ResNet-50 — every parameter gradient node IS the loss's derivative in that parameter #

r50_net_tiedB says each of the 161 parameter gradient nodes denotes its layer's parameter Jacobian contracted with the cotangent the emitted backward chain threads to it, from a loss cotangent g it takes as a binder; the r50*CotIn_eq_vjp lemmas say the block-input cotangents are certified VJP backwards. r50_net_lossGrad composes them: for any loss L of the logits whose gradient at the net's output is g, every node is ∂L/∂θ of the WHOLE net with that one parameter varied. The two losses the artifacts ship discharge hL:

• r50_net_lossGrad_smoothedCE — the bce := false artifacts: L = smoothedBatchLoss, g the six-op label-smoothed chain (smoothedBatchLoss_grad).
• r50_net_lossGrad_bce — the bce := true artifacts, resnet50in160_lambaccdp8x64bce among them: L = bceBatchLoss, the mean over B×K, g the three-op chain (bceBatchLoss_grad).

How. ResNet34ParamGrad's three layers, with R34's stem and head bundles reused verbatim (the stem and head ARE R34's functions at R50's widths):

• Per block kind (at variable widths): the loss read at each internal activation of an identity / stride-1 projection / strided projection bottleneck (r50IdG*, r50ProjG*, r50DownG*), its gradient the chain's own cotangent, pulled back one certified stage at a time. The bundles r50IdLossTiedB / r50ProjLossTiedB / r50DownLossTiedB state the block's nodes against Φ, the loss as a function of the block's weight record — the same slots as r50IdTiedB / r50ProjTiedB / r50DownTiedB, no conv bias (ResNet50RenderB emits none).
• Per net: the loss read after each block (r50Suf*), its gradient pulled back through the sixteen certified bottleneck VJPs, and Φ identified with the whole net at updated weights (r50_factor_*), each a standalone theorem.

Hypotheses. 0 < q, R50PosB (every BN ε > 0), R50SmoothAtB (every relu off its kink and the stem pool tie-free at the real activations); for the smoothed loss also every example's target summing to one and 0 < nCls. The BCE corollary takes no hypothesis on the target.