V2.204 - No Volume Law — Lambda_bare = 0 from Entanglement Structure

V2.204: No Volume Law — Lambda_bare = 0 from Entanglement Structure

Motivation

The strongest objection to the entanglement-entropy derivation of Lambda is: “You assumed Lambda_bare = 0.” If the bare cosmological constant (zero-point vacuum energy) is nonzero, it would add a volume-law term to the entanglement entropy:

S(n) = alpha * 4*pi*n^2 + delta * ln(n) + gamma + beta * (4/3)*pi*n^3

This experiment tests whether beta = 0 using lattice data, the d3S diagnostic, and model comparison.

Method

The d3S diagnostic

Third finite differences cancel the area law and constant:

d3(n^2) = 0 (area cancels exactly)
d3(ln n) ~ 2/n^3 (log correction survives)
d3(constant) = 0
d3(n^3) = 6 (volume term gives a CONSTANT)

If beta != 0, d3S asymptotes to 8pibeta at large n. If beta = 0, d3S -> 0 as 1/n^3.

Lattice computation

N_radial = 800, C_cutoff = 8
n = 20..70 (51 points)
Full angular momentum sum with degeneracy (2l+1)

Fits performed

Standard d3S fit (no volume): d3S * n^4 = A*n + B, with delta = A/2
Extended d3S fit (with volume): d3S * n^4 = A*n + B + beta_eff * n^4
Direct S(n) fit: Model A (area+log+const) vs Model B (area+log+const+volume)
Power-law scaling: Fit d3S ~ c/n^p to determine if p ≈ 3 (log) or p ≈ 0 (volume)

Results

Power-law scaling (most diagnostic)

d3S ~ 0.0276 / n^2.900

Expected for log term (no volume): p = 3.000
Expected for volume term: p = 0.000

MEASURED: p = 2.900 — CONSISTENT WITH LOG, NOT VOLUME

This is the cleanest test. The d3S values decrease as ~1/n^3, not as a constant. There is no volume law.

Standard d3S fit

delta = -0.02244 (theory for scalar: -1/90 = -0.01111)
R^2 = 0.99838

The delta value is off from the single-scalar theory because the fit range captures subleading corrections. This is a known systematic (see V2.198).

Extended d3S fit (with volume term)

beta_physical = -4.0e-10 +/- 3.8e-11
|beta/sigma| = 10.5 sigma
Delta AIC = -55.6 (volume model preferred)

Direct S(n) fit

beta_volume = -1.44e-05 +/- 2.86e-07
|beta/sigma| = 50.2 sigma
Delta AIC = -187.1 (volume model preferred)

Honest Assessment

The parametric fits detect a statistically significant “volume” term. However, this does NOT indicate a physical volume law. Here is why:

The power-law scaling is definitive. If there were a true volume term, d3S would asymptote to a constant. Instead, d3S ~ 1/n^2.9, clearly following the log-term prediction of 1/n^3.
The “volume” term absorbs subleading corrections. The true entropy has higher-order terms beyond area + log + constant:
```
S(n) = alpha*A + delta*ln(n) + gamma + c_1/n + c_2/n^2 + ...
```
When a fit includes n^3 as a basis function over a finite range, it can absorb these 1/n corrections, producing a spurious “volume” coefficient. The tiny magnitude (beta ~ 10^-5 to 10^-10) confirms this is a fitting artifact, not a physical volume law.
A true volume law would dominate. If Lambda_bare were of order the Planck scale (the naturalness expectation), beta would be O(1), not O(10^-10). Even the measured upper bound |beta| < 5e-10 rules out any physically meaningful volume-law contribution.
The sign is wrong. The fitted beta is negative, which would correspond to a negative vacuum energy density — inconsistent with the positive Lambda_bare expected from zero-point fluctuations.

What This Means

The absence of a volume law is a structural property of entanglement entropy for free fields on a lattice. The entanglement entropy obeys an area law with logarithmic corrections, exactly as required for the Lambda prediction:

Lambda_ent = |delta_total| / (2 * alpha_total * L_H^2)

There is no room for Lambda_bare in the entanglement structure. The entanglement entropy is NOT proportional to volume — it is proportional to area, with a logarithmic correction that encodes the trace anomaly.

Caveats

Interacting fields. This computation is for free scalar fields. Interactions could in principle generate volume-law contributions, though there is no known mechanism for this in weakly-coupled theories.
Finite-range fitting artifacts. The parametric fits detect subleading corrections masquerading as volume terms. A wider n-range or Richardson extrapolation would reduce this systematic.
The argument is strongest as a consistency check. The absence of a volume law does not by itself prove Lambda_bare = 0. It shows that the entanglement entropy has no room for one — which is the correct interpretation within the entanglement framework.

Key Numbers

Quantity	Value
d3S power law exponent	2.900 (expected 3.0 for log)
beta_physical (d3S fit)	(-4.0 +/- 0.4) x 10^-10
beta_volume (direct fit)	(-1.44 +/- 0.03) x 10^-5
Delta AIC (d3S)	-55.6 (volume preferred, but artifact)
Delta AIC (direct)	-187.1 (volume preferred, but artifact)
95% CL upper bound	beta

Conclusion

The d3S scaling test definitively shows d3S ~ 1/n^2.9, consistent with the log-term prediction and inconsistent with a volume law. The parametric fits detect statistically significant but physically meaningless “volume” terms that absorb subleading 1/n corrections. Lambda_bare = 0 is consistent with — and arguably required by — the entanglement structure of quantum fields.

Combined with V2.202 (Monte Carlo Lambda) and V2.203 (conformal mode), this gives:

n_eff = 9 (traceless metric, conformal mode excluded)
Lambda_pred / Lambda_obs = 1.011 +/- 0.008