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1 \clearpage
2 \section{The Polylogarithmic Three-Point Path Packet}
3 
4 \subsection{Root-distribution compression}
5 Let $u^M=z$ and let $\zeta_M$ be a primitive $M$th root of unity. For $n\ge1$,
6 \[
7  \boxed{
8  \sum_{j=0}^{M-1}
9  \operatorname{Li}_n(\zeta_M^ju)
10  =
11  M^{1-n}\operatorname{Li}_n(u^M).
12  }
13 \]
14 Inside the disk of convergence, this follows by summing the defining series and using the root-of-unity filter. Analytic continuation extends the identity with compatible branches.
15 
16 The inverse synthesis gain at weight $n$ is $M^{n-1}$. Through weights one, two, and three, the total gain is
17 \[
18  \boxed{1\cdot M\cdot M^2=M^3.}
19 \]
20 The displayed coordinate rescaling has determinant $M^3$. It has not been identified here with an actual integral motivic or period-lattice map. That identification would require the missing arithmetic realization.
21 
22 \subsection{Three-point functional closure}
23 Set
24 \[
25  z=\frac ac,
26  \qquad
27  1-z=\frac bc,
28  \qquad
29  \beta=1-\frac1z=-\frac ba.
30 \]
31 The Rogers dilogarithm
32 \[
33  L_2(z)
34  =
35  \operatorname{Li}_2(z)
36  +
37  \frac12\log z\log(1-z)
38 \]
39 satisfies
40 \[
41  \boxed{
42  L_2(z)+L_2(1-z)=\frac{\pi^2}{6}.
43  }
44 \]
45 At weight three,
46 \[
47  \boxed{
48  \begin{aligned}
49  &\operatorname{Li}_3(z)
50  +\operatorname{Li}_3(1-z)
51  +\operatorname{Li}_3(\beta)\\
52  &\qquad=
53  \zeta(3)
54  +\frac16\log^3z
55  +\frac{\pi^2}{6}\log z
56  -\frac12\log^2z\log(1-z).
57  \end{aligned}
58  }
59 \]
60 For $0<z<1$, all three displayed arguments use their real branches. The dilogarithm complement identity proves the Rogers relation. The auxiliary identity
61 \[
62  \Li_2(1-1/z)+\Li_2(1-z)=-\tfrac12\log^2z
63 \]
64 follows by differentiation and the limit at $z=1$. Using $\dd\Li_3(w)=\Li_2(w)\,\dd w/w$, the derivative of the displayed trilogarithm sum is
65 \[
66  \frac{\pi^2}{6z}-\frac{\log z\log(1-z)}z+
67  \frac{\log^2z}{2z(1-z)}.
68 \]
69 It is also the derivative of the right side. Both sides tend to $\zeta(3)$ as $z\to1$, proving the formula. The balanced Kummer coordinate is therefore exactly the third argument in the depth-three functional closure.
70 
71 A fixed path packet is
72 \[
73  \Pi_{\le3}(z)
74  =
75  \left(
76  \log z,
77  \log(1-z),
78  L_2(z),
79  \operatorname{Li}_3(z),
80  \operatorname{Li}_3(1-z)
81  \right).
82 \]
83 Its coordinate count is independent of $M$.
84 
85 \subsection{Archimedean envelope}
86 For $0<z<1$,
87 \[
88  |\log z|,
89  |\log(1-z)|
90  \le\log c.
91 \]
92 The positive dilogarithm and trilogarithm values on $(0,1)$ are bounded by $\zeta(2)$ and $\zeta(3)$. The three-point identity gives
93 \[
94  \left|\operatorname{Li}_3\left(-\frac ba\right)\right|
95  \le
96  3\zeta(3)
97  +
98  \frac{\pi^2}{6}\log c
99  +
100  \frac23(\log c)^3.
101 \]
102 Thus the complex packet has only cubic logarithmic growth. The surrounding motivic and regulator language is informed by the polylogarithmic complexes of Goncharov, the regulator framework discussed by Zagier, and Borel's rank theorem~\cite{Goncharov,ZagierPolylog,Borel}.
103 
104 This is a genuine nonlinear compression of the displayed complex data. It is not yet an integral compression. A fixed-rank integer lattice carrying a large torsion quotient must pay through covolume, denominator, algebraic degree, or a short vector.
105 
106 \subsection{Fixed-rank integral uncertainty}
107 Let $A\in M_d(\Z)$ be nonsingular and suppose $D\mid\det A$. If every entry satisfies $|A_{ij}|\le B$, then Hadamard's inequality gives
108 \[
109  D\le|\det A|\le d^{d/2}B^d.
110 \]
111 Hence
112 \[
113  \boxed{
114  B\ge\frac{D^{1/d}}{\sqrt d}.
115  }
116 \]
117 A small complex period packet does not imply a small integral realization.
118 
119 \subsection{Exact finite-place notch}
120 At an output prime $p^e\parallel c$,
121 \[
122  \beta=1-\frac ca.
123 \]
124 Since $p\nmid a$,
125 \[
126  v_p(\beta-1)=e.
127 \]
128 For odd $p$,
129 \[
130  \boxed{
131  v_p(\log_p\beta)=e.
132  }
133 \]
134 The corresponding weighted dyadic identity holds on the principal branch when $e\ge2$.
135 
136 Writing
137 \[
138  \beta=-\prod_{q\mid ab}q^{\nu_q},
139 \]
140 we obtain the cross-prime logarithmic filter relation
141 \[
142  \boxed{
143  \log_p\beta
144  =
145  \sum_{q\mid ab}
146  \nu_q\log_p\langle q\rangle_p.
147  }
148 \]
149 The same depth-three path therefore has a small complex envelope and exact deep finite notches.
150 
151 \subsection{Weight-three motivation and limitation}
152 Over $\Q$, Borel's rank theorem gives $\operatorname{rank}K_3(\Q)=0$ and $\operatorname{rank}K_5(\Q)=1$, the rational weights two and three. This makes weight three a plausible rational regulator layer. It does not assert the absence of all weight-two periods or of regulator directions over arbitrary number fields. This motivates the depth-three path packet. It does not prove that weight three suffices uniformly. A complete proof may require adaptive higher weights or a different motivic quotient.
153 
154 \subsection{Conditional motivic transfer}
155 Let $K_\beta$ be the cyclic group generated by the balanced Kummer class. Suppose that for every primitive positive triple there exists an integral finite object $\mathcal H_3(a,b,c)$ and a canonical map
156 \[
157  \Theta_3:K_\beta\longrightarrow\mathcal H_3(a,b,c)
158 \]
159 such that, for every $\eps>0$, constants independent of the triple give
160 \[
161  |\ker\Theta_3|\le C_\eps R^\eps
162 \]
163 and
164 \[
165  |\mathcal H_3(a,b,c)|\le C'_\eps R_aR_bR^\eps.
166 \]
167 Then the strong $abc$ conjecture follows after absorbing the logarithmic Kummer kernel.
168 
169 The missing object must combine finite Kummer depth, the three-point additive relation, an integral motivic or syntomic lattice, and an archimedean period norm.
170 

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