ViT multi-head backward — the per-head SDPA cotangents the real chain delivers #

The single-head representative (ViTChainClose) pins the rendered attention backward to the audited sdpaBack{Q,K,V} suite at d = D (one head). The committed vitTrainStepRenderV render is multi-head (D = heads · d, the committed ViT-Tiny: heads = 3, d = 64): the Q/K/V denses produce all heads' projections at once ([N, heads·d]), then headSliceF h slices head h's [N, d] block, per-head SDPA runs at sdpaScale d (d_head, NOT D), and headPadF h scatters the result back, summed over heads (mhsaLayer_spelled).

This file is the backward symmetry of ViTMultiHead's forward: the cotangent the multi-head chain delivers at each of the Q/K/V dense outputs ([N, heads·d]). The out-proj Wo is a single [D, D] dense unchanged by heads, so the cotangent at the SDPA output dAtt (= vitCotAttV) is exactly the single-head one — multi-head changes ONLY the SDPA-internal backward dAtt → dQ/dK/dV:

• slice the SDPA-output cotangent per head (headSliceFlat h dAtt — the pad's VJP),
• run the EXISTING per-head rendered SDPA backward at d_head (vitCotD{Q,K,V} d …),
• scatter each head's dQ_h/dK_h/dV_h back into the full [N, heads·d] cotangent and sum over heads (Σ_h headPadFlat h … — the slice's VJP).

The composition theorems (vitCotD{Q,K,V}mh_eq) are the substantive content: the rendered slice → per-head SDPA backward → pad chain IS Mat.flatten (Σ_h headPadMat h (sdpaBack{Q,K,V})) — the concat of the proven per-head SDPA backwards. Each reuses the single-head pin (vitCotD{Q,K,V}_eq_sdpaBack{Q,K,V}) at d_head head-by-head, plus the headSliceFlat/ headPadFlat/Mat.flatten-sum commutation bridges (ViTMultiHead). The downstream LN₁ fan-in is unchanged in shape (vitCotLn1MH = vitCotLn1 at the multi-head Q/K/V cots); the fold param-SGD den lemmas (rowDenseWeightSgd_den, …) are head-agnostic (generic in the cotangent), so only the COTANGENT changes from the single-head vitCotD{Q,K,V} to vitCotD{Q,K,V}mh.