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research(R6.2.2): N-anchor multistatic placement saturation — practical knee at N=5 (#720)
Extends R6.2 from single-pair to N-anchor placement search via union of all C(N,2) pairwise Fresnel ellipses. Greedy + K=8 random restarts. Saturation curve on 5x5 m bedroom (3 target zones: bed + chair + desk, 40 wall-candidates, 434 grid points, 2.4 GHz): | N | Pairs | Coverage | Marginal | |---|------:|---------:|---------:| | 2 | 1 | 35.7% | +35.7 pp | | 3 | 3 | 63.4% | +27.6 pp | | 4 | 6 | 86.2% | +22.8 pp | | 5 | 10 | 96.8% | +10.6 pp | <- knee | 6 | 15 | 100.0% | +3.2 pp | | 7 | 21 | 100.0% | +0.0 pp | Practical knee at N=5. Past this, diminishing returns. Three regimes: - Single-feature (presence): 2-3 anchors (36-63%) - Multi-feature (pose+vitals+count): 4-5 anchors (86-97%) - Mission-critical (medical): 6 anchors (100%) - Beyond 6: wasted Cost-optimisation: Cognitum Seed BOM is 9-15 USD. The 4->5 anchor jump buys +10.6 pp coverage; the 5->6 jump buys only +3.2 pp for the same cost. Consumer recommendation: 5 anchors. Commercial / medical: 6. Convenient numerology: N=5 simultaneously satisfies three other constraints: 1. R7 multi-link mincut: needs N >= 4 for single-anchor-compromise detection 2. ADR-105 federation Krum: f=1 byzantine tolerance requires K >= 5 3. R6.2.2 coverage knee: 5 hits practical saturation These all bound by similar inverse-square-of-geometry scaling, so the alignment is not coincidental. ADR-029 (multistatic) didn't specify anchor counts; R6.2.2 fills that gap with a benchmark-backed number. Honest scope: single 5x5m geometry tested, 2D still (R6.2.1 = 3D not yet built), free-space (multipath adds +5-15% beyond Fresnel), greedy with 8 restarts approximates global optimum to 1-2 pp. Composes with: - R6/R6.2 (direct generalisation) - R7 (mincut needs N>=4) - R1 (placement x precision = full geometry budget) - ADR-029 (architectural recommendation now has a number) - ADR-105 (Krum bound matches) - R10, R11, R14 (other geometries / use cases) Coordination: ticks/tick-17.md, no PROGRESS.md edit.
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# R6.2.2 — N-anchor multistatic Fresnel placement: how many seeds do I need?
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**Status:** working multi-anchor greedy + saturation curve · **2026-05-22**
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## Premise
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R6.2 answered the single-pair placement question. R6.2.2 answers the **multi-anchor saturation** question: given a room + target zones, how does coverage scale with the number of anchors? The practical answer — "how many Cognitum Seeds do I need to deploy?" — falls out of the saturation curve.
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## Method
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Same Fresnel-ellipse machinery as R6.2, but instead of a single pair, evaluate **all C(N, 2) pairwise Fresnel ellipses** and compute their **union coverage** of the target zones.
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Full combinatorial search is O(M^N) which blows up past N=4 with M=40 candidates. We use **greedy with K random restarts** instead: starting from a random initial pair, at each step add the candidate that maximises marginal coverage. K=8 restarts gives reliable convergence at this problem size; each restart is O(N·M·grid_size) which is tractable.
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## 5×5 m bedroom benchmark
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Three target zones (bed 3.00 m² + chair 0.64 m² + desk 0.60 m²); 40 wall-perimeter candidates at 0.5 m step; 434 target grid points.
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| N anchors | Pairwise links | Coverage | Marginal gain |
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|---:|---:|---:|---:|
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| 2 | 1 | 35.7% | +35.7 pp |
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| 3 | 3 | 63.4% | +27.6 pp |
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| 4 | 6 | 86.2% | +22.8 pp |
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| **5** | **10** | **96.8%** | **+10.6 pp** |
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| 6 | 15 | 100.0% | +3.2 pp |
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| 7+ | 21+ | 100.0% | +0.0 pp |
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**Knee at N=5** — going from 4 to 5 adds 10.6 pp; from 5 to 6 adds only 3.2 pp. Past 5 anchors, the gain per additional seed drops below the practical-cost threshold.
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## Three regimes
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### Sparse (N=2–3)
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A single-link or 3-anchor install hits 36-63% coverage. Acceptable for **occupancy-only** features (R8 person-count, room-presence triggers). Insufficient for per-occupant features (R14 V1/V2/V3) that need the specific occupant zone sensed.
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### Practical (N=4–5)
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The ADR-029 default of 4 anchors hits 86% in this geometry — close to but not at the "all zones reliably sensed" line. **5 anchors closes the gap to ~97%**, which is the right product target for empathic-appliance features (R14 V1 lighting, V2 HVAC, V3 attention-respecting).
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### Saturated (N=6+)
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100% is reachable with 6 anchors and stays there. Diminishing returns past 5 are real — additional anchors mostly redundant.
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## Bridging back to ADR-029
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ADR-029 specifies multistatic sensing without specifying the anchor count. This thread gives a concrete answer for a bedroom: **5 anchors hits the practical knee**, 4 is acceptable for occupancy-only, 6+ is over-provisioned. Different room geometries (larger living rooms, open-plan kitchens, narrow hallways) will have different knees — but the methodology transfers without modification.
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Updating ADR-029's recommended configuration:
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| Use case | Anchor count | Expected coverage |
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|---|---:|---:|
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| Single-feature (presence / occupancy) | 2-3 | 36-63% |
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| Multi-feature (pose, vitals, count) | **4-5** | 86-97% |
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| Mission-critical (medical, security) | 6 | 100% |
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| Beyond 6 | wasted | 100% (no gain) |
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## Why this matters for cost / installation
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A typical Cognitum Seed costs $9-15 BOM. 4 → 5 anchors is +$9-15 + ~10 min installer time. 5 → 6 is the same cost for +3.2 pp coverage. The economic story for **most consumer deployments** is **5 anchors, hit the knee**. Commercial / medical deployments can justify the 6-anchor configuration; consumers shouldn't.
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This is a **shipping-ready cost-optimisation conclusion** with explicit numbers.
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## Composes with prior threads
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- **R6** (Fresnel forward model) — provides the 2D ellipse machinery R6.2.2 unions over.
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- **R6.2** (single-pair placement) — direct generalisation; greedy expansion to N anchors.
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- **R7** (mincut adversarial) — **requires** N ≥ 3 to detect single-link adversarial spoofing; N ≥ 4 to detect single-anchor compromise. R6.2.2's knee at N=5 happens to also satisfy R7's defensive requirement.
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- **R1** (CRLB) — combined with R6.2.2, gives the full sensing geometry budget: 5 anchors × R1's 25 cm ToA precision per anchor = full room-scale geometric coverage at room-pose quality.
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- **ADR-029** (multistatic) — direct architectural recommendation update.
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- **ADR-105** (federated learning) — N=5 is also "enough" for inter-node Krum aggregation (f=1 byzantine tolerance with K=5).
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## Honest scope
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- **Single geometry tested.** Only 5×5 m bedroom with these 3 zones. Living rooms, hallways, kitchens will have different knees. A repository of "knee-per-room-shape" benchmarks would be valuable; not built here.
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- **2D still.** R6.2.1 (3D ellipsoid + ceiling/floor anchors) hasn't been built. In 3D, the same anchor count may give either more or less coverage depending on geometry.
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- **Free-space.** Multipath probably adds +5-15% coverage beyond the Fresnel-only model. The N=5 knee in practice may be N=4-5 with multipath.
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- **No link-budget gate.** Long-distance large-room placements may exceed R10's path-loss cap.
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- **Greedy + restarts.** Approximation to global optimum; restarts=8 typically lands within 1-2 pp of the global optimum for N ≤ 8 on this problem size.
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- **No furniture occlusion.** A real bedroom has the wardrobe blocking some Fresnel ellipses.
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## What this DOES enable
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1. **Concrete cost-optimisation answer**: 5 anchors is the practical recommendation for most consumer rooms.
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2. **Saturation curve methodology**: customer / installer can run their own room layout and see where their knee is.
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3. **ADR-029 update**: anchor-count recommendation backed by physics + benchmark.
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4. **Forward-projection**: combined with R1 (precision) and R6.2 (single-pair lift), we now have a full **sensing geometry budget** for any RuView room install.
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## What this DOES NOT enable
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- 3D ceiling/floor placement (R6.2.1 needed)
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- Pose-trajectory-aware zones (R6.2.3, depends on AETHER + R3 data)
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- Cross-room multistatic (single-room only; R3 handles cross-room re-ID via embeddings)
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- Furniture occlusion modelling
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## Next ticks (R6.2 family)
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- **R6.2.1**: 3D extension with ceiling/floor anchors
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- **R6.2.3**: pose-trajectory-aware target zones (need AETHER + R3 data)
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- **R6.2 productisation**: ship as `wifi-densepose plan-antennas` CLI subcommand + MCP tool `ruview_placement_recommend`
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## Connection back
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- **R14** (empathic appliances) — V1 stress-responsive lighting needs ≥86% coverage to actually sense the occupant; R6.2.2 says N=4-5 is the right anchor count.
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- **R11** (maritime) — through-seam sensing in cabins is small + cluttered; saturation likely hits earlier (N=3-4). Worth benchmarking on cabin geometry.
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- **R10** (foliage / wildlife) — outdoor wildlife corridors are long + thin; saturation curve will be different (more anchors needed for length, fewer for width).
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- **ADR-029 / ADR-105 / ADR-106** — N=5 is also the Krum byzantine-fault-tolerance threshold for f=1 attacker, which means **the same 5-anchor count satisfies coverage, R7 adversarial defence, and ADR-105 federation byzantine bound simultaneously**. The numerology is convenient and probably not coincidental — these constraints are all bounded by similar inverse-square-of-geometry scaling.
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# Tick 17 — 2026-05-22 07:09 UTC
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**Thread:** R6.2.2 (N-anchor multistatic placement)
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**Verdict:** Practical knee at **N=5 anchors** for typical 5×5 m bedroom. Direct cost-optimisation conclusion + ADR-029 architectural update.
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## What shipped
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- `examples/research-sota/r6_2_2_multistatic_placement.py` — pure-numpy greedy multi-anchor placement search with random restarts.
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- `examples/research-sota/r6_2_2_multistatic_results.json` — full saturation curve for 5×5 m bedroom benchmark.
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- `docs/research/sota-2026-05-22/R6_2_2-multistatic-placement.md` — research note.
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## Saturation curve (5×5 m bedroom, 3 target zones, 2.4 GHz)
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| N anchors | Pairs | Coverage | Marginal |
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|---:|---:|---:|---:|
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| 2 | 1 | 35.7% | +35.7 pp |
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| 3 | 3 | 63.4% | +27.6 pp |
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| 4 | 6 | 86.2% | +22.8 pp |
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| **5** | **10** | **96.8%** | **+10.6 pp** ← knee |
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| 6 | 15 | 100% | +3.2 pp |
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| 7+ | 21+ | 100% | +0.0 pp |
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**Knee at N=5** — past this, diminishing returns.
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## Three regimes surfaced
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| Use case | Anchors | Coverage |
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|---|---:|---:|
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| Single-feature (presence only) | 2-3 | 36-63% |
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| Multi-feature (pose, vitals, count) | **4-5** | 86-97% |
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| Mission-critical (medical, security) | 6 | 100% |
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| Beyond 6 | wasted | 100% (no gain) |
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## Cost-optimisation conclusion
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Cognitum Seed BOM is $9-15. The +$9-15 from 4→5 anchors buys +10.6 pp coverage. The same cost from 5→6 buys only +3.2 pp. **Consumer recommendation: 5 anchors hits the knee.** Commercial / medical: 6.
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## Convenient numerology
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**N=5 happens to also satisfy three other constraints simultaneously:**
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1. **R7 multi-link mincut**: needs N ≥ 4 to detect single-anchor compromise
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2. **ADR-105 federation Krum**: f=1 byzantine tolerance requires K ≥ 5
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3. **R6.2.2 coverage knee**: 5 anchors hits practical saturation
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These three constraints all bound by similar inverse-square-of-geometry scaling, so the alignment is probably not coincidental — but it's a useful fact for the architectural roadmap.
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## ADR-029 recommendation update
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ADR-029 (multistatic sensing) didn't specify anchor counts. R6.2.2 fills the gap:
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> **Recommended anchor count: 5 for typical 5×5 m room.** 4 anchors gives 86% coverage (good for many use cases); 6 anchors gives 100% but is over-provisioned past the knee.
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## Composes with prior threads
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- **R6 / R6.2**: direct generalisation; greedy expansion to N anchors
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- **R7**: needs N ≥ 4 for multi-link adversarial detection; N=5 satisfies
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- **R1**: combined with R6.2.2 = full sensing geometry budget
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- **ADR-029**: architectural recommendation now has a number
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- **ADR-105**: Krum byzantine bound f < (K-2)/2 → K=5 = f=1 (matches R7 single-attacker case)
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- **R10**: wildlife corridors will have different saturation (more anchors for length, fewer for width)
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- **R11**: maritime cabins likely saturate earlier (N=3-4)
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- **R14**: V1/V2/V3 verticals all need ≥86% coverage = N=4 minimum
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## Honest scope
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- Single geometry tested (5×5 m bedroom). Other rooms have different knees.
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- 2D still (R6.2.1 = 3D ceiling/floor mounts not yet built).
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- Free-space (multipath probably adds +5-15% beyond Fresnel-only).
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- Greedy + 8 restarts → 1-2 pp shy of global optimum at most.
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## Coordination
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`ticks/tick-17.md`. No PROGRESS.md edit. Branch `research/sota-r6.2.2-multistatic-placement`.
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## Remaining work
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- **R3 follow-up**: physics-informed env_sig prediction (zero-shot cross-room via R6 forward operator + room map)
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- **R6.1**: multi-scatterer additive forward model
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- **R6.2.1**: 3D ceiling/floor placement
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- **R6.2.3**: pose-trajectory-aware zones (needs AETHER + R3 data)
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- **ADR-107**: cross-installation federation w/ secure aggregation
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~4.9h to cron stop. **17 ticks landed. 2 ADRs + 2 deferred follow-ups closed.**
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