The 38-parameter signature of the cifar8 BN net, cifar8BnVerified.toSpecs order: (conv W, conv b, BN γ, BN β) ×8, then the three dense (W, b). ONE source for the order and the shapes — the arg signature, the packed m/v blocks and the return types all read it.

The 16 running-stat slots — 8 BN layers × (μ, var), each [c], in BN-forward order n1 … n8, μ before var per layer (bnStatSlots). This is exactly the order cifar8wbBbnVerified.bnChannels lists (#[16,16,16,16,32,32,32,32]), which is how the driver packs runningBnStats. The train step takes them as passthrough inputs (%n1mui, …) and returns the batch statistics in the same order; @cifar8wb_bbn_fwd_eval reads them as %n1mu, ….

One conv→BN→relu layer at batch statistics, batched op family: batchOp (convAt bf16 …) (the bf16 twin when bf16), then bnFwdSite (bnBatchF at one replica, the sync-BN composition above it), then batchOp relu. Parameters %W{i}/%cb{i}/%g{i}/%bt{i}; the sync-BN collectives are tagged n{i}, which is no parameter's name.

The batch-BN training forward, batched op family, at N := B — the ONE chain both @cifar8wb_bbn_fwd and every cifar8wb_bbn_*_train_step emit, so the forward artifact is a byte prefix of the train step by construction. (conv→BN→relu)×2→pool, ×4, then (dense→relu)×2→dense. h,w are the final pooled spatials; stage spatials double from s4 = 2h back up to s1 (the input).

The parameter block of a signature: %{nm}{suf}: {ty} for each entry of cifar8BnSigList.

@<slug>_fwd — the batch-BN TRAINING forward (BN over the batch's own statistics), cifar8BbnFwdChainB and a return. Not what the driver scores through — that is @<slug>_fwd_eval — but what LEAN_MLIR_EVAL_BATCHSTATS reads and what the BN-world check in trainAdamSched compares against the train step (dimensions = [0, 2, 3] in both). A byte prefix of every <slug>_*_train_step body from %v0 on.

@<slug>_fwd_eval — the inference forward with FROZEN running statistics. The per-example chain (flatConvF / maxPoolF / denseF / reluF under pretty B) with every BN site bnEvalSite (bnPerChannelEvalF: affine BN at the %n{i}mu/%n{i}var graph inputs, no reduction), so this forward is class-batch-independent. Inputs: %x, the 38 parameters, then the 16 stat slots of cifar8BbnStatSigList — the layout trainAdamSched packs (paramShapes ++ bnStatShapes). Returns logits [B, nClasses]. f32 only: BN is f32 in every arm, and the bf16 train steps score through this same artifact, as the ImageNet nets do.

One layer's backward at the BN: the relu mask on the BN output (selectPosB), then bnBackSite (bnBatchBack at one replica, the sync-BN backward above it) reading the layer's conv output and packed statistics. Returns (code, dBn, dHc): the cotangent at the BN output (which the γ/β gradients read) and at the conv output (which the conv gradients read).

One layer's four parameter gradients and optimizer tails, in cifar8BnSigList order (W, conv bias, γ, β): the batched conv weight gradient (bf16 twin when bf16) and bias gradient at the layer's INPUT xIn, bnGammaSite and bnBetaGradB at the BN-output cotangent. Returns (code, θ' names, m' names, v' names).

One BN layer's returned batch statistics — μ then var, each [oc] — from that layer's BN INPUT (bnBatchMeanB / bnBatchVarB: the SAME bnMean/bnVar the forward normalised by). At replicas > 1 read off the all-reduced packed vector (bnStatsMeanB / bnStatsVarB), so the host EMAs the GLOBAL batch statistics.

cifar8BbnTrainStepBText — the cifar8 BN train step at BATCH statistics.

cifar8BnTrainStepBText with its eight bnPerChannelF/bnPerChannelBack/bnGammaGrad sites replaced by the batch-coupled bnFwdSite/bnBackSite/bnGammaSite (bnBatchF / bnBatchBack / bnGammaGradB, reduce over [0,2,3]) and bnBetaGradB, plus the running-statistics protocol: 16 passthrough stat inputs after %bc2 and 16 batch-statistic outputs after %bc2 (cifar8BbnStatSigList order). Signature otherwise the packed [θ|m|v] one every cifar8 optimizer render shares — %x, 38 θ, 38 m, 38 v, %lr/%bc1/%bc2, 16 stats, %onehot → 38 θ', 38 m', 38 v', %loss/%bc1/%bc2, 16 stats — so VerifiedNet.trainAdamSched drives it unchanged, as it drives ResNet-34's.

Three optimizers from one body (optTail); bf16 routes the 23 convolutions through their bf16 twins and leaves BN in f32, as cifar8BnTrainStepBText and every ImageNet render do. Entry @{slug}_{variant}_train_step, the driver's convention.