ConvNeXt-T — every parameter gradient node IS the loss's derivative in that parameter #

cnx_net_tiedGB says each of the 182 parameter gradient nodes denotes its layer's parameter Jacobian contracted with the cotangent the emitted backward chain threads to it, the chain's top being the smoothed-loss cotangent. cnx_net_lossGrad composes that with the chain: for any loss L of the logits whose gradient at the net's output is g, the loss of the WHOLE net cnxNetB with that one parameter varied is differentiable in it and every node is its gradient (HasGradAt). cnxNetB is the tie's forward, batched, at its stochastic-depth sites sd; at none it is batchMap N of the canonical convNextForwardTCh (cnxNetB_eq_convNextForwardTCh). cnx_net_lossGrad_smoothedCE discharges hL for the label-smoothed loss the artifacts ship (smoothedBatchLossDiv, whose gradient is the softmaxDiv cotangent the render emits).

How. No ConvNeXt op couples examples, so the work is per example and lifted once:

• Per example (at variable widths): the loss read at each activation inside a block, a downsample, the stem and the head has the tie's own per-example cotangent as its gradient (cnxBlk_hasGradAt, cnxDown_hasGradAt, …). Every stage VJP is global — GELU has no kink and LayerNorm needs only 0 < ε — so each step is one HasGradAt.comp_global, and the channel-LN step is chanLNTensor3Back_eq_chanLN_vjp.
• Lifted (HasGradAt.param_batchMapIdx_through, Foundation.Batched.Indexed, for a block; param_batchMap_through for the stem, downsamples and head): read against the linear loss ⟨·, dyₙ⟩ per example, those gradients turn each tied node's Σ_n Σ_j ∂per/∂θ · cotₙ into ∂G/∂θ of the batched block, G the loss at the block's output. Example n's block suffix ends in its own drop site (cnxPostP's siteScale), so every block cotangent is the drop-free one at s ⊙ dy.
• Per net: the loss read after each stage (cnxSuf*), pulled back through the certified batched block, downsample and head VJPs (cnxBlockCotInB_eq_vjp, cnxDownCotInB_eq_vjp, cnxHeadDyB_eq_vjp), and each Φ identified with the whole net at updated weights by a standalone cnx_factor_* theorem.

The stem's bias node is emitted as a stride-1 convBiasGradB at the output resolution, and reads no input; its Jacobian in the bias is the channel indicator whatever the conv, so it is the patchify conv's (GradNodeB.pdiv_flatConvStride4_bias_eq_conv2d), as in the tie.

Hypotheses. 0 < ε (the LayerNorms' VJPs); no smoothness hypothesis. For the smoothed loss, every example's target sums to one and 0 < nC. Stochastic depth is the tie's sd binder, none the drop-free chain.

Scope. One replica, at either precision: every conv, depthwise, strided-conv and patchify weight node is stated on the renderers' switch (convWeightGradBAt bf16 id …, depthwiseWeightGradBAt bf16 id …, convStridedWeightGradBAt bf16 id …, convStride4WeightGradBAt bf16 id …; StableHLO.PrecisionSwitch), so bf16 := true is the bf16 kind the convnextin_*bf16 artifacts emit, read over ℝ at the identity rounding (Bf16Erasure), and bf16 := false the f32 artifacts'. The bias, LayerNorm, layer-scale and classifier nodes carry no flag because no render switches them.