Φ writes zeros on the padding

Φ writes zeros on the padding.

The following holds (Lean statement, in the scope of module Solutions.KFUPhiMath):

theorem NLogN.KFU.PhiD_zero (h d : ℕ) (bL : ℕ → Bool) (sF μ : ℕ → ℕ) (xs : List ℤ) (b : List ℕ)
    (hbT : Valid (shR d (NLogN.KFU.tP d h bL)) b) (hbS : ¬ Valid (shR d (sP d sF)) b) :
    (PhiD h d bL sF μ xs).getD (encode (shR d (NLogN.KFU.tP d h bL)) b) 0 = 0

Context. The lemma belongs to the part KFU §10: what Φ and Φ⁻¹ compute (against RCRT.upd).

Formalization Note. This lemma belongs to the item-language (IL) layer of the integer-multiplication-kappa campaign: tape-level programs (register programs over 64 item stacks of the NLogN multitape framework, definition bundles IntMul_HvdH_M_Defs1–IntMul_HvdH_M_Defs8) that implement the data-movement primitives of the OpenAI preprint Integer multiplication below $n\log n$ (field moves, row split/merge, shifts, the uniform Proposition 11 interchange, Lemma 12 with fine $O(1)$-per-element costs, Gaussian record primitives, Proposition 14 pieces) with exact list denotations from the array-model bundles IntMul_Kappa_K_Defs* and explicit cost bounds. All names refer to the definition bundles up to IntMul_Kappa_K_Defs18.