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

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1 \hypertarget{cont9:paper-ix-compatible-towers-with-prescribed-parameters}{%
2 \subsection{Paper IX: compatible towers with prescribed
3 parameters}\label{cont9:paper-ix-compatible-towers-with-prescribed-parameters}}
4 
5 \textbf{Author: Komorebi P. --- 8 September 2026.}
6 
7 This block constructs actual compatible polynomial towers from initial
8 solved-block surjectivity, for every compatible integer parameter path.
9 For a prime base, it constructs such a path for every p-adic parameter
10 in the initial ball and proves an exact polynomial endpoint with all
11 retained precisions. The six new mathematical modules compile; the
12 release verification report records the authoritative fresh-build and
13 axiom-audit results.
14 
15 \hypertarget{cont9:statement-and-notation}{%
16 \subsubsection{Statement and notation}\label{cont9:statement-and-notation}}
17 
18 Let \(q>0\), \(L\ge1\), and \[
19 F\in\mathbb Z[Y_1,\ldots,Y_m,T_1,\ldots,T_k]^e,
20 \qquad x=(y_0,t_0)\in\mathbb Z^{m+k},
21 \qquad F(x)\in q^L\mathbb Z^e.
22 \] Write \(S_q=\mathbb Z/q\mathbb Z\), and let the actual reduced
23 Jacobian blocks be \[
24 J_x=D_YF(x)\bmod q:S_q^m\longrightarrow S_q^e,
25 \qquad K_x=D_TF(x)\bmod q:S_q^k\longrightarrow S_q^e.
26 \] Assume \(J_x\) is surjective. A vector divisibility condition below
27 is coordinatewise.
28 
29 \textbf{Integer prescribed-path theorem.} For every supplied path
30 \(t:\mathbb N\to\mathbb Z^k\) satisfying \[
31 t(0)=t_0,\qquad t(n+1)-t(n)\in q^{L+n}\mathbb Z^k,
32 \] there is a single tower \(z:\mathbb N\to\mathbb Z^{m+k}\) with \[
33 z(0)=x,\qquad \operatorname{free}(z(n))=t(n),
34 \] \[
35 F(z(n))\in q^{L+n}\mathbb Z^e,\qquad
36 z(n+1)-z(n)\in q^{L+n}\mathbb Z^{m+k}.
37 \] The parameter equality is exact in integers. For \(a\le b\), \[
38 z(b)-z(a)\in q^{L+a}\mathbb Z^{m+k},\qquad
39 DF(z(n))\bmod q=DF(x)\bmod q.
40 \] Consequently \(J_{z(n)}=J_x\) at every level. No later surjectivity
41 or per-step solvability is assumed. This theorem includes positive
42 composite bases and \(q=1\).
43 
44 \textbf{Arbitrary p-adic target theorem.} Now let \(q=p\) be prime. For
45 every \[
46 \tau\in t_0+p^L\mathbb Z_p^k,
47 \] there are a compatible integer tower \(z\) as above and
48 \(b\in\mathbb Z_p^{m+k}\) such that \[
49 z(n)\longrightarrow b,\qquad F(b)=0,\qquad
50 \operatorname{free}(b)=\tau,
51 \qquad b-z(n)\in p^{L+n}\mathbb Z_p^{m+k}\quad(n\ge0).
52 \] A compatible integer parameter path converging to \(\tau\) is
53 constructed in the proof, with its initial representative fixed exactly.
54 A separate theorem accepts every supplied compatible integer path with
55 limit \(\tau\) and retains its exact coordinates in the constructed
56 tower.
57 
58 \textbf{Exact branching.} At any stage \(n\), put \(N=L+n\). For every
59 parameter digit \(v\in\mathbb F_p^k\), let \[
60 \mathcal L_{n,v}=\left\{\bar w\in(\mathbb Z/p^{N+1}\mathbb Z)^{m+k}:
61 \begin{array}{l}
62 \bar w\equiv z(n)\pmod{p^N},\\
63 F(\bar w)=0,\\
64 \operatorname{free}(\bar w)=t(n)+p^Nv\pmod{p^{N+1}}
65 \end{array}\right\}.
66 \] Here \(p^Nv\) means any integer representative of \(v\), multiplied
67 by \(p^N\); the residue class does not depend on the representative.
68 Then \[
69 \#\mathcal L_{n,v}=p^{m-e}\qquad\text{for every }n\text{ and every }v.
70 \] Surjectivity implies \(e\le m\). This is a count of distinct actual
71 next-level residue classes. It does not count integer representatives or
72 entire infinite towers.
73 
74 \textbf{Square solved-block uniqueness.} If \(m=e\), every prescribed
75 \(\tau\) has exactly one root in the original ball: \[
76 \exists!\,b\in x+p^L\mathbb Z_p^{e+k}:\quad
77 F(b)=0,\qquad\operatorname{free}(b)=\tau.
78 \] Thus all valid integer parameter approximations and all lifting
79 choices with the same target have the same endpoint. This does not
80 assert uniqueness of the integer representatives in the tower. All prime
81 statements include \(p=2\).
82 
83 \hypertarget{cont9:proof}{%
84 \subsubsection{Proof}\label{cont9:proof}}
85 
86 At depth \(N\ge1\), suppose \(F(a)=q^Nr\) and a desired next integer
87 parameter target satisfies \(t'-\operatorname{free}(a)=q^Nw\), with
88 \(w\in\mathbb Z^k\). Surjectivity supplies a solved digit \(u\in S_q^m\)
89 satisfying \[
90 J_a u=-\bar r-K_a\bar w.
91 \] Choose an integer representative \(\widetilde u\), and set \[
92 a'=a+q^N(\widetilde u,w).
93 \] The parameter quotient is the supplied integer \(w\). Therefore
94 \(\operatorname{free}(a')=t'\) exactly. The checked polynomial Taylor
95 identity gives \[
96 F(a')\equiv F(a)+q^NDF(a)(\widetilde u,w)
97 \equiv0\pmod{q^{N+1}}.
98 \] Its quadratic remainder is divisible by \(q^{2N}\), and
99 \(2N\ge N+1\). Also \(a'\equiv a\pmod q\), so evaluation of every
100 polynomial partial derivative gives the same reduced Jacobian. Induction
101 on the level, carrying the root, exact parameters, and fixed solved
102 Jacobian as its state, constructs the entire integer tower. The
103 existence proof uses ordinary classical choice; it does not supply an
104 optimized computational lifting algorithm.
105 
106 For a p-adic target \(\tau\), use \(t(0)=t_0\) and, for \(n\ge1\), the
107 canonical integer representatives of \(\tau\bmod p^{L+n}\). The initial
108 congruence hypothesis and the truncation precision imply \[
109 \tau-t(n)\in p^{L+n}\mathbb Z_p^k.
110 \] Subtracting successive precision relations gives compatibility in
111 \(\mathbb Z_p\); the checked faithful integer-divisibility theorem gives
112 compatibility in \(\mathbb Z\). These precision relations also give
113 convergence to \(\tau\).
114 
115 The compatible vector-tower completion theorem constructs \(b\), proves
116 convergence, and retains every stage precision. Continuous polynomial
117 evaluation and the increasing root precision give \(F(b)=0\). Each
118 parameter coordinate has both the corresponding coordinate of \(b\) and
119 \(\tau\) as its limit, so uniqueness of limits gives exact parameter
120 equality.
121 
122 For every proposed next parameter digit, the previously checked
123 obstruction is \([\bar r+K_av]\in S_p^e/\operatorname{im}J_a\). The
124 fixed solved Jacobian is surjective, so this class vanishes for every
125 digit at every stage. The actual lift fiber is a torsor for
126 \(\ker J_a\). Rank-nullity gives its cardinality \(p^{m-e}\).
127 
128 When \(m=e\), the new solved-Jacobian definition is definitionally the
129 existing square block used by the checked continuous parameter chart.
130 Every root in the initial ball with the given parameters is that chart
131 point. Applying this to any two roots proves equality, including
132 endpoints obtained by different constructions.
133 
134 \hypertarget{cont9:scope-controls}{%
135 \subsubsection{Scope controls}\label{cont9:scope-controls}}
136 
137 The inherited checked example \(F(Y,T)=T\), \(p=2\), \(x=0\), \(L=1\)
138 has a surjective full Jacobian but zero solved block. Prescribing
139 parameter digit \(1\) forces \(T=2\pmod4\), so no next root exists.
140 Full-Jacobian surjectivity cannot replace the hypothesis on the
141 prescribed solved block.
142 
143 The new checked example \(F(Y_1,Y_2,T)=Y_1\) has a surjective solved
144 block \((1,0)\). Nevertheless \((0,0,0)\) and \((0,2,0)\) are distinct
145 exact roots in the same modulo-two ball with identical parameters.
146 Uniqueness therefore requires the square solved case. The corresponding
147 negative control deliberately attempts the stronger false conclusion and
148 must be rejected by Lean.
149 
150 Depth \(L\ge1\) is essential for the Taylor step used here. The general
151 singular case can have a nonzero next obstruction after an earlier zero
152 obstruction. Neither the finite one-step criterion nor this smooth
153 theorem establishes unrestricted singular persistence.

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