The cifar8-bn (Chapter 4 deeper, 8-conv per-channel BN) TIE #

cifar8's tie (Cifar8Tie.cifar8_train_step_tied_certified) + a BN-back at every conv. The backward chain alternates BN-output cotangent dyBnᵢ (relu-masked — fed to the γ/β ops) and conv-output cotangent cotCᵢ (bnPerChannelTensor3GradInput of dyBnᵢ — fed to the conv W/b ops), repeated over 4 conv→conv→pool stages, crossing each pool as conv-back then maxpool-back.

Zero new ops/bridges/constructors. Conv ties reuse SgdNode.convW_den/convB_den; BN ties reuse SgdNode.bnGamma_den/bnBeta_den; the dense head reuses SgdNode.denseW_den/denseB_den, the loss cotangent cifar8's. All 38 parameter tensors (8 conv W/b, 8 BN γ/β, 3 dense W/b) are stated in cifar8Bn_train_step_tied_certified, each with the generics.

Not the trained artifact. The committed cifar8_bn_* / cifar8w_bn_* arms are this renderer with a separate optimizer, feeding the same chain to *Grad nodes; they are tied in Cifar8BnTieG.cifar8Bn_train_step_tiedG.

Scope (same as the rest of the suite) #

• Below the output layer the cotangents are the rendered chain (cotCᵢ, dyBnᵢ in cifar8Bn_train_step_tied_certified, and mlpCotOut1/mlpCotOut0 in the head). The loss-gradient statement is Cifar8BnTieG.cifar8Bn_net_lossGrad, at the trained arms' *Grad nodes and with each pool's cotangent routed to one maximal cell, as the rendered select_and_scatter does; this chain routes it to the first maximal cell (maxPool2Argmax), the op's own choice, so each pool step is the capstone's scatter at that selection (SmallParamGrad.maxpool_flatDenote_eq_selScatter).
• Conv/BN backward rendered hand-written (cotangent SSA ↔ chain-cot per-op trust); per-op pretty lexing; ℝ → Float32.