Paper IX: Arithmetic Packets, Dyadic Towers, and Local Polynomial Deformation

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1 \hypertarget{cont8:paper-ix-prescribed-parameter-polynomial-lifting}{%
2 \subsection{Paper IX: prescribed-parameter polynomial
3 lifting}\label{cont8:paper-ix-prescribed-parameter-polynomial-lifting}}
4 
5 \textbf{Komorebi P. --- 8 September 2026.}
6 
7 \hypertarget{cont8:setup-and-exact-lift-set}{%
8 \subsubsection{Setup and exact lift
9 set}\label{cont8:setup-and-exact-lift-set}}
10 
11 Let \(q>0\), \(L\ge1\), and \(m,k,e\ge0\). Write
12 \(S=\mathbb Z/q\mathbb Z\). Let \[
13 F\in\mathbb Z[Y_1,\ldots,Y_m,T_1,\ldots,T_k]^e,
14 \qquad x=(y_0,t_0)\in\mathbb Z^{m+k},
15 \qquad F(x)\equiv0\pmod{q^L}.
16 \] The actual lower root supplies the unique integral residual \[
17 r=q^{-L}F(x)\in\mathbb Z^e,\qquad \bar r=r\bmod q\in S^e.
18 \] Let the actual reduced Jacobian blocks be \[
19 J=(D_YF)(x)\bmod q:S^m\to S^e,
20 \qquad K=(D_TF)(x)\bmod q:S^k\to S^e.
21 \] For a prescribed parameter digit \(v\in S^k\), choose its canonical
22 integral representative \(\widehat v\). Define \(\mathcal L_v\) to be
23 the set of actual residue vectors
24 \(z\in(\mathbb Z/q^{L+1}\mathbb Z)^{m+k}\) satisfying \[
25 z\equiv x\pmod{q^L},\qquad F(z)\equiv0\pmod{q^{L+1}},\qquad
26 z_T\equiv t_0+q^L\widehat v\pmod{q^{L+1}}.
27 \] The formal predicate uses canonical integer representatives for
28 evaluation and the lower congruence. The previously checked polynomial
29 congruence theorem and digit-to-residue equivalence establish the actual
30 residue-class interpretation. Changing an integral representative of a
31 parameter digit by a multiple of \(q\) does not change its prescribed
32 residue modulo \(q^{L+1}\).
33 
34 \hypertarget{cont8:the-obstruction-theorem}{%
35 \subsubsection{The obstruction theorem}\label{cont8:the-obstruction-theorem}}
36 
37 Define the finite solved-block cokernel and its parameter-dependent
38 class by \[
39 \mathcal C_x=S^e/\operatorname{im}J,
40 \qquad o_x(v)=[\bar r+Kv]\in\mathcal C_x.
41 \] Then \[
42 \boxed{\mathcal L_v\ne\varnothing\iff o_x(v)=0.}
43 \] More precisely, the checked digit-to-lift equivalence is \[
44 \boxed{\mathcal L_v\cong\{u\in S^m:Ju=-\bar r-Kv\}.}
45 \] It sends a solved digit \(u\) to the actual next residue represented
46 by \(x+q^L(\widehat u,\widehat v)\). The obstruction depends on the
47 solved block \(J\), not on the range of the full map \((J\ K)\).
48 
49 Every nonempty \(\mathcal L_v\) has a checked free and transitive action
50 of \(\ker J\). The action is defined without choosing a base lift.
51 Choosing one valid solved digit then gives an explicit bijection
52 \(\mathcal L_v\cong\ker J\). In particular, for every positive modulus,
53 \[
54 o_x(v)=0\quad\Longrightarrow\quad\#\mathcal L_v=\#\ker J.
55 \] These results permit composite \(q\), the degenerate modulus \(q=1\),
56 and zero dimensions. Neither \(J\) nor the full reduced Jacobian is
57 assumed surjective.
58 
59 \hypertarget{cont8:exact-prime-field-counts}{%
60 \subsubsection{Exact prime-field
61 counts}\label{cont8:exact-prime-field-counts}}
62 
63 When \(q=p\) is prime, put \(\rho=\operatorname{rank}_{\mathbb F_p}J\).
64 Then \[
65 \boxed{\#\mathcal L_v=
66 \begin{cases}
67 p^{m-\rho},&o_x(v)=0,\\
68 0,&o_x(v)\ne0,
69 \end{cases}}
70 \qquad
71 \boxed{\#\mathcal C_x=p^{e-\rho}.}
72 \] The count is of distinct actual residue classes at level \(L+1\). The
73 solved dimension \(m\), rather than the total number \(m+k\) of
74 variables, appears in the exponent because the parameter digit is fixed.
75 These rank formulas are asserted only over prime fields.
76 
77 The main wrapper assumes only the original lower polynomial congruence.
78 It constructs one residual \(r\) and supplies the criterion and count
79 uniformly for every \(v\), together with the exact obstruction
80 cardinality.
81 
82 \hypertarget{cont8:proof-and-dependency-chain}{%
83 \subsubsection{Proof and dependency
84 chain}\label{cont8:proof-and-dependency-chain}}
85 
86 The actual derivative calculation gives \[
87 (DF(x)\bmod q)(u,v)=Ju+Kv.
88 \] Taylor expansion at depth \(L\ge1\) makes the quadratic remainder
89 divisible by \(q^{L+1}\). Therefore the digit \((u,v)\) is a polynomial
90 lifting correction exactly when \[
91 \bar r+Ju+Kv=0.
92 \] The quotient class \([\bar r+Kv]\) vanishes exactly when this
93 equation admits \(u\). This uses additive inverses in the image of
94 \(J\), so the sign agrees with the actual Taylor equation.
95 
96 The earlier unrestricted correction-digit equivalence is then restricted
97 to the predicate that the parameter digit equals \(v\). Taking free
98 coordinates commutes with forming digit residues, and digit residues are
99 injective in the digit. Consequently the restricted equivalence enforces
100 exactly the actual parameter residue condition on \(\mathcal L_v\).
101 
102 Translation by \(\ker J\) acts freely and transitively on the
103 solved-digit fiber. Transporting this explicit action through the
104 residue equivalence yields the torsor on \(\mathcal L_v\). Rank-nullity
105 over \(\mathbb F_p\) then gives the two displayed cardinalities. No
106 numerical enumeration is used to establish the general theorem.
107 
108 \hypertarget{cont8:why-the-solved-block-matters}{%
109 \subsubsection{Why the solved block
110 matters}\label{cont8:why-the-solved-block-matters}}
111 
112 Take \(F(Y,T)=T\), \(q=2\), \(L=1\), and \(x=(0,0)\). The full Jacobian
113 \((0\ 1)\) is surjective, but \(J=0\). Prescribing \(v=1\) forces
114 \(T\equiv2\pmod4\), which cannot satisfy \(F=0\pmod4\). Prescribing
115 \(v=0\) gives exactly two lifts, corresponding to \(Y\equiv0,2\pmod4\).
116 
117 The Lean controls check this actual polynomial example. Thus
118 full-Jacobian surjectivity alone does not make every prescribed
119 coordinate choice admissible. The automatic coordinate selection in the
120 earlier chart theorem is compatible with this obstruction.

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