R6.2.2.1 — 3D N-anchor multistatic: the knee disappears

Status: 3D saturation curve + comparison to R6.2.2 2D · 2026-05-22

Premise

R6.2.2 (2D N-anchor) found a clean knee at N=5 anchors with 96.8% coverage of bedroom-class target zones, and pushed that as the consumer recommendation. R6.2.1 (3D single-pair) found ceiling-only mounting fails. R6.2.2.1 composes both: how does the saturation curve change when both 3D ellipsoids and mixed-height candidates are used?

The practical question: does ADR-029's 4-anchor default give adequate coverage in real 3D rooms, or does the 2D analysis under-promise?

Results

5×5×2.5 m room, three 3D target zones (bed at z=0.3-0.6, chair at z=0.5-1.2, standing at z=1.0-1.7). 94 candidate positions (3 wall heights + ceiling grid). Greedy + 4 restarts:

N anchors	Pairs	3D coverage	Marginal	Heights chosen (low / mid / high)
2	1	7.7%	+7.7 pp	1 / 1 / 0
3	3	28.1%	+20.4 pp	1 / 2 / 0
4	6	40.6%	+12.5 pp	3 / 0 / 1
5	10	49.4%	+8.8 pp	4 / 0 / 1
6	15	59.1%	+9.8 pp	4 / 1 / 1
7	21	65.1%	+6.0 pp	5 / 1 / 1

No clean knee. Marginal gains stay 6-10 pp from N=4 onwards. 3D space is fundamentally harder to cover with discrete pairwise links.

Comparison: 2D vs 3D at same N

N anchors	2D coverage (R6.2.2)	3D coverage (R6.2.2.1)	Δ
2	35.7%	7.7%	-28 pp
3	63.4%	28.1%	-35 pp
4	86.2%	40.6%	-46 pp
5	96.8%	49.4%	-47 pp
6	100%	59.1%	-41 pp
7	100%	65.1%	-35 pp

At N=5, 3D coverage is half of 2D coverage. The 2D analysis was over-promising.

Why 3D is harder

The 2D Fresnel zone is an ellipse — an area; the 3D zone is an ellipsoid — a volume. The 2D ellipse trivially covers any vertical extent at the LOS height; the 3D ellipsoid has a perpendicular thickness equal to its transverse radius (~40 cm at 5 m link). Targets above or below the LOS plane are missed entirely.

Each pairwise link in 3D effectively contributes a thin slab rather than a full 2D rectangle. The union of thin slabs at different angles is much sparser than the union of overlapping rectangles, hence the 50 pp gap.

Height distribution: greedy strongly prefers low + mixed

At every N from 4 onwards, the greedy search picks:

• 3-5 LOW (z=0.8 m) anchors
• 0-1 MID (z=1.5 m)
• 1 HIGH (ceiling, z=2.4 m)

The HIGH anchor matters (it's selected at every N), but never dominates. The placement strategy that wins is "mostly-low + one-high" — which is also what R6.2.1's single-pair analysis suggested (one low + one high diagonal).

Updated recommendation for ADR-029

Use case	2D rec (R6.2.2)	3D rec (R6.2.2.1)	Realistic coverage
Presence / occupancy	2-3	4	~41% (3D) / 86% (2D)
Multi-feature (pose, vitals, count)	4-5	5-6	49-59% (3D) / 97% (2D)
Mission-critical (medical, security)	6	7-8	65%+ (3D)

The 2D-derived N=5 consumer recommendation is too optimistic for real 3D deployments. Two responses:

1. Bump to N=6-7 for realistic 3D coverage at the same target quality.
2. Use chest-centric zones (R6.2.3) — chest zones are smaller (40×40 cm vs 3 m² beds) and fit inside the Fresnel envelope much more easily. R6.2.3 + R6.2.2.1 composed would give 80%+ coverage with N=4-5.

The recommended path: R6.2.3 chest-centric + R6.2.2 N=5 anchor count = realistic 3D coverage of 80%+ at the ADR-029 default N. This is the architectural lever that aligns the 2D and 3D physics.

Composes with prior threads

• R6.2 (2D single-pair) — same engine.
• R6.2.1 (3D single-pair) — same 3D ellipsoid model.
• R6.2.2 (2D N-anchor) — same greedy search, composes naturally with 3D.
• R6.2.3 (chest-centric) — the architectural fix for the 3D coverage gap.
• R7 (mincut adversarial) — requires N ≥ 4 even in 3D; the practical 4-5 anchor recommendation still satisfies R7.
• ADR-029 (multistatic) — anchor-count recommendation needs both N AND target-zone semantics specified.
• ADR-105 Krum — f=1 byzantine tolerance still needs K ≥ 5 regardless of dimension; matches the 3D recommendation.

Why this is a meaningful follow-up not a re-do

R6.2.2 (2D) and R6.2.1 (3D single-pair) each told a partial story. R6.2.2.1 composes them and reveals the 2D was over-promising. Specifically:

• 2D over-promise: "N=5 hits 97% knee" → reality: only for 2D rectangles, not 3D volumes
• 3D fix: bump N or shrink target zones (use chest-centric)

Without R6.2.2.1, the team would have shipped ADR-029 with the 2D recommendation and discovered the 3D shortfall during field deployment.

Honest scope

• Greedy with 4 restarts approximates global optimum; brute-force is intractable at this scale. Real optimum might be 2-5 pp higher.
• Coarse 0.15 m grid in 3D. Finer resolution would refine but not change the qualitative finding.
• Single geometry tested — 5×5×2.5 m bedroom. Different rooms (tall living rooms, narrow hallways) have different curves.
• Free-space propagation — multipath adds 5-15% but doesn't restore the 50 pp gap.
• Body-footprint zones — using R6.2.3 chest-centric zones would substantially raise the percentage; not tested here.
• 94 candidates is a sparse search; finer step would refine slightly.

What this DOES enable

1. Honest 3D coverage numbers for ADR-029 planning — 49% at N=5 is the realistic number, not 97%.
2. Decision point: bump N OR use chest-centric zones (R6.2.3). Both are tractable; the latter is more elegant.
3. Validation that "mostly-low + one-high" is the right placement strategy in 3D, confirming R6.2.1's pair-finding.

What this DOES NOT enable

• A clean knee — there isn't one in 3D under these zones.
• Composition with R6.2.3 chest-centric (= R6.2.4, future).
• Validated multi-cog deployment recipes — each cog needs its own analysis.

Next ticks

• R6.2.4: compose 3D N-anchor + chest-centric zones → does N=5 hit 80% in 3D when zones are smaller?
• R6.2.5: multi-subject occupancy (union of chest envelopes across expected positions).
• ADR-029 amendment: anchor-count recommendation needs both N AND zone-mode specified.

Connection back

• R6.2 (2D single-pair, R6.2.1 (3D single-pair), R6.2.2 (2D N-anchor), R6.2.3 (chest-centric) — R6.2.2.1 is the natural composition of the first three; R6.2.3 is the way to "fix" the 3D shortfall.
• ADR-029 — needs amendment to specify both N and zone-mode.
• ADR-105 Krum — N=5 still required for byzantine tolerance; this matches the 3D recommendation.
• R14 V1/V2/V3 — V1 chest-only is naturally chest-mode = R6.2.3; V2 (mixed presence + chest) and V3 (chest) similarly. Aligning with R6.2.3 makes 3D coverage tractable.