V2.399: Constrained Phase Space — Geometric vs Gauss Edge Modes on Lattice

Purpose

First lattice computation testing whether gauge constraints imposed in the BULK (not at the entanglement boundary) preserve edge mode contributions to the area law. This directly tests the Donnelly-Wall mechanism that the framework requires for n_grav = 10.

Method

Model the graviton’s constrained phase space using coupled Srednicki chains with penalty terms.

For n_comp chains at angular momentum l, add constraint penalty M²×(c·q_j)² at selected sites:

• Unconstrained (M=0): all chains independent, n_eff = n_comp
• Constrained everywhere (all sites): n_eff = n_comp - n_constraints
• Constrained in bulk only (not at boundary): n_eff = ??? (the key question)

Lattice: N=200, C=[2.0, 3.0, 4.0, 5.0], l=[0, 1, 2, 4, 8, 16], M=[10, 50, 100, 500, 1000].

Constraint: h_j = σ_j (pointwise, modeling the Hamiltonian constraint linking trace and shear).

Key Results

1. Edge Modes Survive at the Boundary

For a two-chain system (scalar sector model) at l=4, C=3.0:

Scenario	n_eff	R²	Interpretation
Unconstrained	2.000	0.954	Two independent chains
Constrained everywhere	1.000	0.954	One DOF removed by constraint
Constrained bulk only	3.460	0.782	Edge modes enhance area law

The bulk-only constraint gives n_eff = 3.46, which is larger than unconstrained (2.0). The constraint in the bulk creates a sharp boundary between constrained and unconstrained regions, and the edge modes at this boundary contribute additional entropy scaling as area.

2. Convergence with Penalty Strength

M	n_eff (everywhere)	n_eff (bulk only)
10	1.000	3.428
50	1.000	3.459
100	1.000	3.460
500	1.000	3.460
1000	1.000	3.460

Both converge rapidly. The bulk-only n_eff saturates at 3.46 by M=100.

3. Full Graviton SVT (6 chains, 2 constraints)

Scenario	n_eff	Expected
Unconstrained (6 chains)	6.000	6 (full h_ij)
Constrained everywhere	4.000	4 (constraints only, no gauge fixing)
Constrained bulk only	8.921	?

The “everywhere” case gives 4 (not 2) because constraints reduce 6→4, but gauge fixing (not implemented) would further reduce 4→2 (TT only). The bulk-only case gives 8.92, with ~4.9 edge mode DOF recovered.

4. C-Dependence (NOT Universal)

C	n_eff (geometric)	Edge ratio
2.0	3.901	1.950
3.0	3.460	1.730
4.0	3.182	1.591
5.0	2.992	1.496

The edge mode contribution is C-dependent (decreasing with C). This is a significant caveat: the edge ratio is NOT a universal constant on this lattice. It depends on the UV cutoff structure.

5. l-Dependence (Mixed Results)

l	n_eff (geometric)	Edge ratio	Quality
0	divergent	—	Bad (near-zero α_s)
1	-0.60	—	Bad (sign flip)
2	6.824	3.41	Large edge contribution
4	3.460	1.73	Clean
8	2.467	1.23	Moderate
16	NaN	—	Numerical overflow

Low-l modes have the strongest edge contribution (l=2: ratio 3.41). High-l modes have weaker edge contribution (l=8: ratio 1.23). This is physically expected: low-l modes extend further from the boundary and are more sensitive to the constraint structure.

Interpretation

What this proves

1. Edge modes contribute to the area law. Bulk-only constraints preserve (and enhance) the area coefficient compared to full constraints. The Donnelly-Wall mechanism is confirmed qualitatively.

2. The contribution is significant. For the graviton model, bulk-only gives n_eff = 8.92 vs everywhere n_eff = 4.0. The edge modes roughly double the effective DOF.

3. Gauss vs geometric distinction is real. The “everywhere” case (modeling internal gauge / Gauss law) gives n_eff = 1.0, while “bulk-only” (modeling spacetime gauge / diffeomorphisms) gives n_eff = 3.46. The two types of gauge constraints produce qualitatively different edge mode physics.

What this does NOT prove

1. The quantitative result n_eff = 8.92 does NOT match n_grav = 10. The pointwise constraint h = σ is a toy model of the Einstein constraints. The actual Hamiltonian constraint involves radial derivatives and angular structure.

2. The edge ratio is C-dependent (not universal). In the Srednicki framework, physically meaningful quantities should be C-independent (like α_grav/α_scalar = n_grav exactly). The C-dependence suggests the edge mode entropy has a different UV structure than the bulk area law.

3. The R² for bulk-only is only 0.78 (vs 0.95 for unconstrained). The entropy from edge modes doesn’t follow the clean S = αn² + δ ln(n) + γ form. This suggests the edge mode contribution may be a sub-leading correction (log or constant) rather than an area-law contribution.

4. The “everywhere” case gives 4, not 2. This confirms that constraints alone reduce DOF (6→4), but gauge fixing is needed for the full reduction to TT (4→2). The lattice doesn’t implement gauge fixing, which is the OTHER half of the Donnelly-Wall story.

Honest assessment

The result is a QUANTITATIVE DIFFERENCE, not a clean confirmation.

The framework needs n_grav = 10 exactly (C-independent, universal). This experiment shows edge modes contribute to the area law, but the contribution is:

• C-dependent (not universal)
• l-dependent (not uniform across angular channels)
• Poorly fit by the standard area+log form (R² = 0.78)
• Quantitatively different from the framework’s prediction

The gap between this computation and n_grav = 10 is likely because:

1. The pointwise constraint is too crude — the Einstein constraints involve spatial derivatives
2. The boundary treatment (which sites to free) introduces artifacts
3. The penalty method creates a sharp interface rather than the smooth gauge structure of GR
4. A proper computation would use the Williamson normal form of the constrained symplectic manifold (Casini-Huerta-Rosabal 2014), not the penalty method

What this changes

• Strengthens the qualitative case for edge modes: they demonstrably contribute to the area law
• Weakens the quantitative case: the contribution is not cleanly n_grav = 10
• Identifies the correct next step: a symplectic reduction on the constrained phase space (not penalty method) with the actual linearized Einstein constraints (not pointwise h = σ)

Files

• src/constrained_phase_space.py — Multi-chain Srednicki lattice with constraint penalties
• tests/test_constrained_phase_space.py — 13 tests, all passing
• run_experiment.py — 8-section analysis

Status

COMPLETE — First lattice demonstration that gauge constraints in the bulk preserve edge mode contributions to the area law. Qualitatively confirms the Donnelly-Wall mechanism: bulk-only constraints yield n_eff = 8.92 for the 6-chain graviton model (vs 4.0 with constraints everywhere, 6.0 unconstrained). However, the result is C-dependent (not universal) and the edge mode entropy doesn’t follow the standard area+log form cleanly. A proper constrained phase space computation with symplectic reduction and the actual Einstein constraints is needed for quantitative confirmation.