The ConvNeXt block body input-gradient VJP at a smooth point — the reverse of convNextBlockBody = layerScale ∘ project ∘ GELU ∘ expand ∘ LN ∘ depthwise:

depthwiseFlatBack Wdw ∘ lnB ∘ convFlatBack Wex ∘ geluB ∘ convFlatBack Wpr ∘ lsB

lsB = diagBack γls (the per-channel layer-scale backward); convFlatBack Wpr the project back; geluB = diagBack (gelu'(saved)); convFlatBack Wex the expand back; lnB the LayerNorm back (= BN-back); depthwiseFlatBack Wdw the depthwise input-VJP. The full block is residual (body), so the block backward is residual (cnxBlockBodyBack …); ConvNeXtBackCertifiedTie.cnxBodyWithChanLNBack_eq_vjp fills the LN slot with the concrete channel-LN chain, pins the other slots to their certified backwards, and shows the chain IS the certified body VJP.

The ConvNeXt downsample input-gradient VJP — the reverse of cnxDownChW = flatConvStride2 W ∘ LN: lnB ∘ flatConvStride2Back W (run the strided-conv backward, then the LayerNorm back).

The whole ConvNeXt-T input-gradient VJP at a smooth point — the exact reverse of convNextForwardTCh: dense ∘ LN ∘ GAP ∘ stage₄ ∘ down₃ ∘ stage₃ ∘ down₂ ∘ stage₂ ∘ down₁ ∘ stage₁ ∘ LN ∘ stem reversed. The stem (flatConvStride4Back sW ∘ lnBstem, the 4×4/s4 patchify backward), GAP and dense endpoints are concrete; the head-LN, stem-LN, 4 stage backwards and 3 downsample backwards are supplied (the stages fold residual (cnxBlockBodyBack …), the downsamples are cnxDownBack). The [3,3,9,3] structure is in the stage maps' depths; the channel/spatial schedule (96→192→384→768, 56→28→14→7) in their dims; the class count nC is a binder (10 at Imagenette, 1000 at the convnextin_* artifacts). The tie fills lnBhead with the head LayerNorm's backward at the GAP output (rowLNVecFlatBack 1 768).

THE BATCHED WHOLE-NET ConvNeXt-T INPUT GRADIENT — convnextInputGrad at each of B examples, stage by stage, as the batched render computes it. The three input-independent leaves (the stem's reversed-kernel conv, GAP, the dense head) are StableHLO.batchMap B of the per-example leaf. Every saved-activation slot is StableHLO.batchMapAux B of a per-example FAMILY saved ↦ (cotangent ↦ gradient) at a batched saved activation: lnBstem at a0 (the stem conv's output), the four stage backwards s1B … s4B at their stage inputs a1 a3 a5 a7, the three downsample backwards d1B … d3B at a2 a4 a6, and lnBhead at a9 (the GAP output). Where the per-example chain fills a slot at ONE saved value, this one fills it at B of them, which is the only way a batch enters — LayerNorm is per-example and no ConvNeXt op couples examples, the honesty argument ConvNeXtStepTieGB makes for the batched T3 tie. B and nC are variables: this chain carries no batch numeral and no class count. convnextInputGradB_eq_batchMap_convNextForwardTCh_vjp (ConvNeXtWholeBackCertifiedTieB.lean) says it IS the certified gradient of batchMap B convNextForwardTCh.