The models, side by side.

Five axes separate the models more than their equations do: what comes out, what labels it needs, whether it travels to a room it has not seen, what kind of interval it gives, and what class of evidence stands behind it. A model that outputs a distribution and needs two anchors in the new room is a different instrument from one that outputs a floor and needs a fitted constant.

Was the floor empty, low, occupied or high in this minute, and how sure is the fleet?

In plain words. Nine microphones and one metronome. An empty room returns the same tick every time; people make it flutter. Each receiver measures its own flutter against its own quiet night, the fleet takes the median, and the median is resampled to see how much it depends on which receivers were listening.

The idea. Per receiver-minute, the standard deviation of the injector-scoped two-chain ratio in dB. Divided by that receiver's floor (the median of its certified-empty night minutes, rule A, else the day's 10th percentile, rule B) and taken in base-two logarithm: octaves, because floors differ threefold between receivers while people multiply them by about two everywhere. The fleet level is the median over receivers with a receiver bootstrap (1,000 resamples, 5 to 95 %). Buckets at 0.25, 0.75 and 1.50 octaves; the bucket probability is the share of bootstrap medians that fall in it.

In: the lake's per-minute blocks of the paced arm, the certified-empty night window. Out: a level in octaves per minute with a band, a bucket with a probability.

Code: web/occupancy_atlas.py: floors_for, level_of, fleet_for · Page: /toolbox/level/ · Evidence: fleet · Status: live on the site; one calibration day

What integer does this minute's level correspond to, and what interval is honest?

In plain words. A ruler with two marks on it: the empty room at zero, and every minute a person stood on the floor and wrote down how many were there. A reading between two marks is read off by straight lines; above the top mark the ruler ends and the answer is 'that many or more'.

The idea. An anchor is a labelled count and the fleet minutes inside its labelled windows (median, p10, p90). Zero is the floor by construction and is never a label. The estimate is piecewise-linear interpolation of count against level median through the anchors, rounded. The interval is the bootstrap band mapped through the same inverse, widened to every anchor whose own spread overlaps it, so two anchors 0.05 oct apart share one interval. Above the top anchor the upper bound is open.

In: fleet levels of the count method; labelled windows written through occupancy_label_write. Out: an integer per minute with an interval; open above the top label.

Code: web/occupancy_atlas.py: anchors_for, count_map, counts_for · Page: /toolbox/occupancy/ · Evidence: fleet · Status: live on the site (IP-148); one anchor

How likely is each count given this minute, and how wide must a bucket be to be honest?

In plain words. Instead of reading one number off the ruler, ask every mark how well it explains the reading. The marks that explain it share the probability. Then ask the ruler itself: how far apart must two marks be before a reading can tell them apart nine times in ten? That distance is the bucket you may quote.

The idea. The level law through the anchors gives each count an expected level and a spread. The minute's likelihood under count N is normal with variance the law's spread squared plus the bootstrap half-width squared; counts above the top anchor share the top's likelihood and are one state, 'more'. Nested sets are the smallest run of counts around the mode reaching 50, 80, 90 and 95 %. The resolvable bucket width at N is the spread divided by the slope of the law, times z; with two labels the saturating law of the scattering model is fitted and gives the slope past the knee.

In: the same anchors and fleet rows as the anchor map. Out: a probability per count, nested sets at four confidences, bucket families, a resolution ladder.

Code: web/counting.py · Page: /toolbox/confidence/ · Evidence: fleet · Status: live on the site (2026-09-05); one anchor

What shape does 'more people, more variance' have, and is it derived or fitted?

In plain words. Each moving person is a small moving mirror. Add mirrors and the signal jitters more, but a room holds only so many mirrors before they shadow each other. The jitter rises fast and then levels off, and that curve falls out of the physics rather than being drawn through the points.

The idea. The channel is a static component plus K independent moving random phasors. The fluctuation power of |H| grows with K at low density (a central-limit argument), and a finite room with mutual occlusion caps the effective number of scatterers, so the law is concave and saturating with a knee K₀. Two predictions follow: concave beats linear on real data, and the shorter 5 GHz wavelength gives a larger phase swing per body displacement, so 5 GHz is the counting band.

In: per-window temporal CV of CSI amplitude against the labelled count (WiMANS 5 GHz, meneghello 80 MHz). Out: a concave link with a knee N₀; the shape the count map should bend into.

Code: notebooks/python/csi_scattering_link.py · Paper: csi-recalibration-plane (§ The mechanism has a shape) · Hypothesis: scattering-saturation-link (chosen, supported) · Evidence: public · Status: supported on public data; hardware test pending (crowd-day staircase)

Is cross-room counting one model with predictable recovery, or a bag of separate fits?

In plain words. 'More people, more wobble' is the same law in every room; only two constants change from place to place. Treat every room as one family sharing that law, and two Bluetooth calibration points in a new room let it borrow the family's knowledge instead of starting from zero. How much you recover with how few points is predictable.

The idea. count = α_e + β_e · log CV per environment e, with (α_e, β_e) drawn from a population whose covariance Σ carries the environment variance, nested inside a platform (WiFi card) level. 'The mechanism travels' is a tight, sign-stable slope population; 'the scale does not' is real variance in α_e and β_e with the card as the top level. For a new room with k anchors the empirical-Bayes posterior shrinks from the population mean toward the local fit, and the error-against-k curve is closed form.

In: the pooled cross-environment corpus: (dataset, environment, count, CV), n = 939, 7 environments, 2 platforms. Out: per-environment slope and intercept with shrinkage; a recovery-against-anchors curve.

Code: notebooks/python/csi_hierarchical_calibration.py · Paper: csi-recalibration-plane · Hypothesis: hierarchical-calibration-shrinkage (chosen, supported) · Evidence: public · Status: supported on public data; the thesis spine

Which likelihood family fits counts conditioned on the CSI feature, and is it calibrated?

In plain words. Given the wobble, how spread out are the true counts? Less spread than a Poisson would give, it turns out, in every room. Poisson cannot say that and the negative binomial can only add spread, so a third family with a dial that goes both ways is the one whose intervals mean what they say.

The idea. Counts conditional on log CV are under-dispersed (conditional variance below conditional mean). Poisson forces equality and NB2 only adds variance, so a mean-parametrised Conway–Maxwell–Poisson with dispersion ν covering both sides is fitted with the same mean structure (log link, per-environment intercept, shared slope). Pre-registered kill lines on the premise, on zero-shot likelihood gain, and on calibration by randomised PIT.

In: the same n = 939 corpus, x = log CV. Out: a predictive distribution over the count per window, with a calibrated central interval.

Code: notebooks/python/csi_com_poisson_head.py · Paper: calibrated-count-intervals · Hypothesis: distributional-output-contract (chosen, supported) · Evidence: public · Status: PUB-5 falsifier run; calibrated in-env, no zero-shot gain

Can a count interval promise its coverage without a distribution, and does the promise survive a new room?

In plain words. A counter that says 'between 9 and 15, and I am right nine times in ten' is a promise that can be checked. Conformal prediction wraps any counter in such a promise using only past residuals. The question is whether the promise survives being carried into a room the residuals never saw.

The idea. Split conformal on the partial-pooling regressor's residuals: the 90 % quantile of calibration nonconformity scores is the half-width. Mondrian per environment gives per-room validity in-room. Under leave-one-environment-out the scores are not exchangeable and coverage collapses; anchoring with k = 2 residuals at weight 1 and the other rooms' anchored residuals at weight 0.5 (beyond-exchangeability weights fixed a priori) repairs it.

In: the partial-pooling model's residuals on the n = 939 corpus; k = 2 anchors per held-out room. Out: an interval per window with a stated coverage; its width as a shift signal.

Code: notebooks/python/conformal_counts.py · Paper: calibrated-count-intervals · Hypothesis: conformal-count-intervals (candidate, plausible) · Evidence: public · Status: candidate; in-room valid, transfer repaired by anchors, not uniform

Why does the count response saturate, and what is the recoverable count in principle?

In plain words. Each occupant adds one faint, coherent voice to a room already full of hiss. A voice is heard only when it rises above the hiss by a fixed margin set by how many microphones listen for how long. Count the voices you can hear and you have the occupancy the array can recover; the ones below the margin are the saturation.

The idea. Model each occupant as a low-rank coherent perturbation of the CSI sample covariance. Random-matrix theory says a scattering mode is resolvable exactly when its signal-to-noise ratio exceeds the Baik–Ben Arous–Péché threshold √γ set by the aspect ratio γ = p/T (bins over snapshots). The recoverable count is the number of sample eigenvalues above the Marchenko–Pastur bulk edge, estimable by the Kritchman–Nadler sequential test; the concave saturation is the BBP transition.

In: per-window CSI amplitude covariance over p = 270 antenna–subcarrier bins, T snapshots (WiMANS 5 GHz, 108 records). Out: a recoverable count and a predicted knee as a function of (γ, per-person SNR).

Code: publications/rmt-csi-counting (figdata, reductions) · Paper: rmt-csi-counting; csi-covariance-theory · Hypothesis: spectral-count-eigenvalue-threshold (candidate, supported) · Evidence: public · Status: paper drafted; the knee location untested directly

How well can any estimator count from one window, and what sets the floor?

In plain words. Before building a better counter, ask what the best possible one could do with this signal. Three floors sit under every estimator: the noise of a finite window, the fact that the occupancy pattern has only a few independent directions, and the room's own day-to-day variability that no window length removes. The last one is the operative floor.

The idea. An occupancy-conditioned Gaussian channel in which each occupant adds diffuse scattered power: R(N) = R₀ + s(N)·B with s(N) = P_d0 + γN. The Slepian–Bangs formula gives the Fisher information of N in closed form, exposing three nested bounds: a full-rank CRB scaling as s²/(npγ²), a rank-r bound where low-rank coherence collapses the aperture gain from p to r, and a structural floor set by session-level variability τ² that no n or p reduces. A Fano converse over the discrete counts lower-bounds the error probability of any estimator.

In: 108 WiMANS 5 GHz classroom records, 18 per count 0 to 5, p = 270 bins. Out: a variance floor per count, a Fano error floor, and the gap decomposition of the achieved estimator.

Code: notebooks/python/csi_crb_limits.py · Paper: csi-counting-limits; csi-covariance-theory · Evidence: public · Status: paper drafted (PUB-7); numeric leg run

Can integrating over time escape the per-window floor, and what should a rare Bluetooth anchor do?

In plain words. People do not teleport: from one minute to the next the count moves by one or stays. A filter that knows this smooths the noisy per-minute guesses into a track that beats any single minute. When a rare Bluetooth check arrives, the right use is not to reset the count but to re-learn how the WiFi reading maps to a count, because that map is what drifted.

The idea. A hidden Markov model over the occupancy state with a physics-grounded birth–death transition (±1 immigration–death) and the per-window emission, run as a log-domain forward filter. A BLE anchor enters as a control input that resets the state and re-identifies the drifted emission by EB shrinkage. A CUSUM on the innovation is the drift trigger. The Kalman reading of the same loop: the gain on the CSI-implied estimate decays between anchors, which is the observed 'staleness switch'.

In: a per-window count feature stream (OPERAnet 2.4 GHz, WiMANS 5 GHz), a periodic anchor. Out: a filtered count track, an anchor policy, a drift trigger.

Code: domain/counting_filter.py; notebooks/python/csi_state_space_filter.py · Paper: csi-calibration-control (PUB-9 proposal) · Hypothesis: state-space-fusion-optimality (candidate, supported) · Evidence: public · Status: candidate; real-feature demonstration, hardware drift pending

Does a periodic Bluetooth anchor bound the drift a CSI-only map suffers across floors?

In plain words. In a simulated building, a WiFi map trained on one floor is carried to another and drifts. A Bluetooth head-count every so often re-fits it. The picture shows the mechanism; the numbers do not survive, because the Bluetooth count was fitted to the same run it then corrected.

The idea. Coupled JuPedSim walk → Sionna CSI and co-registered BLE RSSI from one solved channel. CSI temporal CV is the high-rate relative signal; BLE aggregate attenuation the low-rate absolute anchor; the anchor periodically re-fits the CSI→count affine. A learned-feature probe (Doppler statistics into a small MLP) tested whether the synthetic CSI was under-featured.

In: synthetic CSI and BLE from three ResPlan floors, two bands. Out: a fused timeline and a cross-floor transfer MAE.

Code: notebooks/python/fusion_count.py; csi_learned_count.py · Evidence: sim · Status: illustration only; superseded by the filter above

The building blocks are textbook: random phasors, partial pooling, Conway–Maxwell–Poisson, split conformal, Marchenko–Pastur, Slepian–Bangs, a hidden Markov model. The corpus already holds counters that output a posterior (the renewal-process estimator of Depatla et al., the fidget-bandwidth estimator of Torun et al.) and a static-emission particle filter for CSI counting. None of that is this project's.

What the corpus does not hold, and these pages do, is the set of statements that tie the blocks together on one occupancy-conditioned covariance: that the count response saturates because of a phase transition whose knee is fixed by the aspect ratio and the per-person SNR; that the same covariance gives a Cramér–Rao floor set by session variability rather than by snapshot noise, so more antennas and longer windows stop helping; that cross-room recovery with two anchors is empirical-Bayes shrinkage with a predicted value rather than a lucky number; that counts conditioned on the feature are under-dispersed in every room measured; that a Bluetooth anchor used to re-identify the emission recovers most of a drift where a count reset recovers little; and that a bucket's honest width at each count follows from the law's slope and spread, before anyone quotes one. The one instrument that reaches the library floor is the last: the detector chain on the Toolbox, calibrated on one day with one label above zero.

Each of those statements has a paper draft or a hypothesis note behind it; the ones with measured support are marked in the table. What none of them has yet is a measured drift on this fleet, which is the leg every calibration claim waits on.