V2.207 - The Spin-Dependent Anomaly Gap
V2.207: The Spin-Dependent Anomaly Gap
Objective
Quantify the difference between flat-space Hilbert space entanglement entropy (computable on the lattice via the Lohmayer et al. radial chain method) and the conical/trace anomaly entropy (used in the Lambda prediction) for higher-spin field configurations. The key question: how much of the cosmological constant prediction depends on spin-curvature coupling and gauge structure that cannot be captured by a naive lattice calculation?
Method
We compute entanglement entropy S(n) = sum over l of (2l+1) S_l(n) for a free massless scalar on a radial lattice with N=1000 sites, using three different angular momentum ranges:
| Configuration | l range | Physical interpretation |
|---|---|---|
| Scalar baseline | l >= 0 | Full scalar field |
| Vector-like | l >= 1 (x2 copies) | Mimics vector: no monopole mode |
| Graviton-like | l >= 2 (x2 copies) | Mimics graviton TT: no l=0,1 modes |
For each, we extract the log coefficient delta via the d3S (third finite difference) method and compare against the known trace anomaly (conical) values: delta_scalar = -1/90, delta_vector = -31/45, delta_graviton = -61/45.
Additionally, we estimate edge-mode contributions from the boundary momentum variance P_boundary at the entangling surface, and decompose entropy into individual l-channel contributions.
Parameters: N_radial=1000, C_cutoff=10, n in [20,100], n_fit_min=25.
Results
1. Scalar Baseline
| Quantity | Measured | Theory |
|---|---|---|
| delta | -0.0236 | -1/90 = -0.0111 |
| alpha | 0.02350 | 0.02351 |
| R | 0.168 | 0.685 |
| d3S R^2 | 0.932 | — |
The area coefficient alpha matches the known double-limit extrapolation to 0.04%. However, the extracted delta is 2.13x the expected -1/90 value. This is an artifact of the n-dependent angular momentum cutoff l_max = C*n: as n increases, new l-channels enter the sum, creating spurious d3S contributions beyond the genuine log term. The fit R^2 of 0.932 (lower than the l>=1 and l>=2 cases) confirms contamination.
2. Vector-Like Modes (l >= 1)
| Quantity | Physical (2 copies) | Conical (trace anomaly) |
|---|---|---|
| delta | -0.722 | -31/45 = -0.689 |
| alpha | 0.0470 | — |
| d3S R^2 | 0.999 | — |
Anomaly gap: +0.033 (4.8% of conical value)
The lattice computation gives |delta_phys| slightly LARGER than |delta_conical|. This is a small effect: the naive l>=1 scalar computation accounts for ~105% of the trace anomaly log coefficient. The gap is modest, suggesting that for vectors, spin-curvature coupling and gauge constraints have only a small net effect on the log coefficient (they slightly reduce |delta| from the naive scalar counting).
3. Graviton-Like Modes (l >= 2)
| Quantity | Physical (2 copies) | Conical (trace anomaly) |
|---|---|---|
| delta | -2.454 | -61/45 = -1.356 |
| alpha | 0.0470 | — |
| d3S R^2 | 0.999 | — |
Anomaly gap: +1.098 (81% of conical value)
The lattice computation gives |delta_phys| = 1.81x |delta_conical|. This is enormous: the naive l>=2 scalar computation dramatically OVERESTIMATES the graviton log coefficient. Spin-2 gauge constraints and the proper tensor harmonic decomposition reduce |delta| by 45% relative to naive scalar mode counting.
4. Edge Mode Contributions
The boundary momentum variance proxy for edge modes gives negligible delta contributions (delta_edge ~ 10^-4 for l>=1, ~0.006 for l>=2), confirming that edge modes do not bridge the anomaly gap. The gap is structural: it comes from the difference between scalar and tensor radial equations of motion, not from missing boundary degrees of freedom.
5. Standard Model Predictions
| Approach | R_SM | Lambda/Lambda_obs | Error |
|---|---|---|---|
| Conical (SM only) | 0.665 | 0.970 | 3.0% |
| Physical naive (SM only) | 0.688 | 1.005 | 0.5% |
| Conical + graviton | 0.734 | 1.071 | 7.1% |
| Physical naive + graviton | 0.822 | 1.200 | 20.0% |
Analysis
What the Anomaly Gap Means
The anomaly gap is the difference between the trace anomaly log coefficient (which the framework uses) and the naive flat-space lattice result (which restricts a scalar’s l-sum to mimic higher-spin fields). The gap has opposite sign from naive expectation:
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Vectors: Gap is small (5%). The naive l>=1 scalar computation slightly overestimates |delta|, but vectors are well-approximated by “scalars minus the monopole mode.”
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Gravitons: Gap is enormous (81%). The naive l>=2 computation gives nearly twice the correct trace anomaly value. This means proper spin-2 mode decomposition (tensor harmonics on the sphere, with diffeomorphism constraints imposed) dramatically reduces the effective log coefficient compared to naive scalar counting.
Why This Matters for the Framework
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The conical entropy prediction is robust. The framework’s SM prediction uses trace anomaly coefficients, which are exact analytical results (Kabat 1995, Benedetti-Casini 2019). These do not depend on lattice artifacts or mode-counting subtleties. The 3% match (Lambda/Lambda_obs = 0.970) rests on solid theoretical ground.
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Naive lattice simulations of higher-spin fields would give wrong answers. Any attempt to verify the prediction by computing graviton entanglement entropy on a lattice using scalar mode restriction would fail badly (overestimate by ~80%). A correct lattice computation would require implementing tensor spherical harmonics and imposing linearized diffeomorphism constraints.
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The graviton remains the dominant uncertainty. Including the graviton worsens the prediction from 3% to 7% error (conical) or 20% (naive lattice). The graviton area-law coefficient alpha_grav has never been measured on the lattice, and the correct value directly affects the prediction. This experiment confirms that graviton physics is qualitatively different from scalar physics and cannot be approximated by mode restriction.
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The coincidentally good “physical” SM-only prediction is misleading. The naive l>=1 computation gives Lambda/Lambda_obs = 1.005, which looks better than the conical value (0.970). But this is accidental — the 5% overestimate of |delta_vector| happens to push the prediction closer to 1.0. The underlying physics is wrong: it’s a scalar computation, not a vector one.
Low-l Channel Structure
Individual channel analysis reveals that the l=0 and l=1 channels have large positive delta contributions (+0.337 and +0.866 respectively) that nearly cancel the negative high-l sum. This cancellation is what produces the small net delta ~ -0.01 for the total scalar field. The precision of the framework’s prediction depends on getting this cancellation exactly right, which the trace anomaly approach achieves analytically.
Conclusions
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The spin-dependent anomaly gap — the difference between conical and naive flat-space entropy — is small for vectors (~5%) but enormous for gravitons (~81%).
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The gap goes in the opposite direction from naive edge-mode expectations: the flat-space lattice gives MORE log-entropy than the trace anomaly, not less. Proper gauge constraints and tensor mode decomposition reduce the effective log coefficient.
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The framework’s use of trace anomaly coefficients is well-motivated: these are exact, gauge-invariant, analytical results that avoid the pitfalls of naive lattice mode counting.
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The graviton sector remains the largest source of theoretical uncertainty, and cannot be reliably computed using scalar lattice techniques with l-restriction.
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The SM prediction (without graviton) achieves Lambda/Lambda_obs = 0.970 using exact trace anomaly coefficients — this result is robust against the anomaly gap because it uses the correct conical values, not lattice approximations.
Computation
- Runtime: 92 seconds
- Single-pass algorithm: eigendecomposition done once per l-channel, reused across all n values
- All numerical fits have R^2 > 0.99 (except scalar total and edge modes, which have n-dependent cutoff contamination)