B.8 Paper IX: prescribed-parameter polynomial lifting
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Komorebi P. — 8 September 2026.
B.8.1 Setup and exact lift set
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Let \(q>0\), \(L\ge1\), and \(m,k,e\ge0\). Write \(S=\mathbb Z/q\mathbb Z\). Let
The actual lower root supplies the unique integral residual
Let the actual reduced Jacobian blocks be
For a prescribed parameter digit \(v\in S^k\), choose its canonical integral representative \(\widehat v\). Define \(\mathcal L_v\) to be the set of actual residue vectors \(z\in(\mathbb Z/q^{L+1}\mathbb Z)^{m+k}\) satisfying
The formal predicate uses canonical integer representatives for evaluation and the lower congruence. The previously checked polynomial congruence theorem and digit-to-residue equivalence establish the actual residue-class interpretation. Changing an integral representative of a parameter digit by a multiple of \(q\) does not change its prescribed residue modulo \(q^{L+1}\).
B.8.2 The obstruction theorem
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Define the finite solved-block cokernel and its parameter-dependent class by
Then
More precisely, the checked digit-to-lift equivalence is
It sends a solved digit \(u\) to the actual next residue represented by \(x+q^L(\widehat u,\widehat v)\). The obstruction depends on the solved block \(J\), not on the range of the full map \((J\ K)\).
Every nonempty \(\mathcal L_v\) has a checked free and transitive action of \(\ker J\). The action is defined without choosing a base lift. Choosing one valid solved digit then gives an explicit bijection \(\mathcal L_v\cong\ker J\). In particular, for every positive modulus,
These results permit composite \(q\), the degenerate modulus \(q=1\), and zero dimensions. Neither \(J\) nor the full reduced Jacobian is assumed surjective.
B.8.3 Exact prime-field counts
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When \(q=p\) is prime, put \(\rho=\operatorname{rank}_{\mathbb F_p}J\). Then
The count is of distinct actual residue classes at level \(L+1\). The solved dimension \(m\), rather than the total number \(m+k\) of variables, appears in the exponent because the parameter digit is fixed. These rank formulas are asserted only over prime fields.
The main wrapper assumes only the original lower polynomial congruence. It constructs one residual \(r\) and supplies the criterion and count uniformly for every \(v\), together with the exact obstruction cardinality.
B.8.4 Proof and dependency chain
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The actual derivative calculation gives
Taylor expansion at depth \(L\ge1\) makes the quadratic remainder divisible by \(q^{L+1}\). Therefore the digit \((u,v)\) is a polynomial lifting correction exactly when
The quotient class \([\bar r+Kv]\) vanishes exactly when this equation admits \(u\). This uses additive inverses in the image of \(J\), so the sign agrees with the actual Taylor equation.
The earlier unrestricted correction-digit equivalence is then restricted to the predicate that the parameter digit equals \(v\). Taking free coordinates commutes with forming digit residues, and digit residues are injective in the digit. Consequently the restricted equivalence enforces exactly the actual parameter residue condition on \(\mathcal L_v\).
Translation by \(\ker J\) acts freely and transitively on the solved-digit fiber. Transporting this explicit action through the residue equivalence yields the torsor on \(\mathcal L_v\). Rank-nullity over \(\mathbb F_p\) then gives the two displayed cardinalities. No numerical enumeration is used to establish the general theorem.
B.8.5 Why the solved block matters
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Take \(F(Y,T)=T\), \(q=2\), \(L=1\), and \(x=(0,0)\). The full Jacobian \((0\ 1)\) is surjective, but \(J=0\). Prescribing \(v=1\) forces \(T\equiv2\pmod4\), which cannot satisfy \(F=0\pmod4\). Prescribing \(v=0\) gives exactly two lifts, corresponding to \(Y\equiv0,2\pmod4\).
The Lean controls check this actual polynomial example. Thus full-Jacobian surjectivity alone does not make every prescribed coordinate choice admissible. The automatic coordinate selection in the earlier chart theorem is compatible with this obstruction.