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

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1 \hypertarget{cont4:polynomial-roots-under-coefficient-perturbation}{%
2 \subsection{Polynomial roots under coefficient
3 perturbation}\label{cont4:polynomial-roots-under-coefficient-perturbation}}
4 
5 Komorebi P. --- 8 September 2026
6 
7 This block formalizes square root uniqueness and nearby-root stability
8 for finite systems of integer polynomials. The polynomials and their
9 partial derivatives are evaluated directly. No root-uniqueness or
10 perturbation-stability premise is assumed.
11 
12 \hypertarget{cont4:notation}{%
13 \subsubsection{Notation}\label{cont4:notation}}
14 
15 Let \(p\) be prime, \(L\ge1\), and \[
16 F,H\in\mathbb Z[X_1,\ldots,X_d]^e,
17 \qquad x\in\mathbb Z^d.
18 \] Write \[
19 A_x=DF(x)\bmod p:\mathbb F_p^d\longrightarrow\mathbb F_p^e,
20 \qquad B_L(x)=x+p^L\mathbb Z_p^d.
21 \] For an integer \(c\), the perturbed system is the actual polynomial
22 list \[
23 G=F+cH.
24 \] There is no restriction on the sign of \(c\), and \(c=0\) is allowed.
25 Divisibility conclusions give lower bounds on precision and allow zero
26 differences.
27 
28 \hypertarget{cont4:unique-roots-for-square-systems}{%
29 \subsubsection{1. Unique roots for square
30 systems}\label{cont4:unique-roots-for-square-systems}}
31 
32 If \(d=e\), \(A_x\) is surjective, and \(F(x)\equiv0\pmod{p^L}\), then
33 \[
34 \boxed{\exists!\,a\in B_L(x):F(a)=0.}
35 \] The stronger comparison lemma proves that any two exact roots of
36 \(F\) in \(B_L(x)\) coincide, without needing a separate lower-root
37 hypothesis.
38 
39 \textbf{Proof implemented.} Map every arbitrary p-adic root into the
40 actual residue rings \(\mathbb Z/p^n\mathbb Z\), and choose its
41 canonical integer representative. Evaluation commutes with reduction, so
42 these representatives are genuine finite polynomial roots. The preceding
43 count theorem gives exactly \(p^{d-d}=1\) next-level residue lift.
44 Inductively any two roots agree at every precision. The p-adic residue
45 maps separate points, so the roots are equal. Existence is the preceding
46 checked tower theorem.
47 
48 This extends uniqueness from the limit of a selected sequence to
49 \textbf{all exact roots in the prescribed ball}.
50 
51 \hypertarget{cont4:coefficient-perturbations-preserve-smooth-liftability}{%
52 \subsubsection{2. Coefficient perturbations preserve smooth
53 liftability}\label{cont4:coefficient-perturbations-preserve-smooth-liftability}}
54 
55 If \(p^L\mid c\), the actual identities give \[
56 G(x)=F(x)+cH(x),\qquad DG(x)\equiv DF(x)\pmod p.
57 \] Thus if \(F(x)\equiv0\pmod{p^L}\) and \(A_x\) is surjective, then \[
58 \exists b\in B_L(x):G(b)=0.
59 \] This theorem allows rectangular systems. The evaluation, derivative
60 and Jacobian identities are proved from the polynomial definition
61 \(G_i=F_i+C(c)H_i\), rather than supplied as hypotheses. Algebraic
62 congruence helpers also permit composite moduli; exact p-adic root
63 existence uses prime \(p\).
64 
65 \hypertarget{cont4:nearby-roots-for-rectangular-systems}{%
66 \subsubsection{3. Nearby roots for rectangular
67 systems}\label{cont4:nearby-roots-for-rectangular-systems}}
68 
69 Let \(M\ge1\) and \(p^M\mid c\). Suppose \(A_x\) is surjective and
70 \(a\in\mathbb Z_p^d\) is an exact root of \(F\) with
71 \(a\equiv x\pmod p\). Then \[
72 \boxed{\exists b\in\mathbb Z_p^d:\quad G(b)=0,
73 \qquad b\equiv a\pmod{p^M}.}
74 \] No uniqueness is claimed when \(d>e\).
75 
76 \textbf{Proof implemented.} Truncate \(a\) to an integer vector \(y\)
77 modulo \(p^M\). It is an actual lower root of \(F\), and its reduced
78 Jacobian agrees with \(A_x\). Apply the coefficient-perturbed existence
79 theorem at precision \(M\), then combine the two congruences through
80 \(y\).
81 
82 \hypertarget{cont4:full-precision-for-the-unique-square-root}{%
83 \subsubsection{4. Full precision for the unique square
84 root}\label{cont4:full-precision-for-the-unique-square-root}}
85 
86 Suppose \(d=e\), \(M\ge L\ge1\), \(p^M\mid c\), and \(A_x\) is
87 surjective. If \[
88 a,b\in B_L(x),\qquad F(a)=0,\qquad G(b)=0,
89 \] then \[
90 \boxed{b-a\in p^M\mathbb Z_p^d.}
91 \] In particular, for \(G=F+p^MH\), no p-adic digits below depth \(M\)
92 are changed in the selected root.
93 
94 The formalization also proves a combined existence theorem. Starting
95 only with \(F(x)\equiv0\pmod{p^L}\), it constructs an original root
96 \(a\in B_L(x)\) and proves that there is exactly one \(b\in B_L(x)\)
97 satisfying \(G(b)=0\) and \(b\equiv a\pmod{p^M}\). The comparison
98 theorem shows that the precision bound holds for every root of \(G\) in
99 that ball, so uniqueness is not obtained merely by adding a restrictive
100 displacement condition.
101 
102 \textbf{Proof implemented.} Section 3 produces a nearby root \(b'\).
103 Since \(M\ge L\), it belongs to \(B_L(x)\). The perturbed Jacobian
104 remains surjective, and square uniqueness identifies \(b'\) with the
105 arbitrary root \(b\).

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