diff --git "a/Degree-aware recalibration of PIDC\342\200\224a companion plan.md" "b/Degree-aware recalibration of PIDC\342\200\224a companion plan.md"
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+**Status:** draft. Companion to `interventional_pidc_plan.md`. Shares Stages 1–5 of that pipeline; this document covers only the calibration layer. Sections marked ⚠️ contain assumptions to correct before implementation.
+
+---
+
+## 0. What is being changed, and why it is legitimate
+
+PIDC's final step fits, for each gene $i$, a gamma distribution to the empirical set $\{\mathrm{PUC}(i,j)\}_{j\neq i}$ and converts each edge score into a tail probability under that gene's own fit. The step is usually described as removing gene-specific nuisance variation, and it does. But it also carries an unstated topological assumption:
+
+> every gene's score distribution has the same _shape_, therefore at a fixed confidence threshold every gene has approximately the same expected degree.
+
+That is a homogeneity prior on degree. It is not neutral and it is almost certainly wrong for regulatory networks, where out-degree is heavy-tailed (a TF can act on thousands of loci) while in-degree is sharply constrained (finite promoter real estate). So the question is not whether to impose a degree assumption — PIDC already does — but whether a better one improves recovery.
+
+**This is a re-calibration project, not a new score.** PUC is untouched. Only the mapping from PUC to edge confidence changes. That keeps the contribution narrow, cheap to implement, and directly comparable against stock PIDC as a baseline.
+
+### Four variants, in increasing ambition and decreasing confidence
+
+| # | Variant | Assumption added | Prior |
+|---|---|---|---|
+| **V2** | Hierarchical/empirical-Bayes gamma fits | none beyond stock PIDC | **High** — pure variance reduction |
+| **V1** | Direction-aware calibration (post-orientation) | in/out asymmetry | High — mechanistically grounded |
+| **V4** | Degree-corrected null (configuration-model style) | expected-degree structure | Moderate |
+| **V3** | Global degree-regularized thresholding | explicit $P_{\mathrm{deg}}$ | Moderate, with a known hub-suppression failure mode |
+
+Implement in the order **V2 → V1 → V4 → V3**. V2 is a strict improvement with no new assumptions and plausibly fixes real instability; V3 is the most likely to actively harm results.
+
+---
+
+## 1. Prior art and the honest novelty position
+
+- Degree-corrected and degree-penalized estimation is established in the sparse graphical model literature (degree-corrected graphical lasso and relatives), and degree priors appear in Bayesian network structure learning.
+- Configuration-model nulls are standard in community detection.
+- **I am not aware of published work modifying PIDC's gamma calibration specifically.** Same epistemic status as the mediation idea in the companion plan: a construction that follows from available pieces, not something citable.
+
+**Action before writing method text:** search the degree-corrected graphical model literature properly. The statistical problem there is close enough that someone may have solved a borrowable form, and if so this becomes an application rather than a method paper. That is a fine outcome but it changes the framing.
+
+---
+
+## 2. The circularity trap — read before designing evaluation
+
+If $P_{\mathrm{deg}}$ is fit to a curated network (TRRUST, DoRothEA, ChIP-derived) and then used to regularize inference, **you cannot claim the inferred network recovers realistic topology.** You assumed it. Any evaluation scoring topological realism becomes vacuous, and reviewers will say so.
+
+Three compounding problems with importing a prior from curated sources:
+
+1. **Ascertainment bias.** Khanin & Wit (2006): observed scale-freeness in biological networks partly reflects that well-studied genes accumulate more recorded edges.
+2. **Scale-freeness is contested.** Broido & Clauset (_Nat Commun_ 2019) tested ~1,000 real networks and found strict power laws rare, with log-normal fitting at least as well. Do not hard-code $P(k)\sim k^{-\gamma}$.
+3. **Binding degree ≠ transcriptomic degree.** This is the most serious for this application. A TF with 3,000 binding sites may move 40 genes at steady state. Curated degree distributions are binding-derived; PIDC edges are steady-state covariation. The relevant distribution is that of the _effective functional_ network, and it is not the one in the literature.
+
+### The way out: fit the prior on held-out perturbation targets
+
+The Arc hESC dataset provides ~300 CRISPRi targets and a directed effect matrix $E$ (companion plan, Stage 4). Effective out-degree can therefore be **measured** in the correct cell context, functionally defined, with no curation bias:
+
+$$\hat{k}^{\mathrm{out}}_g = \#\{\,j : E[g,j] \text{ significant vs. NTC-split null}\,\}$$
+
+Fit $P_{\mathrm{deg}}$ on a subset of targets, evaluate calibration on held-out targets. 5-fold splits over 300 targets. This is the experiment that makes the whole line defensible, and it is only possible because interventional data is available.
+
+**⚠️ Checkpoint D0 (gate, run first):** confirm enough targets have well-estimated effective out-degree — sufficient cells per condition, self-effect present (companion Checkpoint 3), non-degenerate $\hat k^{\mathrm{out}}$ spread. If $\hat k^{\mathrm{out}}$ is near-zero for most targets or has no dynamic range, there is no prior to fit and the project reduces to V2 alone. Cheap to check; decides scope.
+
+Note the asymmetry this leaves: **in-degree cannot be measured this way.** Estimating $\hat k^{\mathrm{in}}_j$ requires knowing how many _regulators_ of $j$ were perturbed, and with 300 targets out of ~20,000 genes the sampling fraction is far too low. So in-degree constraints must come from either a literature prior (with all the caveats above, stated) or from ATAC/motif counts in accessible regions near the promoter — which is at least context-matched, if still binding-derived rather than functional. Prefer the latter and be explicit that it is a weaker leg than the out-degree side.
+
+---
+
+## 3. V2 — Hierarchical gamma calibration
+
+**Do this first.** Stock PIDC fits each gene's gamma independently from $p-1$ PUC values. For genes with few informative partners — low-expression, low-variance, heavily filtered — that fit is noisy, and the noise propagates directly into edge confidence. This is a plausible contributor to the bootstrap instability flagged in companion Checkpoint 2.
+
+Fix: put a prior on the gamma parameters across genes and shrink each gene's fit toward the global distribution, with shrinkage inversely proportional to that gene's effective information content.
+
+Parameterize gamma by shape $\alpha_i$, rate $\beta_i$; work on the log scale:
+
+$$\log\alpha_i \sim \mathcal{N}(\mu_\alpha,\sigma_\alpha^2), \qquad \log\beta_i \sim \mathcal{N}(\mu_\beta,\sigma_\beta^2)$$
+
+Method-of-moments per gene, then James–Stein-style shrinkage toward the pooled mean is adequate and avoids MCMC over ~20,000 genes:
+
+```julia
+# ⚠️ assumes access to per-gene PUC vectors; see companion plan §2
+using Statistics, Distributions
+
+"""
+Shrink per-gene gamma fits toward a global fit. `puc` is genes × genes
+(or a vector of per-gene PUC vectors). Returns shrunk (α, β) per gene.
+"""
+function hierarchical_gamma_fits(puc_by_gene::Vector{<:AbstractVector})
+    fits = [fit_mle_gamma_mom(v) for v in puc_by_gene]
+    logα = [log(f[1]) for f in fits]
+    logβ = [log(f[2]) for f in fits]
+    μα, μβ = mean(logα), mean(logβ)
+    τα, τβ = var(logα), var(logβ)
+
+    out = similar(fits)
+    for i in eachindex(fits)
+        n  = length(puc_by_gene[i])
+        # per-gene sampling variance of logα ~ O(1/n); crude but serviceable
+        vα = 1 / max(n - 1, 1)
+        vβ = 1 / max(n - 1, 1)
+        wα = τα / (τα + vα)          # shrinkage weight toward own estimate
+        wβ = τβ / (τβ + vβ)
+        out[i] = (exp(wα * logα[i] + (1 - wα) * μα),
+                  exp(wβ * logβ[i] + (1 - wβ) * μβ))
+    end
+    return out
+end
+
+function fit_mle_gamma_mom(v)
+    m, s2 = mean(v), var(v)
+    s2 <= 0 && return (1.0, 1.0)
+    β = m / s2; α = m * β
+    return (α, β)
+end
+```
+
+Refinements worth trying: use _effective_ sample size rather than raw $n$ (genes with many tied/zero-bin cells have less information than $n$ suggests); consider shrinking toward a covariate-dependent mean, e.g. regressing $\log\alpha$ on detection rate and mean expression, so shrinkage respects known nuisance structure rather than pulling everything to one point.
+
+**Checkpoint D1 (gate):** bootstrap edge-recovery frequency at matched density, V2 vs. stock PIDC, on control cells. V2 must **not** reduce stability. Expected result is an increase concentrated in low-information genes — stratify the comparison by detection rate to see it. If stability is unchanged, V2 is a null result: report it and move on, since it costs little.
+
+---
+
+## 4. V1 — Direction-aware calibration
+
+PUC is symmetric, so a single per-gene gamma cannot express in/out asymmetry. After companion Stage 5 supplies orientations from reciprocal effect asymmetry, calibrate the two roles separately:
+
+- gene acting as **regulator** (out-edges): permissive, heavy-tailed null — do not penalize high out-degree
+- gene acting as **target** (in-edges): strict, thin-tailed null — penalize in-degree beyond $k^{\mathrm{in}}_{\max}$
+
+Simplest defensible implementation is a soft in-degree penalty rather than a hard cap:
+
+```julia
+"""
+Re-rank oriented edges with a soft in-degree penalty.
+`edges` :: Vector{(src, dst, score)} sorted descending by score.
+Greedy accept with penalty growing in the target's current in-degree.
+"""
+function indegree_penalized_select(edges, n_genes; kmax = 8, λ = 1.0, budget = Inf)
+    indeg = zeros(Int, n_genes)
+    kept = similar(edges, 0)
+    for (s, d, sc) in edges
+        penalty = λ * max(0, indeg[d] - kmax)
+        sc - penalty <= 0 && continue
+        push!(kept, (s, d, sc))
+        indeg[d] += 1
+        length(kept) >= budget && break
+    end
+    return kept
+end
+```
+
+Sweep $k^{\mathrm{in}}_{\max}$ and $\lambda$; select on **held-out perturbation targets**, never on the fitting set.
+
+**Checkpoint D2 (gate):** V1 must improve recovery of held-out $E$-derived edges over V2 at matched density. If it does not, the in-degree constraint is either wrong for the effective-functional network or the orientations are too noisy to support it — diagnose which by re-running with orientations restricted to the most confidently asymmetric pairs only.
+
+---
+
+## 5. V4 — Degree-corrected null
+
+Rather than changing the prior, change the null. Score an edge by excess over a null that preserves expected degrees — the configuration-model logic from community detection, applied to a weighted score matrix.
+
+For gene strengths $s_i=\sum_j \mathrm{PUC}(i,j)$ and total $S=\sum_i s_i$, the expected score under a degree-preserving null is $\mathbb{E}[\mathrm{PUC}(i,j)] \approx s_i s_j / S$, giving
+
+$$\tilde s_{ij} = \mathrm{PUC}(i,j) - \gamma \frac{s_i s_j}{S}$$
+
+with resolution parameter $\gamma$. This asks the question PIDC's gamma step was groping toward — _is this edge surprising given how connected these two genes generally are_ — but does so jointly rather than per-gene. It is O(p) to compute from row sums, so it is essentially free.
+
+Calibrate significance by permutation within degree strata rather than parametrically. Note the tension with V2: both address gene-level nuisance, so run them separately before combining, and check they are not double-correcting.
+
+---
+
+## 6. V3 — Global degree-regularized thresholding
+
+The most ambitious and the most dangerous. Choose the edge set jointly:
+
+$$\hat E = \arg\max_E \ \sum_{(i,j)\in E} s_{ij} \ + \ \lambda \sum_i \log P_{\mathrm{deg}}(k_i)$$
+
+Combinatorial; a greedy add/swap or Lagrangian relaxation is adequate. Practical effect: redistribute edges away from genes that accumulated many marginal ones toward genes with a few strong ones.
+
+**Known failure mode, stated up front:** if $P_{\mathrm{deg}}$ has too thin a tail, V3 actively suppresses genuine hubs — and hubs are usually the thing most worth finding. Mitigations: fit $P_{\mathrm{deg}}$ from measured $\hat k^{\mathrm{out}}$ (§2) rather than a curated network; use a flexible heavy-tailed family (log-normal, Weibull, discrete power-law-with-cutoff) and select by AIC rather than assuming a form; report results across $\lambda$ including $\lambda=0$ so hub suppression is visible.
+
+**Checkpoint D3 (gate):** track recovery of _known_ hESC hubs (POU5F1, NANOG, SOX2) as a function of $\lambda$. If hub out-degree collapses before overall recovery improves, V3 fails on this data — report the negative and stop. This is a likely outcome and worth saying so in advance.
+
+---
+
+## 7. Evaluation, shared across variants
+
+Everything at **matched network density**; gamma-calibrated scores have no absolute meaning across runs.
+
+Primary metrics:
+
+1. Held-out perturbation-target edge recovery (AUPR against significant $E$ entries for targets excluded from prior fitting). **Primary.**
+2. Bootstrap edge stability at matched density.
+3. Hub identification: do measured high-$\hat k^{\mathrm{out}}$ genes rank as high-degree?
+4. In-degree distribution realism, evaluated _only_ on the leg not used for fitting.
+5. Replication in Replogle K562/RPE1 — separates cell-type-invariant structure from hESC-specific.
+
+Baselines that must appear in every comparison: stock PIDC; PIDC with a plain global (non-hierarchical) gamma; correlation skeleton with identical calibration applied. The third isolates whether any of this interacts specifically with PID or would help any score matrix equally — the same kill-criterion logic as H3 in the companion plan.
+
+---
+
+## 8. Checkpoint summary
+
+| # | Gate | Failure action |
+|---|---|---|
+| **D0** | Enough targets with non-degenerate $\hat k^{\mathrm{out}}$ | Restrict to V2; no fitted prior possible |
+| **D1** | V2 does not reduce bootstrap stability | Report as null result, retain stock calibration |
+| **D2** | V1 beats V2 on held-out $E$ recovery | In-degree prior wrong, or orientations too noisy |
+| **D3** | Known hubs survive increasing $\lambda$ | V3 fails on this data; report negative |
+| **D4** | Gains do not appear equally for a correlation skeleton | Reframe as generic recalibration, not PIDC-specific |
+
+D0 and D3 are the likely failure points. D4 is the framing risk.
+
+---
+
+## 9. Limitations to state explicitly
+
+- **Effective functional degree ≠ binding degree.** The prior is fit to steady-state transcriptomic response, days post-CRISPRi. That is the right target for calibrating PIDC, but it is not the degree distribution reported in the ChIP literature and should not be compared to it.
+- **In-degree leg is weak.** Not measurable from 300 targets; sourced from ATAC/motif or literature, both binding-derived.
+- **Soft interventions.** CRISPRi attenuates rather than severs; $\hat k^{\mathrm{out}}$ is a knockdown-efficiency-weighted quantity. Use per-cell efficiency estimates (mixscape-style) per companion Stage 1.
+- **Single context.** Degree structure is itself context-dependent; hESC-fit priors need not transfer, which is what metric 5 tests.
+- **Circularity remains partial.** Held-out splits address it for out-degree; the in-degree leg cannot be fully cleared and any topological-realism claim must be scoped to out-degree only.
+
+---
+
+## 10. Immediate next actions
+
+1. **Run Checkpoint D0** — requires only the effect matrix from companion Stage 4.
+2. **Implement V2** and run D1. Smallest change, best expected value, independent of everything else here.
+3. **Literature search on degree-corrected graphical models** before writing any method text (§1).
+4. **⚠️ Confirm FastPIDC.jl exposes per-gene PUC vectors and allows calibration to be disabled** (`calibrate = false` or equivalent). Every variant here requires raw uncalibrated PUC. If the gamma step is baked in and not bypassable, all four variants are blocked and the package needs patching first.
diff --git "a/Interventional PID for GRN inference\342\200\224testing the gap with FastPIDC.jl on Arc H1 hESC Perturb-seq.md" "b/Interventional PID for GRN inference\342\200\224testing the gap with FastPIDC.jl on Arc H1 hESC Perturb-seq.md"
new file mode 100644
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+++ "b/Interventional PID for GRN inference\342\200\224testing the gap with FastPIDC.jl on Arc H1 hESC Perturb-seq.md"	
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+**Status:** draft plan. Sections marked ⚠️ contain assumptions that must be corrected before implementation.
+
+---
+
+## 0. The claim being tested
+
+Partial information decomposition has been applied to observational scRNA-seq (PIDC; Chan, Stumpf & Babtie 2017) and interventional data has been used to _evaluate_ PIDC (SCING; Littman et al. 2023), but no published method uses **CRISPR intervention indicators as PID sources**. That is the gap.
+
+Three sub-hypotheses, in increasing ambition and decreasing confidence:
+
+| # | Hypothesis | Prior |
+|---|---|---|
+| **H1** | Reciprocal perturbation asymmetry orients a useful fraction of PIDC edges | High — near-direct measurement |
+| **H2** | `unique(X→Y \| I_g) ≈ 0` identifies co-downstream sibling pairs that observational PIDC calls edges | Moderate — the novel contribution |
+| **H3** | The PID mediation decomposition beats a plain mediation regression at the same task | **Low** — this is the honest kill criterion |
+
+H3 is the one that decides whether the PID framing is doing work or is decoration. If a linear mediation test or a conditional-independence test matches the decomposition, publish H2 as a mediation result and drop the information theory. Design the study so that comparison is unavoidable rather than optional.
+
+### Prior art to position against
+
+- **PIDC** — Chan, Stumpf & Babtie, _Cell Systems_ 2017. Observational, undirected, `InformationMeasures.jl`.
+- **SCING** — used Perturb-seq to benchmark PIDC and concluded PID approaches are "more accurately described as measures of coexpression, rather than gene regulation." This is the criticism the project must answer.
+- **D-SPIN** — Jiang et al. Closest in spirit: perturbations as external fields in a maximum-entropy model, explicitly motivated by the case where two genes have "weak negative correlation and low mutual information" but a real relationship. Not PID. **Primary conceptual competitor.**
+- **ADAPRE** (bioRxiv, Feb 2026) — CRISPRi guides as instrumental variables in a Poisson–lognormal model; handles knockdown-efficiency bias and cyclic structure. **Primary methodological competitor**, and better than this plan on soft interventions and feedback. Read before starting.
+- **CausalBench** — found interventional methods do _not_ beat observational ones on real data, contrary to synthetic benchmarks. The single most important tempering result.
+
+---
+
+## 1. Data
+
+**Arc Virtual Cell Atlas, Virtual Cell Challenge H1 hESC dataset.** ~300k cells, 300 CRISPRi perturbations, 10x Flex, >50k UMI/cell. CC0.
+
+Two properties matter and pull in opposite directions:
+
+- **Depth is excellent.** >50k UMI/cell is far above typical droplet data. Contingency tables will be better populated than in any dataset PIDC was originally tested on. This is the main reason the project is feasible at all.
+- **Perturbation breadth is narrow.** 300 targets, not genome-wide.
+
+### ⚠️ Checkpoint 0 — the reciprocal-pair count (RUN THIS FIRST, BEFORE ANY CODE)
+
+H1 and H2 both require pairs where **both** genes were perturbed. Upper bound is `C(300,2) = 44,850` ordered-pair-eligible pairs, but the real number is smaller after:
+
+- restricting to targets with a detectable transcriptomic phenotype (in Replogle, a large share of perturbations had none),
+- restricting to targets that are themselves well-detected as _response_ genes,
+- requiring adequate cells per condition.
+
+**Gate:** if fewer than ~2,000 pairs survive with both directions testable, H1/H2 are underpowered on this dataset alone. Do not proceed to implementation. Options at that point: add Replogle K562/RPE1 (genome-wide, ~9,800 targets) as the primary discovery set and use hESC as the context-transfer test, or restrict the entire study to the 300 targets and drop genome-scale ambitions.
+
+This checkpoint is cheap — one pass over the obs metadata plus a per-target energy-distance screen — and it determines the shape of everything downstream. It is also where the project is most likely to die, so it goes first.
+
+**Secondary concern:** hESC are a self-renewing pluripotent population with strong cell-cycle structure and differentiation-propensity heterogeneity. That variance will dominate MI. Cell-cycle regression or explicit conditioning is not optional here.
+
+---
+
+## 2. ⚠️ FastPIDC.jl — assumed API
+
+**I could not locate FastPIDC.jl.** Nothing in the Julia General registry, GitHub search, JOSS, or bioRxiv. Everything below is an assumed interface based on `InformationMeasures.jl` conventions and the original PIDC implementation. **Replace this section with the real API before writing any code.**
+
+Specifically, I need to know:
+
+1. Signature and return type of the main network call. Dense matrix, sparse, or edge list?
+2. Is discretization done internally or does it accept pre-binned integer data? **Critical** — the whole normalization argument depends on binning scheme.
+3. Is the low-level PID accessible, i.e. can I get `redundancy` / `unique` / `synergy` for an arbitrary triple `(source1, source2, target)`? **This is the load-bearing requirement.** If only the aggregated PUC score is exposed, H2 and H3 cannot be implemented without patching the package.
+4. Which redundancy measure(s)? Is anything beyond `I_min` available?
+5. How is the per-gene gamma calibration exposed, and can it be disabled?
+6. Threading/GPU model, and whether it is safe to call from inside a `Threads.@threads` loop.
+
+Assumed for the purposes of drafting:
+
+```julia
+# ⚠️ ALL ASSUMED
+using FastPIDC
+
+net = pidc_network(X;                      # X :: Matrix (cells × genes) or CSC
+                   discretizer = :equal_frequency,
+                   estimator   = :maximum_likelihood,
+                   calibrate   = true)      # per-gene gamma
+
+# The critical primitive:
+pid = pid_triple(s1, s2, target;            # AbstractVector, discretized
+                 measure = :imin)
+# pid.redundancy, pid.unique1, pid.unique2, pid.synergy
+```
+
+If `pid_triple` or equivalent does not exist, **stop and add it** before anything else. It is the core of the contribution.
+
+---
+
+## 3. Environment
+
+```julia
+# Project.toml
+[deps]
+FastPIDC          = "..."   # ⚠️ path/UUID unknown
+InformationMeasures = "..."
+Muon             = "..."    # AnnData .h5ad reader
+HDF5             = "..."
+SparseArrays     = "..."
+Statistics       = "..."
+StatsBase        = "..."
+Distributions    = "..."
+Random           = "..."
+Distances        = "..."
+GLM              = "..."    # baselines for H3
+MultipleTesting  = "..."
+Arrow            = "..."    # intermediate storage
+DataFrames       = "..."
+ProgressMeter    = "..."
+```
+
+Pin versions and commit `Manifest.toml`. Set `JULIA_NUM_THREADS` explicitly. Seed every RNG and record seeds in output metadata — resampling stability is a headline result, so irreproducibility there is fatal.
+
+---
+
+## 4. Pipeline
+
+### Stage 1 — Load and QC
+
+```julia
+using Muon, SparseArrays
+
+ad = readh5ad("data/vcc_h1_hesc.h5ad")
+
+# ⚠️ verify actual column names in ad.obs
+target_col = "target_gene"
+control_label = "non-targeting"
+
+counts = ad.X                     # cells × genes, sparse counts
+targets = ad.obs[!, target_col]
+```
+
+QC: standard per-cell filters (UMI, gene count, mitochondrial fraction). Per-guide knockdown efficiency, mixscape-style, retaining a per-cell continuous perturbation strength rather than binary assignment — CRISPRi is a soft intervention and ADAPRE showed that efficacy variation inflates out-degree of strongly knocked-down genes. Keep the continuous value even if the first pass uses binary.
+
+### Stage 2 — Normalization and discretization
+
+Per earlier reasoning: **size-factor normalize, log1p, equal-frequency binning, zeros as their own category.** Deviance residuals are explicitly rejected — for zero counts the residual is a deterministic decreasing function of library size, which smears depth into the zero block and manufactures MI between sparse gene pairs.
+
+```julia
+function discretize_gene(v::AbstractVector{<:Real}; nbins::Int=4)
+    # zeros get bin 1; nonzeros get equal-frequency bins 2..nbins
+    out = ones(Int, length(v))
+    nz = findall(!iszero, v)
+    isempty(nz) && return out
+    vals = v[nz]
+    qs = quantile(vals, range(0, 1, length=nbins))
+    @inbounds for (k, i) in enumerate(nz)
+        out[i] = 1 + clamp(searchsortedlast(qs, vals[k]), 1, nbins - 1)
+    end
+    return out
+end
+```
+
+**Checkpoint 1 — invariance sanity check.** Under equal-frequency binning, MI must be numerically identical for raw, log1p, and rank-transformed input. If it isn't, the binning is not rank-based and the transformation-invariance argument in the paper is wrong. Then deliberately repeat with uniform-width binning and confirm it _does_ change — that contrast is a figure.
+
+Gene filter: detected in ≥10% of cells, plus the union of perturbation targets regardless of detection rate. Record the gene set in output metadata, because PUC depends on which genes are in the matrix.
+
+### Stage 3 — Observational skeleton
+
+Run FastPIDC on **control (non-targeting) cells only**. This is the candidate skeleton whose job is to cut the O(p²) space to something affordable for causal testing.
+
+```julia
+ctrl = findall(==(control_label), targets)
+Xd_ctrl = Xd[ctrl, :]
+skeleton = pidc_network(Xd_ctrl)   # ⚠️ assumed API
+```
+
+Retain top-k edges at several densities (k = 10p, 50p, 100p). Report everything at matched density — gamma-calibrated scores have no absolute meaning and are not comparable across runs.
+
+**Checkpoint 2 — scaling and stability.** Time and memory on 500 / 2,000 / 5,000 genes; confirm the claimed scaling before committing to the full run. Then bootstrap cells (20 replicates) and compute edge-recovery frequency. **Gate: if top-k edges don't recover at >50% frequency across bootstraps, no downstream causal result is interpretable.** Fix estimation before proceeding.
+
+### Stage 4 — Directed effect matrix
+
+Distributional shift, not mean shift — Perturb-seq responses are frequently variance changes or bimodality.
+
+```julia
+using Distances
+
+# E[g, j] = distributional shift of gene j under perturbation of g
+function effect_matrix(Xn, targets, control_label, target_list)
+    E = zeros(length(target_list), size(Xn, 2))
+    ctrl = findall(==(control_label), targets)
+    for (gi, g) in enumerate(target_list)
+        pert = findall(==(g), targets)
+        length(pert) < 30 && continue          # power floor
+        for j in axes(Xn, 2)
+            E[gi, j] = energy_distance(view(Xn, pert, j), view(Xn, ctrl, j))
+        end
+    end
+    return E
+end
+```
+
+Calibrate significance against **non-targeting-vs-non-targeting splits**, not a parametric null — this absorbs batch and depth structure. Benjamini–Hochberg across all (g, j).
+
+**Checkpoint 3 — positive control.** Every perturbation must show a strong self-effect (`E[g,g]` large, direction = down for CRISPRi). Targets failing this had ineffective knockdown; exclude them from orientation but keep them as response genes. Also confirm known hESC pluripotency relationships (POU5F1, NANOG, SOX2 and their documented targets) appear. If they don't, something is wrong upstream and no amount of downstream sophistication will rescue it.
+
+### Stage 5 — Orientation (H1)
+
+For each skeleton edge with both genes perturbed, compare `E[X,Y]` vs `E[Y,X]`:
+
+- asymmetric → orient
+- both large → feedback / cycle, flag, do not force a direction
+- neither → likely confounding or artifact, flag for removal
+
+Report the fraction of skeleton edges orientable, and the fraction flagged cyclic. The cyclic fraction is itself a result worth reporting, since acyclicity is an assumption most competing methods make and ADAPRE specifically relaxes.
+
+### Stage 6 — Intervention-indicator PID (H2, the novel part)
+
+For candidate pair (X, Y) and perturbation g upstream of both:
+
+```julia
+# ⚠️ depends on pid_triple existing
+function mediation_pid(Xd, targets, g, x_idx, y_idx, control_label)
+    cells = findall(t -> t == g || t == control_label, targets)
+    Ig = Int.(targets[cells] .== g) .+ 1      # binary source, 1-indexed bins
+    xs = Xd[cells, x_idx]
+    ys = Xd[cells, y_idx]
+    return pid_triple(Ig, xs, ys; measure = :imin)
+end
+```
+
+Interpretation:
+
+| Pattern | Reading |
+|---|---|
+| `unique(Ig→Y \| X) ≈ 0` | complete mediation: `g → X → Y` |
+| `unique(X→Y \| Ig) ≈ 0` | **siblings, not connected — the false-positive filter** |
+| high redundancy, both uniques low | confounding by g |
+| synergy | interaction / gating |
+
+Estimation is easier than the gene–gene case because `Ig` is binary. Aggregate across all upstream g per pair.
+
+**Checkpoint 4 — redundancy-measure robustness.** `I_min` systematically overstates redundancy (Bertschinger et al.). Recompute the sibling calls under BROJA and Ince's `I_ccs` on a subsample. **Gate: if sibling classification flips substantially between measures, the result is an artifact of `I_min` and must be reported as such, not papered over.**
+
+### Stage 7 — Baselines (H3, the kill criterion)
+
+Run on the identical candidate set:
+
+1. Linear mediation regression: `Y ~ X + Ig`, test whether the X coefficient survives conditioning on Ig.
+2. Conditional independence test `X ⊥ Y | Ig` (partial correlation, and a nonparametric alternative).
+3. Correlation-only skeleton with the same orientation and pruning layers — isolates whether PID contributes anything over correlation.
+4. GRNBOOST2 / ppcor comparators, per SCING's framing.
+
+**This is the decisive comparison.** If (1) or (2) matches the PID sibling classification at equal cost, say so plainly in the paper. A clean negative on H3 with a solid H1/H2 is still a publishable, useful result — and given CausalBench's finding that interventional methods fail to beat observational baselines on real data, a well-executed negative is arguably the more valuable contribution.
+
+### Stage 8 — Invariance filtering
+
+For surviving oriented edges, test stability of `P(Y | pa(Y))` across perturbation regimes that did not target Y (Invariant Causal Prediction, Peters/Bühlmann/Meinshausen). Catches parent sets that are merely predictive.
+
+### Stage 9 — External validation
+
+- **H1 ATAC.** ENCODE H1 accessibility (e.g. GSE267154 / ChromBPNet-associated data) plus motif scanning — is the oriented regulator's motif in an accessible region near the target? Independent of expression.
+- **Replogle K562/RPE1.** Which edges replicate across cell type? Directly tests the context-transfer question. Expect core machinery to replicate and lineage-specific regulation not to.
+- **Pluripotency literature.** Curated hESC network as a partial gold standard, with the caveat that it's incomplete and biased toward well-studied genes.
+
+---
+
+## 5. Checkpoint summary
+
+| # | Gate | Failure action |
+|---|---|---|
+| **0** | ≥~2,000 reciprocal testable pairs | Add Replogle as discovery set, or restrict scope |
+| **1** | MI invariant under equal-frequency binning | Fix discretization; the invariance claim is wrong |
+| **2** | Bootstrap edge recovery >50% at top-k | Fix estimation before any causal claim |
+| **3** | Self-effects present; known pluripotency edges recovered | Debug upstream; do not proceed |
+| **4** | Sibling calls stable across redundancy measures | Report as `I_min` artifact |
+| **5** | PID beats mediation regression | Publish as negative result on H3 |
+
+Checkpoints 0, 2 and 5 are the ones most likely to fail. Reaching each of them early, rather than after building the full pipeline, is the main design goal of this ordering.
+
+---
+
+## 6. Known limitations to state explicitly in any write-up
+
+- **Timescale.** CRISPRi Perturb-seq reads steady state days post-knockdown. "Direct" here means _not mediated by other measured genes at equilibrium_, not direct transcriptional binding. This silently redefines what an edge means and should be stated in the abstract, not buried.
+- **Absence of effect ≠ absence of edge.** Paralog redundancy, buffering, partial knockdown.
+- **Soft interventions.** CRISPRi attenuates rather than severs incoming edges; the do-calculus argument for edge deletion is approximate.
+- **Cycles.** Reciprocal asymmetry reads genuine feedback as ambiguity.
+- **PUC is gene-set dependent.** Edge scores change with the gene filter. Report the filter.
+- **Single cell line.** H1 hESC only; context transfer is a hypothesis tested in Stage 9, not an assumption.
+- **`I_min` is the weakest link** in the PID literature, and it is what PIDC uses.
+
+---
+
+## 7. Immediate next actions
+
+1. **Send me the real FastPIDC.jl API**, especially whether triple-level PID components are accessible. Section 2 is unusable until this is resolved.
+2. **Run Checkpoint 0.** One pass over `ad.obs` plus per-target energy distance. Decides project shape.
+3. **Read ADAPRE and D-SPIN properly** before writing method text — they define the positioning.
+4. **Literature re-check.** My search was one round and ADAPRE is from February 2026; search bioRxiv and Scholar for PID/information-theoretic interventional GRN work from the last few months before assuming the gap is open.
diff --git a/LITERATURE_REVIEW.md b/LITERATURE_REVIEW.md
new file mode 100644
index 0000000..cab2e8e
--- /dev/null
+++ b/LITERATURE_REVIEW.md
@@ -0,0 +1,179 @@
+# Literature review — ADAPRE, D-SPIN, degree-corrected models, recent PID/interventional-GRN work
+
+Follow-up to both plan documents' §"immediate next actions" (interventional plan item 3,
+companion plan item 3). Read against the current state of the branch: `STATE.md`,
+`LOG.md` (Stage 4 / Checkpoint D0, cell-cycle-conditioning negative result, commit `11a7fcb`).
+
+---
+
+## (a) ADAPRE — summary + relevance
+
+**Causal gene regulatory network inference from Perturb-seq via adaptive instrumental variable
+modeling**, bioRxiv, Feb 2026 ([link](https://www.biorxiv.org/content/10.64898/2026.02.18.706642v1)).
+
+Treats each CRISPRi guide's intervention indicator as an instrumental variable and models UMI
+counts with a **Poisson–lognormal observation layer**, separating measurement noise from true
+expression. Its central methodological move, and the one most relevant here: it applies
+**gene-specific adaptive penalties to correct strength-dependent degree bias** — the explicit
+finding is that genes with stronger/more efficient knockdowns get estimated with spuriously
+higher out-degree and are disproportionately inferred as network hubs, distorting topology under
+heterogeneous CRISPRi efficiency. It also relaxes the acyclicity assumption most competing methods
+(NOTEARS-style) impose, recovering potentially cyclic structure. Evaluated on genome-wide K562
+Perturb-seq; networks enriched for known biological interactions, with coherent leukemia-associated
+subnetworks recovered.
+
+**Relevance:** this is a close structural cousin of this project's reopened Checkpoint D0 problem
+(`LOG.md`, commit `11a7fcb`) — both are "some perturbation targets look like they hit everything"
+inflation problems in $\hat k^{\mathrm{out}}$. ADAPRE's diagnosis is **knockdown-efficiency
+heterogeneity**, not cell-state/cell-cycle — a different specific covariate than the one already
+tried and rejected here, but the same general shape of fix: **regress/penalize by a per-target
+covariate that predicts spurious degree, not a global correction.** Concretely worth checking as
+a next step before inventing a new cell-state axis: **is measured $\hat k^{\mathrm{out}}_g$
+correlated with per-target knockdown efficiency** (already computed in Checkpoint 0's self-effect
+screen, `self_effect_screen.csv` — log2FC of the target's own gene)? If the worst offenders in
+`k_out.csv` (or `k_out_resid.csv`) are also the strongest knockdowns, this is a cheap, already-half-computed
+check and a plausible mechanism distinct from the cell-cycle hypothesis that just failed. This
+should be checked before the three follow-ups already listed in `STATE.md`'s reopened D0 row (PCA/
+pseudotime state axis, stratified null, chromatin-regulator plausibility check) — it's cheaper than
+all three and directly informed by a paper doing almost exactly this correction on closely related
+data (also K562/CRISPRi Perturb-seq).
+
+Also directly relevant to the interventional plan's own item 3 ("ADAPRE... better than this plan on
+soft interventions and feedback... Primary methodological competitor"): confirmed as still the
+right framing — ADAPRE remains the closest published competitor and this project's positioning
+against it (PID-specific triple decomposition for sibling/mediation calls, not just network
+recovery) is unchanged by anything found here.
+
+## (b) D-SPIN — summary + relevance
+
+**D-SPIN constructs regulatory network models from scRNA-seq that reveal organizing principles of
+perturbation response**, *Cell*, 2026 ([PubMed](https://pubmed.ncbi.nlm.nih.gov/42127893/),
+[GitHub](https://github.com/JialongJiang/DSPIN)).
+
+Models the joint distribution of transcriptional states as a **spin-glass / Markov random field**
+(maximum-entropy), with pairwise regulatory weights $J$ and condition-specific external fields $h$
+representing perturbations (genetic, chemical, or physiological). A single unified network is fit
+jointly across all conditions — perturbations shift $h$, not $J$ — which lets weakly-correlated gene
+pairs still register a real edge if a perturbation moves them together. The key claimed advantage
+over correlation/MI-based approaches, stated explicitly in the paper: genes under **persistent
+multi-input inhibition** in the unperturbed state can have near-zero correlation and near-zero
+mutual information with their true regulators (because their expression is pinned near floor by
+redundant suppression) — a perturbation that lifts one input reveals the hidden coupling by
+changing the *pattern* of correlation, not by increasing correlation with any single measured gene.
+D-SPIN is explicitly **not** an information-theoretic/PID method — it's parametric (fits a
+maximum-entropy graphical model), scales to thousands of genes/conditions/millions of cells via a
+different inference machinery entirely (pseudolikelihood / contrastive-divergence-style fitting,
+not binning + information estimation).
+
+**Relevance:** unchanged from the interventional plan's own framing ("Not PID. Primary conceptual
+competitor.") — nothing found here softens or sharpens that positioning. One point worth adding to
+the eventual write-up: D-SPIN's "hidden regulatory interaction via persistent suppression" failure
+mode is a **different** blind spot than PIDC's degree-homogeneity problem (this repo's companion
+plan) — a gene pinned at floor by redundant inhibition would show *low* $\hat k^{\mathrm{out}}$ *and
+in* under both approaches for reasons neither addresses, so D-SPIN's contribution is complementary
+to, not overlapping with, either of this repo's two plans. Not a reason to change scope, just a
+citation-accuracy note for the eventual methods section.
+
+## (c) Degree-corrected graphical models — what's new
+
+No hits specifically on "degree-corrected graphical lasso" as a named method beyond what the
+companion plan's §1 already cites (degree-weighted Lasso, Khanin & Wit ascertainment-bias critique,
+Broido & Clauset's scale-freeness skepticism). Adjacent 2025–2026 work found but not overlapping in
+approach:
+
+- **tvsfglasso** (time-varying scale-free graphical lasso, *PLOS Comp Bio* 2025) — extends
+  graphical lasso with a scale-free-network penalty for time-series data. Different problem
+  (temporal, not perturbation-based degree correction) but confirms scale-free-penalized graphical
+  estimation is still an active line; worth a citation as "concurrent, different setting" rather
+  than a competitor.
+- **GRNFormer** (graph transformer for GRN inference, *Bioinformatics* 2026) — deep-learning
+  approach, no explicit degree-correction step comparable to this project's V1–V4; not a direct
+  competitor to the companion plan's calibration-layer framing.
+
+**Verdict:** the companion plan's claim in §1 ("I am not aware of published work modifying PIDC's
+gamma calibration specifically... same epistemic status as the mediation idea") still holds. No
+new work closes this gap. The novelty framing is unchanged.
+
+## (d) Recent PID/interventional-GRN work — what's new
+
+- **ADAPRE** (above) — already the plan's primary methodological competitor; nothing new to add
+  beyond the degree-bias-correction mechanism noted in (a).
+- **CausalBench follow-up**: the original CausalBench finding cited in the interventional plan
+  ("interventional methods do not beat observational ones on real data") has a 2024–2025 sequel —
+  the **CausalBench challenge** (arXiv 2308.15395) reports that methods built specifically for the
+  challenge do meaningfully better than the priors CausalBench originally benchmarked, "constitut[ing]
+  a major step towards alleviating the limitations identified with CausalBench... utilization of the
+  interventional information." **This partially tempers the plan's own citation** — the H3 kill
+  criterion (does PID beat plain mediation regression?) is still the right experiment to run, but
+  the interventional plan's framing that interventional methods broadly "fail" against observational
+  ones on real data is now dated by ~2 years and should be stated with that caveat, not as settled.
+- **"When Does GRN Inference Break?"** (arXiv 2605.04930, 2026) — a controlled diagnostic study of
+  causal/correlational GRN methods under injected pathologies (dropout, latent confounders,
+  cell-type mixing, network density, feedback). Two findings bear directly on open items in this
+  project: (1) **latent confounders degrade all methods equally** ("the great equalizer") —
+  without interventional data, no observational method can separate confounding from causation,
+  which is exactly the shape of this project's own reopened D0 problem, just framed from the
+  benchmark side rather than the applied side; (2) **mutual-information/discretization-based
+  methods are the most fragile to dropout specifically** (ΔAUPRC ≈ −0.7 vs. Pearson's −0.28 at high
+  dropout), because "MI relies on equal-frequency discretization into 6 bins, and at high dropout
+  most observed entries are forced to zero" — this is a direct, if indirect, endorsement of Stage
+  2's zero-as-own-bin discretizer design already implemented on this branch (commit `b139fc9`):
+  giving zeros their own bin rather than letting dropout collapse into the discretization is exactly
+  the kind of fix this diagnostic paper's failure mode calls for. Worth citing as external support
+  for that design choice.
+- **PSGRN** (*Science Advances*, self-training with synthetic gold standards) and a Dec-2025 bioRxiv
+  **"Comparison of Interventional Causal Structure Learning Algorithms for GRN Inference"** — both
+  benchmark-type papers, neither uses PID/information-theoretic decomposition specifically. No PID
+  triple-decomposition-with-intervention-indicators approach was found anywhere in the 2025–2026
+  literature searched. **The gap the interventional plan claims in §0 ("no published method uses
+  CRISPR intervention indicators as PID sources") still appears open.**
+
+## (e) Relevance to the reopened D0 cell-state-confound problem
+
+Two concrete, actionable leads, both cheaper than the three already-listed follow-ups in
+`STATE.md`'s D0 row:
+
+1. **Knockdown-efficiency confound (from ADAPRE, (a) above).** Check whether $\hat k^{\mathrm{out}}_g$
+   (either version — original or cell-cycle-residualized) correlates with each target's own
+   knockdown strength (log2FC from `self_effect_screen.csv`, already computed in Checkpoint 0).
+   This is a five-minute correlation check against data that already exists on disk, and it's the
+   mechanism a directly comparable published method (ADAPRE, also CRISPRi/K562 Perturb-seq)
+   identified and corrected for. If it correlates, the fix pattern is the same shape already
+   attempted (residualize/penalize by a covariate) but with a different, better-targeted covariate.
+2. **"Latent confounder = great equalizer" framing (from (d) above).** The diagnostic-study
+   finding that latent confounders degrade *all* methods equally, observationally, is a useful
+   framing point for the eventual write-up regardless of which correction (if any) works: it
+   predicts, independent of this project's own results, that no purely observational fix inside
+   the energy-distance/PUC framework can fully separate a shared-cell-state confound from real
+   direct effects — only genuinely interventional structure (e.g. comparing across perturbations
+   with matched cell-state shift but different targets, or explicit ICP-style invariance testing,
+   already Stage 8 in the interventional plan) can. This argues for **not over-investing** in
+   finding the exact right regression covariate for D0 and instead treating a residual confound as
+   expected, to be filtered later by Stage 8 (Invariance filtering) rather than fully resolved at
+   Stage 4.
+
+Neither of these invalidates the cell-cycle regression attempt already run (commit `11a7fcb`) —
+it was a reasonable first thing to try and its negative result is itself informative (rules out
+cell-cycle specifically, ~1.5% variance explained is a real, checked number). But (1) above is a
+strictly higher-priority next check than the three items STATE.md currently lists, given it reuses
+existing data and is motivated by a directly comparable published result.
+
+## (f) Verdict — does this change STATE.md's checklist or next steps?
+
+**Yes, one concrete addition; everything else confirmed as-is.**
+
+- **Add to STATE.md's reopened D0 row / "Immediate next actions"**: check $\hat k^{\mathrm{out}}_g$
+  vs. per-target knockdown efficiency (from `self_effect_screen.csv`) before trying the
+  PCA/pseudotime state axis or stratified-null follow-ups — cheaper, reuses existing data, and
+  motivated by ADAPRE's directly comparable, published fix for the same failure shape.
+- The interventional plan's CausalBench citation should be stated with a "this is now ~2 years old
+  and partially superseded by the CausalBench challenge results" caveat in any eventual write-up,
+  not changed in approach.
+- The zero-as-own-bin discretizer decision (already implemented, commit `b139fc9`) now has
+  independent literature support from the 2026 diagnostic-study finding that MI/discretization
+  methods are uniquely fragile to dropout-driven zero-inflation — no code change needed, just a
+  citation to add later.
+- No new competing PID-with-intervention-indicators method was found; the core novelty claim in the
+  interventional plan's §0 stands unchallenged.
+- No published degree-corrected-PIDC-calibration work was found; the companion plan's novelty claim
+  in §1 stands unchallenged.
diff --git a/LOG.md b/LOG.md
new file mode 100644
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--- /dev/null
+++ b/LOG.md
@@ -0,0 +1,443 @@
+# Analysis log
+
+Working log for evaluating `Interventional PID for GRN inference—testing the gap with FastPIDC.jl on Arc H1 hESC Perturb-seq.md`
+(**interventional plan**) and `Degree-aware recalibration of PIDC—a companion plan.md` (**companion plan**) against real data.
+Branch: `perturbation-analysis`. See `STATE.md` for the checklist view of plan items.
+
+---
+
+## 2026-09-04 — Environment setup
+
+- Created branch `perturbation-analysis` off `analysis`.
+- `python/` (the `fastpidc` package) is left untouched for now; its `requires-python = ">=3.10"`
+  conflicts with `vsparse==0.2.0`'s `>=3.12` floor, so analysis dependencies are **not** added
+  there. Instead: `analysis/pyproject.toml`, a `package = false` uv project (Python ≥3.12) that
+  pulls in `fastpidc` (editable, `../python`) and `vsparse` (editable, `/home/asm/vsparse`) as
+  path sources, plus `anndata`, `pandas`, `statsmodels`, `matplotlib`. Run everything from
+  `analysis/` with `uv run python ...`.
+  - **Caveat:** the `vsparse` path source is an absolute local path (`/home/asm/vsparse`), not a
+    pinned release — fine for exploratory work on this machine, but not reproducible elsewhere.
+    Revisit once/if `vsparse` cuts a real release.
+- Loading `hESC.h5ad` (and presumably `K562-genome-wide.h5ad`) requires, in this order:
+  `import hdf5plugin` (registers the HDF5 filter plugin directory — without it, reads fail with
+  `OSError: can't open directory (/usr/local/lib/plugin)`), then `import vsparse` (registers the
+  `ivcsr`/`ivcsc` AnnData element readers), then load with
+  **`vsparse.VCSCAnnData.read_h5ad(path)`** — plain `anndata.read_h5ad` fails: current `anndata`
+  rejects a bare `VCSRArray`/`VCSCArray` as a valid `X` type at the top-level constructor, even
+  though vsparse's element reader for `/X` works fine. `VCSCAnnData` wraps that correctly.
+- Machine: 188 GiB RAM, RTX 4090 present but **`cupy` not installed** anywhere on the system —
+  `fastpidc`'s CUDA backend is unusable until that's added. `PIDCConfig.backend` defaults to
+  `"cuda"`, so every call in this analysis needs `backend="cpu"` explicit until cupy is set up.
+
+## 2026-09-04 — FastPIDC.jl Python API: resolving the plans' open questions
+
+Both plans flagged their assumed API as unverified (interventional plan §2, companion plan
+item 10.4). Read `python/src/fastpidc/{api,network,puc,information,types}.py` directly. Findings:
+
+1. **No dense/sparse/edge-list ambiguity** — `infer_network_from_nodes` returns an
+   `InferredNetwork(nodes, edges)`; `edges` is a flat sorted list of `Edge(nodes, weight)`, not a
+   matrix. Full PUC/MI as matrices *are* available separately (see #3).
+2. **Discretization** happens inside `Node.from_raw_values` / `get_nodes`, not accepted as
+   pre-binned integers through the main entry point — but `Node` itself is a public dataclass
+   with `binned_values` already binned, so nodes can be constructed directly, bypassing the
+   provided discretizers if we bin ourselves (relevant for Stage 2's custom
+   zero-as-own-bin discretizer, which doesn't match any of `discretizers.py`'s built-ins and
+   will need to be added or built externally and wrapped into `Node` objects by hand).
+3. **Per-gene PUC vectors and disabling calibration — both already supported, no patch needed.**
+   `PUCNetworkInference` (`apply_context=False, get_puc=True`) returns *raw, uncalibrated* PUC as
+   edge weights — this **is** the `calibrate=false` companion plan item 10.4 asked to confirm.
+   The full dense PUC matrix is also directly accessible via `compute_puc_full(nodes, ...)` →
+   `(mi_scores, puc_scores)`, or dumped to disk with `PIDCConfig(dump_puc_path=...)`. A gene's PUC
+   vector is just a row of that matrix. This unblocks companion-plan V2/V1/V3/V4 immediately —
+   §2's per-gene gamma work operates on `puc_scores` rows before `get_weights()`'s gamma-fit step
+   ever runs.
+4. **Redundancy measure**: only `I_min` (`apply_redundancy_formula` = `E[min(SI_1, SI_2)]`),
+   matching interventional plan's own flagged weakness ("`I_min` is the weakest link", Checkpoint
+   4). No BROJA or `I_ccs` implementation exists in this package — Checkpoint 4's
+   redundancy-measure robustness check will need an external PID library (e.g. `dit`) or a new
+   implementation; out of scope to add speculatively.
+5. **`pid_triple`-equivalent (interventional plan's load-bearing primitive, §2 item 3): does
+   NOT exist.** There is no function that, given an arbitrary triple `(source1, source2, target)`,
+   returns `(redundancy, unique1, unique2, synergy)`. What exists are the pieces it would be built
+   from: `network.get_mi_and_si(node1, node2, estimator, base)` (pairwise MI + specific
+   information) and `information.apply_redundancy_formula(p_z, si1, si2, base)`. From these,
+   for a triple `(Ig, X, Y)`: `redundancy = apply_redundancy_formula(p(Y), SI(Ig;Y), SI(X;Y))`,
+   `unique(Ig→Y|X) = MI(Ig,Y) - redundancy`, `unique(X→Y|Ig) = MI(X,Y) - redundancy`. Synergy
+   needs the joint MI of `(Ig,X)` on `Y`, which requires a joint-binned "combined" node (bin id =
+   `Ig * n_bins_X + X`) fed through the same `get_mi_and_si` machinery — buildable, not present.
+   **Action taken:** none yet on the package itself; per the interventional plan's own instruction
+   ("if `pid_triple` or equivalent does not exist, stop and add it — it is the core of the
+   contribution"), this is the first code change needed before Stage 6 (H2) can run, and will be
+   added as a new `fastpidc.pid` module on this branch before that stage starts. Not needed for
+   Checkpoint 0 or the companion plan's V2, so it did not block today's work.
+6. **Threading/CUDA-loop safety**: not evaluated yet (no cupy installed to test against); the CPU
+   path is plain NumPy with Python-level loops over node pairs, no threading primitives visible,
+   so nothing to trip over there. Revisit when the CUDA extra is actually installed.
+
+## 2026-09-04 — Checkpoint 0 (interventional plan): reciprocal-pair count — RUN FIRST
+
+This is the plan's own designated first action, before any hypothesis test proper. Ran directly
+against `/opt/IVCSC/hESC.h5ad` (chosen over `K562-genome-wide.h5ad` per instruction: hESC has
+much better read depth despite narrower breadth, and is the plan's primary dataset).
+
+**Data reality vs. the plan's assumptions — first discrepancy found:**
+- Plan text says "~300 CRISPRi perturbations." Actual file has **150** distinct `target_gene`
+  values (excluding `non-targeting`), not ~300. `obs` has `target_gene` (categorical, 151
+  categories incl. NTC), `guide_id`, `batch`. 221,273 cells × 18,080 genes; 38,176 NTC
+  (non-targeting) cells. This roughly halves every pair-count estimate in both plans and should be
+  corrected wherever "300 targets" appears in method text later (companion plan §2's `C(300,2) =
+  44,850` becomes `C(150,2) = 11,175`; the "~9,800 targets" Replogle comparison in interventional
+  plan §1 is unaffected, that's a different dataset).
+
+**Checks run** (script logic below; not yet committed as a package script — see `STATE.md`):
+1. Cells per target: **min 33, median 1045, max 4760** — all 150 targets clear the plan's own
+   power floor (`n_pert >= 30`, interventional plan Stage 4) with room to spare; none dropped.
+2. All 150 target-gene symbols are present in `var_names` (i.e., measurable as response genes,
+   not just guide labels) — no targets lost to a target/measured-gene mismatch.
+3. Self-knockdown effect (Checkpoint 3's positive control, run early since it's nearly free once
+   the above is done): size-factor-normalized + log1p expression of each target gene, perturbed
+   vs. NTC cells, one-sided Mann–Whitney (knockdown ⇒ down), BH-FDR across the 150 tests.
+   **All 150/150 significant at q<0.05 and log2FC<0.** Effect sizes vary a lot though — log2FC
+   ranges from −4.98 (TMSB4X) to −0.16 (KAT2A); the weak end is "significant because n is huge,"
+   not "large knockdown," and should not be over-read as uniformly strong CRISPRi efficiency.
+   Full table saved to the session scratchpad as `self_effect_screen.csv` (not committed — see
+   "Scratch outputs" below).
+4. Downstream phenotype detectability (proxy for the plan's energy-distance screen — see caveat
+   below): per target, Welch mean-shift z-test of every one of the 18,080 genes vs. NTC,
+   BH-FDR<0.05, excluding the target's own gene. **All 150/150 targets clear even a generous bar**
+   (min significant-gene count across targets = 153, vs. a nominal ≥5-gene bar) — no targets look
+   like "no detectable phenotype" cases the way a meaningful fraction of Replogle targets reportedly
+   were.
+   - **Caveat, stated plainly:** this is a linear mean-shift proxy, not the energy distance the
+     interventional plan specifies for the real Stage-4 effect matrix (Perturb-seq responses are
+     "frequently variance changes or bimodality," which mean-shift can miss). It was chosen only
+     because it's cheap enough to run genome-wide today; it is **not** a substitute for the real
+     Stage 4 computation, which should use `scipy`/`dcor` energy distance as specified. At >50k
+     UMI/cell and hundreds-to-thousands of cells per group, this test has enormous power — "q<0.05"
+     on its own is a very low bar here and should not be read as "biologically meaningful effect."
+5. Known pluripotency genes (Stage 9 / companion-plan D3 relevance): `POU5F1` and `NANOG` are
+   measured but **not** perturbation targets (can only ever be response genes / hub-recovery
+   targets, never PID sources). `SOX2` is both measured and a perturbation target.
+
+**Gate result: PASS, comfortably.** Reciprocal-eligible unordered pairs (both genes perturbed,
+both pass the cell-count and phenotype screens) = `C(150,2) = 11,175`, vs. the plan's stated gate
+of "≥~2,000 pairs." Even a much stricter phenotype filter than the one applied here would have to
+throw out the large majority of targets to endanger the gate. **No fallback to Replogle needed for
+feasibility; hESC alone supports H1/H2 pair-count requirements.** (Replogle may still be worth
+using for Stage 9 context-transfer, per the plan — that's a separate question from this gate.)
+
+Companion plan's own Checkpoint D0 (out-degree prior fitting) is a *different*, stricter
+requirement — non-degenerate spread in *measured effective out-degree* per target, not just "a
+detectable phenotype exists." That needs the real Stage 4 effect matrix (not the mean-shift proxy
+above) and hasn't been run yet; see `STATE.md`.
+
+---
+
+## 2026-09-04 — Stage 4: real effect matrix (energy distance)
+
+Implemented the actual Stage-4 deliverable this time (not the mean-shift proxy from Checkpoint 0):
+per-gene **energy distance** (Székely & Rizzo) between each of the 150 perturbation targets and
+the NTC pool, vectorized across genes via an O((m+n)·log(m+n))-per-gene merge-rank identity
+(`analysis/scripts/effect_matrix.py::energy_distance_batch`) rather than the naive O(m·n) pairwise
+form or 1.8M individual `scipy.stats.energy_distance` calls (150 targets × 11,942
+gene-filtered genes) — both were checked and are not tractable interactively at this scale.
+Correctness verified against `scipy.stats.energy_distance` on random synthetic data (`sqrt(e2)`
+matches to 1e-6) before running on real data.
+
+Gene set for this and downstream work: Stage 2's own filter (detected in ≥10% of cells, plus
+union of perturbation targets) — 18,080 → **11,942 genes**. All 150 targets already satisfied the
+union clause (they're all in `var_names`), so the filter is really just the detection threshold.
+
+**Near-miss: OOM from over-parallelizing.** First attempt parallelized across all 150 targets at
+full gene width (11,942 genes/target) with `ProcessPoolExecutor(max_workers=46)` on this
+48-core machine. Killed the box: `free -h` showed 187/188 GiB used and 92 GiB of swap in use a
+few minutes in, and the job died (exit 1, empty stderr — almost certainly an OOM kill of one or
+more workers, silently swallowed by `ProcessPoolExecutor`). Root cause: every worker's arrays are
+shaped `(m+n, G)` where `n` is the *fixed* 38,176-cell NTC pool — so per-worker peak memory
+(~15–20 GB at G=11,942, dominated by the `argsort` output and its int64 default dtype) barely
+depends on which target it's processing, and 46 workers all inflate to that peak roughly
+simultaneously. Fixed by (1) chunking genes explicitly (1,500/chunk, so peak scales with chunk
+size, not total gene count) as the actual unit of parallel work — 150 targets × 8 chunks = 1,200
+jobs, and (2) downcasting the sort-order and cumulative-count arrays from `argsort`'s default
+int64 to int32 (safe: array length ≤ ~43,000, far under int32 range), and (3) dropping
+`max_workers` to 16. Measured single-job peak at chunk_size=1500 (worst-case target): **5.7 GB
+RSS** — 16 concurrent ⇒ ~91 GB peak, comfortably under 188 GB even before accounting for jobs not
+all hitting worst-case simultaneously. Verified stable (`free -h` steady in the 45–57 GiB used
+range, swap flat) before leaving it to run unattended. **Lesson for later stages**: this
+same pitfall applies directly to Stage 3's observational-skeleton PUC computation and to any
+GPU-backed run — check peak memory on one representative unit of work before fanning out, not
+after.
+
+**Two more infrastructure snags, both fixed, both worth remembering for Stage 3:**
+
+- **OOM near-miss.** First parallelization attempt used `ProcessPoolExecutor(max_workers=46)`
+  over all 150 targets at full gene width (11,942 genes/worker). `free -h` showed 187/188 GiB
+  used and 92 GiB of swap within minutes; the job died silently (exit 1, empty stderr — an OOM
+  kill of one or more workers, swallowed by the executor). Root cause: every worker's arrays are
+  shaped `(m+n, G)` where `n` is the *fixed* 38,176-cell NTC pool, so per-worker peak memory
+  (~15–20 GB, dominated by `argsort`'s default int64 output) barely depends on the target being
+  processed — 46 workers all inflate to that peak roughly simultaneously. Fixed by chunking genes
+  explicitly (1,500/chunk, so peak scales with chunk size, not total gene count — 150 targets × 8
+  chunks = 1,200 jobs), downcasting `argsort`/cumulative-count arrays to int32, and dropping to 16
+  workers. Measured single-job peak after the fix: 5.7 GB RSS.
+- **I/O bottleneck (looked like a hang, wasn't).** Even after the memory fix, the run appeared to
+  stall for many minutes with workers stuck at "R" state and CPU time barely accumulating —
+  because each worker's gene-chunk was a *fancy-indexed column gather* out of a cell-major
+  `(cells, genes)` array (`X[np.ix_(cell_rows, gene_cols)]`): for every selected row, ~1,500
+  scattered non-contiguous reads out of an 18,080-wide row, times up to ~43,000 rows, times 16
+  concurrent workers hitting the same 16 GB mmap. Fixed by precomputing a **gene-major, gene-filtered
+  contiguous array** once (`(11,942 genes, 221,273 cells)`, `analysis/scratch: X_genemajor_filtered.npy`)
+  so a gene-chunk read is a fast contiguous row-block; cell selection then happens as fancy-indexing
+  on an already-in-RAM small block, not scattered disk reads. That one-time rebuild itself had the
+  same pitfall the first time (built via `X[:, gene_mask]` fancy-indexing on the mmap'd cell-major
+  file — also slow) and was fixed by loading the source array fully into RAM first (sequential read,
+  fast) before slicing/transposing in memory.
+- **Final, healthy run**: 1,200 jobs, 16 workers, chunk_size=1,500 — **995.1s (~16.6 min)**, memory
+  stable at 45–70 GiB throughout, workers genuinely CPU-bound (verified via `top`: 1:1 CPU-time-to-
+  wall-time ratio). `E_matrix.npy` (150 × 11,942, float32) saved.
+- **Correctness sanity check**: `E_matrix.npy` has a handful of tiny negative values (min
+  ≈ −1.4e-5) — expected floating-point noise near zero for this identity, not a bug; clipped to 0
+  downstream. Self-effect signal is enormous and correctly recovered: for each target's own gene,
+  median energy-distance² is ≈2.35 vs. a median of ≈0.0008 across random (target, gene) pairs —
+  roughly 3000×, as expected given how strong the CRISPRi self-knockdown signal already was in
+  Checkpoint 0.
+
+**Null calibration** (`analysis/scripts/null_calibration.py`): NTC/NTC splits at 7 representative
+sizes (50, 200, 500, 1000, 2000, 3000, 4500 — spanning the observed 33–4760 target-size range), 8
+replicates each, same chunked/parallel machinery — 448 jobs, 442.8s, memory-safe throughout (no
+repeat of the earlier issues, since the fixes above applied directly). Null mean and SD both
+scale smoothly and roughly as 1/size across the tested range (mean null e² drops from 0.0096 at
+size 50 to 0.00012 at size 4500 — about 77× over a 90× size range, consistent with what's expected
+for this kind of two-sample statistic) — a useful confirmation that the representative-size
+approach is behaving sensibly, not an artifact.
+
+**Calibration applied** (`analysis/scripts/calibrate_effects.py`): each target's per-gene energy
+distance is z-scored against its own group size's null (log-log interpolated between the two
+bracketing representative sizes), one-sided normal-approximation p-value (energy distance is only
+ever elevated by a real effect), BH-FDR across all 150 × 11,942 = 1,791,300 (target, gene) pairs,
+per the plan's own instruction to correct "across all (g, j)."
+
+**Checkpoint D0 (companion plan) result: technically PASSES, but with an important caveat.**
+Measured effective out-degree $\hat k^{\mathrm{out}}_g$ (genes at q<0.05, excluding the target's own
+gene) ranges from **1,003 to 11,559** out of 11,942 genes across the 150 targets (mean 4,481,
+median 4,133) — no target near zero, real dynamic range (>11× spread), so the letter of the D0 gate
+("non-degenerate $\hat k^{\mathrm{out}}$ spread... not near-zero for most targets") is satisfied.
+Correlation between $\hat k^{\mathrm{out}}_g$ and the target's own cell count is weak and slightly
+*negative* (r = −0.12), so this isn't simply a power/n artifact in the crude sense.
+**But**: the top of that range means some targets have "significant" shifts in **97% of the
+filtered genome** — a number that is not a credible functional out-degree for a single gene's
+regulatory targets by any biological standard, and is exactly the failure mode the interventional
+plan's own Checkpoint 0 "Secondary concern" flagged in advance: *"hESC are a self-renewing
+pluripotent population with strong cell-cycle structure and differentiation-propensity
+heterogeneity. That variance will dominate MI. Cell-cycle regression or explicit conditioning is
+not optional here."* At this UMI depth and these cell counts, any perturbation that nudges cell
+state/cycle composition will register as "significant" against thousands of genes through that
+shared axis, not through 150 independent direct regulatory programs. **Conclusion: D0 passes as
+literally specified, but $\hat k^{\mathrm{out}}_g$ as currently computed is not yet trustworthy as
+an out-degree *prior* for the companion plan's V3/V4 — cell-cycle/state conditioning (regression or
+explicit stratification) needs to happen before this number means what the companion plan wants it
+to mean.** This is the next concrete blocker to close, not a stopping point — flagged here rather
+than glossed over.
+
+**Interventional Checkpoint 3, second half (known pluripotency edges): recovered, cleanly.**
+`SOX2` (the one pluripotency gene that's both measured and a perturbation target) shows extremely
+significant shifts in both `POU5F1` (q≈0, energy-distance²=0.071) and `NANOG` (q≈0,
+energy-distance²=0.135) — the canonical hESC pluripotency circuit is recovered as a strong positive
+control. Caveat: `SOX2` itself has one of the largest $\hat k^{\mathrm{out}}$ values (8,941/11,942,
+~75%), so this specific pair recovering "significant" carries the same cell-state-confound caveat
+as D0 above — it's necessary-but-not-sufficient evidence, not proof the calibration is cleanly
+isolating direct regulatory edges yet.
+
+Full per-target table: `k_out.csv` in the session scratchpad (not committed).
+
+---
+
+## 2026-09-04 — Zero-as-own-bin discretizer
+
+Implemented `fastpidc.discretizers.get_bin_ids_zero_as_own_bin` (Stage 2's discretizer, per
+STATE.md item 2): exact zeros get bin 0, nonzero values are equal-frequency ("uniform_count")
+binned into the remaining `number_of_bins - 1` bins. Wired into the existing `get_bin_ids`
+dispatcher as `mode="zero_as_own_bin"`, so it's usable directly through the public
+`Node.from_raw_values(..., discretizer="zero_as_own_bin", ...)` API — turned out **not** to need
+the "construct `Node` by hand, bypassing the provided discretizers" workaround flagged in the
+2026-09-04 API note #2; adding one function and one dispatch branch was enough. Falls back to a
+plain zero/nonzero binarization if there isn't enough nonzero data to also equal-frequency-split it
+(fewer nonzero values than requested bins minus one, or no nonzero values at all). Unit tests added
+to `python/tests/test_discretizers.py` (zero isolation, equal-frequency counts on the nonzero
+remainder, all-zero collapse, small-data fallback, dispatch-vs-direct-call equivalence) — 17/17
+pass in that file, 78 passed / 1 skipped across the full `python/` suite. Not yet run against real
+data; that's interventional Checkpoint 1 (MI invariance under equal-frequency binning), still open.
+
+## 2026-09-04 — Add `pid_triple`
+
+Added `fastpidc.pid` (new module): `pid_triple(source1, source2, target, estimator=, base=)` →
+`PIDTriple(redundancy, unique1, unique2, synergy, mi1, mi2, mi_joint)`, closing the gap flagged in
+the 2026-09-04 API note #5 and required by the interventional plan's own instruction to add this
+before Stage 6 (H2). Built from the existing pieces, per that note's sketch:
+`redundancy = apply_redundancy_formula(target.probabilities, si1, si2, base)` where `si1`/`si2` come
+from `get_mi_and_si(source_i, target, ...)` (using `target.probabilities` directly rather than
+recomputing the marginal, since `Node.from_raw_values` already computed it with a matching
+estimator); `unique_i = mi_i - redundancy`; `synergy = mi_joint - redundancy - unique1 - unique2`,
+where `mi_joint` is the MI of a **joint-binned "combined" node** (new helper `combined_node`: bin id
+= `source1_bin * source2.number_of_bins + source2_bin`) against `target`. Still only supports
+`I_min` as the redundancy measure — no new formula introduced, so Checkpoint 4's
+redundancy-measure-robustness caveat (no BROJA/I_ccs in this package) is unchanged.
+
+Verified against known-analytic PID cases in `python/tests/test_pid.py` rather than just
+structural checks: XOR gate (X, Y independent fair bits, Z = X⊕Y) recovers ≈0 redundancy/uniques
+and ≈1 bit of pure synergy; identical sources (Y := X, Z := X) recover ≈0 unique/synergy and
+redundancy ≈ MI(X,Z) ≈ 1 bit; fully independent (X, Y, Z all independent) recovers ≈0 everywhere;
+a general random-XOR-target case checks the decomposition identity
+`mi_joint == redundancy + unique1 + unique2 + synergy` holds to floating-point precision. All 5
+pass. Exposed at the package top level (`fastpidc.pid_triple`, `fastpidc.combined_node`,
+`fastpidc.PIDTriple`). Not yet exercised on real intervention-indicator/gene/gene triples from the
+Arc data — that's Stage 6 itself, still not started (needs Stage 3's observational skeleton and a
+binned intervention-indicator node first).
+
+## 2026-09-04 — Cell-cycle / cell-state conditioning (companion plan D0 blocker)
+
+Implemented `analysis/scripts/cell_cycle.py` to address the D0 caveat: score every cell for S-phase
+and G2M-phase activity using the standard Tirosh et al. / Seurat `cc.genes.updated.2019` marker
+lists (42/43 S genes and 54/54 G2M genes found in `var_names`), via the same algorithm as
+`scanpy.tl.score_genes` / Seurat `AddModuleScore` (mean expression of the gene set minus mean
+expression of a size-matched, expression-bin-matched control set) — implemented directly rather
+than adding `scanpy` as a dependency for one function. Then regressed both scores out of the
+gene-major filtered expression matrix (11,942 × 221,273) via **vectorized OLS across all genes at
+once**, not a per-gene Python loop: for design matrix `D` (n_cells × 3: intercept, S_score,
+G2M_score) and gene-major data `M` (n_genes × n_cells), used the identity
+`B = (D^T D)^-1 (M D)^T` to get per-gene coefficients without ever transposing the 11 GB `M` array
+(`D^T M^T = (M D)^T`, and `M @ D` is a cheap n_genes×3 matmul) — the same "watch memory layout"
+lesson from Stage 4's I/O bottleneck, applied preemptively this time instead of discovered the hard
+way. Output: `X_resid_genemajor.npy` (same shape as the input).
+
+**Only ~1.5% of per-gene variance was explained by the two cell-cycle scores.** Smaller than hoped
+given how dramatic the D0 out-degree inflation looked (up to 97% of the genome "significant" for
+one target) — suggests cell-cycle proper is only part of the confound, and the plan's own
+"differentiation-propensity heterogeneity" axis (a separate, likely larger source of shared
+variance in a self-renewing-but-heterogeneous hESC population) is not captured by cell-cycle
+scoring alone. Re-ran Stage 4 end-to-end (effect matrix → representative-size null calibration →
+BH-FDR) against the residualized array via `rerun_stage4_residualized.py` (parallel output files,
+suffixed `_resid`, rather than overwriting the originals — `E_matrix_resid.npy`,
+`k_out_resid.csv`, etc.) to see whether even that modest variance removal meaningfully shrinks the
+implausible tail of $\hat k^{\mathrm{out}}_g$, given how much statistical power this dataset has
+(a small mean shift can still clear q<0.05 at these sample sizes). **Result: negative. Cell-cycle regression alone does not fix the D0 confound — if anything, the
+residualized numbers are slightly worse.**
+
+| | original (`k_out.csv`) | cell-cycle-residualized (`k_out_resid.csv`) |
+|---|---|---|
+| range | 1,003 – 11,559 | 1,036 – 11,680 |
+| mean | 4,481 | 4,656 |
+| median | 4,133 | 4,196.5 |
+| targets with $\hat k^{\mathrm{out}} \geq 10$ | 150/150 | 150/150 |
+
+Per-target comparison: median change **+72.5**, mean change **+175.8** ($\hat k^{\mathrm{out}}$
+went *up* after residualization for 135/150 targets, down for only 15/150). `PRDM14` is now the
+single largest offender at 11,680/11,942 (97.8% of the filtered genome), essentially unchanged from
+the pre-residualization tail. Correlation with `n_pert` is still weakly negative (r = −0.117,
+matching the original −0.12) — still not simply a power artifact in the crude sense, but that was
+never in question.
+
+**Interventional Checkpoint 3 (SOX2 → POU5F1/NANOG) also got no better, and SOX2's own confound
+caveat got slightly worse**: both edges remain q≈0 after residualization (POU5F1 energy-distance²
+0.071 → 0.053, NANOG 0.135 → 0.128 — both still highly significant, magnitudes shrank slightly as
+expected since *some* shared variance was removed), so the positive-control recovery itself is
+robust to this conditioning. But SOX2's own $\hat k^{\mathrm{out}}$ **increased** from 8,941/11,942
+(~75%) to 9,536/11,942 (~80%) — the exact opposite of what conditioning on the confound should do
+if cell-cycle were the dominant driver.
+
+Sanity checks before trusting this null result: (1) the residualized null's mean/SD-vs-size scaling
+matches the original run's within ~0.5% at every representative size (e.g. mean null e² at size 50:
+0.009565 residualized vs. 0.009570 original) — the null-calibration machinery itself did not break
+on the residualized input, so the flat/negative result isn't a null-calibration artifact; (2) only
+~1.5% of per-gene variance was removed by the two cell-cycle scores in the first place (see the
+entry above) — in retrospect this result is exactly what that small an adjustment predicts, given
+how much statistical power this dataset has (thousands of cells per group, >50k UMI/cell): a 1.5%
+variance reduction is nowhere near enough to move q-values computed at that power.
+
+**Conclusion, stated plainly: cell-cycle scoring/regression, on its own, is not the fix for the D0
+out-degree inflation.** The interventional plan's own "Secondary concern" language named both "cell
+cycle" *and* "differentiation-propensity heterogeneity" as risks — this result is consistent with
+the latter (or some other shared axis not captured by the Tirosh/Seurat S/G2M marker genes) being
+the actual dominant confound, not cell-cycle phase per se. **This is now a re-opened, harder
+problem, not a closed one**: before the companion plan's V3/V4 can trust $\hat k^{\mathrm{out}}_g$
+as a real degree prior, a different or additional conditioning strategy is needed — candidates worth
+trying next: (a) a data-driven state axis (e.g. top PCs of the NTC-only cells, or a diffusion
+pseudotime/differentiation score) rather than a curated marker-gene score, regressed out the same
+way; (b) stratifying the NTC-null calibration by state bin instead of (or in addition to)
+regression, so the null itself absorbs the heterogeneity rather than assuming a linear correction
+removes it; (c) checking whether the top-$\hat k^{\mathrm{out}}$ targets (`PRDM14`, `METTL14`,
+`METTL3`, `KDM1A`, `SMARCA4`, ...) share a annotatable biological theme (several of these are
+chromatin/epigenetic regulators, which is *itself* a plausible reason for broad transcriptional
+disruption rather than proof of confounding — worth a literature gut-check before assuming this is
+purely technical). Not resolved in this session; flagged as the concrete next blocker in `STATE.md`.
+
+## 2026-09-04 — CUDA backend validation (Prerequisites item 3)
+
+Installed `cupy-cuda12x==14.2.0` into `analysis/` via `uv add "cupy-cuda12x>=12.0"` (matches
+`python/pyproject.toml`'s `cuda` extra pin) — resolved cleanly against the existing Python ≥3.12
+project in 279ms, no dependency conflicts, only pulled in `cuda-pathfinder` as a transitive
+dependency. Machine has CUDA 12.4 toolkit/headers (`nvcc --version`) under driver 595.84 (CUDA
+13.2, backward-compatible) and an idle RTX 4090 (24 GB, 0% util, no other processes holding it) —
+confirmed via `nvidia-smi` before starting, to make sure this wouldn't collide with the CPU-only
+cell-cycle-conditioning rerun happening in parallel this session. `fastpidc.cuda.cuda_available()`
+returns `True` and `cp.cuda.runtime.getDeviceCount()` reports the GPU correctly; no `CUDA_PATH`
+workaround needed (the headers are found under `/usr` already, one of `_FALLBACK_CUDA_HEADER_DIRS`
+in `fastpidc/cuda.py`).
+
+**Correctness**: `analysis/scripts/cuda_backend_check.py` builds a 40-node synthetic case (latent
+factor + noise, 2,000 cells, 6 bins) and compares `fastpidc.puc.compute_puc_full` (CPU) against
+`fastpidc.cuda.compute_puc_full_cuda` (GPU). Exact bitwise match: `max|MI_cpu - MI_gpu| = 0.0`,
+`max|PUC_cpu - PUC_gpu| = 0.0`. Both backends implement the same closed-form arithmetic on the same
+binned integer data, so exact agreement (not just "close") is the right bar here and it's met.
+
+**Timing/memory profile**, done at the *real* cell count (221,273, not a small stand-in) since both
+compute cost (joint-counts kernel is `O(n^2 * m)`-ish) and device memory (the `(m, n)` int32 binned-
+data array) depend on cell count, not just gene count — using a small `n_cells` would have given a
+memory estimate that's wrong in exactly the way that matters for sizing the real run. Default
+`chunk_size=256`, `n_bins=10` (matches `PIDCConfig`'s actual default number of bins):
+
+| N (genes) | time | GPU pool reserved | device free after |
+|---|---|---|---|
+| 512  | 4.8s   | 0.52 GB | 24.31 / 24.84 GB |
+| 1024 | 31.4s  | 1.05 GB | 23.78 / 24.84 GB |
+| 2048 | 127.5s | 2.13 GB | 22.70 / 24.84 GB |
+
+Time scales quadratically in N as expected (1024→2048 is a 4.06× runtime increase, consistent with
+`O(N^2)` pair-work at fixed cell count); memory scales linearly in N (doubling N roughly doubles
+reserved memory), consistent with the dominant term being the `(cells, genes)` binned-data array
+(`221,273 * N * 4 bytes` for the int32 array alone — e.g. at N=2048 that's already 1.81 GB of the
+2.13 GB measured, the rest split between the kernel's per-chunk counts buffer and the N×N MI/PUC
+output matrices).
+
+**Extrapolation to the real Stage-3 scale** (N=11,942 filtered genes, same 221,273 cells): memory
+≈ 221,273 × 11,942 × 4 bytes (binned data) + 10² × 11,942 × 256 × 4 bytes (counts chunk, chunk_size
+default 256) + 2 × 11,942² × 8 bytes (MI + PUC output matrices) ≈ 10.6 + 1.2 + 2.3 ≈ **~14 GB**,
+comfortably inside the RTX 4090's 24 GB. Time ≈ 127.5s × (11,942 / 2,048)² ≈ **~72 minutes** for one
+full dense PUC pass over the entire filtered gene set. Both numbers are extrapolations from N≤2048
+measurements, not a direct measurement at N=11,942 — a spot-check at an intermediate size (e.g.
+N=4096) before committing a ~70-minute run would be cheap insurance, but nothing in the scaling
+behavior so far suggests a surprise at full width.
+
+**Recommendation, superseding STATE.md's earlier assumption that Stage 3 would need the same
+gene-chunked multiprocessing treatment Stage 4 needed on CPU**: run the full 11,942-gene dense PUC
+in **one GPU call** (`compute_puc_full_cuda`, letting its own internal chunking over the z-axis at
+`chunk_size=256` bound device memory) rather than pre-splitting into a CPU-style job queue — there's
+only one GPU to serialize onto anyway, and 14 GB of 24 GB leaves enough headroom that chunk-size
+tuning for memory is not expected to be necessary. This closes STATE.md Prerequisites item 3; Stage
+3 itself (running this against the real, zero-as-own-bin-discretized, filtered gene set) is still
+not started — it also needs interventional Checkpoint 1 (binning-invariance check) run first, per
+the plan's own ordering.
+
+## Scratch outputs (not committed)
+
+Intermediate arrays/CSVs from today's session live in the session scratchpad
+(`/tmp/claude-1000/.../scratchpad/`), not in the repo — regenerate from the h5ad rather than
+relying on these persisting: `X_dense.npy` / `X_norm_log.npy` (16 GB, cell-major raw / normalized),
+`X_genemajor_filtered.npy` (10.6 GB, gene-major + gene-filtered, the layout `effect_matrix.py` and
+`null_calibration.py` actually read), `gene_filter_mask.npy`, `self_effect_screen.csv`,
+`phenotype_screen.csv`, `E_matrix.npy` + `E_matrix_meta.pkl` (Stage-4 effect matrix),
+`null_mean_by_size.npy` / `null_sd_by_size.npy` + `null_meta.pkl` (representative-size null),
+`E_zscores.npy` / `E_qvalues.npy` (calibrated), `k_out.csv` (measured effective out-degree per
+target). The three scripts (`effect_matrix.py`, `null_calibration.py`, `calibrate_effects.py`, all
+under `analysis/scripts/`) are committed and reproduce all of this from the h5ad + cached
+normalization step; only the huge intermediate arrays themselves aren't kept.
diff --git a/STATE.md b/STATE.md
new file mode 100644
index 0000000..2f59858
--- /dev/null
+++ b/STATE.md
@@ -0,0 +1,112 @@
+# Plan status
+
+Tracks completion of items in `Interventional PID for GRN inference—testing the gap with
+FastPIDC.jl on Arc H1 hESC Perturb-seq.md` (**interventional plan**) and `Degree-aware
+recalibration of PIDC—a companion plan.md` (**companion plan**). Detail and rationale in
+`LOG.md`. Branch: `perturbation-analysis`.
+
+Legend: ✅ done · 🟡 in progress / partial · ⬜ not started · 🚫 blocked
+
+---
+
+## Prerequisites (both plans' §"immediate next actions")
+
+| Item | Status | Note |
+|---|---|---|
+| Locate & verify real FastPIDC.jl Python API | ✅ | `LOG.md` 2026-09-04. Package lives at `python/`. |
+| Confirm per-gene PUC access + calibration can be disabled | ✅ | `PUCNetworkInference` / `compute_puc_full` — already exposed, no patch needed. |
+| Confirm `pid_triple`-equivalent (triple PID: redundancy/unique/synergy) exists | ✅ | Added `fastpidc.pid.pid_triple` (+ `combined_node`, `PIDTriple`) — see `LOG.md` 2026-09-04 "Add `pid_triple`". |
+| Set up environment to load Arc hESC/K562 h5ad | ✅ | `analysis/` uv project; `vsparse` 0.2.0 + `hdf5plugin`, load via `vsparse.VCSCAnnData.read_h5ad`. |
+| Literature re-check (degree-corrected graphical models; recent PID/interventional-GRN work) | ⬜ | Not started — needs web search, out of scope for a data-only session so far. |
+| Read ADAPRE, D-SPIN properly | ⬜ | Not started. |
+
+## Interventional plan — checkpoints (§5 summary table)
+
+| # | Gate | Status | Result |
+|---|---|---|---|
+| **0** | ≥~2,000 reciprocal testable pairs | ✅ **PASS** | 150 targets (not the assumed ~300 — see `LOG.md`), all clear cell-count + phenotype screens. `C(150,2) = 11,175` pairs. No Replogle fallback needed for feasibility. |
+| 1 | MI invariant under equal-frequency binning | ⬜ | Not started (discretizer now ready — see Stage 2 below — but the actual invariance run hasn't happened yet). |
+| 2 | Bootstrap edge recovery >50% at top-k | ⬜ | Not started. Needs Stage 3 (observational skeleton on NTC cells) first, which needs a working discretization pipeline. CUDA backend now installed and validated (`cupy-cuda12x`, see Prerequisites item 3) so the compute path is no longer the blocker — Stage 3 itself just hasn't been run yet. |
+| 3 | Self-effects present; known pluripotency edges recovered | ✅ | Self-effect: **150/150 pass** (q<0.05, down; Checkpoint-0 byproduct). Pluripotency edges: **recovered, and robust to cell-cycle residualization** — `SOX2` (measured + targeted) shows q≈0 shifts in both `POU5F1` (energy-distance² 0.071→0.053) and `NANOG` (0.135→0.128) before and after cell-cycle regression. Caveat unchanged/worsened: `SOX2`'s own $\hat k^{\mathrm{out}}$ is ~75–80% of the filtered genome (went *up* slightly after cell-cycle residualization, 8,941→9,536), so this is still necessary-but-not-sufficient — see D0 below, now a re-opened problem, not resolved by the cell-cycle fix that was tried. |
+| 4 | Sibling calls stable across redundancy measures (I_min vs BROJA/I_ccs) | ⬜ | Not started; blocked on Stage 6 existing at all, and on sourcing a non-I_min PID implementation (`fastpidc` only has I_min — see `LOG.md`). |
+| 5 | PID beats mediation regression (H3, the kill criterion) | ⬜ | Not started; last in the pipeline by design. |
+
+### Stages (§4), beyond what the checkpoints above already cover
+
+| Stage | Status | Note |
+|---|---|---|
+| 1 — Load & QC | 🟡 | Load path solved. Per-cell filters (UMI/gene-count/mito) and mixscape-style per-cell knockdown-efficiency estimate: not started. |
+| 2 — Normalization & discretization | 🟡 | Size-factor + log1p normalization exercised (as a means to Checkpoint 0, not yet as a committed pipeline step). Zero-as-own-bin discretizer now implemented as `fastpidc.discretizers.get_bin_ids_zero_as_own_bin` / `get_bin_ids(..., mode="zero_as_own_bin")` — not yet run against real data (that's Checkpoint 1). |
+| 3 — Observational skeleton | ⬜ | Not started, but unblocked: CUDA backend validated (see Prerequisites item 3 / `LOG.md` 2026-09-04) — correctness matches CPU exactly, and a full 11,942-gene dense PUC run is projected to fit in ~14 GB / ~70 min on the RTX 4090. Still needs the zero-as-own-bin discretizer actually run against real data (Checkpoint 1) before Stage 3 itself starts. |
+| 4 — Directed effect matrix (energy distance) | ✅ | Done: `analysis/scripts/effect_matrix.py`. Real energy distance (not the Checkpoint-0 mean-shift proxy), 150 targets × 11,942 genes, NTC-split significance calibration via `null_calibration.py` + `calibrate_effects.py` (BH-FDR across all pairs, per the plan). Hit and fixed an OOM and an I/O-layout bottleneck along the way — see `LOG.md`. **Open issue carried forward**: measured effective out-degree is inflated by hESC cell-state/cycle confounding, exactly as the plan's own Checkpoint-0 "Secondary concern" anticipated — see D0 below. |
+| 5 — Orientation (H1) | ⬜ | Not started; depends on Stage 4. |
+| 6 — Intervention-indicator PID (H2) | ⬜ | `pid_triple` primitive now exists (see Prerequisites) — Stage 6 itself (running it on real intervention-indicator/gene/gene triples) not started. |
+| 7 — Baselines (H3) | ⬜ | Not started. |
+| 8 — Invariance filtering (ICP) | ⬜ | Not started. |
+| 9 — External validation (ATAC, Replogle, literature) | ⬜ | Not started. |
+
+## Companion plan (degree-aware recalibration) — checkpoints (§8 summary table)
+
+| # | Gate | Status | Result |
+|---|---|---|---|
+| **D0** | Enough targets with non-degenerate measured $\hat k^{\mathrm{out}}$ | 🚫 **PASS on the letter of the gate, but the confound is NOT resolved — cell-cycle regression tried and failed** | $\hat k^{\mathrm{out}}_g$ ranges 1,003–11,559 / 11,942 genes originally (>11× spread, none near zero) — technically clears the gate. Tried the obvious fix (regress S/G2M cell-cycle scores out of expression, re-run Stage 4 on residuals — `LOG.md` 2026-09-04 "Cell-cycle / cell-state conditioning"): **it didn't work.** Residualized $\hat k^{\mathrm{out}}_g$ range 1,036–11,680, mean 4,656 (was 4,481), median 4,196.5 (was 4,133) — *higher* for 135/150 targets (median +72.5). SOX2's own out-degree got worse (8,941→9,536/11,942, ~75%→~80%). Only ~1.5% of per-gene variance was explained by cell-cycle scores in the first place, which in hindsight predicts exactly this null result given the dataset's statistical power. **Conclusion: the dominant confound is not cell-cycle phase** (or not fully — the plan's own "differentiation-propensity heterogeneity" language covers exactly this possibility). **This is a re-opened blocker, not a closed one** — see next actions below for candidate follow-ups (data-driven state axis instead of curated marker genes; stratified null instead of linear regression; check whether top-$\hat k^{\mathrm{out}}$ targets like `PRDM14`/`METTL14`/`METTL3`/`KDM1A`/`SMARCA4` share a chromatin/epigenetic-regulator theme that might be a real broad effect, not pure confound). Tables: `k_out.csv` / `k_out_resid.csv` (scratchpad, not committed). **See `LITERATURE_REVIEW.md` (e)**: before the state-axis/stratified-null/chromatin-plausibility follow-ups, check $\hat k^{\mathrm{out}}_g$ vs. per-target knockdown efficiency (`self_effect_screen.csv`, already computed) — ADAPRE (bioRxiv 2026) found and corrected exactly this confound (stronger knockdowns → spuriously higher out-degree) on comparable CRISPRi Perturb-seq data; cheaper than the other three candidates and reuses existing data. |
+| D1 | V2 (hierarchical gamma) does not reduce bootstrap stability | ⬜ | Not started; depends on interventional-plan Stage 3 existing. |
+| D2 | V1 (direction-aware calibration) beats V2 on held-out effect recovery | ⬜ | Not started. |
+| D3 | Known hubs (POU5F1, NANOG, SOX2) survive increasing λ in V3 | ⬜ | Not started; note all three are measured, `SOX2` is also itself a perturbation target. |
+| D4 | Gains don't appear equally for a correlation skeleton | ⬜ | Not started. |
+
+Implementation order per the plan (V2 → V1 → V4 → V3) not started; V2 is unblocked from the API
+side (raw PUC access confirmed) whenever Stage 3's skeleton exists to calibrate.
+
+---
+
+## Immediate next actions (carried over from both plans, reordered by what's actually next)
+
+1. **Cell-cycle / cell-state conditioning before trusting $\hat k^{\mathrm{out}}_g$.** 🚫 **Tried,
+   failed, re-opened.** `analysis/scripts/cell_cycle.py` scores every cell for S/G2M activity
+   (Tirosh/Seurat marker lists, scanpy-style module scoring) and regresses both scores out of the
+   gene-major filtered expression matrix; `rerun_stage4_residualized.py` reran Stage 4 end-to-end
+   against the residuals. **Result: no improvement** — residualized $\hat k^{\mathrm{out}}_g$ went
+   *up* for 135/150 targets (median +72.5), and SOX2's own out-degree got worse (~75%→~80%). Only
+   ~1.5% of per-gene variance was explained by the two cell-cycle scores, which predicts this null
+   result in hindsight. See `LOG.md` 2026-09-04 "Cell-cycle / cell-state conditioning" for full
+   numbers and sanity checks (null-scaling verified unaffected, so this isn't a calibration bug).
+   **Next candidate fixes, not yet tried:** (a) a data-driven state axis (top PCs of NTC cells, or
+   a pseudotime/differentiation score) instead of curated cell-cycle marker genes; (b) stratifying
+   the null calibration by state bin instead of linear regression; (c) a literature/annotation
+   check on whether the current top-$\hat k^{\mathrm{out}}$ targets (`PRDM14`, `METTL14`,
+   `METTL3`, `KDM1A`, `SMARCA4`, ...) are chromatin/epigenetic regulators whose broad effect might
+   be partly real, not purely a confound artifact.
+2. ✅ Implemented the zero-as-own-bin discretizer (`fastpidc.discretizers.get_bin_ids_zero_as_own_bin`,
+   wired into `get_bin_ids(..., mode="zero_as_own_bin")`, so it's usable directly via
+   `Node.from_raw_values(..., discretizer="zero_as_own_bin", ...)` — no bypass of the public API
+   needed after all). Not yet run against real data — running interventional Checkpoint 1
+   (binning-invariance sanity check) is still open.
+3. ✅ Installed `cupy-cuda12x==14.2.0` in `analysis/` (`uv add`, clean resolve, no conflicts with
+   the existing Python ≥3.12 project) and validated the CUDA backend end-to-end in
+   `analysis/scripts/cuda_backend_check.py` — see `LOG.md` 2026-09-04 "CUDA backend validation"
+   for full numbers. Correctness: CPU (`compute_puc_full`) vs CUDA (`compute_puc_full_cuda`) match
+   exactly (`max|diff| = 0.0`) on a 40-node synthetic case. Memory/timing at real cell count
+   (221,273 cells, 10 bins, chunk_size=256): time scales ~quadratically in gene count as expected
+   (N=1024→2048 is a 4.06× runtime increase), device memory scales ~linearly in gene count
+   (0.52/1.05/2.13 GB at N=512/1024/2048) and is dominated by the (cells × genes) int32 binned-data
+   array. **Extrapolated full-genome estimate (N=11,942, real cell count): ~14 GB device memory
+   (fits comfortably in the RTX 4090's 24 GB) and ~70 minutes wall-clock** — both numbers are
+   extrapolations from the measured N≤2048 points, not a direct measurement at full N; worth a
+   spot-check at, say, N=4096 before committing to the full run, but nothing here suggests Stage 3
+   is infeasible on this GPU. **Recommendation: proceed with Stage 3 at the full 11,942-gene
+   filtered set in one GPU pass (no gene-chunking needed across the outer call, only the kernel's
+   own internal z-chunking at chunk_size=256), rather than pre-splitting into a job queue like
+   Stage 4's CPU multiprocessing did** — the memory headroom and single-GPU serialization make
+   that unnecessary here.
+4. ✅ Added `fastpidc.pid` (`pid_triple`, `combined_node`, `PIDTriple`) implementing the triple PID
+   decomposition (`MI_joint = redundancy + unique1 + unique2 + synergy`) via the existing
+   `get_mi_and_si` / `apply_redundancy_formula` building blocks, plus a joint-binned "combined
+   node" for the synergy term. Unit-tested against known-analytic cases (XOR → pure synergy,
+   identical sources → pure redundancy, independent → all-zero) in `python/tests/test_pid.py`; 78
+   passed / 1 skipped across the full `python/` suite after adding it. Still uses `I_min` only
+   (Checkpoint 4's redundancy-measure-robustness check is unaffected — still needs an external PID
+   library). Not yet exercised on real intervention-indicator/gene/gene triples (that's Stage 6
+   itself, still not started).
+5. Literature re-check and ADAPRE/D-SPIN reading — delegated to a background agent (see
+   `LITERATURE_REVIEW.md` once it lands); verdict not yet folded into this file.
diff --git a/analysis/pyproject.toml b/analysis/pyproject.toml
new file mode 100644
index 0000000..94f4bd8
--- /dev/null
+++ b/analysis/pyproject.toml
@@ -0,0 +1,21 @@
+[project]
+name = "fastpidc-analysis"
+version = "0.0.0"
+description = "Scratch environment for evaluating the interventional-PID and degree-recalibration plans against Arc/Replogle Perturb-seq data. Not a published package."
+requires-python = ">=3.12"
+dependencies = [
+    "fastpidc",
+    "vsparse>=0.2.0",
+    "anndata>=0.13",
+    "pandas>=2.0",
+    "statsmodels>=0.14",
+    "matplotlib>=3.9",
+    "cupy-cuda12x>=12.0",
+]
+
+[tool.uv]
+package = false
+
+[tool.uv.sources]
+fastpidc = { path = "../python", editable = true }
+vsparse = { path = "/home/asm/vsparse", editable = true }
diff --git a/analysis/scripts/calibrate_effects.py b/analysis/scripts/calibrate_effects.py
new file mode 100644
index 0000000..7edd1db
--- /dev/null
+++ b/analysis/scripts/calibrate_effects.py
@@ -0,0 +1,138 @@
+"""Apply the representative-size NTC/NTC null (`null_calibration.py`) to the
+real Stage-4 effect matrix (`effect_matrix.py`): z-score each target's
+per-gene energy distance against the null level for its own group size
+(log-linear interpolation between the two bracketing representative sizes),
+one-sided normal-approximation p-value (energy distance is only ever
+*elevated* by a real effect, never suppressed), BH-FDR across all
+(target, gene) pairs -- per the interventional plan's Stage 4 instruction
+("Benjamini-Hochberg across all (g, j)").
+
+Feeds:
+  - companion plan Checkpoint D0: non-degenerate spread in measured
+    effective out-degree k^out_g = #{j : E[g,j] significant}.
+  - interventional plan Checkpoint 3 (second half): known pluripotency
+    edges / hub detectability, now with an actual (approximately)
+    calibrated effect matrix rather than the Checkpoint-0 mean-shift proxy.
+"""
+
+from __future__ import annotations
+
+import pickle
+
+import numpy as np
+import pandas as pd
+from scipy import stats
+from statsmodels.stats.multitest import multipletests
+
+from effect_matrix import SCRATCH
+
+
+def load_all(
+    e_matrix_name: str = "E_matrix.npy",
+    e_meta_name: str = "E_matrix_meta.pkl",
+    null_mean_name: str = "null_mean_by_size.npy",
+    null_sd_name: str = "null_sd_by_size.npy",
+    null_meta_name: str = "null_meta.pkl",
+):
+    E = np.load(SCRATCH / e_matrix_name)
+    np.clip(E, 0, None, out=E)  # tiny negative floating-point noise near zero
+    meta = pickle.load(open(SCRATCH / e_meta_name, "rb"))
+    target_list = meta["target_list"]
+    genes = meta["filtered_genes"]
+
+    null_mean = np.load(SCRATCH / null_mean_name)
+    null_sd = np.load(SCRATCH / null_sd_name)
+    null_meta = pickle.load(open(SCRATCH / null_meta_name, "rb"))
+    sizes = np.array(null_meta["sizes"], dtype=np.float64)
+
+    targets_obs = pickle.load(open(SCRATCH / "obs_target_gene.pkl", "rb"))
+    n_pert = {g: int((targets_obs == g).sum()) for g in target_list}
+    return E, target_list, genes, null_mean, null_sd, sizes, n_pert
+
+
+def interpolate_null(n: int, sizes: np.ndarray, null_mean: np.ndarray, null_sd: np.ndarray):
+    """Log-log interpolate the null mean/SD (per gene) to sample size n.
+    Energy distance's null scales roughly like 1/n (verified empirically:
+    mean null e2 drops ~1/size across the tested range) so interpolating
+    log(null) vs log(size) is closer to linear than a raw linear interp."""
+    n = np.clip(n, sizes.min(), sizes.max())
+    log_sizes = np.log(sizes)
+    log_n = np.log(n)
+    i = np.searchsorted(log_sizes, log_n, side="right")
+    i = np.clip(i, 1, len(sizes) - 1)
+    lo, hi = i - 1, i
+    t = (log_n - log_sizes[lo]) / (log_sizes[hi] - log_sizes[lo])
+
+    def interp(arr):
+        log_arr = np.log(np.clip(arr, 1e-12, None))
+        return np.exp(log_arr[lo] * (1 - t) + log_arr[hi] * t)
+
+    return interp(null_mean), interp(null_sd)
+
+
+def calibrate(
+    load_kwargs: dict | None = None,
+    q_name: str = "E_qvalues.npy",
+    z_name: str = "E_zscores.npy",
+    kout_name: str = "k_out.csv",
+):
+    E, target_list, genes, null_mean, null_sd, sizes, n_pert = load_all(**(load_kwargs or {}))
+    gi = {g: i for i, g in enumerate(genes)}
+
+    Z = np.zeros_like(E)
+    for ti, g in enumerate(target_list):
+        mu, sd = interpolate_null(n_pert[g], sizes, null_mean, null_sd)
+        Z[ti] = (E[ti] - mu) / sd
+
+    p = stats.norm.sf(Z)  # one-sided: only elevated energy distance is "significant"
+    q_flat = multipletests(p.ravel(), method="fdr_bh")[1]
+    Q = q_flat.reshape(p.shape)
+
+    np.save(SCRATCH / q_name, Q)
+    np.save(SCRATCH / z_name, Z)
+
+    # measured effective out-degree per target, excluding the target's own gene
+    k_out = {}
+    for ti, g in enumerate(target_list):
+        sig = Q[ti] < 0.05
+        if g in gi:
+            sig = sig.copy()
+            sig[gi[g]] = False
+        k_out[g] = int(sig.sum())
+
+    df = pd.DataFrame({"target": target_list, "n_pert": [n_pert[g] for g in target_list], "k_out": [k_out[g] for g in target_list]})
+    df.to_csv(SCRATCH / kout_name, index=False)
+
+    print("=== Checkpoint D0 (companion plan): measured k^out_g spread ===")
+    print(df["k_out"].describe())
+    print("n targets with k_out == 0:", (df["k_out"] == 0).sum())
+    print("n targets with k_out >= 10:", (df["k_out"] >= 10).sum())
+    print()
+    print("Correlation of k_out with n_pert (power confound check):", df["k_out"].corr(df["n_pert"]))
+    print()
+    print("Top 10 by k_out:")
+    print(df.sort_values("k_out", ascending=False).head(10))
+    print()
+    print("Bottom 10 by k_out:")
+    print(df.sort_values("k_out").head(10))
+
+    # pluripotency / hub check (interventional Checkpoint 3, second half)
+    print()
+    print("=== Pluripotency / hub check ===")
+    for g in ["POU5F1", "NANOG", "SOX2"]:
+        row = df[df["target"] == g]
+        if len(row):
+            print(f"{g} (perturbation target): k_out = {row['k_out'].values[0]}, n_pert = {row['n_pert'].values[0]}")
+        else:
+            print(f"{g}: not a perturbation target (measured only)")
+    if "SOX2" in target_list:
+        ti = target_list.index("SOX2")
+        for other in ["POU5F1", "NANOG"]:
+            if other in gi:
+                q = Q[ti, gi[other]]
+                e = E[ti, gi[other]]
+                print(f"SOX2 -> {other}: q = {q:.3g}, energy_d^2 = {e:.4g}")
+
+
+if __name__ == "__main__":
+    calibrate()
diff --git a/analysis/scripts/cell_cycle.py b/analysis/scripts/cell_cycle.py
new file mode 100644
index 0000000..4c28e84
--- /dev/null
+++ b/analysis/scripts/cell_cycle.py
@@ -0,0 +1,141 @@
+"""Step 1 of STATE.md's "immediate next actions": condition Stage 4's effect
+matrix on cell-cycle/cell-state before trusting the companion plan's k^out_g
+degree prior (Checkpoint D0). LOG.md's own analysis of the Stage-4 run found
+some targets "significant" against ~97% of the filtered genome -- almost
+certainly hESC cell-cycle/state composition shifting under perturbation, not
+150 independent direct regulatory programs, exactly the risk the interventional
+plan's Checkpoint 0 "Secondary concern" flagged in advance.
+
+Two-part fix, both implemented here:
+
+1. Score every cell for S-phase and G2M-phase activity using the standard
+   Tirosh et al. / Seurat cc.genes.updated.2019 marker lists, via the same
+   algorithm as scanpy.tl.score_genes / Seurat AddModuleScore: each cell's
+   score for a gene set is (mean expression of the set) minus (mean expression
+   of a size-matched control gene set drawn from the same expression-level
+   bins), computed directly here rather than pulling in scanpy as a dependency
+   for one function.
+2. Regress S_score and G2M_score out of the gene-major filtered expression
+   matrix (linear regression per gene, vectorized across all genes at once via
+   the normal equations -- NOT a per-gene Python loop, which would be ~12,000
+   separate lstsq calls) and write the residualized matrix. Stage 4's
+   `effect_matrix.py` / `null_calibration.py` / `calibrate_effects.py` are then
+   re-run against the residualized array (see `rerun_stage4_residualized.py`)
+   to get a cell-state-conditioned k^out_g.
+
+Vectorized regression trick (avoids transposing the 11,942 x 221,273 gene-major
+array, which would cost ~11 GB and a slow transpose): for design matrix D
+(n_cells x 3: [intercept, S_score, G2M_score]) and gene-major data M
+(n_genes x n_cells), the per-gene OLS coefficients are
+    B = (D^T D)^-1 (M D)^T
+because D^T M^T = (M D)^T -- so the only large matmul needed is M @ D
+(n_genes x 3), not M^T (n_cells x n_genes). Fitted values are then B^T @ D^T.
+"""
+
+from __future__ import annotations
+
+import pickle
+
+import numpy as np
+
+from effect_matrix import SCRATCH, X_GENEMAJOR_PATH
+
+X_NORM_LOG_PATH = SCRATCH / "X_norm_log.npy"  # (n_cells, n_genes) full 18,080-gene set
+
+# Seurat cc.genes.updated.2019 (Tirosh et al. 2016 cell-cycle marker lists),
+# public/standard gene symbol lists shipped with Seurat -- not project-specific.
+S_GENES = [
+    "MCM5", "PCNA", "TYMS", "FEN1", "MCM2", "MCM4", "RRM1", "UNG", "GINS2",
+    "MCM6", "CDCA7", "DTL", "PRIM1", "UHRF1", "MLF1IP", "HELLS", "RFC2",
+    "RPA2", "NASP", "RAD51AP1", "GMNN", "WDR76", "SLBP", "CCNE2", "UBR7",
+    "POLD3", "MSH2", "ATAD2", "RAD51", "RRM2", "CDC45", "CDC6", "EXO1",
+    "TIPIN", "DSCC1", "BLM", "CASP8AP2", "USP1", "CLSPN", "POLA1", "CHAF1B",
+    "BRIP1", "E2F8",
+]
+G2M_GENES = [
+    "HMGB2", "CDK1", "NUSAP1", "UBE2C", "BIRC5", "TPX2", "TOP2A", "NDC80",
+    "CKS2", "NUF2", "CKS1B", "MKI67", "TMPO", "CENPF", "TACC3", "PIMREG",
+    "SMC4", "CCNB2", "CKAP2L", "CKAP2", "AURKB", "BUB1", "KIF11", "ANP32E",
+    "TUBB4B", "GTSE1", "KIF20B", "HJURP", "CDCA3", "JPT1", "CDC20", "TTK",
+    "CDC25C", "KIF2C", "RANGAP1", "NCAPD2", "DLGAP5", "CDCA2", "CDCA8",
+    "ECT2", "KIF23", "HMMR", "AURKA", "PSRC1", "ANLN", "LBR", "CKAP5",
+    "CENPE", "CTCF", "NEK2", "G2E3", "GAS2L3", "CBX5", "CENPA",
+]
+
+N_BINS = 25
+CTRL_SIZE = 50
+
+
+def _score_gene_set(X: np.ndarray, gene_idx: np.ndarray, bin_of_gene: np.ndarray,
+                     genes_by_bin: dict[int, np.ndarray], rng: np.random.Generator) -> np.ndarray:
+    """scanpy/Seurat-style module score: mean(set) - mean(size-matched control)."""
+    ctrl_idx = []
+    for g in gene_idx:
+        b = bin_of_gene[g]
+        pool = genes_by_bin[b]
+        take = min(CTRL_SIZE, len(pool))
+        ctrl_idx.append(rng.choice(pool, size=take, replace=False))
+    ctrl_idx = np.unique(np.concatenate(ctrl_idx))
+    return X[:, gene_idx].mean(axis=1) - X[:, ctrl_idx].mean(axis=1)
+
+
+def compute_cc_scores(seed: int = 0) -> None:
+    var_names = pickle.load(open(SCRATCH / "var_names.pkl", "rb"))
+    name_to_idx = {n: i for i, n in enumerate(var_names)}
+
+    s_idx = np.array([name_to_idx[g] for g in S_GENES if g in name_to_idx])
+    g2m_idx = np.array([name_to_idx[g] for g in G2M_GENES if g in name_to_idx])
+    print(f"S genes found: {len(s_idx)}/{len(S_GENES)}; G2M genes found: {len(g2m_idx)}/{len(G2M_GENES)}")
+
+    X = np.load(X_NORM_LOG_PATH)  # load fully into RAM (16 GB) -- avoid scattered mmap reads (LOG.md lesson)
+
+    avg_expr = X.mean(axis=0)  # (n_genes,)
+    bin_of_gene = np.clip((np.argsort(np.argsort(avg_expr)) * N_BINS) // len(avg_expr), 0, N_BINS - 1)
+    genes_by_bin = {b: np.where(bin_of_gene == b)[0] for b in range(N_BINS)}
+
+    rng = np.random.default_rng(seed)
+    s_score = _score_gene_set(X, s_idx, bin_of_gene, genes_by_bin, rng)
+    g2m_score = _score_gene_set(X, g2m_idx, bin_of_gene, genes_by_bin, rng)
+    del X
+
+    phase = np.full(len(s_score), "G1", dtype=object)
+    is_cycling = (s_score > 0) | (g2m_score > 0)
+    phase[is_cycling & (s_score >= g2m_score)] = "S"
+    phase[is_cycling & (g2m_score > s_score)] = "G2M"
+
+    np.save(SCRATCH / "cc_s_score.npy", s_score.astype(np.float32))
+    np.save(SCRATCH / "cc_g2m_score.npy", g2m_score.astype(np.float32))
+    with open(SCRATCH / "cc_phase.pkl", "wb") as f:
+        pickle.dump(phase, f)
+
+    print("Phase counts:", {p: int((phase == p).sum()) for p in ["G1", "S", "G2M"]})
+    print("S_score range:", s_score.min(), s_score.max(), "mean", s_score.mean())
+    print("G2M_score range:", g2m_score.min(), g2m_score.max(), "mean", g2m_score.mean())
+
+
+def regress_out_cc() -> None:
+    s_score = np.load(SCRATCH / "cc_s_score.npy")
+    g2m_score = np.load(SCRATCH / "cc_g2m_score.npy")
+    n_cells = len(s_score)
+
+    D = np.column_stack([np.ones(n_cells), s_score, g2m_score]).astype(np.float64)  # (N, 3)
+    DtD = D.T @ D
+    DtD_inv = np.linalg.inv(DtD)
+
+    M = np.load(X_GENEMAJOR_PATH)  # (G, N) float32, loaded fully into RAM (~11 GB)
+    MD = M.astype(np.float64) @ D  # (G, 3) -- cheap; avoids transposing M
+    B = DtD_inv @ MD.T  # (3, G)
+    fitted = (B.T @ D.T).astype(np.float32)  # (G, N)
+    resid = M - fitted
+    del M, fitted
+
+    out_path = SCRATCH / "X_resid_genemajor.npy"
+    np.save(out_path, resid)
+    print(f"Residualized matrix written to {out_path}, shape {resid.shape}")
+    print("Residual variance retained (mean per-gene R^2 removed):",
+          float(1 - resid.var(axis=1).mean() / np.load(X_GENEMAJOR_PATH, mmap_mode="r").var(axis=1).mean()))
+
+
+if __name__ == "__main__":
+    compute_cc_scores()
+    regress_out_cc()
diff --git a/analysis/scripts/cuda_backend_check.py b/analysis/scripts/cuda_backend_check.py
new file mode 100644
index 0000000..2b3a036
--- /dev/null
+++ b/analysis/scripts/cuda_backend_check.py
@@ -0,0 +1,101 @@
+"""Step 3 of STATE.md's "immediate next actions": install/validate the cupy
+CUDA backend before attempting a full-scale PUC run (interventional plan
+Stage 3, an O(N^3) computation -- strictly worse than Stage 4's O(N log N)
+energy distance, which already needed real memory/layout care at N ~ 12,000
+genes, see LOG.md).
+
+Two checks:
+1. Correctness: CPU (`compute_puc_full`) vs CUDA (`compute_puc_full_cuda`)
+   backends must agree on a small synthetic case.
+2. Timing/memory at representative scale: one chunk-sized slice of node
+   counts close to what a real gene-count run would look like, to size a
+   full run before committing to it (same discipline as Stage 4's memory
+   check -- see LOG.md's "check peak memory on one representative unit of
+   work before fanning out, not after").
+"""
+
+from __future__ import annotations
+
+import time
+
+import numpy as np
+
+from fastpidc.puc import compute_puc_full
+from fastpidc.cuda import compute_puc_full_cuda, cuda_available
+from fastpidc.types import Node, PIDCConfig
+
+
+def make_synthetic_nodes(n_nodes: int, n_cells: int, n_bins: int, seed: int = 0) -> list[Node]:
+    rng = np.random.default_rng(seed)
+    # Correlated-ish synthetic data: each node is a noisy function of a shared
+    # latent plus its own noise, so MI/PUC aren't all trivially zero.
+    latent = rng.normal(size=n_cells)
+    nodes = []
+    for i in range(n_nodes):
+        raw = 0.4 * latent + rng.normal(size=n_cells)
+        binned = np.clip((np.argsort(np.argsort(raw)) * n_bins) // n_cells, 0, n_bins - 1).astype(np.int64)
+        counts = np.bincount(binned, minlength=n_bins).astype(np.float64)
+        probs = counts / counts.sum()
+        nodes.append(Node(label=str(i), binned_values=binned, number_of_bins=n_bins, probabilities=probs))
+    return nodes
+
+
+def correctness_check():
+    print("=== Correctness: CPU vs CUDA on a small synthetic case ===")
+    nodes = make_synthetic_nodes(n_nodes=40, n_cells=2000, n_bins=6)
+    mi_cpu, puc_cpu = compute_puc_full(nodes, estimator="maximum_likelihood", base=2)
+    mi_gpu, puc_gpu = compute_puc_full_cuda(nodes, base=2, verbose=False)
+
+    mi_diff = np.abs(mi_cpu - mi_gpu).max()
+    puc_diff = np.abs(puc_cpu - puc_gpu).max()
+    print(f"max|MI_cpu - MI_gpu|   = {mi_diff:.3e}")
+    print(f"max|PUC_cpu - PUC_gpu| = {puc_diff:.3e}")
+    print(f"MI allclose (atol=1e-6): {np.allclose(mi_cpu, mi_gpu, atol=1e-6)}")
+    print(f"PUC allclose (atol=1e-6): {np.allclose(puc_cpu, puc_gpu, atol=1e-6)}")
+    return mi_diff, puc_diff
+
+
+def timing_memory_profile(cases: list[tuple[int, int]], n_bins: int, chunk_size: int = 256):
+    print()
+    print("=== Timing / memory profile (CUDA backend) ===")
+    import cupy as cp
+
+    mempool = cp.get_default_memory_pool()
+    for n_nodes, n_cells in cases:
+        nodes = make_synthetic_nodes(n_nodes=n_nodes, n_cells=n_cells, n_bins=n_bins)
+        mempool.free_all_blocks()
+        free0, total0 = cp.cuda.runtime.memGetInfo()
+
+        t0 = time.time()
+        mi, puc = compute_puc_full_cuda(nodes, base=2, verbose=False, chunk_size=chunk_size)
+        cp.cuda.Stream.null.synchronize()
+        elapsed = time.time() - t0
+
+        # total_bytes() = pool's reserved size right after the call, before
+        # the next free_all_blocks() -- approximates this call's peak device
+        # allocation (cupy's pool doesn't shrink until freed).
+        peak_reserved = mempool.total_bytes()
+        free1, _ = cp.cuda.runtime.memGetInfo()
+        print(
+            f"n_nodes={n_nodes:6d} n_cells={n_cells:7d} n_bins={n_bins} chunk={chunk_size}: "
+            f"{elapsed:.3f}s, pool reserved={peak_reserved / 1e9:.2f} GB, "
+            f"device free before/after={free0/1e9:.2f}/{free1/1e9:.2f} GB"
+        )
+        del nodes, mi, puc
+
+
+if __name__ == "__main__":
+    if not cuda_available():
+        raise SystemExit("cupy / GPU not available")
+    correctness_check()
+    # Representative sizes: the real Stage-3 run is ~11,942 genes x ~221,273
+    # cells. Probe increasing N at real (or near-real) cell count -- cell
+    # count dominates device memory via the (m, n) binned-data array and
+    # dominates compute via the joint_counts_kernel's O(n^2 * m / chunk)
+    # inner loop -- rather than a small n_cells stand-in, to get a timing/
+    # memory extrapolation that's actually trustworthy at Stage-3 scale.
+    timing_memory_profile(
+        cases=[(512, 221_273), (1024, 221_273), (2048, 221_273)],
+        n_bins=10,
+        chunk_size=256,
+    )
diff --git a/analysis/scripts/effect_matrix.py b/analysis/scripts/effect_matrix.py
new file mode 100644
index 0000000..28452b6
--- /dev/null
+++ b/analysis/scripts/effect_matrix.py
@@ -0,0 +1,162 @@
+"""Stage 4 of the interventional plan: the directed effect matrix E[g, j] --
+the distributional shift (energy distance) of every gene j under CRISPRi
+perturbation of target g, vs. non-targeting (NTC) control cells.
+
+Energy distance (Szekely & Rizzo), not a mean-shift test, because Perturb-seq
+responses are frequently variance changes or bimodality rather than pure mean
+shifts (plan's own framing). Implemented with an O((m+n) log(m+n))-per-gene
+sort trick (see `energy_distance_batch`), vectorized across genes and
+parallelized across targets with multiprocessing -- a naive O(m*n) pairwise
+implementation, or a Python-level loop calling scipy.stats.energy_distance
+once per (target, gene) pair (150 x ~12,000 = ~1.8M calls), is not tractable
+at this scale in an interactive session.
+
+Significance: the plan specifies calibrating against non-targeting-vs-
+non-targeting splits rather than a hardcoded parametric null, to absorb
+batch/depth structure. A literal permutation p-value with per-target,
+per-gene resolution would need thousands of same-size NTC/NTC splits per
+target (each costing about as much as one real target's computation) --
+not tractable today. Approximation actually used, and why it's still in the
+spirit of "empirical, not assumed" calibration: draw NTC/NTC splits at a
+handful of representative sample sizes spanning the observed target-group-size
+range, several replicates each, to get an empirically *estimated* (not
+assumed-functional-form) null mean/SD of energy distance as a function of
+group size, then z-score each target's observed per-gene energy distance
+against the null level interpolated for that target's actual group size.
+This is documented explicitly in LOG.md as an approximation to revisit with
+more compute if literal permutation p-values are needed for a specific
+downstream claim (e.g. a small shortlist of hits worth a dedicated
+permutation test).
+"""
+
+from __future__ import annotations
+
+import pickle
+import time
+from concurrent.futures import ProcessPoolExecutor, as_completed
+from pathlib import Path
+
+import numpy as np
+
+SCRATCH = Path("/tmp/claude-1000/-home-asm-FastPIDC-jl/564c6d04-ee7a-44e7-b8c1-702a37701a63/scratchpad")
+X_PATH = SCRATCH / "X_norm_log.npy"  # (n_cells, n_genes) float32, size-factor normalized + log1p
+# (n_genes_filtered, n_cells) float32, contiguous, rows in the same order as the
+# gene-filter mask's True entries -- see genemajor_rebuild step in LOG.md. Built
+# once from X_PATH so that a contiguous gene-chunk read (this file's natural
+# row-major layout) doesn't have to gather ~1,500 scattered columns out of an
+# 18,080-wide row per cell, which is what made the first two attempts at this
+# script I/O-bound (workers pegged "R" in `ps` but barely accumulating CPU
+# time -- symptomatic of mmap page-fault-bound scattered reads, not compute).
+X_GENEMAJOR_PATH = SCRATCH / "X_genemajor_filtered.npy"
+
+
+def energy_distance_batch(
+    a: np.ndarray, b: np.ndarray, b_within_mean: np.ndarray | None = None
+) -> tuple[np.ndarray, np.ndarray]:
+    """Energy-distance^2 between group `a` (m, G) and group `b` (n, G),
+    one value per gene (column), via the merge-rank trick (verified against
+    `scipy.stats.energy_distance` on random data: `sqrt(e2) == scipy result`).
+
+    Pass `b_within_mean` (from a prior call) to skip recomputing b's
+    within-group term when `b` (e.g. the full NTC pool) is reused across
+    many calls.
+
+    Memory note: every array below is shaped ((m+n), G) -- this is the thing
+    that blew up a first attempt at parallelizing over 150 targets at full
+    gene width (G ~ 12,000): each worker's peak was ~15-20 GB (dominated by
+    the fixed n=38,176-cell NTC block, not by the much smaller target group),
+    and 46 concurrent workers pushed the machine into swap thrashing and an
+    OOM kill. `compute_effect_matrix` now chunks G explicitly so a worker's
+    peak is bounded by `chunk_size`, independent of `max_workers`.
+    """
+    m, G = a.shape
+    n = b.shape[0]
+    combined = np.concatenate([a, b], axis=0)
+    order = np.argsort(combined, axis=0).astype(np.int32)  # (m+n) fits int32; halves this array's footprint vs default int64
+    sorted_vals = np.take_along_axis(combined, order, axis=0)
+    label = np.concatenate([np.ones(m, dtype=np.int8), np.full(n, -1, dtype=np.int8)])
+    sorted_label = label[order]
+    is_a = sorted_label == 1
+    is_b = ~is_a
+    cumB = np.cumsum(is_b, axis=0, dtype=np.int32)
+    cumBval = np.cumsum(np.where(is_b, sorted_vals, 0), axis=0)
+    totalBval = cumBval[-1]
+    contrib = np.where(is_a, sorted_vals * (2 * cumB - n) - 2 * cumBval + totalBval, 0.0)
+    mean_cross = contrib.sum(axis=0) / (m * n)
+
+    a_sorted = np.sort(a, axis=0)
+    ranks = np.arange(1, m + 1)[:, None]
+    mean_within_a = 2 * np.sum((2 * ranks - m - 1) * a_sorted, axis=0) / (m * m)
+
+    if b_within_mean is None:
+        b_sorted = np.sort(b, axis=0)
+        ranksb = np.arange(1, n + 1)[:, None]
+        b_within_mean = 2 * np.sum((2 * ranksb - n - 1) * b_sorted, axis=0) / (n * n)
+
+    e2 = 2 * mean_cross - mean_within_a - b_within_mean
+    return e2, b_within_mean
+
+
+def _worker(args):
+    target, cell_indices, ntc_indices, chunk_start, chunk_end, chunk_id, genemajor_path = args
+    # Genes-major layout: a contiguous row-block read (fast, sequential), then
+    # cell selection is fancy-indexing on an already-in-RAM small block instead
+    # of scattered reads against the full 16 GB mmap.
+    Xg = np.load(genemajor_path, mmap_mode="r")
+    block = np.asarray(Xg[chunk_start:chunk_end, :])  # (chunk_genes, n_cells), contiguous read
+    a = block[:, cell_indices].T  # (m, chunk_genes)
+    b = block[:, ntc_indices].T  # (n, chunk_genes)
+    e2, _ = energy_distance_batch(a, b)
+    return target, chunk_id, e2
+
+
+def compute_effect_matrix(
+    max_workers: int = 16,
+    chunk_size: int = 1500,
+    genemajor_path: Path = X_GENEMAJOR_PATH,
+    out_name: str = "E_matrix.npy",
+    meta_name: str = "E_matrix_meta.pkl",
+) -> None:
+    """`genemajor_path`/`out_name`/`meta_name` let this be re-run against a
+    residualized array (see `cell_cycle.py` / `rerun_stage4_residualized.py`)
+    without duplicating this file."""
+    targets_obs = pickle.load(open(SCRATCH / "obs_target_gene.pkl", "rb"))
+    var_names = pickle.load(open(SCRATCH / "var_names.pkl", "rb"))
+    gene_mask = np.load(SCRATCH / "gene_filter_mask.npy")
+    gene_cols = np.where(gene_mask)[0]
+    filtered_genes = [var_names[i] for i in gene_cols]
+    n_genes = len(filtered_genes)
+    chunk_bounds = [(i, min(i + chunk_size, n_genes)) for i in range(0, n_genes, chunk_size)]
+
+    ntc_indices = np.where(targets_obs == "non-targeting")[0]
+    target_list = sorted(set(targets_obs) - {"non-targeting"})
+
+    jobs = []
+    for g in target_list:
+        cell_indices = np.where(targets_obs == g)[0]
+        for ci, (start, end) in enumerate(chunk_bounds):
+            jobs.append((g, cell_indices, ntc_indices, start, end, ci, genemajor_path))
+
+    E = np.zeros((len(target_list), len(filtered_genes)), dtype=np.float32)
+    target_to_row = {g: i for i, g in enumerate(target_list)}
+
+    t0 = time.time()
+    with ProcessPoolExecutor(max_workers=max_workers) as ex:
+        futs = [ex.submit(_worker, job) for job in jobs]
+        n_done = 0
+        for fut in as_completed(futs):
+            g, chunk_id, e2 = fut.result()
+            start, _ = chunk_bounds[chunk_id]
+            E[target_to_row[g], start : start + len(e2)] = e2
+            n_done += 1
+            if n_done % 50 == 0:
+                print(f"[{n_done}/{len(jobs)}] elapsed {time.time() - t0:.1f}s", flush=True)
+    print(f"done in {time.time() - t0:.1f}s, {len(jobs)} jobs total")
+
+    np.save(SCRATCH / out_name, E)
+    with open(SCRATCH / meta_name, "wb") as f:
+        pickle.dump({"target_list": target_list, "filtered_genes": filtered_genes}, f)
+
+
+if __name__ == "__main__":
+    compute_effect_matrix()
diff --git a/analysis/scripts/null_calibration.py b/analysis/scripts/null_calibration.py
new file mode 100644
index 0000000..ff1074f
--- /dev/null
+++ b/analysis/scripts/null_calibration.py
@@ -0,0 +1,108 @@
+"""Significance calibration for the Stage-4 effect matrix (`effect_matrix.py`),
+per the interventional plan: "Calibrate significance against non-targeting-
+vs-non-targeting splits, not a parametric null -- this absorbs batch and
+depth structure."
+
+A literal permutation p-value with per-target, per-gene resolution would need
+many same-size NTC/NTC splits *for every one of the 150 targets' exact group
+sizes* -- each split costs about as much as one real target's computation
+(dominated by the fixed ~38k-cell NTC pool, not the smaller draw), so that's
+not tractable in an interactive session (it would roughly multiply Stage 4's
+~1000s runtime by however many splits-per-target resolution requires, e.g.
+20-50x). What's implemented instead: NTC/NTC splits at a handful of
+*representative* sample sizes spanning the actually-observed target-group-size
+range (33 to 4760 cells), several replicates each, giving an *empirically
+estimated* (not assumed-functional-form) null mean/SD of energy distance as a
+function of group size. Each real target is then scored against the null
+level for its nearest representative size. This is an approximation to the
+plan's literal instruction, made explicit here and in LOG.md -- revisit with
+per-target-exact-size permutation if a specific downstream claim needs it
+(e.g. a short list of top hits worth a dedicated test).
+"""
+
+from __future__ import annotations
+
+import pickle
+import time
+from concurrent.futures import ProcessPoolExecutor, as_completed
+from pathlib import Path
+
+import numpy as np
+
+from effect_matrix import SCRATCH, X_GENEMAJOR_PATH, energy_distance_batch
+
+REPRESENTATIVE_SIZES = [50, 200, 500, 1000, 2000, 3000, 4500]
+N_REPLICATES = 8
+CHUNK_SIZE = 1500
+
+
+def _null_worker(args):
+    size, rep_seed, chunk_start, chunk_end, chunk_id, genemajor_path = args
+    targets_obs = pickle.load(open(SCRATCH / "obs_target_gene.pkl", "rb"))
+    ntc_indices = np.where(targets_obs == "non-targeting")[0]
+
+    rng = np.random.default_rng(rep_seed)
+    perm = rng.permutation(ntc_indices)
+    pseudo_target = perm[:size]
+    pseudo_control = perm[size:]  # rest of the NTC pool
+
+    Xg = np.load(genemajor_path, mmap_mode="r")
+    block = np.asarray(Xg[chunk_start:chunk_end, :])
+    a = block[:, pseudo_target].T
+    b = block[:, pseudo_control].T
+    e2, _ = energy_distance_batch(a, b)
+    return size, rep_seed, chunk_id, e2
+
+
+def compute_null(
+    max_workers: int = 16,
+    genemajor_path: Path = X_GENEMAJOR_PATH,
+    mean_name: str = "null_mean_by_size.npy",
+    sd_name: str = "null_sd_by_size.npy",
+    meta_name: str = "null_meta.pkl",
+) -> None:
+    """`genemajor_path`/`*_name` let this be re-run against a residualized
+    array without duplicating this file -- see `cell_cycle.py` /
+    `rerun_stage4_residualized.py`."""
+    gene_mask = np.load(SCRATCH / "gene_filter_mask.npy")
+    n_genes = int(gene_mask.sum())
+    chunk_bounds = [(i, min(i + CHUNK_SIZE, n_genes)) for i in range(0, n_genes, CHUNK_SIZE)]
+
+    jobs = []
+    for size in REPRESENTATIVE_SIZES:
+        for rep in range(N_REPLICATES):
+            seed = hash((size, rep)) % (2**31)
+            for ci, (start, end) in enumerate(chunk_bounds):
+                jobs.append((size, seed, start, end, ci, genemajor_path))
+
+    # e2_by_size[size] -> (N_REPLICATES, n_genes) array
+    e2_by_size = {s: np.zeros((N_REPLICATES, n_genes), dtype=np.float32) for s in REPRESENTATIVE_SIZES}
+    rep_index = {s: {} for s in REPRESENTATIVE_SIZES}  # seed -> rep row index, assigned on first sight
+
+    t0 = time.time()
+    with ProcessPoolExecutor(max_workers=max_workers) as ex:
+        futs = [ex.submit(_null_worker, job) for job in jobs]
+        n_done = 0
+        for fut in as_completed(futs):
+            size, seed, chunk_id, e2 = fut.result()
+            if seed not in rep_index[size]:
+                rep_index[size][seed] = len(rep_index[size])
+            row = rep_index[size][seed]
+            start, _ = chunk_bounds[chunk_id]
+            e2_by_size[size][row, start : start + len(e2)] = e2
+            n_done += 1
+            if n_done % 50 == 0:
+                print(f"[{n_done}/{len(jobs)}] elapsed {time.time() - t0:.1f}s", flush=True)
+    print(f"null calibration done in {time.time() - t0:.1f}s")
+
+    null_mean = np.stack([e2_by_size[s].mean(axis=0) for s in REPRESENTATIVE_SIZES])  # (n_sizes, n_genes)
+    null_sd = np.stack([e2_by_size[s].std(axis=0, ddof=1) for s in REPRESENTATIVE_SIZES])
+
+    np.save(SCRATCH / mean_name, null_mean)
+    np.save(SCRATCH / sd_name, null_sd)
+    with open(SCRATCH / meta_name, "wb") as f:
+        pickle.dump({"sizes": REPRESENTATIVE_SIZES, "n_replicates": N_REPLICATES}, f)
+
+
+if __name__ == "__main__":
+    compute_null()
diff --git a/analysis/scripts/rerun_stage4_residualized.py b/analysis/scripts/rerun_stage4_residualized.py
new file mode 100644
index 0000000..46b05c0
--- /dev/null
+++ b/analysis/scripts/rerun_stage4_residualized.py
@@ -0,0 +1,57 @@
+"""Re-run Stage 4 (effect matrix -> null calibration -> BH-FDR calibration)
+against the cell-cycle-residualized expression matrix produced by
+`cell_cycle.py`, to test whether conditioning on cell-cycle state shrinks the
+implausible k^out_g values flagged in LOG.md's Checkpoint D0 discussion
+(some targets "significant" against ~97% of the filtered genome).
+
+Writes a parallel set of outputs (suffixed `_resid`) rather than overwriting
+the originals, so the raw and residualized k^out_g can be compared directly.
+"""
+
+from __future__ import annotations
+
+from pathlib import Path
+
+from effect_matrix import SCRATCH, compute_effect_matrix
+from null_calibration import compute_null
+from calibrate_effects import calibrate
+
+RESID_PATH = SCRATCH / "X_resid_genemajor.npy"
+
+
+def main():
+    if not RESID_PATH.exists():
+        raise SystemExit(f"{RESID_PATH} not found -- run cell_cycle.py first")
+
+    print("=== Stage 4 (residualized): effect matrix ===")
+    compute_effect_matrix(
+        genemajor_path=RESID_PATH,
+        out_name="E_matrix_resid.npy",
+        meta_name="E_matrix_meta_resid.pkl",
+    )
+
+    print("=== Stage 4 (residualized): null calibration ===")
+    compute_null(
+        genemajor_path=RESID_PATH,
+        mean_name="null_mean_by_size_resid.npy",
+        sd_name="null_sd_by_size_resid.npy",
+        meta_name="null_meta_resid.pkl",
+    )
+
+    print("=== Stage 4 (residualized): BH-FDR calibration + k_out ===")
+    calibrate(
+        load_kwargs=dict(
+            e_matrix_name="E_matrix_resid.npy",
+            e_meta_name="E_matrix_meta_resid.pkl",
+            null_mean_name="null_mean_by_size_resid.npy",
+            null_sd_name="null_sd_by_size_resid.npy",
+            null_meta_name="null_meta_resid.pkl",
+        ),
+        q_name="E_qvalues_resid.npy",
+        z_name="E_zscores_resid.npy",
+        kout_name="k_out_resid.csv",
+    )
+
+
+if __name__ == "__main__":
+    main()
diff --git a/analysis/uv.lock b/analysis/uv.lock
new file mode 100644
index 0000000..79d0dba
--- /dev/null
+++ b/analysis/uv.lock
@@ -0,0 +1,1376 @@
+version = 1
+revision = 3
+requires-python = ">=3.12"
+resolution-markers = [
+    "python_full_version >= '3.14' and sys_platform == 'win32'",
+    "python_full_version >= '3.14' and sys_platform == 'emscripten'",
+    "python_full_version >= '3.14' and sys_platform != 'emscripten' and sys_platform != 'win32'",
+    "python_full_version == '3.13.*' and sys_platform == 'win32'",
+    "python_full_version < '3.13' and sys_platform == 'win32'",
+    "python_full_version == '3.13.*' and sys_platform == 'emscripten'",
+    "python_full_version < '3.13' and sys_platform == 'emscripten'",
+    "python_full_version == '3.13.*' and sys_platform != 'emscripten' and sys_platform != 'win32'",
+    "python_full_version < '3.13' and sys_platform != 'emscripten' and sys_platform != 'win32'",
+]
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+]
+
+[[package]]
+name = "zarr"
+version = "3.3.0"
+source = { registry = "https://pypi.org/simple" }
+dependencies = [
+    { name = "donfig" },
+    { name = "google-crc32c" },
+    { name = "numcodecs" },
+    { name = "numpy" },
+    { name = "packaging" },
+    { name = "typing-extensions" },
+]
+sdist = { url = "https://files.pythonhosted.org/packages/34/15/436cb1d3bbe86173bd44ce7a34ecb210d0c0416946e337858149a905ef5a/zarr-3.3.0.tar.gz", hash = "sha256:cd0c8cf738b4bb4807815bc1255acad5bdf1a7b7264b606c5a1bc0d0392a306b", size = 943626, upload-time = "2026-07-30T16:35:10.491Z" }
+wheels = [
+    { url = "https://files.pythonhosted.org/packages/6d/c6/6b726ddf4c3ac5a123f285c3650fa8268902c28ea541a0282867f1336e65/zarr-3.3.0-py3-none-any.whl", hash = "sha256:323bf5366d4f909052ef6e2e03e7481a7434c3ee75d3a981eb3a71fc1ae22cef", size = 363685, upload-time = "2026-07-30T16:35:08.794Z" },
+]
diff --git a/python/src/fastpidc/__init__.py b/python/src/fastpidc/__init__.py
index 7cd0d6c..024cb88 100644
--- a/python/src/fastpidc/__init__.py
+++ b/python/src/fastpidc/__init__.py
@@ -21,6 +21,7 @@
     PIDCNetworkInference,
     PUCNetworkInference,
 )
+from .pid import PIDTriple, combined_node, pid_triple
 from .types import Edge, InferredNetwork, Node, PIDCConfig
 
 __all__ = [
@@ -42,6 +43,10 @@
     "read_network_file",
     "get_adjacency_matrix",
     "infer_network",
+    # Triple partial information decomposition
+    "PIDTriple",
+    "combined_node",
+    "pid_triple",
 ]
 
 __version__ = "0.1.0"
diff --git a/python/src/fastpidc/discretizers.py b/python/src/fastpidc/discretizers.py
index 52f388d..7d4fc6d 100644
--- a/python/src/fastpidc/discretizers.py
+++ b/python/src/fastpidc/discretizers.py
@@ -262,6 +262,44 @@ def _encode_uniform_width(values: np.ndarray, number_of_bins: int) -> tuple[np.n
     return LinearDiscretizer(edges).encode(values), number_of_bins
 
 
+def get_bin_ids_zero_as_own_bin(values: np.ndarray, number_of_bins: int) -> tuple[np.ndarray, int]:
+    """Discretize ``values`` giving exact zeros their own dedicated bin (bin
+    0), then equal-frequency ("uniform_count") binning the nonzero values
+    into the remaining ``number_of_bins - 1`` bins.
+
+    Motivated by dropout-dominated single-cell count/expression data, where
+    "detected vs. not detected" is itself informative and a naive
+    equal-frequency discretizer run on all values (zero and nonzero mixed)
+    would otherwise split the zero mass arbitrarily across multiple bins
+    whenever zeros exceed one bin's worth of the data, or absorb nonzero
+    values into a zero-dominated bin -- either way conflating "not detected"
+    with "detected but low" in a way that depends on the zero fraction
+    rather than reflecting a real difference in expression level.
+
+    Falls back to a plain two-bin zero/nonzero split if ``number_of_bins`` is
+    too small to also split the nonzero values, or if there are no nonzero
+    values at all (equivalent then to the ``get_bin_ids`` ``lo == hi`` case).
+    """
+    values = np.asarray(values, dtype=np.float64)
+    zero_mask = values == 0
+    nonzero = values[~zero_mask]
+
+    if nonzero.size == 0:
+        return np.zeros(values.shape, dtype=np.int64), 1
+
+    if number_of_bins <= 1 or nonzero.size < number_of_bins - 1:
+        # Not enough room/data for equal-frequency splitting of the nonzero
+        # values; fall back to a single "nonzero" bin alongside the zero bin.
+        bin_ids = zero_mask.astype(np.int64) ^ 1  # 0 if zero, 1 if nonzero
+        return bin_ids, 2
+
+    nonzero_ids, nonzero_nbins = get_bin_ids(nonzero, "uniform_count", number_of_bins - 1)
+
+    bin_ids = np.zeros(values.shape, dtype=np.int64)
+    bin_ids[~zero_mask] = nonzero_ids + 1
+    return bin_ids, nonzero_nbins + 1
+
+
 def get_bin_ids(values: np.ndarray, mode: str, number_of_bins: int) -> tuple[np.ndarray, int]:
     """Discretize ``values`` into bin ids using discretization method ``mode``.
 
@@ -300,6 +338,9 @@ def get_bin_ids(values: np.ndarray, mode: str, number_of_bins: int) -> tuple[np.
             return _encode_uniform_width(values, number_of_bins)
         return LinearDiscretizer(edges).encode(values), number_of_bins
 
+    if mode == "zero_as_own_bin":
+        return get_bin_ids_zero_as_own_bin(values, number_of_bins)
+
     if mode == "bayesian_blocks":
         try:
             edges = binedges_bayesian_blocks(values)
diff --git a/python/src/fastpidc/pid.py b/python/src/fastpidc/pid.py
new file mode 100644
index 0000000..0d42833
--- /dev/null
+++ b/python/src/fastpidc/pid.py
@@ -0,0 +1,108 @@
+"""Triple partial information decomposition (PID): redundancy, unique and
+synergistic information that two source variables carry about a target.
+
+This is the primitive the interventional plan calls ``pid_triple`` (see
+``Interventional PID for GRN inference...md`` and ``LOG.md``'s 2026-09-04
+entry #5): given an arbitrary triple ``(source1, source2, target)``, decompose
+``MI((source1, source2); target)`` into
+
+    MI_joint = redundancy + unique1 + unique2 + synergy
+
+following the Williams & Beer (2010) lattice with the ``I_min`` redundancy
+measure already implemented in :mod:`fastpidc.information`
+(:func:`apply_redundancy_formula` = ``E[min(SI_1, SI_2)]``) -- this package
+does not implement any other redundancy measure (no BROJA, no ``I_ccs``); see
+the interventional plan's own Checkpoint 4 caveat about ``I_min`` being the
+weakest link in this approach.
+
+Built entirely from existing building blocks (:func:`fastpidc.network.get_mi_and_si`
+and :func:`fastpidc.information.apply_redundancy_formula`); no new information
+formula is introduced here. ``redundancy`` and ``unique*`` come directly from
+those; ``synergy`` additionally needs the joint mutual information of a
+"combined" source built by joint-binning ``source1`` and ``source2``
+(:func:`combined_node`), a step this package's discretizers don't do for you
+since the two sources are typically bin-compatible pre-existing nodes, not raw
+values to be jointly rediscretized.
+"""
+
+from __future__ import annotations
+
+from dataclasses import dataclass
+
+import numpy as np
+
+from .information import apply_redundancy_formula, get_frequencies_from_bin_ids, get_probabilities
+from .network import get_mi_and_si
+from .types import Node
+
+__all__ = ["PIDTriple", "combined_node", "pid_triple"]
+
+
+def combined_node(node1: Node, node2: Node, estimator: str, label: str | None = None) -> Node:
+    """Joint-bin ``node1`` and ``node2`` into a single :class:`Node` whose bin
+    id encodes both nodes' bin ids (``node1_bin * node2.number_of_bins +
+    node2_bin``), for use as the "combined source" in :func:`pid_triple`'s
+    synergy term. The combined node's marginal probabilities are estimated
+    with ``estimator``, consistent with how :meth:`Node.from_raw_values`
+    would compute them for a single discretized variable.
+    """
+    n2 = node2.number_of_bins
+    combined_bins = node1.binned_values.astype(np.int64) * n2 + node2.binned_values.astype(np.int64)
+    number_of_bins = node1.number_of_bins * n2
+    frequencies = get_frequencies_from_bin_ids(combined_bins, number_of_bins)
+    probabilities = get_probabilities(estimator, frequencies)
+    return Node(label or f"({node1.label},{node2.label})", combined_bins, number_of_bins, probabilities)
+
+
+@dataclass(frozen=True, slots=True)
+class PIDTriple:
+    """Partial information decomposition of ``MI((source1, source2); target)``.
+
+    ``mi_joint == redundancy + unique1 + unique2 + synergy`` (up to floating
+    point error) by construction.
+    """
+
+    redundancy: float
+    unique1: float
+    """Information about ``target`` unique to ``source1`` (not available from ``source2``)."""
+    unique2: float
+    """Information about ``target`` unique to ``source2`` (not available from ``source1``)."""
+    synergy: float
+    mi1: float
+    """``MI(source1, target)``."""
+    mi2: float
+    """``MI(source2, target)``."""
+    mi_joint: float
+    """``MI((source1, source2), target)``, the quantity being decomposed."""
+
+
+def pid_triple(
+    source1: Node,
+    source2: Node,
+    target: Node,
+    *,
+    estimator: str = "maximum_likelihood",
+    base: float = 2.0,
+) -> PIDTriple:
+    """Decompose the information ``source1`` and ``source2`` jointly carry
+    about ``target`` into redundant, unique and synergistic components.
+
+    ``target.probabilities`` is used directly as the target's marginal
+    distribution for the redundancy formula (rather than recomputing it), so
+    ``target`` should have been constructed (e.g. via
+    :meth:`Node.from_raw_values`) with an estimator matching ``estimator``
+    here -- consistent with how :func:`fastpidc.network.infer_network_from_nodes`
+    itself only supports ``estimator="maximum_likelihood"`` for PUC/PIDC
+    without violating shared-marginal assumptions (see its docstring).
+    """
+    mi1, si1, _ = get_mi_and_si(source1, target, estimator, base)
+    mi2, si2, _ = get_mi_and_si(source2, target, estimator, base)
+    redundancy = apply_redundancy_formula(target.probabilities, si1, si2, base)
+    unique1 = mi1 - redundancy
+    unique2 = mi2 - redundancy
+
+    joint = combined_node(source1, source2, estimator)
+    mi_joint, _, _ = get_mi_and_si(joint, target, estimator, base)
+    synergy = mi_joint - redundancy - unique1 - unique2
+
+    return PIDTriple(redundancy, unique1, unique2, synergy, mi1, mi2, mi_joint)
diff --git a/python/tests/test_discretizers.py b/python/tests/test_discretizers.py
index 63ca4e0..f39bf61 100644
--- a/python/tests/test_discretizers.py
+++ b/python/tests/test_discretizers.py
@@ -7,6 +7,7 @@
     binedges_uniform_count,
     binedges_uniform_width,
     get_bin_ids,
+    get_bin_ids_zero_as_own_bin,
 )
 
 
@@ -98,3 +99,47 @@ def test_get_bin_ids_bin_range():
         ids, nbins = get_bin_ids(values, mode, 8)
         assert ids.min() >= 0
         assert ids.max() < nbins
+
+
+def test_zero_as_own_bin_puts_all_zeros_in_bin_zero():
+    rng = np.random.default_rng(2)
+    nonzero = rng.exponential(size=180) + 0.1
+    values = np.concatenate([np.zeros(20), nonzero])
+    rng.shuffle(values)
+
+    ids, nbins = get_bin_ids_zero_as_own_bin(values, 5)
+
+    np.testing.assert_array_equal(ids[values == 0], 0)
+    assert np.all(ids[values != 0] >= 1)
+    assert nbins == 5
+    assert ids.max() == nbins - 1
+
+
+def test_zero_as_own_bin_nonzero_values_equal_frequency():
+    data = np.concatenate([np.zeros(10), np.arange(1, 11, dtype=np.float64)])
+    ids, nbins = get_bin_ids_zero_as_own_bin(data, 3)  # bin 0 = zero, 2 nonzero bins
+    assert nbins == 3
+    nonzero_ids = ids[data != 0]
+    counts = np.bincount(nonzero_ids - 1, minlength=2)
+    np.testing.assert_array_equal(counts, [5, 5])
+
+
+def test_zero_as_own_bin_all_zero_collapses_to_single_bin():
+    ids, nbins = get_bin_ids_zero_as_own_bin(np.zeros(5), 5)
+    assert nbins == 1
+    np.testing.assert_array_equal(ids, 0)
+
+
+def test_zero_as_own_bin_falls_back_to_two_bins_when_too_few_nonzero():
+    data = np.array([0.0, 0.0, 0.0, 1.0, 2.0])
+    ids, nbins = get_bin_ids_zero_as_own_bin(data, 10)  # nonzero.size (2) < number_of_bins - 1 (9)
+    assert nbins == 2
+    np.testing.assert_array_equal(ids, [0, 0, 0, 1, 1])
+
+
+def test_get_bin_ids_dispatches_zero_as_own_bin():
+    data = np.array([0.0, 0.0, 1.0, 2.0, 3.0, 4.0])
+    ids_direct, nbins_direct = get_bin_ids_zero_as_own_bin(data, 3)
+    ids_dispatch, nbins_dispatch = get_bin_ids(data, "zero_as_own_bin", 3)
+    assert nbins_direct == nbins_dispatch
+    np.testing.assert_array_equal(ids_direct, ids_dispatch)
diff --git a/python/tests/test_pid.py b/python/tests/test_pid.py
new file mode 100644
index 0000000..8402cdb
--- /dev/null
+++ b/python/tests/test_pid.py
@@ -0,0 +1,100 @@
+import numpy as np
+import pytest
+
+from fastpidc.pid import combined_node, pid_triple
+from fastpidc.types import Node
+
+
+def _node(label: str, bin_ids: np.ndarray, number_of_bins: int, estimator="maximum_likelihood") -> Node:
+    from fastpidc.information import get_frequencies_from_bin_ids, get_probabilities
+
+    freq = get_frequencies_from_bin_ids(bin_ids.astype(np.int64), number_of_bins)
+    probs = get_probabilities(estimator, freq)
+    return Node(label, bin_ids.astype(np.int64), number_of_bins, probs)
+
+
+def test_combined_node_bin_ids_encode_both_sources():
+    n1 = _node("A", np.array([0, 0, 1, 1]), 2)
+    n2 = _node("B", np.array([0, 1, 0, 1]), 2)
+    joint = combined_node(n1, n2, "maximum_likelihood")
+    assert joint.number_of_bins == 4
+    np.testing.assert_array_equal(joint.binned_values, [0, 1, 2, 3])
+    assert joint.probabilities.sum() == pytest.approx(1.0)
+
+
+def test_pid_triple_decomposition_sums_to_joint_mi():
+    rng = np.random.default_rng(0)
+    n = 4000
+    x = rng.integers(0, 2, size=n)
+    y = rng.integers(0, 2, size=n)
+    z = x ^ y  # target: pure synergy between x and y
+
+    nx = _node("X", x, 2)
+    ny = _node("Y", y, 2)
+    nz = _node("Z", z, 2)
+
+    result = pid_triple(nx, ny, nz)
+    assert result.mi_joint == pytest.approx(
+        result.redundancy + result.unique1 + result.unique2 + result.synergy, abs=1e-9
+    )
+
+
+def test_pid_triple_xor_is_pure_synergy():
+    # Classic XOR gate: X, Y independent fair bits, Z = X XOR Y. Individually
+    # X and Y carry zero information about Z; jointly they determine it
+    # exactly (MI_joint = 1 bit). Redundancy and both uniques should be ~0,
+    # synergy should be ~1 bit.
+    rng = np.random.default_rng(1)
+    n = 20000
+    x = rng.integers(0, 2, size=n)
+    y = rng.integers(0, 2, size=n)
+    z = x ^ y
+
+    nx = _node("X", x, 2)
+    ny = _node("Y", y, 2)
+    nz = _node("Z", z, 2)
+
+    result = pid_triple(nx, ny, nz)
+    assert result.mi1 == pytest.approx(0.0, abs=1e-2)
+    assert result.mi2 == pytest.approx(0.0, abs=1e-2)
+    assert result.redundancy == pytest.approx(0.0, abs=1e-2)
+    assert result.unique1 == pytest.approx(0.0, abs=1e-2)
+    assert result.unique2 == pytest.approx(0.0, abs=1e-2)
+    assert result.synergy == pytest.approx(1.0, abs=1e-2)
+
+
+def test_pid_triple_identical_sources_is_pure_redundancy():
+    # X == Y (identical copies), Z = X. All information both sources carry
+    # about Z is fully redundant: unique1 == unique2 == synergy ~= 0, and
+    # redundancy == mi1 == mi2 == MI(X, Z) == 1 bit for a fair bit.
+    rng = np.random.default_rng(2)
+    n = 4000
+    x = rng.integers(0, 2, size=n)
+    z = x.copy()
+
+    nx = _node("X", x, 2)
+    ny = _node("Y", x.copy(), 2)  # identical to nx
+    nz = _node("Z", z, 2)
+
+    result = pid_triple(nx, ny, nz)
+    assert result.unique1 == pytest.approx(0.0, abs=1e-9)
+    assert result.unique2 == pytest.approx(0.0, abs=1e-9)
+    assert result.synergy == pytest.approx(0.0, abs=1e-9)
+    assert result.redundancy == pytest.approx(result.mi1, abs=1e-9)
+    assert result.redundancy == pytest.approx(1.0, abs=0.05)
+
+
+def test_pid_triple_independent_of_everything_is_all_zero():
+    rng = np.random.default_rng(3)
+    n = 4000
+    x = rng.integers(0, 2, size=n)
+    y = rng.integers(0, 2, size=n)
+    z = rng.integers(0, 2, size=n)  # independent of both x and y
+
+    nx = _node("X", x, 2)
+    ny = _node("Y", y, 2)
+    nz = _node("Z", z, 2)
+
+    result = pid_triple(nx, ny, nz)
+    for value in (result.redundancy, result.unique1, result.unique2, result.synergy, result.mi_joint):
+        assert value == pytest.approx(0.0, abs=0.05)