Prime-Contact Coherence Calculus

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1 \PartChapter{Objects and Maps}{Objects, Maps, and Theorem Boundaries}
2 
3 \section{Construction at a glance}
4 The theory is organized by a chain of objects. Each arrow is exact, conditional, or blocked by a stated structural theorem.
5 
6 \[
7 \boxed{
8 \begin{aligned}
9 &\text{primitive sparse relation}\\
10 &\quad\Downarrow\\
11 &\text{prime-packet differential and Wronskian}\\
12 &\quad\Downarrow\\
13 &\text{valuation-word and shell refinement}\\
14 &\quad\Downarrow\\
15 &\text{joint obstruction complex and exact search}\\
16 &\quad\Downarrow\\
17 &\text{prime-shell signal and Frey transform}\\
18 &\quad\Downarrow\\
19 &\text{divisor-polyphase persistence and Hecke filters}\\
20 &\quad\Downarrow\\
21 &\text{filtered cusp-contact and Witt packet}\\
22 &\quad\Downarrow\\
23 &\text{global Kummer and Fermat character packet}\\
24 &\quad\Downarrow\\
25 &\text{polylogarithmic path and cross-prime logarithms}\\
26 &\quad\Downarrow\\
27 &\text{adelic coherence character and kernel lattice}.
28 \end{aligned}}
29 \]
30 
31 The first three arrows preserve local valuation data. The middle arrows seek global arithmetic carriers. The final arrows isolate the remaining conductor-scale problem.
32 
33 \section{Object comparison table}
34 \begingroup\footnotesize\setlength{\tabcolsep}{3pt}
35 \begin{longtable}{L{0.14\textwidth}L{0.21\textwidth}L{0.27\textwidth}L{0.27\textwidth}}
36 \toprule
37 Layer & Object & Retained information & Uncontrolled information\\
38 \midrule
39 \endfirsthead
40 \toprule
41 Layer & Object & Retained information & Uncontrolled information\\
42 \midrule
43 \endhead
44 Prime derivative & $\mathcal T_{a,b,c}$ and $W_x$ & prime support, valuation weights, additive tangent, repeated-root divisor & uniform small integral transversality\\
45 Polyphase & $V_e(T)$ and phase packets & valuation-shell residues, dyadic and odd-adic primary lifts & global small-height synchronization\\
46 Obstruction complex & $\mathsf M_{a,b,c}$, $L$, $L_0$ & exact torsion $abc/R$, primitive obstruction class & metric size of a section\\
47 Moving search & $H(k)$ and interval certificates & exact optimum in every obstruction coset & no gain below the real first-order floor\\
48 Prime-shell signal & $\Psi_n$ & support energy and persistence energy & nonlinear additive arithmetic\\
49 Frey packet & $(E,f_E,N_E,\Delta_E)$ & conductor, discriminant depth, automorphic eigenchannel & marked integral representative within isogeny or twist class\\
50 Divisor packets & $B_d(n)$ & exact-order persistence spectrum & a global inequality controlling all output packets\\
51 Hecke filter & $A_\varepsilon$, $e_f$, $C_f$ & local signs and integral eigenline separation & output contact depth by itself\\
52 Filtered contact & $\mathbf J_c$ & cyclic prime-power depth with carries & geometry-induced global transfer\\
53 Witt packet & $W_{e-1}(\F_p)$ & typed Frobenius and Verschiebung carry structure & conductor-scale global synthesis\\
54 Kummer packet & $\kappa_c$ or $[\beta]$ & almost all contact order in one global Galois class & additive relation unless paired with another object\\
55 Fermat packet & $(P_M,\mu_M^2)$ & additive relation and exact character bank & geometric degree grows with $M$\\
56 Polylog path & $\Pi_{\le3}(z)$ & fixed complex packet and three-point closure & integral lattice denominator or covolume\\
57 Coherence object & $(\chi_{\mathrm{coh}},\Lambda_{\mathrm{coh}})$ & exact residual simultaneous logarithmic conductor & radical-scale arithmetic bound\\
58 \bottomrule
59 \end{longtable}
60 \endgroup
61 
62 \section{Exact theorem dependency chain}
63 The core exact implication chain is:
64 
65 \begin{enumerate}[leftmargin=1.5em]
66 \item Pairwise coprimality of a primitive triple separates prime sectors.
67 \item The prime-packet differential produces one additive row.
68 \item The arithmetic Wronskian produces one transverse row.
69 \item Local valuation divisibility gives $abc/R\mid W$.
70 \item Smith reduction gives the exact obstruction torsion and the lattice extension.
71 \item Sector compression and CRT reduce exact search to one moving integer parameter.
72 \item The continuous relaxation gives the exact stable height and finite stopping.
73 \item The no-escape identity closes the first-order search as a proof route.
74 \item Filtered contact restores the carry data lost by split shell packets.
75 \item Kummer theory globalizes the cyclic contact packet with logarithmic loss.
76 \item The additive relation yields a Fermat point and exact character bank.
77 \item Polylogarithmic distribution compresses the complex branch bank but leaves integral gain.
78 \item Simultaneous local logarithms produce the coherence character and exact final factorization.
79 \end{enumerate}
80 
81 Every unproved global theorem occurs after this chain.
82 
83 \section{Summary of established and conditional statements}
84 \begingroup\footnotesize\setlength{\tabcolsep}{3pt}
85 \begin{longtable}{L{0.31\textwidth}L{0.15\textwidth}L{0.43\textwidth}}
86 \toprule
87 Claim & Status & Reason\\
88 \midrule
89 \endfirsthead
90 \toprule
91 Claim & Status & Reason\\
92 \midrule
93 \endhead
94 $abc/R$ divides every additive Wronskian & Exact & local valuation proof\\
95 Polyphase block has one nontrivial invariant factor & Exact & circulant determinant and Smith form\\
96 Every shell tangent lifts formally & Exact & primitive gradient and integral implicit recursion\\
97 Exact first-order optimum can be found finitely & Exact & moving intervals plus continuous stopping\\
98 No independent $abc$ gain from first-order search & Structural limitation & no-escape identity\\
99 No reduction of the persistence ratio by unitary shell transforms & Structural limitation & Parseval and exact synthesis\\
100 Filtered contact source is cyclic & Exact & ideal quotient $R_c\Z/c\Z$\\
101 No near-injective map from split persistence to one cyclic target & Structural limitation & $p$-rank obstruction\\
102 Filtered cyclic source can transfer by common depth & Exact & gcd transducer\\
103 Contact packet globalizes through Kummer theory & Exact up to logarithmic kernel & valuation order formula\\
104 No free intrinsic height saving from the full Fermat cover & Structural limitation & full-plane line-bundle cost\\
105 Fixed complex polylog packet gives integral saving & Open and constrained & integral uncertainty lower bound\\
106 No coherence-index bound from relation length alone & Structural limitation & arbitrary-index annihilator theorem\\
107 Coherence conductor is radical-scale & Open & final arithmetic theorem\\
108 Sparse motivic quotient is radical-scale & Open & no construction with all required bounds\\
109 General prime-contact functor glues relation diagrams & Open & local pieces defined, global derived gluing unproved\\
110 \bottomrule
111 \end{longtable}
112 \endgroup
113 
114 \section{Normalization rules}
115 Several normalization choices are mathematically essential.
116 
117 \begin{enumerate}[leftmargin=1.5em]
118 \item Remove the common gcd of the additive relation before constructing prime sectors.
119 \item Preserve the dyadic sign layer separately from the principal-unit logarithm.
120 \item Remove common Kummer powers before measuring local logarithmic degeneration.
121 \item Divide local rows by their common coefficient depth before forming coherence.
122 \item Carry exact synthesis norms through every polyphase refinement.
123 \item Distinguish the marked Frey representative from its rational newform or minimal twist.
124 \item Distinguish a finite torsion packet from the real regulator of an extension containing it.
125 \item Use arbitrary-precision integers before every multiplication or determinant.
126 \end{enumerate}
127 
128 Failure to respect these rules produces double counting, false rank obstructions, lost integral depth, or numerical overflow.
129 
130 \section{Four recurring proof patterns}
131 The proofs repeatedly use four patterns.
132 
133 \subsection{Local-to-global divisibility}
134 Prove a prime-power divisor at every active place. Pairwise coprimality then lets the local factors multiply.
135 
136 \subsection{Compress-then-lift}
137 Use Smith form to reduce a large prime block to a few aggregate channels. Retain an exact local lift gauge for reconstructing integral coordinates.
138 
139 \subsection{Analysis-synthesis normalization}
140 Whenever an analysis map lowers coordinate size, include the norm or determinant of exact synthesis. This prevents coordinate compression from being mistaken for intrinsic height reduction.
141 
142 \subsection{Finite packet plus archimedean metric}
143 A finite torsion packet alone cannot carry a real regulator. The correct global object is often an extension, determinant line, or metrized lattice combining finite and infinite realizations.
144 
145 \section{Two principal conjectural bounds}
146 The theory has two direct arithmetic endpoints.
147 
148 \begin{conjecture}[Combined logarithmic--coherence bound]
149 For every $\eps>0$,
150 \[
151  WC\le C_\eps R_aR_bR^\eps.
152 \]
153 \end{conjecture}
154 
155 \begin{conjecture}[Integral adelic regulator]
156 There exists a canonical global object carrying the balanced contact packet whose finite kernel and integral archimedean synthesis cost are bounded by $R^\eps$ and $R_aR_bR^\eps$, respectively.
157 \end{conjecture}
158 
159 Either the combined bound or the stated integral-regulator theorem, together with the exact factorization and logarithmic Kummer bound, yields strong $abc$.
160 
161 \section{Logical economy}
162 All layers use one normalization and one chain of maps. Negative theorems are retained only when they impose a necessary condition on the final object. Computational evidence is separated from uniform arithmetic statements, and every conditional implication identifies its missing estimate explicitly.

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