V2.307 - Vector (Spin-1) Double-Counting Identity
V2.307: Vector (Spin-1) Double-Counting Identity
Motivation
All prior double-counting tests (V2.285, V2.295, V2.301, V2.303, V2.304) used scalar fields only. But the SM field content is:
- 4 real scalars (Higgs doublet)
- 45 Weyl fermions
- 12 gauge vectors (8 gluons + W±, Z, γ)
Vectors contribute 90 of 118 effective components to α (76%), making them the dominant contributor. If tr(P)/ρ fails for vectors, Approach B is incomplete for the full SM.
A Maxwell vector on the Srednicki lattice decomposes into TE + TM polarizations, each equivalent to a scalar with l ≥ 1. The coupling matrix K_l is spin-independent — only the degeneracy weighting 2(2l+1) and l-range (l ≥ 1, no monopole) differ.
Key Results
1. Per-Channel Identity Holds for All Vector Channels
| l | ratio | deviation |
|---|---|---|
| 1 | 1.00436 | 0.436% |
| 5 | 1.00239 | 0.239% |
| 10 | 1.00118 | 0.118% |
| 20 | 1.00036 | 0.036% |
| 40 | 1.00006 | 0.006% |
Identical to scalar channels at the same l — as expected since K_l is spin-independent.
2. Weighted Totals: Scalar = Vector = SM
| Field type | Weighted ratio | Deviation |
|---|---|---|
| Scalar (Σ(2l+1)) | 1.00022945 | 0.0229% |
| Vector (Σ 2(2l+1), l≥1) | 1.00022876 | 0.0229% |
| Weyl proxy (Σ 2(2l+1)) | 1.00022945 | 0.0229% |
| SM total (4s+12v+45w) | 1.00022931 | 0.0229% |
The vector weighted total matches the scalar to 5 significant figures. Skipping l=0 makes negligible difference (l=0 contributes <0.001% of the total weight at l_max=40).
3. N-Independence Is Exact for Both
| Field | CV across N=100–800 |
|---|---|
| Scalar | 0.000000% |
| Vector | 0.000000% |
Exact N-independence at all lattice sizes, confirming the identity is UV-cutoff independent regardless of spin weighting.
4. Cumulative Convergence
As l_max increases, vector and scalar ratios track each other:
| l_max | Scalar dev | Vector dev |
|---|---|---|
| 5 | 0.302% | 0.297% |
| 20 | 0.081% | 0.080% |
| 40 | 0.023% | 0.023% |
The vector deviation is consistently slightly smaller because it skips l=0 (the worst channel).
Why This Result Was Expected (But Still Needed)
The per-channel identity tr(P_l)/ρ_A(l) = 1 depends only on K_l, which is the same tridiagonal matrix for all bosonic spins. The Benzi-Golub mechanism (V2.295, V2.304) applies to any tridiagonal positive-definite matrix regardless of how the channels are summed.
However, empirical verification was necessary because:
- The degeneracy weighting changes which channels dominate
- The l=0 exclusion for vectors changes the low-l structure
- No prior experiment had explicitly tested non-scalar field content
Honest Assessment
Achieved
- Confirmed tr(P)/ρ = 1 identity for vector (spin-1) field content
- Showed scalar, vector, and SM-weighted totals agree to 5 significant figures
- Demonstrated N-independence is exact for vector weighting
- Extended Approach B from “proven for scalars” to “proven for all bosonic SM fields”
Limitations
- The result is structurally guaranteed (same K_l) — this is a verification, not a discovery
- Weyl fermions use a different (fermionic) entanglement structure; only tested as a proxy with bosonic chain and fermionic degeneracy
- Entropy fit (Part 5) shows site-level α/δ — the log coefficient δ is badly extracted at C=6 (known convergence issue, not a physics problem)
- Fermions contribute 24/118 = 20% of effective components; their double-counting remains untested
For the programme
Closes the vector gap in Approach B. Combined with V2.285 (massless scalar), V2.301 (massive scalar), V2.295 (mechanism), and V2.304 (bound), the double-counting identity is now proven for all bosonic SM fields (scalar + vector, 94/118 = 80% of α). The remaining 20% (Weyl fermions) requires a separate fermionic analysis using the correlation matrix formalism.