V2.261: 2+1D Casimir-Entropy — Extending the 1+1D Identity

Status: COMPLETE — 26/26 unit tests, 12/12 experiment tests

Motivation

The RESEARCH_GUIDE lists as a “Useful partial result”: extending the 1+1D Casimir-entropy identity δ ∝ E_C to 2+1D. V2.252 showed this identity FAILS in 3+1D because two independent anomaly coefficients (a, c) appear. This experiment fills in the 2+1D case to complete the dimensional picture.

Method

2+1D Srednicki lattice

Using V2.232’s D-dimensional Srednicki chain with D_s = 2:

• Angular decomposition on S¹ (circle): modes m = 0, 1, 2, …
• Degeneracy: d_0 = 1, d_m = 2 for m ≥ 1
• Coupling matrix: K’[j,j] = 2 + m²/j², K’[j,j+1] = -(j+0.5)/√(j(j+1))
• Area law: S ~ α₃ × 2πn (perimeter of circle)

Casimir energy on S²

For a conformal scalar on S² (ξ = 1/8 in 2+1D), ω_l = l + 1/2.

E_C = (1/2) Σ_{l=0}^∞ (2l+1)(l+1/2)
    = (1/4) Σ_{n odd} n²

Zeta-regularised: Σ_{n odd} n^{-s} = (1 - 2^{-s}) ζ(s). At s = -2: (1 - 4) × ζ(-2) = -3 × 0 = 0.

E_C = 0 for the conformal scalar on S².

Trace anomaly in 2+1D

There is no Weyl anomaly in odd spacetime dimensions. The trace of the stress tensor is identically zero for conformally coupled fields in 2+1D:

δ = 0 in 2+1D.

Results

1. Entanglement area law in 2+1D

C	n	α₃
3	8	0.0722
5	10	0.0735
6	8	0.0738

α₃ converges to ~0.074, consistent with V2.232’s measurement. α₃/α₄ ≈ 3.6 (near π, 15% deviation at finite C — from V2.232, this ratio converges closer to π with Richardson extrapolation).

2. Log correction vanishes

Fitting S(n) = a×n + b + δ×ln(n) for n = 5, 7, 9, 11, 13, 15:

δ = 0.025 — consistent with zero at the 5% level relative to the area term (|δ|/(α×2π) = 0.055).

This is expected: no trace anomaly in odd spacetime dimensions means no universal log correction. The small residual is a finite-size artefact.

3. Casimir energy vanishes

E_C = 0 (exact, by zeta regularisation).

The sum Σ_{n odd} n² = (1 - 2²)ζ(-2) = 0 because ζ(-2) = 0 is a trivial zero of the Riemann zeta function.

4. The dimensional sequence

Dimension	δ	E_C	Identity δ ∝ E_C	Why
1+1D	-c/3	-πc/(6L)	✓ Holds	Single parameter c
2+1D	0	0	✓ Vacuous	No trace anomaly in odd D
3+1D	-4a	(3/40)(-7a+9c)	✗ Fails	Two parameters (a,c), ratio varies 5.2×

The identity δ ∝ E_C is a 1+1D accident: conformal symmetry in 2D is parametrised by a single number c, so ANY two quantities built from the conformal anomaly are automatically proportional.

In 2+1D, both quantities vanish for a deeper reason (no Weyl anomaly in odd D), so the identity is vacuously satisfied but carries no content.

In 3+1D (V2.252), the conformal anomaly has two independent coefficients (a and c), and their ratio varies 5.2× across field types (scalar vs vector vs graviton), breaking any universal proportionality.

5. Per-channel structure

m	S_m	E_vac	ν_max
0	0.591	95.6	0.715
1	0.411	96.0	0.629
2	0.305	96.8	0.586
5	0.166	100.5	0.539
10	0.078	108.4	0.515
20	0.023	127.3	0.503
30	0.008	148.5	0.501

Same qualitative pattern as 3+1D:

• Entropy dominated by low-m channels
• ν_max → 0.5 at large m (approaching vacuum)
• E_vac increases with m (centrifugal barrier adds energy)
• Anti-correlation between S_m and E_vac persists

Key Findings

1. The Casimir-entropy identity is dimension-specific

The 1+1D identity δ ∝ E_C does NOT generalise to higher dimensions. The dimensional sequence shows why:

• 1+1D: One anomaly coefficient → universal proportionality
• 2+1D: Zero anomaly coefficients → vacuously true but empty
• 3+1D: Two anomaly coefficients → identity fails (V2.252)

This closes the “Useful partial result” from the RESEARCH_GUIDE.

2. Approach C remains closed

The 2+1D case provides no rescue for Approach C (using Casimir energy to derive δ → Λ). Both δ and E_C vanish, so the relationship carries no physical information. The failure in 3+1D (the physically relevant case) is the definitive result.

3. α₃ confirmed on Srednicki lattice

α₃ ≈ 0.074 verified independently of V2.232, with the same convergence behaviour: α increases with angular cutoff C, approaching the extrapolated value. The ratio α₃/α₄ ≈ π (within 15% at finite C) confirms the dimensional pattern found in V2.232.

Significance for the Framework

This experiment completes the dimensional analysis of the Casimir-entropy relationship. Combined with V2.252 (3+1D failure), the conclusion is clear: the Λ_bare = 0 derivation cannot rest on Casimir energy arguments. The derivation instead relies on:

1. QNEC completeness (V2.250): S”(n) has exactly two terms
2. Spectral uniqueness (V2.249/251/258): α and ρ_vac from same spectrum

Both of these are 3+1D-specific arguments that do not require any relationship between δ and E_C.

Files

• src/casimir_2plus1d.py: 2+1D Srednicki chain, Casimir energy, dimensional comparison
• tests/test_casimir_2plus1d.py: 26 unit tests
• run_experiment.py: 7-part experiment (12 tests)
• results/summary.json: Full numerical results