Realization Calculus

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

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