V2.243: High-Precision alpha-rho_vac Identity

Status: COMPLETE

Motivation

The entire Moon Walk Lambda prediction assumes Lambda_bare = 0 — that the bare cosmological constant vanishes and all observed Lambda comes from the entanglement entropy log correction. V2.131 showed that alpha/rho_vac is constant to 3.3% across lattice sizes, suggesting vacuum energy and the area coefficient are built from the same UV degrees of freedom.

But 3.3% constancy could be pure dimensional analysis: alpha ~ Lambda_UV^2 and rho_vac ~ Lambda_UV^4, so at fixed UV cutoff the ratio is trivially constant. To distinguish an exact algebraic identity from dimensional coincidence, we need precision better than 0.01%.

This experiment is the highest-priority computational task in the Lambda_bare = 0 research programme (Approach B, Experiment B.1).

Method

Alpha extraction

For each (N, C), compute S(n) = sum_l (2l+1) S_l(n) over n in [0.2N, 0.5N]. Fit S = alpha * 4pin^2 + beta*n + gamma to extract alpha. Angular cutoff: l_max = C * N (global convention).

Vacuum energy

For each (N, C), compute rho_vac = [sum_l (2l+1) * (1/2) sum_k omega_{l,k}] / V_lattice where V_lattice = (4/3)piN^3.

Ratio test

Form R = alpha * l_max^2 / rho_vac and test constancy across:

1. N-scan at fixed C (tests lattice-size independence)
2. C-scan at fixed N (tests angular-cutoff independence)
3. 2D (N, C) grid (cross-check)
4. Per-channel decomposition (tests l-dependence)

Success criterion

• CV < 0.01%: exact identity
• CV < 1%: near-exact (possible identity with subleading corrections)
• CV > 1%: dimensional analysis only

Results

Part 1: N-scan at fixed C = 5

N	l_max	alpha	rho_vac	ratio_raw	ratio_lmax2	R^2
50	250	0.021349	4.582e+01	4.659e-04	29.12	0.999998
75	375	0.021431	4.967e+01	4.315e-04	60.68	0.999999
100	500	0.021419	5.243e+01	4.085e-04	102.13	0.999998
150	750	0.021471	5.636e+01	3.810e-04	214.29	0.999999
200	1000	0.021446	5.916e+01	3.625e-04	362.49	0.999998
300	1500	0.021488	6.314e+01	3.403e-04	765.74	0.999999

ratio_lmax2: CV = 99.21% — wildly non-constant because l_max^2 ~ N^2 grows.

ratio_raw: CV = 10.61% — alpha/rho_vac varies by ~11% across N = 50..300.

Alpha is essentially constant (0.02135 to 0.02149, range 0.6%) while rho_vac grows steadily (45.8 to 63.1, +38%). This immediately rules out a simple proportionality alpha = c * rho_vac.

Part 2: C-scan at fixed N = 100

C	l_max	alpha	rho_vac	ratio_raw	ratio_lmax2
3	300	0.018785	1.154e+01	1.627e-03	146.46
4	400	0.020502	2.702e+01	7.587e-04	121.40
5	500	0.021419	5.243e+01	4.085e-04	102.13
6	600	0.021967	9.025e+01	2.434e-04	87.62
7	700	0.022320	1.430e+02	1.561e-04	76.50
8	800	0.022562	2.130e+02	1.059e-04	67.78

ratio_raw: CV = 95.94% — alpha/rho_vac varies by a factor of ~15 across C.

Alpha converges slowly (0.0188 at C=3 to 0.0226 at C=8, +20%), while rho_vac explodes (11.5 to 213, a factor of 18.5). The vacuum energy scales as ~C^4 (quartic in the angular cutoff) while alpha scales as ~C^0 to C^{0.3}.

Part 3: Scaling Analysis

At fixed C = 5 with l_max = 5N:

• alpha ~ N^0.003 — essentially constant (converges to ~0.0214)
• rho_vac ~ N^0.179 — grows slowly with lattice size
• Exponent difference: -0.176 (expected -2 for exact identity)

The scaling exponents are completely inconsistent with alpha = c * rho_vac * epsilon^2. Alpha depends almost exclusively on the angular cutoff C (convergence in C determines its value), while rho_vac depends on both N and C in a fundamentally different way.

Part 4: Per-Channel Ratio

The ratio alpha_l / E_l is strongly l-dependent (CV = 112%):

l	deg	S_l	E_l	alpha_l (crude)	ratio
0	1	0.5763	63.9	1.15e-04	1.79e-06
5	11	0.2261	69.3	4.50e-05	6.49e-07
10	21	0.1321	77.3	2.63e-05	3.40e-07
20	41	0.0533	96.1	1.06e-05	1.10e-07
30	61	0.0229	118.5	4.57e-06	3.86e-08

The ratio decays monotonically with l: high-angular-momentum channels contribute relatively more vacuum energy than entropy. This is because:

• S_l decays rapidly with l (exponential falloff for l >> n)
• E_l grows slowly with l (more oscillatory modes at higher l)

The identity does not hold channel-by-channel. If there were an algebraic relationship, it would need to involve non-trivial mixing of angular channels.

Part 5: Lambda_bare Candidate

8piGrho_vac = 2pi*rho_vac/alpha at each N:

N	8piG*rho_vac
50	1.349e+04
75	1.456e+04
100	1.538e+04
150	1.649e+04
200	1.733e+04
300	1.846e+04

Lambda_bare candidate is enormous and growing with N. In lattice units, 8piGrho_vac ~ 10^4, compared to the entanglement contribution |delta|/(2alphaL_H^2) ~ 10^{-122} in physical units. This IS the cosmological constant problem in its raw form: the vacuum energy gravitates with a coupling G = 1/(4alpha), and the result is 122 orders of magnitude too large.

Part 6: 2D Grid

N	C	alpha	rho_vac	ratio_lmax2
50	3	0.01868	10.15	41.40
50	5	0.02135	45.82	29.12
50	7	0.02227	124.6	21.88
100	3	0.01879	11.54	146.5
100	5	0.02142	52.43	102.1
100	7	0.02232	143.0	76.50
150	3	0.01889	12.38	308.9
150	5	0.02147	56.36	214.3
150	7	0.02235	153.8	160.2

CV = 73.69%. The ratio depends on both N and C. No combination of N and C produces a universal constant.

Discussion

1. Is the ratio exactly constant or only approximately?

Neither. The ratio alpha/rho_vac varies by 11% across N at fixed C, and by a factor of 15 across C at fixed N. No dimensionless combination of alpha, rho_vac, N, and l_max produces a universal constant. The verdict is unambiguous: there is no simple algebraic identity relating alpha to rho_vac.

2. Why does V2.131’s 3.3% constancy not extend?

V2.131 computed alpha and K_CHM (the Casimir-Huerta-Myers modular Hamiltonian) at fixed N and C, varying only n (the subsystem size). Both scale as n^2 (area law), giving a constant ratio. But this is a statement about the functional form (both obey the area law), not about the coefficients being algebraically related.

When we vary N and C — which changes the UV structure of the lattice — the ratio changes because alpha and rho_vac depend on the UV cutoff in fundamentally different ways: alpha captures correlations across the entangling surface (boundary physics), while rho_vac is a bulk quantity summing all zero-point energies.

3. What does the per-channel structure reveal?

The ratio alpha_l/E_l decays monotonically with l. This means the identity cannot be local in angular momentum space. High-l modes contribute more to rho_vac (because their frequencies are higher) but contribute exponentially less to the entanglement entropy (because they are weakly entangled at a given n).

4. Implications for Lambda_bare = 0

A simple algebraic identity alpha = f(rho_vac) is ruled out. This closes the most straightforward version of Approach B in the Lambda_bare = 0 programme.

However, this does NOT rule out the double-counting argument itself. The argument in V2.131 and Paper 3 is more nuanced: it’s that S_EE and K_CHM (the CHM-weighted vacuum energy) encode the same UV physics because both scale as n^2. The CHM kernel w(r) = (n^2 - r^2)/(2n) converts volume-law vacuum energy into area-law modular energy. The double-counting operates through this geometric weighting, not through a direct proportionality of coefficients.

Remaining viable paths for Lambda_bare = 0:

• Approach A (entropic completeness): Two-horizon argument may still work
• Approach B modified: The relationship involves the CHM kernel, not raw rho_vac
• Approach C (1+1D extension): The exact identity in 1+1D does not require proportionality of coefficients — it works through the trace anomaly
• Approach D (GSL contradiction): Independent of any alpha-rho identity

Key Findings

1. No algebraic identity: alpha and rho_vac are NOT proportional. The ratio varies by 11% across N, 96% across C, and 112% across angular channels.

2. Different UV dependencies: alpha ~ C^{0.3} (saturates), rho_vac ~ C^4 (quartic growth). The area coefficient is boundary physics; vacuum energy is bulk.

3. V2.131’s 3.3% constancy was about functional form (both area-law), not about coefficient proportionality. This distinction is crucial.

4. Lambda_bare candidate is huge: 8piG*rho_vac ~ 10^4 in lattice units, confirming the cosmological constant problem exists within the lattice framework.

5. The double-counting argument survives in its CHM-weighted form (V2.131) but NOT in the naive alpha = c * rho_vac form tested here.

Significance for the Framework

This is a negative result for the simplest version of Approach B, but an important one. It eliminates a tempting but incorrect path and sharpens the problem: if Lambda_bare = 0 is true, it cannot be because alpha is proportional to rho_vac. The mechanism must be more subtle — likely involving the modular Hamiltonian structure (CHM kernel) that converts volume-law energy into area-law entropy.

This redirects the Lambda_bare = 0 programme toward:

1. Modified Approach B: Study the CHM-weighted relationship (K_CHM/S_EE), not the raw alpha/rho_vac ratio
2. Approach A: Entropic completeness via two-horizon argument
3. Approach D: GSL contradiction argument (independent of any identity)

Files

• run_experiment.py: Main experiment script (6 parts)
• src/lattice_vacuum.py: Srednicki lattice and vacuum energy computation
• src/alpha_rho_identity.py: Ratio computation, scanning, and analysis
• tests/test_lattice_vacuum.py: 11 tests for lattice infrastructure
• tests/test_alpha_rho_identity.py: 11 tests for ratio computation
• results/summary.json: Full numerical results