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

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1 \hypertarget{cont10:paper-ix-simultaneous-coefficient-and-parameter-deformation}{%
2 \subsection{Paper IX: simultaneous coefficient and parameter
3 deformation}\label{cont10:paper-ix-simultaneous-coefficient-and-parameter-deformation}}
4 
5 \textbf{Author: Komorebi P. --- 8 September 2026.}
6 
7 This block combines coefficient perturbation with prescribed parameter
8 motion. The original solved Jacobian controls existence, local
9 uniqueness, displacement and the first-order error. The four new
10 mathematical modules compile. The fresh verification report and
11 independent audit record the authoritative release status and exact
12 source revisions.
13 
14 \hypertarget{cont10:the-theorem}{%
15 \subsubsection{The theorem}\label{cont10:the-theorem}}
16 
17 Let \(p\) be any prime, \(R=\mathbb Z_p\), and \(Q=\mathbb Q_p\). Let \[
18 F,H\in R[Y_1,\ldots,Y_e,T_1,\ldots,T_k]^e,
19 \qquad a=(y_0,t_0)\in R^{e+k},\qquad F(a)=0.
20 \] Here \(e,k\ge0\) are arbitrary finite dimensions. There are as many
21 solved coordinates as equations. Define the actual derivative blocks at
22 the original root: \[
23 J=D_YF(a)\in M_e(R),\qquad K=D_TF(a)\in M_{e\times k}(R).
24 \] Suppose, for natural numbers \(s,M\), that \[
25 J\in\operatorname{GL}_e(Q),\qquad p^sJ^{-1}\in M_e(R),\qquad M>2s.
26 \] For every \(v\in R^k\), set \[
27 \tau=t_0+p^Mv,\qquad G=F+p^MH.
28 \] Then there is exactly one \(y\in y_0+p^{s+1}R^e\) with \[
29 G(y,\tau)=0.
30 \] It satisfies \[
31 y-y_0\in p^{M-s}R^e
32 \] and there is an integral vector \(\varepsilon\in p^{2M-3s}R^e\) such
33 that, in \(Q^e\), \[
34 \boxed{\displaystyle
35  y-y_0=-p^MJ^{-1}\bigl(H(a)+Kv\bigr)+\varepsilon.
36 }
37 \] The same original exponent \(s\) appears throughout. No inverse or
38 rank assumption on the perturbed Jacobian is imposed. The statement
39 includes arbitrary integral p-adic coefficients and centers, the dyadic
40 prime, and zero dimensions.
41 
42 The full-point theorem constructs \(b=(y,\tau)\) and also proves \[
43 b-a\in p^{M-s}R^{e+k}\cap p^{s+1}R^{e+k}.
44 \] Parameter equality \(b_T=\tau\) is exact. Uniqueness concerns every
45 root in the stated solved-coordinate ball with these parameters, not
46 only roots obtained by the construction. No global uniqueness is
47 claimed.
48 
49 \hypertarget{cont10:actual-augmented-system}{%
50 \subsubsection{Actual augmented system}\label{cont10:actual-augmented-system}}
51 
52 Define polynomial systems with \(e+k\) equations and variables by \[
53 \mathcal F(Y,T)=\bigl(F(Y,T),T-t_0\bigr),\qquad
54 \mathcal H(Y,T)=\bigl(H(Y,T),-v\bigr).
55 \] Both are actual polynomials over \(R\). Direct evaluation proves \[
56 \mathcal F(a)=0,
57 \qquad
58 (\mathcal F+p^M\mathcal H)(b)=0
59 \iff F(b)+p^MH(b)=0\ \text{and}\ b_T=t_0+p^Mv.
60 \] Actual polynomial partial differentiation gives \[
61 D\mathcal F(a)=B=\begin{pmatrix}J&K\\0&I_k\end{pmatrix}.
62 \] These identities are proved from evaluation and partial derivatives,
63 rather than supplied as additional hypotheses.
64 
65 Choose the integral matrix \(A=p^sJ^{-1}\). The existing
66 inverse-certificate equivalence gives \[
67 AJ=JA=p^sI_e.
68 \] Put \(c=p^s\) and define an integral block matrix \[
69 C=\begin{pmatrix}A&-AK\\0&cI_k\end{pmatrix}.
70 \] Then \[
71 CB=\begin{pmatrix}AJ&AK-AK\\0&cI_k\end{pmatrix}=cI_{e+k},
72 \] \[
73 BC=\begin{pmatrix}JA&-JAK+cK\\0&cI_k\end{pmatrix}=cI_{e+k}.
74 \] The second equality uses \(JA=cI_e\). The generic block proof works
75 over any commutative ring. The p-adic specialization also explicitly
76 proves that \(B\) is invertible over \(Q\) and that \(p^sB^{-1}\) is
77 integral. There is no denominator loss from appending the parameter
78 equations.
79 
80 \hypertarget{cont10:construction-error-and-uniqueness}{%
81 \subsubsection{Construction, error and
82 uniqueness}\label{cont10:construction-error-and-uniqueness}}
83 
84 Apply the previously checked weighted coefficient-perturbation
85 construction to \(\mathcal F,\mathcal H,a,C\). Its hypotheses follow
86 from the actual augmented identities and the two-sided certificate. It
87 constructs \(b\), proves the full displacement bound \(p^{M-s}\), and
88 proves uniqueness throughout \(a+p^{s+1}R^{e+k}\). The augmented root
89 equivalence gives the required original equations and exact parameter
90 values.
91 
92 The checked integral first-order estimate gives \[
93 b-a+p^{M-s}C\mathcal H(a)\in p^{2M-3s}R^{e+k}.
94 \] The decisive block action is \[
95 C\binom{H(a)}{-v}
96 =
97 \binom{A\bigl(H(a)+Kv\bigr)}{-p^sv}.
98 \] Projecting the solved coordinates therefore gives \[
99 y-y_0+p^{M-s}A\bigl(H(a)+Kv\bigr)
100 \in p^{2M-3s}R^e.
101 \] The existing rational-conversion theorem turns \(p^{M-s}A\) into
102 \(p^MJ^{-1}\) and retains an integral error vector. This proves the
103 displayed first-order formula with the original derivative blocks.
104 
105 For any competing solved-coordinate root \(z\in y_0+p^{s+1}R^e\), the
106 joined point \((z,\tau)\) lies in the augmented uniqueness ball:
107 \(M>2s\) implies \(M\ge s+1\), so \(\tau-t_0=p^Mv\in p^{s+1}R^k\).
108 Full-point uniqueness gives \((z,\tau)=b\), hence \(z=y\). This is the
109 additional comparison needed to obtain uniqueness in exactly the ball
110 stated in the solved-coordinate theorem.
111 
112 \hypertarget{cont10:checked-controls-and-precision-boundary}{%
113 \subsubsection{Checked controls and precision
114 boundary}\label{cont10:checked-controls-and-precision-boundary}}
115 
116 For \(p=2\), \(s=0\), \(F(Y,T)=Y+T\), \(H=1\), \(a=0\), and \(v=1\),
117 both original blocks satisfy \(J=K=1\). At \(M=1\), the exact parameter
118 is \(\tau=2\), and the unique solved value is \(y=-4\). The combined
119 correction is \(-2(1+1)=-4\), with zero error.
120 
121 The sign controls use \(M=2\): the actual augmented root is
122 \((y,\tau)=(-8,4)\), and the required error ideal is \(16\mathbb Z_2\).
123 Omitting \(Kv\) predicts \(-4\), with error \(-4\notin16\mathbb Z_2\).
124 Reversing its sign predicts \(0\), with error \(-8\notin16\mathbb Z_2\).
125 Both failures are checked. At binary depth \(M=1\), the reversed-sign
126 difference would lie in the allowed error ideal, so that shallower
127 example alone would not distinguish the sign. This control-design
128 limitation was identified and corrected before the release gate.
129 
130 For parameter-only motion, use \[
131 F(Y,T)=Y+Y^2+T,\quad H=0,\quad a=0,\quad v=1,\quad p=2,\ s=0,\ M=1.
132 \] A checked transfer from the prior scalar quadratic root constructs an
133 even \(y\) with \[
134 y+y^2+2=0,\qquad 4\mid y+2,\qquad 8\nmid y+2.
135 \] Thus the error exponent \(2M-3s=2\) is attained even when only
136 parameters move. No uniform extra error digit can be added to the
137 general theorem. This example establishes that limitation; it does not
138 assert optimality at every choice of \(p,s,M\).
139 
140 The inherited controls continue to record the necessity of the strict
141 weighted threshold and other earlier scope boundaries. General
142 rationally rank-deficient systems are outside the inverse hypothesis.

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