Dense-grid k* study — how many ceiling APs does the mall actually need?
Campaign: c-mall-archcad · Session: 01KY0CPRNYVMDR9EF27A0DET3P · Runs: 5 coupled walk→ray-traced-CSI seeds on mall-archcad-floor-0, experiment csi-mall-grid-dense (40 ceiling candidates)
Executive summary
The first mall study found that 5 access points reach only 57% of where the crowd walks, and asked: is that the floor being hard to cover, or just too few candidate mounting spots to choose from? We answered it by tripling the candidate grid — from 14 to 40 ceiling positions — and asking how coverage grows as we add APs. The result: you can cover ≥90% of the footfall, but it takes about 11 access points on this 84×39 m mall; five is simply not enough, no matter where you put them. Meanwhile the tension the first study found — that the best spots for radio coverage are not the best spots for counting the crowd — gets stronger, not weaker, once you have more mounts to choose from.
Why this ran
The sealed 14-mount session refuted the hoped-for "a handful of APs saturates coverage" (H3), but with only 14 candidate positions it couldn't separate "the floor needs many APs" from "we didn't offer the optimizer enough choices." A denser 40-candidate grid disambiguates: if coverage still plateaus far below 90%, the floor is the bottleneck; if it climbs to 90% at some k*, that k* is the deployment answer.
What we measured
For each candidate count k (2…12) we take the coverage-optimal set of k APs (greedy submodular selection) and record the fraction of crowd footfall within 8 m line-of-sight of at least one chosen AP. k* is the smallest k whose coverage crosses 0.90. We separately compute, per candidate, the measured mutual information I(C;Φ) between occupancy count and CSI amplitude (multi-seed mean + t-CI), and compare the counting-optimal AP set to the coverage-optimal one (1 − Jaccard).
Results
Coverage vs AP count → k*. Footfall coverage climbs cleanly and submodularly: k=5 → 0.63, k=8 → 0.80, k=10 → 0.87, k=11 → 0.908 (crosses 90%), k=12 → 0.934. So k* ≈ 11 for ≥90% footfall; ≥95% is not reached within 12. (At the original k=5 budget the dense grid already does better than the sparse one — 0.63 vs 0.57 — because it has finer positions to pick from, but 5 APs remain far short of 90%.) k* is a greedy upper bound (n=40, k≤12 exceeds exhaustive search), so read it as "about 11".
Cover ≠ count, sharpened. At k=12 the coverage-optimal set (footfall 0.93, count-info 1.44) and the counting-optimal set (footfall 0.60, count-info 2.29) share only ~26% of their mounts: 1 − Jaccard = 0.737, well above the 14-grid's 0.571. With more candidates the two objectives pull further apart, and the counting-optimal set sacrifices more footfall (down to 0.60) to chase informativeness. Seed-rank stability is ρ = 0.672 — better than the sparse grid's 0.474, because a wider candidate field (MI spread 0.284 nats) yields a clearer ordering. The top count hub is the concourse-centre mount (cand 7, I(C;Φ)=0.337).
What it means
For a real deployment on a mall floor of this size and construction, plan for roughly a dozen ceiling APs if the goal is to blanket footfall for comms — five is a coverage-sensing compromise, not a coverage solution. And the more APs you can mount, the more the crowd-counting objective diverges from the coverage objective: a genuinely count-optimised sensing deployment will look materially different from a coverage-optimised one, and will trade away a large slice of physical coverage to do it. This is the dual-objective placement problem stated quantitatively (thesis/system-design, thesis/csi-sensing).
Confidence & caveats
k* is a greedy result (exhaustive search is infeasible at n=40), so ≈11 is an upper bound on the true minimum. "≥90% footfall coverage" is an 8 m-LOS geometry statement, not a 90% counting-accuracy claim. Single floor, same crowd design as the sealed 14-grid session, seed is the unit of replication (n=5), sim-only — the ranking and the k* both remain hypotheses until a real AX210/Pi5 floor (IP-106/IP-112) confirms them.
Figure: fig_placement_oracle_mall-archcad-floor-0_dense (40-candidate grid · footfall · coverage · measured informativeness · chosen APs), at the session artefacts prefix.