Realization Calculus

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

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