Prime-Contact Coherence Calculus

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1 \chapter{Primewise Derivation of the Final Coherence Factorization}
2 
3 \section{Sequential normalization}
4 Let
5 \[
6  p^e\parallel c.
7 \]
8 The local output contact packet has order
9 \[
10  p^{e-1}.
11 \]
12 The final factorization separates this depth into distinct mechanisms. The separation must be sequential because the same prime-power depth can otherwise be counted more than once.
13 
14 Start with the signed input exponent vector
15 \[
16  \nu=(\nu_q)_{q\mid ab},
17  \qquad
18  \nu_q=v_q(b)-v_q(a).
19 \]
20 Let $K_p$ be the $p$-primary part of the common Kummer factor. Divide the exponent vector and the local modulus by this common power. The normalized vector is primitive relative to the remaining modulus.
21 
22 For odd $p$, form the logarithmic row
23 \[
24  \ell_{p,e}(q)
25  =
26  \frac1p\log_p\langle q\rangle_p
27  \pmod{p^{e-1}}.
28 \]
29 Let
30 \[
31  w_p
32  =
33  \min_{q\mid ab}
34  v_p(\ell_{p,e}(q))
35 \]
36 after Kummer normalization. Divide the row and the modulus by $p^{w_p}$. The remaining row is primitive modulo the residual power $p^{\chi_p}$.
37 
38 The local residual character is
39 \[
40  \chi_p(x)
41  =
42  \exp\left(
43  \frac{2\pi i}{p^{\chi_p}}g_p\cdot x
44  \right).
45 \]
46 Its kernel contains the normalized exponent vector. The local factors are therefore
47 \[
48  K_p,
49  \qquad
50  W_p=p^{w_p},
51  \qquad
52  C_p=p^{\chi_p}.
53 \]
54 At two, one sign layer $\epsilon_{2,p}$ is removed before the weighted logarithmic row is formed.
55 
56 Multiplying over output primes gives
57 \[
58  \boxed{
59  M=\epsilon_2KWC.
60  }
61 \]
62 The equality follows because the local exponents partition the original contact depth:
63 \[
64  e-1
65  =
66  v_p(\epsilon_2)
67  +
68  v_p(K)
69  +
70  v_p(W)
71  +
72  v_p(C).
73 \]
74 
75 \section{Why the Kummer layer comes first}
76 Suppose every input exponent is divisible by $p^k$. Then the balanced rational packet is already a $p^k$th power in the valuation lattice. The corresponding output contact depth should not also be attributed to a local logarithm defect.
77 
78 Dividing the exponent vector first removes this preload. The logarithmic row is then evaluated on a primitive packet. Its common coefficient depth measures a genuine degeneration of the logarithm operator rather than a common power already present in the input.
79 
80 This is the same normalization principle used in regulator theory: separate the algebraic divisibility of the vector before measuring the determinant of the linear form.
81 
82 \section{Odd-prime coefficient depth}
83 Let $q\neq p$ be prime. Write
84 \[
85  q=\omega(q)\langle q\rangle_p,
86 \]
87 where $\omega(q)$ is the Teichm\"uller factor. Since
88 \[
89  \langle q\rangle_p^{p-1}=q^{p-1},
90 \]
91 we have
92 \[
93  \log_p\langle q\rangle_p
94  =
95  \frac1{p-1}\log_p(q^{p-1}).
96 \]
97 If
98 \[
99  s=v_p(q^{p-1}-1)>0,
100 \]
101 then
102 \[
103  v_p(\log_p(q^{p-1}))=s.
104 \]
105 Therefore
106 \[
107  v_p\left(\frac1p\log_p\langle q\rangle_p\right)
108  =s-1.
109 \]
110 Truncating at the available contact depth gives
111 \[
112  v_p(\ell_{p,e}(q))
113  =
114  \min\{e-1,s-1\}.
115 \]
116 
117 \section{Dyadic coefficient depth}
118 For odd $q$, separate the sign by writing
119 \[
120  q=(-1)^{(q-1)/2}\langle q\rangle_2,
121  \qquad
122  \langle q\rangle_2\in1+4\Z_2.
123 \]
124 Then
125 \[
126  \langle q\rangle_2^2=q^2.
127 \]
128 The weighted coefficient is
129 \[
130  \ell_{2,e}(q)
131  =
132  \frac14\log_2\langle q\rangle_2.
133 \]
134 If
135 \[
136  s=v_2(q^2-1),
137 \]
138 then
139 \[
140  v_2(\ell_{2,e}(q))=s-3,
141 \]
142 truncated at $e-2$. The missing first layer is the separate sign bit. This is the final form of the weighted dyadic branch.
143 
144 \section{CRT synthesis of the global row}
145 The local residual moduli $C_p$ are pairwise coprime. Let
146 \[
147  C=\prod_{p\mid c}C_p.
148 \]
149 For each input coordinate $q$, the Chinese remainder theorem gives a unique residue
150 \[
151  g_q\pmod C
152 \]
153 whose reduction modulo $C_p$ is the local coefficient $g_{p,q}$.
154 
155 The resulting row
156 \[
157  g=(g_q)_{q\mid ab}
158 \]
159 is primitive modulo $C$ because every local row is primitive. Its character
160 \[
161  \chi_{\mathrm{coh}}(x)
162  =
163  e^{2\pi i g\cdot x/C}
164 \]
165 has exact order $C$. The coherence lattice is its kernel.
166 
167 \section{A determinant viewpoint}
168 When several independent exponent packets are available, replace the one-row character by an exterior-power regulator. Let $L$ be a normalized cross-prime logarithm matrix. A nonzero $r\times r$ minor
169 \[
170  \det L_{I,J}
171 \]
172 measures the failure of $r$ input packets to cancel simultaneously across $r$ output channels.
173 
174 The one-row coherence quotient is the rank-one case. A higher-rank determinant theorem may be stronger because it can use several relations or several independent row constraints at once.
175 
176 \chapter{Exact Smith and CRT Calculations}
177 
178 \section{One-row Smith normal form}
179 For an integer row
180 \[
181  a=(a_1,\ldots,a_s),
182 \]
183 let
184 \[
185  d=\gcd(a_1,\ldots,a_s).
186 \]
187 There exists a unimodular matrix $U\in\mathrm{GL}_s(\Z)$ such that
188 \[
189  aU=(d,0,\ldots,0).
190 \]
191 Therefore the image of the row is $d\Z$, the kernel has rank $s-1$, and the quotient by the kernel is infinite cyclic scaled by $d$.
192 
193 This elementary fact drives sector compression, local lift gauges, and the primitive-gradient formal-smoothness theorem.
194 
195 \section{Two-row Smith data}
196 For a two-row matrix $M$, the first Smith factor is the gcd of all entries. The product of the two factors is the gcd of all nonzero $2\times2$ minors. This allows the obstruction torsion to be computed without constructing the full unimodular reduction.
197 
198 For the exponent-free phase obstruction matrix, every nonzero cross-sector minor is
199 \[
200  \pm\frac{abc}{pq}
201  =
202  \pm\frac{abc}{R}\frac{R}{pq}.
203 \]
204 All such minors are divisible by $abc/R$. For each prime $\ell\mid R$, choose one column with $p=\ell$ and one column from a different nonempty sector. The reduced minor $R/(pq)$ is then not divisible by $\ell$. Hence the gcd of the reduced minors is one and the second Smith factor is exactly $abc/R$.
205 
206 \section{Local CRT lift in closed form}
207 Let
208 \[
209  r=\prod_{i=1}^{m}p_i,
210  \qquad
211  w_i=r/p_i.
212 \]
213 The equation
214 \[
215  \sum_iw_ix_i=t
216 \]
217 has a solution because
218 \[
219  \gcd(w_1,\ldots,w_m)=1.
220 \]
221 Modulo $p_i$,
222 \[
223  x_i\equiv t w_i^{-1}\pmod{p_i}.
224 \]
225 Let $\xi_i$ be the centered residue. Then
226 \[
227  \sum_iw_i\xi_i-t
228 \]
229 is divisible by $r$. Set
230 \[
231  q=\frac{t-\sum_iw_i\xi_i}{r}.
232 \]
233 Writing $x_i=\xi_i+p_i z_i$ gives
234 \[
235  \sum_i z_i=q.
236 \]
237 
238 For a box $|x_i|\le H$, the permitted $z_i$ form intervals. The sum of independent integer intervals is every integer between the sum of lower endpoints and the sum of upper endpoints. This proves the exact feasibility test.
239 
240 \section{Binary search certificate}
241 The lower and upper endpoint sums are monotone in $H$. Therefore the least feasible height is found by binary search. The result includes both:
242 
243 \begin{enumerate}[leftmargin=1.5em]
244 \item a witness at height $H$
245 \item an interval-separation certificate at height $H-1$
246 \end{enumerate}
247 
248 This is stronger than an optimization routine returning a candidate. It certifies global minimality for the local lift.
249 
250 \section{Polyphase circulant determinant}
251 For the valuation word
252 \[
253  V_e(T)=1+T+\cdots+T^{e-1}
254 \]
255 in $A_N=\Z[T]/(T^N-1)$, multiplication is circulant. Its determinant is
256 \[
257  \prod_{\zeta^N=1}V_e(\zeta).
258 \]
259 Let $I_N$ be the augmentation ideal of $A_N$. When $\gcd(e,N)=1$, the nontrivial roots are permuted by $\zeta\mapsto\zeta^e$, so multiplication by $V_e$ has determinant one on the integral lattice $I_N$ and is therefore an automorphism of it. On $A_N/I_N\cong\Z$, the map is multiplication by $e$. The exact sequence for $I_N\subset A_N$ then gives cokernel $\Z/e\Z$ and Smith form $\operatorname{diag}(1,\ldots,1,e)$.
260 
261 \section{Synthesis-normalized refinement}
262 Suppose a coarse integer $t$ is split into $e$ shell coordinates. The minimum child sup norm is roughly $|t|/e$. The shell-sum synthesis map has norm $e$. The product of analysis norm and synthesis norm remains at least one. Any claimed gain must therefore be stated in a synthesis-normalized norm.
263 
264 \section{Exact ratio semantics}
265 Three ratios recur in the computation:
266 \[
267  \frac{\Phi}{\overline H},
268  \qquad
269  \frac{\Phi}{k_*\overline H},
270  \qquad
271  \frac{H(1)}{\Phi}.
272 \]
273 They are all at least one by definition. A value greater than one is a discrete overhead, not an error. Exact rational strings should accompany decimal displays when the integers are large.
274 
275 \chapter{Finite Data and Interpretation Rules}
276 
277 \section{What the finite data record}
278 A useful data row contains at least:
279 
280 \begin{longtable}{p{0.24\textwidth}p{0.65\textwidth}}
281 \toprule
282 Field & Meaning\\
283 \midrule
284 $a,b,c$ & primitive additive triple\\
285 prime factorizations & exact valuation data for all three terms\\
286 $R$ & global radical\\
287 $q$ & standard quality\\
288 $M=c/R_c$ & output contact modulus\\
289 $\Theta,\Phi$ & coarse and phase packet heights when computed\\
290 $H(1),k_*,H(k_*)$ & exact moving-search data\\
291 $\epsilon_2,K,W,C$ & sequential final factorization\\
292 $\nu_0$ & normalized input exponent vector\\
293 local rows & normalized odd and dyadic logarithmic filters\\
294 coherence row & CRT-combined primitive character\\
295 certificates & witnesses, empty-box bounds, and exact residual checks\\
296 \bottomrule
297 \end{longtable}
298 
299 \section{What a high-quality row does not prove}
300 A large $C$, a short relation vector, or a striking phase resonance in one row does not imply a uniform law. The row can suggest a mechanism. A theorem requires a family definition and a proof that the mechanism persists or is rare.
301 
302 \section{Recommended finite experiments}
303 The most informative next experiments are not simply larger bounds on $c$. They should stratify by mechanism.
304 
305 \begin{enumerate}[leftmargin=1.5em]
306 \item Fix the number of input primes and vary output depth.
307 \item Fix the output prime and search for large local coefficient defects.
308 \item Separate $W>1$ rows from $W=1$ coherence rows.
309 \item Measure normalized logarithmic minors rather than only one coherence quotient.
310 \item Search families generated by recurrences or perfect powers.
311 \item Compare exact lattice invariants before and after gluing two relations.
312 \end{enumerate}
313 
314 \section{Diagnostic interpretation}
315 Every computational test should state what conclusion it does and does not support.
316 
317 \begin{longtable}{p{0.28\textwidth}p{0.61\textwidth}}
318 \toprule
319 Failure type & Diagnostic\\
320 \midrule
321 Kummer preload & common gcd of modulus and valuation vector\\
322 entrywise logarithmic defect & large generalized Wieferich coefficient depth\\
323 collective coherence & large residual quotient with ordinary coefficients\\
324 rank collapse & small logarithmic minors\\
325 archimedean cost & large period, degree, or synthesis norm\\
326 integral cost & large denominator, covolume, or congruence exponent\\
327 out-of-scope object & contact depth not represented in the chosen global carrier\\
328 \bottomrule
329 \end{longtable}
330 
331 \section{From finite observation to theorem}
332 A finite observation should pass through four mathematical levels.
333 
334 \begin{enumerate}[leftmargin=1.5em]
335 \item Numerical observation on a declared finite range.
336 \item Exact symbolic identity on each tested row.
337 \item Structural conjecture stated independently of the data.
338 \item Proof or a precise counterexample family.
339 \end{enumerate}
340 
341 This protocol prevents a finite pattern from being described as an established theorem.
342 
343 \chapter{Theorem Summary and Structural Limitations}
344 
345 \section{Exact theorem summary}
346 The exact core can be compressed to the following list.
347 
348 \begin{enumerate}[leftmargin=1.5em]
349 \item Prime-packet Leibniz rule.
350 \item One-row additive lattice.
351 \item Arithmetic Wronskian and radical divisibility.
352 \item Three-sector Smith compression.
353 \item Polyphase Smith form with one aggregate invariant.
354 \item Joint obstruction torsion $abc/R$.
355 \item Exact lattice extension $0\to L_0\to L\to\Z\to0$.
356 \item Exact real stable height.
357 \item Formal smoothness and all-order integral jet extension.
358 \item Exact moving-interval search and finite stopping.
359 \item Prime-shell energy identities.
360 \item Divisor-polyphase perfect reconstruction.
361 \item Atkin--Lehner scaled Walsh bank.
362 \item Integral projector denominator theorem.
363 \item Filtered contact quotient and carry correction.
364 \item Exact cyclic gcd transfer.
365 \item Witt realization of contact depth.
366 \item Exact Kummer kernel and logarithmic bound.
367 \item Fermat height compression and character bank.
368 \item Full-plane Fermat line-bundle cost.
369 \item Polylogarithmic distribution and three-point closure.
370 \item Exact finite-place logarithmic notch.
371 \item Exact Kummer--Wieferich--coherence factorization.
372 \item Coherence lattice index and cyclic quotient.
373 \item Arbitrary-index limitation theorem.
374 \item Direct positive divisibility lower bound.
375 \item Mean-preserving Hecke cascade.
376 \end{enumerate}
377 
378 \section{Conditional implications}
379 The strongest conditional routes are:
380 
381 \begin{enumerate}[leftmargin=1.5em]
382 \item small transverse packet implies $abc$
383 \item filtered cusp-to-Hecke transfer with two size bounds implies $abc$
384 \item sparse Kummer--Jacobi quotient with two size bounds implies $abc$
385 \item coherence conductor bound implies $abc$
386 \item suitable adelic logarithmic minor bound implies $abc$
387 \end{enumerate}
388 
389 The final two are the most direct endpoints.
390 
391 \section{Structural limitations}
392 The following routes are closed without a new ingredient:
393 
394 \begin{enumerate}[leftmargin=1.5em]
395 \item more exact search of the same first-order invariant
396 \item more local lift existence
397 \item unitary shell transforms
398 \item raw coordinate splitting
399 \item direct split persistence into one cyclic quotient
400 \item the full Fermat character bank
401 \item small complex periods without integral control
402 \item abstract lattice reduction without arithmetic row structure
403 \item direct positive contact relations
404 \item linear Hecke averaging
405 \end{enumerate}
406 
407 \section{Arithmetic endpoint}
408 The arithmetic endpoint is the coherence character
409 \[
410  \chi_{\mathrm{coh}}:
411  \Z^{S_{ab}}\to\Z/C\Z
412 \]
413 whose row is built from simultaneous logarithms of the actual input primes and whose kernel contains the normalized exponent vector forced by $a+b=c$.
414 
415 The missing theorem is not generic. It must exploit the shared arithmetic origin of all local rows and the exact identity of the output primes as divisors of the sum.

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