ConvNeXt — the block cotangent chain #

The param bridges (ConvNeXtFold and the shared conv and dense bridges) certify each ConvNeXt param output for any cotangent dy at that layer's output. This file defines the cotangent the actual backward chain delivers at each layer, and the step ties (ConvNeXtStepTie, ConvNeXtStepTieGB) feed those cotangents to the bridges at the real forward. The definitions are per example; the channel LayerNorm is per-example separable, so the batched tie lifts them with batchMapAux.

The chain through a ConvNeXt block composes the rendered backward denotations — layer-scale back (layerScale γls applied to the cotangent: the input-VJP γ ⊙ dy is the forward map itself, the layerScaleF-on-the-cotangent trick the render uses), the 1×1 conv input-VJP (conv2dHasVJP3 via the flatten bridge, = convBack's denotation), the GELU mask at either form of the GELU (dy ⊙ gf.scalarDeriv, = GeluForm.hasVJP's backward; geluScalarDeriv_eq and geluErfScalarDeriv_eq_erfc certify the closed forms geluBack emits at .tanh and .erf) — back through layerScale → project → gelu → expand to the LN output, where the channel-LN (chanLNTensor3, Vec c γ/β) grads read it:

block: o = addV( layerScale γls (conv₁ₓ₁ₚᵣ( gelu( conv₁ₓ₁ₑₓ( LN( dw₇ₓ₇(x) ))))), x )

The residual addV is the outermost op and passes the block cotangent dyOut straight through to the layer-scale output (ConvNeXt has no post-add activation — the r34 relu(add(…)) mask never appears), and the identity skip adds dyOut back at the block input. Unlike MNV2/r34 there is no stride split: ConvNeXt blocks keep resolution (stride-1 7×7 depthwise), so one set of cotangent definitions covers every block, including the cotangents of the two ConvNeXt-signature families (layer-scale γ, channel-LN (chanLNTensor3, Vec c γ/β)), which MNV2/r34 had no analogue of. The step ties carry the chain on through the LN input-VJP and the depthwise.