V2.282 - Vacuum Energy Spatial Profile — Boundary vs Bulk Duality in P_sub
V2.282: Vacuum Energy Spatial Profile — Boundary vs Bulk Duality in P_sub
Status: 13/13 tests passed | 6 experiments completed
Motivation
V2.243 found α/ρ_vac is NOT constant across channels (Approach B failure). V2.279 found tr(P_sub)/ρ_A ≈ 1 (R² = 0.9999996). The contradiction: if tr(P) determines ρ, why can’t it determine α too?
This experiment resolves the puzzle by spatially decomposing P_sub. The hypothesis: α comes from the BOUNDARY of P while ρ comes from the BULK, so their ratio varies with the spatial profile shape.
Key Results
Finding 1: Vacuum Energy is BULK-Distributed (Not Boundary-Localized)
The diagonal P_{ii} shows the vacuum kinetic energy density at each radial site. For l=0, the profile is nearly flat (P_{ii}/P_{nn} ≈ 1.0-1.2). For high l, the profile rises steeply toward the origin (P_1/P_n ≈ 12× at l=29).
Boundary fraction: P_{nn}/tr(P) = 3.5% (mean), range [2.4%, 6.5%]
This is the exact opposite of entropy, where 98% comes from the boundary (V2.268). Vacuum energy density is concentrated at the origin (small r), not the entangling surface.
Physical reason: the centrifugal potential l(l+1)/r² pushes mode energy toward small r, while entanglement entropy comes from correlations across the cut at r = n_sub.
Finding 2: Neither Boundary Nor Bulk Alone Predicts α
| Predictor | → α_l (R²_LOO) | → ρ_A (R²_LOO) |
|---|---|---|
| P_{nn} (boundary) | 0.355 | 0.990 |
| tr(P) (bulk) | 0.460 | 0.99999965 |
| {P_{nn}, tr(P)} | 0.996 | 0.99999998 |
Single quantities fail for α: neither boundary energy nor bulk energy alone captures the area-law coefficient. But the TWO-PARAMETER combination works:
α ≈ −0.068 × P_{nn} + 0.001 × tr(P) + 0.033The negative coefficient on P_{nn} is crucial: more boundary energy means LESS entanglement area coefficient. This is consistent with the centrifugal barrier decoupling the interior from exterior at high l.
Finding 3: Why V2.243 Failed — Ratio Decomposition
The ratio α/ρ can be decomposed as:
α/ρ = (α/P_{nn}) × (P_{nn}/tr(P)) × (tr(P)/ρ)Coefficient of variation for each factor:
| Factor | CV |
|---|---|
| α/ρ (total) | 35.4% |
| α/P_{nn} | 37.4% (dominant) |
| P_{nn}/tr(P) | 34.3% |
| tr(P)/ρ | 0.17% (negligible) |
tr(P)/ρ is essentially constant (CV = 0.17%), confirming V2.279. The variation in α/ρ comes entirely from the geometric factors α/P_{nn} and P_{nn}/tr(P). V2.243’s failure is a geometric artifact, not a physical obstruction to Λ_bare = 0.
Finding 4: α/ρ Scales as 1/n_sub
At fixed l=10:
| n_sub | P_{nn}/tr(P) | α/ρ |
|---|---|---|
| 5 | 0.0995 | 8.8×10⁻⁴ |
| 10 | 0.0489 | 4.8×10⁻⁴ |
| 20 | 0.0283 | 9.9×10⁻⁵ |
| 30 | 0.0211 | 3.1×10⁻⁵ |
Both P_{nn}/tr(P) ~ 1/n and α/ρ ~ 1/n². This is the scaling puzzle: α ~ n⁰ (area law) while ρ ~ n¹ (extensive), so α/ρ → 0. But this is exactly as expected: α determines G ~ 1/M_Pl² (fixed), while ρ ~ Λ_UV⁴ (cutoff-dependent). The ratio α/ρ is NOT the physical observable — R = |δ|/(6α) is, and R is n-independent.
Finding 5: CHM Weighting Doesn’t Help
Five weighting schemes for Σ w_i P_{ii} → α:
| Weight | R²_LOO |
|---|---|
| Boundary only | 0.355 |
| Linear (boundary-peaked) | 0.418 |
| Uniform | 0.460 |
| CHM sphere | 0.470 |
| Rindler | 0.475 |
Best is Rindler weight w_i ∝ (n−i)/n, but all are poor (R²_LOO < 0.5). No single weighting of P_{ii} captures α. The entanglement area coefficient is not a weighted energy density — it involves the full CORRELATION structure (off-diagonal elements of P_sub and X_sub), not just the diagonal.
Finding 6: {P_{nn}, tr(P)} Jointly Determine Both α and ρ
| Target | {P_{nn}, tr(P)} R²_LOO |
|---|---|
| α | 0.9957 |
| ρ | 0.99999998 |
Two numbers from the momentum covariance matrix determine BOTH gravitational constants {G, Λ}:
- G = 1/(4α) from the entanglement area law
- Λ from ρ_vac via the Friedmann equation
Since both come from the SAME matrix P_sub, adding Λ_bare would require modifying P_sub, which simultaneously changes G. You cannot add vacuum energy without changing the gravitational coupling — this is double-counting.
Implications for Λ_bare = 0
The spatial duality argument
Entropy and vacuum energy are complementary projections of the same covariance matrix P_sub:
-
Entropy ← symplectic eigenvalues of (X_sub ⊗ P_sub) ← boundary- dominated (98% from entangling surface, V2.268)
-
Vacuum energy ← tr(K_int^{1/2}) ≈ tr(P_sub) ← bulk-distributed (only 3.5% from boundary)
-
Both are fully determined by P_sub (and X_sub). The same matrix that gives G = 1/(4α) also gives ρ_vac = tr(P). No independent Λ_bare can exist without modifying P_sub, which would change G.
Resolving the V2.243 puzzle
V2.243’s approach B failure (α/ρ not constant) is now understood:
- Not a failure of double-counting — tr(P)/ρ IS constant (CV = 0.17%)
- A geometric effect — α depends on boundary structure of P, not its trace
- The ratio α/ρ varies because α encodes boundary correlations while ρ encodes the bulk sum — different projections of the same matrix
The hierarchy
| Experiment | What it showed | Approach B status |
|---|---|---|
| V2.243 | α/ρ not constant | Apparent failure |
| V2.279 | tr(P)/ρ ≈ 1 | Identity found |
| V2.280 | Correction vanishes at cosmo scale | Identity exact |
| V2.282 | α/ρ varies due to geometry, not physics | Failure explained |
Connection to previous experiments
- V2.268 (entanglement contour): 98.4% boundary — now complemented by vacuum energy at 3.5% boundary (dual localization)
- V2.279 (tr(P) = ρ): confirmed here, extended with spatial decomposition
- V2.243 (α/ρ failure): resolved — geometric artifact, not physical
- V2.250 (QNEC completeness): the formal argument is now supported by the constructive mechanism: P_sub determines both G and Λ
Summary
| Test | Result | Implication |
|---|---|---|
| P_{nn}/tr(P) | 3.5% at boundary | Vacuum energy is BULK, not boundary |
| P_{nn} → α | R²_LOO = 0.36 | Boundary energy alone insufficient |
| tr(P) → α | R²_LOO = 0.46 | Bulk energy alone insufficient |
| {P_{nn}, tr(P)} → α | R²_LOO = 0.996 | Two P-projections sufficient |
| {P_{nn}, tr(P)} → ρ | R²_LOO = 1.0000 | Vacuum energy fully determined |
| tr(P)/ρ CV | 0.17% | tr(P) = ρ confirmed |
| α/ρ CV | 35.4% | Variation is geometric |
Bottom line: Vacuum energy and entanglement entropy are complementary — boundary-localized (entropy) vs bulk-distributed (energy) — projections of the same momentum covariance P_sub. The V2.243 α/ρ failure is a geometric artifact: two projections of P_sub ({P_{nn}, tr(P)}) jointly determine both G and Λ with no room for Λ_bare. Adding bare vacuum energy would modify P_sub, simultaneously changing G — the double-counting is inescapable.