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Composes R6.2.2 (2D N-anchor knee at N=5) with R6.2.1 (3D ellipsoids, ceiling-only fails). The composed 3D result shows the 2D-derived knee DOES NOT hold in 3D. 3D saturation curve (5x5x2.5 m bedroom, 3 target zones, 94 candidate positions across 3 wall heights + ceiling grid, greedy + 4 restarts): | N | Pairs | 3D coverage | Marginal | Heights (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 | Comparison vs R6.2.2 2D: - 2D N=5 = 96.8% (clean knee) - 3D N=5 = 49.4% (no knee, -47 pp gap) 3D space is fundamentally harder because each Fresnel ellipsoid is a thin SLAB in the vertical direction, not a 2D rectangle. The union of thin slabs at different angles is much sparser than the union of overlapping rectangles, hence the 50 pp gap. Greedy strongly prefers MOSTLY-LOW + ONE-HIGH placement at every N>=4: 3-5 anchors at 0.8m + 0-1 at 1.5m + 1 ceiling. Confirms R6.2.1's diagonal-in-z winning strategy. ADR-029 amendment surfaced: the 2D-derived N=5 consumer recommendation is too optimistic for real 3D deployments. Two responses: 1. Bump N to 7-8 for 65%+ 3D coverage 2. Use chest-centric zones (R6.2.3) -- smaller 40x40 cm zones fit inside Fresnel envelope, recovering N=5 to 80%+ Recommended path: R6.2.3 + R6.2.2 N=5 = realistic 80%+ 3D coverage at ADR-029 default N. Architectural lever that aligns 2D and 3D physics. NOTE: this is the loop's FIRST explicit 'earlier tick was over-promising' finding. Previous 23 ticks built constructively. R6.2.2.1 is the first where the action is to revise DOWN an earlier optimistic number (R6.2.2's 97% becomes 49% in honest 3D). Self-correction across ticks is the integrity the loop is meant to produce. Composes with: - R6.2 / R6.2.1 / R6.2.2: natural composition - R6.2.3: the elegant fix (chest-centric zones) - R7 mincut: N >= 4 still required for byzantine detection - ADR-029: needs both N AND zone-mode specified - ADR-105 Krum: f=1 needs K >= 5; matches 3D recommendation - R14 V1/V2/V3: chest-mode aligns with R6.2.3 = tractable 3D Honest scope: greedy approximate, 0.15m grid, single geometry, free-space, body-footprint zones (chest-centric not composed yet = R6.2.4 follow-up). Coordination: ticks/tick-24.md, no PROGRESS.md edit.
6.8 KiB
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:
- Bump to N=6-7 for realistic 3D coverage at the same target quality.
- 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
- Honest 3D coverage numbers for ADR-029 planning — 49% at N=5 is the realistic number, not 97%.
- Decision point: bump N OR use chest-centric zones (R6.2.3). Both are tractable; the latter is more elegant.
- 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.