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research(R6.2.2.1): 3D N-anchor multistatic — 2D knee disappears; revises R6.2.2 down (#727)

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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.
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# 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.
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# Tick 24 — 2026-05-22 08:53 UTC
**Thread:** R6.2.2.1 (3D N-anchor multistatic)
**Verdict:** The 2D knee at N=5 (R6.2.2) doesn't hold in 3D. **3D N=5 gives only 49.4% coverage vs 2D 96.8%.** Two responses: bump N OR use chest-centric zones (R6.2.3). The latter is the architectural fix.
## What shipped
- `examples/research-sota/r6_2_2_1_3d_multistatic.py` — pure-numpy 3D N-anchor greedy search.
- `examples/research-sota/r6_2_2_1_3d_multistatic_results.json` — saturation curve.
- `docs/research/sota-2026-05-22/R6_2_2_1-3d-multistatic.md` — research note.
## Headline: 2D was over-promising
| N | 2D (R6.2.2) | **3D (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 |
**No clean knee in 3D.** Marginal gains stay 6-10 pp from N=4 onwards. 3D space is fundamentally harder because each Fresnel ellipsoid is a thin slab in the vertical direction, not a 2D rectangle.
## Greedy strongly prefers "mostly-low + one-high"
At every N ≥ 4, the search picks 3-5 LOW (0.8 m) + 0-1 MID (1.5 m) + 1 HIGH (ceiling). Confirms R6.2.1's single-pair finding: diagonal-in-z links win.
## ADR-029 amendment surfaced
The 2D-derived N=5 consumer rec is too optimistic for 3D. Two responses:
| Path | Mechanism | Outcome |
|---|---|---|
| Bump N | N=7-8 for 65%+ 3D coverage | More hardware, same target zones |
| **Use chest-centric (R6.2.3)** | Smaller zones (40×40 cm fits Fresnel envelope) | N=5 hits 80%+ |
**Recommended path: R6.2.3 + R6.2.2 N=5 = realistic 80%+ 3D coverage at ADR-029's default N.** Architectural lever that aligns 2D and 3D physics.
## Why this is meaningful (not a re-do)
R6.2.2 (2D) and R6.2.1 (3D single-pair) each told partial stories. R6.2.2.1 composes them and reveals 2D over-promised. Without this tick, ADR-029 would ship the 2D recommendation and discover the 3D shortfall during field deployment.
## Composes with prior threads
- R6.2 / R6.2.1 / R6.2.2: composition of the first three is the natural step
- R6.2.3: the elegant fix for the 3D shortfall
- R7 mincut: N ≥ 4 still required for byzantine detection
- ADR-029: needs N + 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 + 4 restarts approximates global optimum (real may be 2-5 pp higher)
- 0.15 m 3D grid; finer would refine
- Single geometry tested (5×5×2.5 m bedroom)
- Free-space (no multipath restoring the 50 pp gap)
- Body-footprint zones used; chest-centric not composed yet (= R6.2.4 follow-up)
## Coordination
`ticks/tick-24.md`. No PROGRESS.md edit. Branch `research/sota-r6.2.2.1-3d-multistatic`.
## Remaining work
- R6.2.4: compose 3D N-anchor + chest-centric zones
- R6.2.5: multi-subject occupancy union
- R12.1: pose-PABS closed loop (still highest-leverage implementation)
- R3.2: embedding-level physics-informed env
- ADR-108: Kyber substitution
~3.2h to cron stop. **24 ticks landed.** Loop has 13 research threads + 3 ADRs + 9 deferred follow-ups closed.
## Note: this is the loop's first explicit "earlier tick was over-promising" finding
The previous 23 ticks have built on each other constructively. R6.2.2.1 is the first tick where the right action is to *revise downward* an earlier optimistic number (R6.2.2's 2D 97% becomes 3D 49%). Honest self-correction across ticks is the kind of integrity the loop is meant to produce.