condition R = 1 with f = 12 has no derivation from first principles. The V2.64-derived condition (f = 6) reduces the gap from 3x to 1.5x, putting the Lambda prediction within a factor of 1.5 of observation.

What would a skeptical physicist say? They would demand an independent derivation of f (not just tracing one experiment’s code). They would want to see the Clausius relation worked out with all factors from scratch, possibly using the Cai-Kim (2005) or Akbar-Cai (2006) formalism directly. They would also point out that 1.5x is still not 1x, and ask what closes the remaining gap.

Is this a breakthrough? Not yet — but it’s an important correction to the self-consistency analysis. If f = 6 is confirmed and the remaining 1.5x gap can be closed (e.g., by graviton + edge modes), the prediction Lambda/Lambda_obs ~ 1 would be remarkable.