Realization Calculus

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1 % Worked exposition inherited from the earlier edition; current navigation and scope are synchronized.
2 
3 \vpage{1}{Three kinds of resolution}{CHAPTER II / A WORKED VISUAL GUIDE}{Chapter II, mathematical text: Definition 3.4, Theorem 5.10, Theorem 7.7}
4 Chapter II asks three different questions that can all sound like ``increase the resolution.'' This guide separates them, then follows small objects far enough that the distinctions can be computed rather than merely named.
5 
6 \vfig[.96]{three_resolutions}
7 \platecaption{Three different operations. On the left, all coefficients survive a regrouping. In the middle, an integer weight controls which operator words can matter modulo a chosen power. On the right, a rational projector needs a denominator cleared before integral precision can be recovered.}
8 
9 \head{Address resolution}
10 A coefficient such as $x_6$ can sit in the even packet, then the residue-$2$ packet modulo $4$, then the residue-$6$ packet modulo $8$. Its value does not become more accurate. Its \emph{address becomes more specific}, while the full object remains exactly reconstructible.
11 
12 \head{Arithmetic precision}
13 An operator word with total filtration weight at least $L$ vanishes in the specified depth-$L$ quotient. Increasing $L$ can make additional words relevant. This is a change in the precision of the ring in which the answer is required, not a lossless rearrangement of one fixed list.
14 
15 \head{Separation precision}
16 For the integral involution used later, $E_+=(I+A)/2$ is an exact rational projector but not an integral one at $2$. One extra input digit suffices to recover the projected difference to the requested precision. The packet is present over the fraction field even when its integral separation is delayed.
17 
18 \takeaway{Keep three quantities distinct: the packet address, the precision cutoff, and the denominator weight. Finer addresses do not automatically improve arithmetic precision or remove a spectral denominator.}
19 
20 \vpage{2}{An atlas of the refinement layer}{ORIENTATION / SYMBOLS WITH JOBS}{Chapter II, mathematical text: Definition 3.2, Definition 3.4, Definition 7.6}
21 The same letter can have different roles in different realizations. The table fixes the roles used here. Every small matrix in the guide is explicitly supplied. No diagonalization or classification theorem is inferred from a picture.
22 
23 {\small
24 \begin{tabularx}{\linewidth}{@{}p{.15\linewidth}Yp{.28\linewidth}@{}}\toprule
25 \textbf{Object} & \textbf{Role} & \textbf{Worked instance}\\\midrule
26 $P_{N,j}$ & Retain coefficients with address $j$ modulo $N$. & Even/odd masks, then mod-$4$ and mod-$8$ masks.\\
27 $\mathcal A_N,\mathcal S_N$ & Split into range-restricted packet vectors, then sum them. & A permutation/interleaving of eight coordinates.\\
28 $J,C$ & Exact refinement and regrouping. In the defect section, $C$ may instead be any supplied coarsening map. & A permutation in Plate II.5 and a shear in Plate II.22.\\
29 $T_{ba}$ & Input packet $a$, output packet $b$: $P_bTP_a$. & Blocks of a finite cyclic filter.\\
30 $D,H,A$ & $T=D+H=D(I+A)$, with $A=D^{-1}H$. & Explicit integral $2\times2$ matrices.\\
31 $w(e),W(\gamma)$ & Positive certified edge gain and its additive word weight. & $w(a)=1$, $w(b)=2$.\\
32 $L$ & Precision depth in a quotient such as $\ZZ/2^L\ZZ$. & $L=5$ gives arithmetic modulo $32$.\\
33 $A(t),Z(t)$ & Polynomial graph adjacency and unsigned walk-count series. & Two parallel edges remain distinct.\\
34 $E,\Lambda,s$ & A rational projector, a specified lattice, and a clearing exponent. & $E_+$ has local weight $1$ at $2$.\\
35 $(s_i)$ & Directional weights of a supplied Smith diagonal. & $(1,2)$ modulo $8$ or $(0,1,3)$ across depths.\\
36 $\Delta$ & The failure of an independently chosen operator to intertwine: $CT_f-T_cC$. & One nonzero entry $1/16$.\\\bottomrule
37 \end{tabularx}}
38 
39 \head{Suggested reading routes}
40 For exact coordinates, read \pagerefV{3} through \pagerefV{8}, then the finite filter on \pagerefV{20}. For weighted inversion and counting, read \pagerefV{9} through \pagerefV{13}. For arithmetic separation and congruence fibers, read \pagerefV{14} through \pagerefV{19}. The final pages distinguish nonlinear chart geometry, error propagation, and the formal contract.
41 
42 \takeaway{Operator words carry signed matrix values. Graph walks carry unsigned counts. Smith normal form concerns linear congruences. The Smith impedance chart on Plate II.21 is a different, nonlinear construction.}
43 
44 \vpage{3}{Follow one coefficient through the tower}{MODEL A / LOSSLESS ADDRESS REFINEMENT}{Chapter II, mathematical text: Theorem 3.3, Theorem 4.6}
45 Take a vector $x=(x_0,\ldots,x_{15})$. A packet is not a single summary number: it retains every coefficient at its selected addresses. For example,
46 \[
47 P_{2,0}x=(x_0,0,x_2,0,\ldots,x_{14},0).
48 \]
49 The rail below lists the addresses in each packet. Its display order groups descendants under their parents. The matrix example on Plate II.5 instead uses increasing residue order.
50 
51 \vfig[.96]{coefficient_tower}
52 \platecaption{Every row contains all sixteen addresses exactly once. A heavier boundary separates packet ranges. The numbers are coefficient addresses, not new coefficient values. There is no interpolation, averaging, or omitted high-frequency tail.}
53 
54 The address $6$ is even, is $2$ modulo $4$, and is $6$ modulo $8$. Its coefficient therefore lies in the successive ranges
55 \[
56 P_{8,6}X\subseteq P_{4,2}X\subseteq P_{2,0}X.
57 \]
58 The corresponding identities hold as operators:
59 \[
60 P_{2,0}=P_{4,0}+P_{4,2},\qquad
61 P_{4,2}=P_{8,2}+P_{8,6}.
62 \]
63 Applying either composite parent selector to $P_{8,6}x$ leaves it unchanged. Applying the odd selector kills it. This is absorption between a parent and a child, rather than an approximate resemblance between nearby scales.
64 
65 \head{Why the reconstruction is unique}
66 If a vector is a sum of components in disjoint packet ranges, apply the projector for one range. All other components vanish and the selected component remains. Thus every child component is recovered uniquely from the sum.
67 
68 \takeaway{Refinement is a sharper address book for the same coefficients. The total number of free coordinates does not change.}
69 
70 \vpage{4}{Regrouping is not averaging}{MODEL A / THE MEANING OF COARSENING}{Chapter II, mathematical text: Definition 3.4, Theorem 3.5}
71 The word \emph{coarsening} can suggest compression. In the exact coordinate theorem it means regrouping complete packet vectors. Replacing each packet by one scalar sum is a different operation and really does lose information.
72 
73 For $x=(1,2,3,4)$, the fine packets are the four coordinate-supported vectors. Regrouping gives $(1,0,3,0)$ and $(0,2,0,4)$. Every original coordinate remains available in one of these two vectors.
74 
75 \vfig[.98]{regrouping_not_averaging}
76 \platecaption{The left and middle panels store the same four independent coordinates. The right panel instead applies a $2\times4$ numerical incidence matrix to four scalar entries. That is not the range-restricted exact coordinate map.}
77 
78 Define
79 \[
80 X_N^{\rm pkt}=\bigoplus_j P_{N,j}X,
81 \qquad
82 \mathcal A_Nx=(P_{N,j}x)_j,
83 \qquad
84 \mathcal S_N(x_j)_j=\sum_jx_j.
85 \]
86 Completeness and mutual annihilation give both inverse laws. Consequently,
87 \[
88 J=\mathcal A_4\mathcal S_2,\qquad C=\mathcal A_2\mathcal S_4,
89 \qquad CJ=I,\quad JC=I.
90 \]
91 The larger list of fine packets has smaller individual packet spaces.
92 
93 \head{A kernel witnesses genuine information loss}
94 For the scalar incidence matrix $B$ drawn on the right,
95 \[
96 B(1,0,-1,0)^\mathsf T=0.
97 \]
98 The zero vector and $(1,0,-1,0)$ therefore have the same two scalar summaries. No inverse can recover the original four scalars from that output alone.
99 
100 \takeaway{Ask what each packet stores. A list of full packet vectors can be exactly invertible. A list of packet means or totals generally is not.}
101 
102 \vpage{5}{The operator changes coordinates too}{MODEL A / COHERENCE AND NATURALITY}{Chapter II, mathematical text: Theorem 3.5, Theorem 3.8}
103 Use a finite periodic carrier $X=\QQ^8$ for this example. Let $Se_j=e_{j+1\bmod8}$ and let
104 \[
105 T=I+2S+S^2.
106 \]
107 This is a concrete cyclic model, not an unannounced boundary convention for the finitely supported signal on Plate II.20. Write $R_N$ for the permutation that groups coordinates in increasing residue order modulo $N$.
108 
109 \vfig[.99]{operator_naturality}
110 \platecaption{Actual $8\times8$ matrices. Dots are zero entries. The other entries are exact integers. Heavy lines mark the packet blocks: size four at level two and size two at level four.}
111 
112 The reordered operators and change of coordinates are
113 \[
114 \widehat T_N=R_NTR_N^{-1},\qquad J_{4,2}=R_4R_2^{-1}.
115 \]
116 Multiplication cancels the intermediate coordinate maps:
117 \[
118 J_{4,2}\widehat T_2=\widehat T_4J_{4,2},\qquad
119 J_{8,4}J_{4,2}=J_{8,2}.
120 \]
121 Thus it makes no difference whether the operator is applied before or after a consistent coordinate change. One must transport the operator, not merely reuse the same numerical matrix under a new ordering.
122 
123 \head{Refined blocks still reconstruct the same map}
124 A coarse block expands as
125 \[
126 P_bTP_a=\sum_{\beta\succ b,\,\alpha\succ a}P_\beta TP_\alpha.
127 \]
128 The terms are maps embedded in the original source and target spaces. Adding their norms, or adding their matrix entries after forgetting the embeddings, is not this operator identity.
129 
130 \takeaway{The fine matrix contains a different block organization of the same operator. An independently guessed coarse matrix is a separate approximation problem, measured later by a defect.}
131 
132 \vpage{6}{Selectors survive without division}{MODEL A / RESIDUE MASKS IN CHARACTERISTIC TWO}{Chapter II, mathematical text: Proposition 4.2}
133 Work first in $\mathbb F_2$, where $1+1=0$. The eight coefficients below are just a finite display of a coefficient sequence. The even and odd projectors are diagonal masks with entries zero and one.
134 
135 \vfig[.95]{division_free_masks}
136 \platecaption{Each nonzero input coefficient is retained by exactly one mask. Reconstruction is exact over $\mathbb F_2$. No average and no root of unity is used.}
137 
138 For a general coefficient ring and a positive integer $N$, the definition is
139 \[
140 (P_{N,j}x)_n=\begin{cases}x_n,&n\equiv j\pmod N,\\0,&\text{otherwise}.\end{cases}
141 \]
142 An address belongs to one residue class, so applying two masks either keeps that same coefficient or kills it. This proves
143 \[
144 P_{N,i}P_{N,j}=\delta_{ij}P_{N,j},\qquad \sum_jP_{N,j}=I.
145 \]
146 The argument is coordinatewise and does not require the coefficients to form a field.
147 
148 \head{Why the usual even/odd average is a different formula}
149 Over a field of characteristic not two, parity extraction can be represented using $F(z)$ and $F(-z)$ and division by $2$. In characteristic two, $-z=z$ and $2=0$, so that formula cannot be used. The coefficient mask, however, still distinguishes even from odd \emph{exponents}.
150 
151 This distinction also explains the later involution example. Failure to reduce a rational spectral projector containing $1/2$ does not make every integral coefficient-address selector disappear.
152 
153 \takeaway{The basic object is the address selector. A Fourier average is one possible realization of it and has additional algebraic hypotheses.}
154 
155 \vpage{7}{A Fourier certificate you can inspect}{MODEL A / AN EXACT NINE-ENTRY TABLE}{Chapter II, mathematical text: Theorem 4.3}
156 Take $N=3$ and work in $\mathbb F_7$. Set
157 \[
158 \omega=2,\qquad \omega^{-1}=4,\qquad \eta=3^{-1}=5.
159 \]
160 Here $2^3=1$ and neither $2$ nor $2^2$ is one. The unnormalized geometric sums are the entries in the first matrix.
161 
162 \vfig[.97]{finite_fourier_table}
163 \platecaption{All entries are exact residues modulo seven. Multiplying the diagonal value $3$ by $5$ produces $1$. The lower calculation is a genuine failed Fourier certificate in a different coefficient ring.}
164 
165 On the monomial $z^n$, define $UF(z)=F(2z)$. The proposed projector acts by
166 \[
167 5\sum_{k=0}^2 4^{jk}2^{nk}.
168 \]
169 The identity $2^3=1$ makes this depend only on $n\bmod3$. The nine-entry table therefore establishes the formula for every exponent once periodicity is used, rather than requiring a separate test at every coefficient.
170 
171 \head{Order alone is not sufficient over every ring}
172 In $\mathbb F_3\times\mathbb F_3$, the element $(-1,1)$ has order two, but one component is a trivial character. Its nontrivial Fourier sum does not vanish in that component. Thus an element of order $N$ in a ring with zero divisors is not, by itself, the full orthogonality certificate.
173 
174 \takeaway{The finite table is a checkable sufficient input. Over a field, a primitive root supplies it. Over a more general ring, inspect the actual table instead of inferring orthogonality from order alone.}
175 
176 \vpage{8}{One address, two independent residues}{MODEL A / THE DYADIC AND ODD RECTANGLE}{Chapter II, mathematical text: Corollary 4.9}
177 Because $4$ and $3$ are coprime, a residue modulo $12$ is specified by a column $a=n\bmod4$ and a row $b=n\bmod3$. Each cell contains the unique combined address.
178 
179 \vfig[.96]{crt_address_grid}
180 \platecaption{The twelve cells are a permutation of $0,\ldots,11$, not twelve extra degrees of freedom. The selected cell contains every integer congruent to $7$ modulo $12$.}
181 
182 An explicit reconstruction formula is
183 \[
184 n\equiv9a+4b\pmod{12},
185 \]
186 because $9$ is one modulo four and zero modulo three, while $4$ has the opposite pair of residues. For $(a,b)=(3,1)$ this gives $31\equiv7$.
187 
188 Applying the two masks in either order gives
189 \[
190 P_{12,7}=P_{4,3}P_{3,1}=P_{3,1}P_{4,3}.
191 \]
192 More generally, an address survives exactly when it satisfies both congruences. The coprime reconstruction identifies this intersection with one combined residue class.
193 
194 \head{The hypothesis can be seen failing}
195 Replace $4$ and $3$ by $4$ and $6$. The requirements $n\equiv0\pmod4$ and $n\equiv1\pmod6$ are incompatible: the first makes $n$ even and the second makes it odd. A full rectangular address product no longer exists. Compatible pairs must agree modulo the greatest common divisor.
196 
197 \takeaway{CRT separates address coordinates. It does not claim that dyadic and odd spectral projectors have the same denominator weights or the same lifting behavior.}
198 
199 \vpage{9}{Which words can survive precision five?}{MODEL B / ORDERED TRANSFERS IN A FILTERED RING}{Chapter II, mathematical text: Theorem 5.10, Corollary 5.11}
200 Let $E_{ij}$ send the $j$th basis vector to the $i$th. In integral $2\times2$ matrices set
201 \[
202 a=2E_{01},\qquad b=4E_{10},\qquad A=a+b.
203 \]
204 Their certified dyadic weights are $1$ and $2$. At precision $L=5$, every word of weight at least five vanishes modulo $32$.
205 
206 \vfig[.97]{weighted_word_cutoff}
207 \platecaption{All nonzero retained terms of the signed inverse sum, followed by the first discarded nonzero word. The empty word contributes $I$. Other words of weight below five can still be zero because $a^2=b^2=0$.}
208 
209 The words are \emph{written matrix products}: in $ab$, the map $b$ acts first. Therefore
210 \[
211 ab=8E_{00},\qquad ba=8E_{11},\qquad aba=16E_{01}.
212 \]
213 The two products $ab$ and $ba$ cannot be identified merely because they have equal weight. Both their order and their source/target information matter.
214 
215 The pruning rule supplies a sufficient reason for a term to vanish. It is not an assertion that every word below the threshold contributes nontrivially. Packet incompatibility and algebraic relations can remove additional words.
216 
217 \takeaway{Length controls the sign. Cumulative weight controls precision. The ordered matrix product controls the actual contribution. None of these three can replace the others.}
218 
219 \vpage{10}{An inverse computed in both orders}{MODEL B / A COMPLETE MODULO-32 CALCULATION}{Chapter II, mathematical text: Theorem 5.10, Corollary 5.11}
220 Keep the matrices from Plate II.9 and choose
221 \[
222 D=\begin{pmatrix}1&1\\0&1\end{pmatrix},\quad
223 D^{-1}=\begin{pmatrix}1&-1\\0&1\end{pmatrix},\quad
224 T=D(I+A)=\begin{pmatrix}5&3\\4&1\end{pmatrix}.
225 \]
226 The six retained contributions sum to
227 \[
228 S=I-a-b+ab+ba-aba
229 =\begin{pmatrix}9&-18\\-4&9\end{pmatrix}.
230 \]
231 The required inverse is $K=SD^{-1}$ in the quotient, not $D^{-1}S$.
232 
233 \vfig[.98]{modular_inverse}
234 \platecaption{The displayed $K$ uses canonical residues from $0$ to $31$. Both left and right products are the identity modulo $32$. These are exact modular equalities, not floating-point residuals.}
235 
236 One can verify the claim without trusting the word expansion:
237 \[
238 TK=\begin{pmatrix}129&64\\64&33\end{pmatrix}\equiv I,
239 \qquad
240 KT=\begin{pmatrix}65&32\\192&97\end{pmatrix}\equiv I\pmod{32}.
241 \]
242 The cancellation of the exterior factor $D$ explains the order. Since $T=D(I+A)$, multiply its inverse by $D^{-1}$ on the \emph{right} of the inverse of $I+A$.
243 
244 \head{What the finite theorem does not require}
245 There is no claim here that the same integer series converges in the ordinary real matrix norm. The quotient kills all sufficiently deep products, so a finite sum suffices. The complete normed-ring theorem is a separate result with its own strict contraction hypothesis.
246 
247 \takeaway{A correct noncommutative formula must work on both sides and preserve the order of its factors. Here the entire result can be checked by two integer matrix products.}
248 
249 \vpage{11}{The residual tells you when to stop}{MODEL B / A CERTIFIED CUTOFF, NOT A HEURISTIC}{Chapter II, mathematical text: Theorem 5.10, Corollary 5.11, Section B.2}
250 For $S_N=\sum_{k=0}^{N-1}(-A)^k$, telescoping gives
251 \[
252 (I+A)S_N=S_N(I+A)=I-(-A)^N.
253 \]
254 Thus the exact residual is known. To certify precision $L$, it is enough to show that every residual entry is divisible by $2^L$.
255 
256 \vfig[.94]{inverse_residual_depth}
257 \platecaption{The plotted values are the minimum dyadic valuations of the nonzero entries of $(-A)^N$. The dashed line is the requested cutoff $L=5$. This is an exact valuation profile, not a measured numerical convergence curve.}
258 
259 In this particular model, $A^2=8I$. Therefore
260 \[
261 v_2(A^{2k})=3k,\qquad v_2(A^{2k+1})=3k+1,
262 \]
263 where the matrix valuation means the minimum entry valuation. Three geometric terms leave a residual of valuation four and do not suffice modulo $32$. Four terms leave valuation six and do suffice.
264 
265 The generic positive-weight estimate is more conservative. If every edge weight is at least $m>0$, a word of length $N$ has weight at least $mN$. Hence the cutoff
266 \[
267 N=\left\lceil\frac Lm\right\rceil
268 \]
269 is sufficient. Here the certified minimum weight is $m=1$, so the generic choice is five. The extra identity $A^2=8I$ justifies stopping at four in this example.
270 
271 \takeaway{A structural identity may improve a cutoff. An observed small real norm or an arbitrarily larger claimed minimum edge weight does not supply that certificate.}
272 
273 \vpage{12}{Walks are not simple paths}{MODEL C / A GRAPH WITH PARALLEL EDGES}{Chapter II, mathematical text: Theorem 6.6}
274 Consider the directed graph below. The loop $\ell$ has weight one, the two distinct edges $a,b$ from $0$ to $1$ each have weight two, and the return edge $r$ has weight one.
275 
276 \vfig[.96]{weighted_multigraph}
277 \platecaption{Every word on the right is read chronologically from left to right as an edge traversal. This is intentionally different from the right-to-left action of a written operator product in Plate II.9.}
278 
279 A walk may revisit a vertex and reuse an edge. At total weight five there are six walks from $0$ to $1$. Two use three loops followed by either $a$ or $b$. Four travel to $1$, return to $0$, and travel to $1$ again, with two independent choices of parallel edge.
280 
281 Their multiplicities enter the polynomial adjacency matrix:
282 \[
283 A(t)=\begin{pmatrix}t&2t^2\\t&0\end{pmatrix}.
284 \]
285 The coefficient $2$ remembers the two distinct edges. Replacing it by $1$ would count a different graph.
286 
287 \head{The first-edge decomposition}
288 Let $a_n(s,t)$ count walks of total weight $n$ from $s$ to $t$. Removing the first edge gives
289 \[
290 a_n(s,t)=\mathbf1_{s=t,\,n=0}
291  +\sum_{\substack{e:\,\mathrm{source}(e)=s\\w(e)\le n}}
292  a_{n-w(e)}(\mathrm{target}(e),t).
293 \]
294 The empty walk is allowed precisely on the diagonal at weight zero. Positive weights ensure that every contributing walk has length at most $n$, so enumeration at each fixed weight is finite.
295 
296 \takeaway{The graph series counts unsigned walks, including repetitions and parallel-edge multiplicity. It does not count simple paths, and it does not know whether an associated operator product vanishes or cancels.}
297 
298 \vpage{13}{A finite graph produces its denominator}{MODEL C / AUTOMATIC RATIONALITY AND RECURRENCE}{Chapter II, mathematical text: Theorem 6.6, Corollary 6.7}
299 For the graph on Plate II.12, compute the determinant and adjugate rather than guessing a recurrence:
300 \[
301 I-A(t)=\begin{pmatrix}1-t&-2t^2\\-t&1\end{pmatrix},
302 \quad Q(t)=1-t-2t^3,
303 \quad \operatorname{adj}(I-A)_{0,1}=2t^2.
304 \]
305 Since $Q(0)=1$, its inverse exists as a formal power series with integer coefficients.
306 
307 \vfig[.99]{zeta_recurrence}
308 \platecaption{The coefficients are independently computed by the first-edge recurrence and checked against the rational series through weight $24$. Literal edge-walk enumeration supplies a third check through weight $10$.}
309 
310 From $QZ_{0,1}=2t^2$ one obtains
311 \[
312 a_0=a_1=0,\quad a_2=2,\qquad a_n=a_{n-1}+2a_{n-3}\quad(n\ge3).
313 \]
314 The numerator is responsible for the initial exception. Omitting it would incorrectly force the entire series to vanish.
315 
316 \head{Three boundaries of this calculation}
317 This is a \emph{formal} series identity. No analytic convergence at a chosen real or complex value of $t$ is needed. The canonical determinant need not be the minimal denominator after cancellation. Finally, positive weights prevent infinitely many walks at a fixed weight: a single weight-zero loop would already give the infinitely many distinct walks $1,\ell,\ell^2,\ldots$ at weight zero.
318 
319 \takeaway{Finite positive-weight graph data determine a rational series and an eventual recurrence. The proof does not begin by assuming a certificate $QZ=P$ that already contains that conclusion.}
320 
321 \vpage{14}{Separate before dividing}{MODEL D / AN EVALUATED B\'EZOUT CERTIFICATE}{Chapter II, mathematical text: Theorem 7.5, Definition 7.6}
322 Use the integral involution
323 \[
324 A=\begin{pmatrix}1&1\\0&-1\end{pmatrix},\qquad A^2=I.
325 \]
326 With $F=A-I$ and $G=A+I$, the identities
327 \[
328 -F+G=2I,\qquad FG=0
329 \]
330 are an explicit evaluated B\'ezout certificate. Here all the required commutations follow because these operators are polynomials in the same matrix.
331 
332 \vfig[.96]{bezout_normalization}
333 \platecaption{The denominator-free numerator is quasi-idempotent: $N^2=2N$. Division by the nonzero scalar $2$ gives an idempotent over the fraction field. It does not yet make the projector integral on the selected lattice.}
334 
335 The numerator $N=G$ acts by $2$ on $\ker(A-I)$ and by zero on $\ker(A+I)$. Thus
336 \[
337 E_+=\frac12N
338 =\begin{pmatrix}1&1/2\\0&0\end{pmatrix}
339 \]
340 is the required separator. The complementary projector is $E_-=I-E_+$.
341 
342 \head{A denominator bound need not be minimal}
343 The same projector can be presented with the inflated certificate $-2F+2G=4I$. Its numerator is $2G$ and its normalization is still $E_+$. This supplies a two-digit clearing bound at $2$, although one digit is enough. Thus the valuation of a supplied certificate gives an upper bound, not automatically the least possible weight.
344 
345 For the original sign factors the concrete resultant is $\operatorname{Res}(x-1,x+1)=2$ and the bound is sharp. That equality is special to this displayed calculation, not a universal minimality theorem for every resultant certificate.
346 
347 \takeaway{First check the evaluated identities and the numerator. Then normalize in the fraction field. Finally ask what denominator is needed on the particular lattice.}
348 
349 \vpage{15}{Where the dyadic digit is lost}{MODEL D / THE EIGENLATTICE HAS INDEX TWO}{Chapter II, mathematical text: Proposition 9.3}
350 The eigenspaces are spanned by $(1,0)$ and $(-1,2)$. Their integral eigenvectors do not form an integral basis of $\ZZ^2$. The change-of-basis determinant is two.
351 
352 \vfig[.94]{dyadic_eigenlattice}
353 \platecaption{A picture of the ordinary integer lattice, not a drawing of the topology of $\QQ_2$. Filled dots have even second coordinate. Unfilled dots lie in the other coset. The hollow point at $(1/2,0)$ records the image of an integral vector under $E_+$.}
354 
355 Write
356 \[
357 B=\begin{pmatrix}1&-1\\0&2\end{pmatrix},\qquad
358 AB=B\diag(1,-1),\qquad
359 E_+=B\diag(1,0)B^{-1}.
360 \]
361 The index-two eigenlattice explains the denominator in $B^{-1}$ and in the projector. Over the localized lattice $\Lambda_p=\ZZ_{(p)}^2$, the number $2$ is a unit for every odd prime $p$. Hence the local denominator weight is zero there and one at $p=2$.
362 
363 This is a \emph{local} assertion. Odd localization does not change the fact that the displayed eigenvectors span an index-two sublattice inside the ordinary integer lattice.
364 
365 \head{The special fiber does not retain the rational splitting}
366 Modulo two, $x-1=x+1$ and $A$ becomes a nontrivial unipotent matrix. The entries $1/2$ in $E_\pm$ cannot be reduced modulo two. By contrast, the coefficient selectors on Plate II.6 use only zero and one and remain available.
367 
368 \takeaway{The obstruction is in integral separation, not in the existence of the two rational eigenspaces. The lattice is part of the question.}
369 
370 \vpage{16}{One extra digit is necessary}{MODEL D / A SHARP EXAMPLE OF PRECISION DELAY}{Chapter II, mathematical text: Theorem 7.7, Proposition 9.3}
371 The projector acts by
372 \[
373 E_+(x,y)=(x+y/2,0).
374 \]
375 Its factor $1/2$ can consume one dyadic digit. The claim concerns differences: if the inputs agree modulo $2^{m+1}\Lambda$, their projected differences lie in $2^m\Lambda$.
376 
377 \vfig[.97]{precision_delay}
378 \platecaption{For $m=3$, inputs differing by $(0,8)$ agree modulo eight, but their projected outputs differ by $(4,0)$ and do not. Raising the input agreement to modulo sixteen makes the projected outputs agree modulo eight.}
379 
380 The example works at every positive depth:
381 \[
382 E_+(0,2^m)=(2^{m-1},0)\notin2^m\Lambda,
383 \qquad
384 E_+(0,2^{m+1})=(2^m,0)\in2^m\Lambda.
385 \]
386 Thus a zero-delay promise fails, while one extra input digit is sufficient. This witnesses sharpness for the specified projector and lattice.
387 
388 \head{The general calculation is just as short}
389 If $\pi^sE\Lambda\subseteq\Lambda$ and $x-y=\pi^{m+s}v$, then
390 \[
391 E(x-y)=\pi^m(\pi^sEv)\in\pi^m\Lambda.
392 \]
393 No rounding procedure is hiding in this identity. Nor does it authorize reducing $E$ modulo $\pi$ when its entries are nonintegral. Individual projected points may lie in a fractional lattice. The theorem controls their difference at the stated integral scale.
394 
395 \head{Accuracy and branching are different views}
396 The later Smith example records how many inputs share one output at finite depth. Here we ask how accurately an input must be known to determine a projected output difference. Both involve directional divisibility, but they are different questions.
397 
398 \takeaway{The phrase ``one lost digit'' has an exact meaning: both a sufficiency theorem and an explicit pair of inputs showing that a smaller delay fails.}
399 
400 \vpage{17}{See every Smith fiber modulo eight}{MODEL E / TWO DIRECTIONAL WEIGHTS}{Chapter II, mathematical text: Theorem 10.6}
401 Consider the actual map of finite sets
402 \[
403 (\ZZ/8\ZZ)^2\longrightarrow(\ZZ/8\ZZ)^2,
404 \qquad (x,y)\longmapsto(2x,4y).
405 \]
406 The first coordinate of the output must be even. The second must be zero or four. Those are precisely the attainable targets in the right panel.
407 
408 \vfig[.95]{smith_fibre_grid}
409 \platecaption{All sixty-four input points and all sixty-four possible target addresses are shown. The eight filled target points are the image. Every attainable target has exactly eight preimages. The other fifty-six targets have none.}
410 
411 For a specific attainable target $(2,4)$,
412 \[
413 2x\equiv2\pmod8\iff x\in\{1,5\},\qquad
414 4y\equiv4\pmod8\iff y\in\{1,3,5,7\}.
415 \]
416 The coordinate choices are independent, giving $2\cdot4=8$ solutions. The entire carrier is partitioned consistently:
417 \[
418 8\text{ attainable targets}\ \times\ 8\text{ inputs per target}=64.
419 \]
420 For an inadmissible target, such as $(1,4)$ or $(2,2)$, the fiber is empty. The same nonzero cardinality formula must not be applied to it.
421 
422 \head{The arithmetic behind the picture}
423 Multiplication by $p^s$ modulo $p^L$ has kernel size $p^{\min(s,L)}$. Its image consists exactly of residues whose canonical representatives are divisible by that same power. The displayed model has $p=2$, $L=3$ and weights $(1,2)$.
424 
425 \takeaway{A directional weight has two consequences: a divisibility condition on the target and a conditional multiplicity of inputs. Solvability comes before counting.}
426 
427 \vpage{18}{Transport the whole solution fiber}{MODEL E / EXPLICIT SMITH COORDINATES}{Chapter II, mathematical text: Theorem 10.6}
428 The preceding diagonal system is connected to a nondiagonal one by the unimodular change of variables
429 \[
430 M=\begin{pmatrix}2&2\\0&4\end{pmatrix},\quad
431 V=\begin{pmatrix}1&-1\\0&1\end{pmatrix},\quad
432 MV=\diag(2,4)=D.
433 \]
434 Since $\det V=1$, $x=Vy$ is bijective modulo eight. Thus $Mx=b$ is equivalent to $Dy=b$.
435 
436 \vfig[.95]{smith_coordinate_transport}
437 \platecaption{The eight solutions for the target $(2,4)$, before and after the coordinate shear. Coordinates are residues in $\{0,\ldots,7\}$. No interpolation on this modular grid is intended.}
438 
439 For example, the diagonal solution $y=(1,3)$ corresponds to
440 \[
441 x=Vy=(-2,3)\equiv(6,3)\pmod8,
442 \]
443 and $M(6,3)^\mathsf T=(18,12)^\mathsf T\equiv(2,4)^\mathsf T$. Applying the inverse shear recovers the original solution.
444 
445 \head{Two boundary cases worth keeping visible}
446 A unit diagonal coordinate has exactly one solution for every target. A literal zero coordinate has all eight possible inputs when its target is zero, and no input when the target is nonzero. A zero coordinate is not inverted or silently omitted.
447 
448 The displayed diagonalization is verified by multiplication. The theorem takes diagonal data as input. A Smith normal form for an arbitrary matrix or general DVR requires a separate structure theorem.
449 
450 \takeaway{An invertible change of coordinates transports existence, every solution, and the fiber count. The target must be transformed as prescribed when checking admissibility.}
451 
452 \vpage{19}{Precision exposes directional freedom}{MODEL E / A FIBER BUDGET AT EACH DEPTH}{Chapter II, mathematical text: Theorem 10.6}
453 Take $p=2$ and the supplied diagonal weights $(s_0,s_1,s_2)=(0,1,3)$. At depth $L$, an admissible target has
454 \[
455 2^{W(L)}\text{ solutions},\qquad
456 W(L)=\min(0,L)+\min(1,L)+\min(3,L).
457 \]
458 Each direction contributes only the amount visible at the current depth.
459 
460 \vfig[.93]{smith_depth_budget}
461 \platecaption{The stacked height records the exponent $W(L)$. The label above each bar records the actual number of solutions. The zero-weight direction contributes no branching. This is a finite cardinality calculation, not an information-theoretic assumption about random inputs.}
462 
463 The values are
464 \[
465 \begin{array}{c|rrrrrr}
466 L&1&2&3&4&5&6\\\hline
467 W(L)&2&3&4&4&4&4\\
468 \#\text{ solutions}&4&8&16&16&16&16.
469 \end{array}
470 \]
471 Once $L\ge3$, the nonzero weights have been fully exposed and the admissible fiber cardinality stabilizes at sixteen. The input space still grows as $2^{3L}$. The number of attainable targets grows accordingly.
472 
473 At depth six, $(1,2,8)$ is admissible and gives sixteen solutions. Replacing the last coordinate by four makes the target inadmissible and gives zero solutions, even though the weights and depth are unchanged.
474 
475 \head{What changes with a zero diagonal factor?}
476 A literal zero factor contributes $L$, not a fixed eventual weight, and requires a zero target coordinate. Its fiber therefore keeps growing with the modulus. That case must not be mistaken for a finite nonzero Smith weight.
477 
478 \takeaway{The scalar sum of weights gives the count, but the separate weights give the admissibility conditions. Keeping only a determinant valuation can hide which target coordinate fails.}
479 
480 \vpage{20}{Filter first or split first: same output}{MODEL F / A FINITE POLYPHASE REALIZATION}{Chapter II, mathematical text: Section B.1}
481 Use the finitely supported signal $x=(1,2,3,4,5,6)$ and the filter $h=(1,2,1)$. Outside the displayed support, the input is zero. Let $e_m=x_{2m}$ and $o_m=x_{2m+1}$.
482 
483 \vfig[.96]{finite_polyphase_filter}
484 \platecaption{Exact integer convolution, including both output tail coefficients. The two bottom rows are the even and odd output phases. Interleaving them reconstructs the full eight-term output.}
485 
486 If $w$ denotes one delay on a phase sequence, the filter becomes
487 \[
488 \begin{pmatrix}y_{\rm even}\\y_{\rm odd}\end{pmatrix}
489 =
490 \underbrace{\begin{pmatrix}1+w&2w\\2&1+w\end{pmatrix}}_{H_{\rm phase}(w)}
491 \begin{pmatrix}e\\o\end{pmatrix}.
492 \]
493 The cross-phase delay is not optional. A unit shift sends an odd sample to the next even position, while it sends an even sample to the odd position in the same phase block.
494 
495 Here $e=1+3w+5w^2$ and $o=2+4w+6w^2$, so direct polynomial multiplication gives
496 \[
497 y_{\rm even}=1+8w+16w^2+17w^3,
498 \qquad
499 y_{\rm odd}=4+12w+20w^2+6w^3.
500 \]
501 This is the same result as ordinary convolution followed by residue extraction. No numerical transform or limiting argument is needed because every sequence is finitely supported.
502 
503 \takeaway{A phase matrix is not just a sparsity pattern. Its delay factors carry the information about quotient and remainder needed when a shift crosses a phase boundary.}
504 
505 \vpage{21}{Two Smith constructions, different roles}{MODEL G / A NONLINEAR PHASE ORBIT}{Chapter II, mathematical text: Section 12.3, Smith-phase identity (p. 43)}
506 Smith \emph{normal form} concerns linear congruences. The Smith \emph{impedance chart} uses the nonlinear coordinate change
507 \[
508 \Gamma=\frac{z-1}{z+1},\qquad z=\frac{1+\Gamma}{1-\Gamma}.
509 \]
510 They share a name, not a mathematical definition. Sample the reflection circle at the exact algebraic points $\Gamma_k=\tfrac12 e^{ik\pi/4}$.
511 
512 \vfig[.96]{smith_chart_orbit}
513 \platecaption{Matching labels identify the same point in the two coordinates. All eight points are computed algebraically. Only their placement on the page is rendered numerically. The outer left circle is $|\Gamma|=1$, not part of the sampled radius-$1/2$ orbit.}
514 
515 The half-turn $\Gamma\mapsto-\Gamma$ gives
516 \[
517 z\longmapsto\frac1z,\qquad z\ne0,-1.
518 \]
519 Thus the labels $0$ and $4$ are $3$ and $1/3$. Label $2$ is $(3+4i)/5$, and the opposite point is its reciprocal. The domain restrictions come from the forward and inverse coordinate denominators.
520 
521 These sampled phase points are not linear idempotent ranges. Refining from two to four to eight samples inserts more points on the same orbit. This does not imply a linear packet decomposition of impedance values.
522 
523 \takeaway{The chart is a geometric realization of phase action. Its nonlinear coordinate sensitivity near $\Gamma=\pm1$ is not the Smith congruence-fiber count and not a new linear projector theorem.}
524 
525 \vpage{22}{A small defect can grow under powers}{MODEL H / EXACT TELESCOPING, EXPLICIT GROWTH}{Chapter II, mathematical text: Theorem 11.1, Corollary 11.3}
526 Choose an invertible coordinate shear but independently keep the same numerical operator at both levels:
527 \[
528 C=\begin{pmatrix}1&\varepsilon\\0&1\end{pmatrix},\quad
529 T_f=T_c=\diag(1,2),\quad \varepsilon=\frac1{16}.
530 \]
531 Then $\Delta=CT_f-T_cC=\varepsilon E_{01}$. Exact coordinate invertibility has not forced these independently selected operators to intertwine.
532 
533 \vfig[.95]{power_defect_growth}
534 \platecaption{Actual defects and certified upper bounds in the induced infinity norm, both divided by $\varepsilon$. All heights are exact integers. The upper estimate is valid but need not be tight.}
535 
536 Direct multiplication gives
537 \[
538 CT_f^n-T_c^nC=\varepsilon(2^n-1)E_{01}.
539 \]
540 The general telescoping formula explains every contribution:
541 \[
542 CT_f^n-T_c^nC
543 =\sum_{k=0}^{n-1}T_c^{n-1-k}\Delta T_f^k.
544 \]
545 Since both operator norms are two, the source theorem bounds the error by $n2^{n-1}\varepsilon$. It does not promise a small error uniformly in $n$ merely because $\varepsilon$ is small.
546 
547 Had we chosen the transported operator $T_c'=CT_fC^{-1}$, the defect would be zero and every polynomial action would intertwine exactly. That is a different choice of coarse operator, not a stronger conclusion from the faulty one.
548 
549 \takeaway{Separate coordinate coherence from operator consistency. A defect from one step is propagated by the dynamics on both sides. The telescoping identity tells you exactly how.}
550 
551 \vpage{23}{Resolvents amplify errors near poles}{MODEL H / A DOMAIN-SENSITIVE COMPARISON}{Chapter II, mathematical text: Theorem 11.5}
552 For the same model, take $z\notin\{1,2\}$ and form the actual inverse matrices
553 \[
554 R_f=(zI-T_f)^{-1},\qquad R_c=(zI-T_c)^{-1}.
555 \]
556 Their off-diagonal intertwining error is
557 \[
558 CR_f-R_cC=R_c\Delta R_f
559 =\frac{\varepsilon}{(z-1)(z-2)}E_{01}.
560 \]
561 Both inverse hypotheses matter. Neither resolvent is asserted at a pole.
562 
563 \vfig[.94]{resolvent_amplification}
564 \platecaption{The bars show exact amplification factors for five rational choices $z>2$. Rows are separate samples, not equally spaced positions on the $z$ axis. The coefficient $\varepsilon=1/16$ has been divided out.}
565 
566 At $z=3$ the amplification factor is only $1/2$. At $z=201/100$, it is $10000/101$, so a small one-step defect can create a much larger resolvent defect. This follows from the exact formula rather than from unstable floating-point subtraction of two nearly singular inverses.
567 
568 The certified general norm estimate is
569 \[
570 \|CR_f-R_cC\|
571 \le \|R_c\|\,\|\Delta\|\,\|R_f\|.
572 \]
573 The chapter also proves a version using one-sided witnesses: a left inverse for the coarse shift and a right inverse for the fine shift suffice for the algebraic identity. That does not turn arbitrary one-sided witnesses into a claim of two-sided resolvent existence.
574 
575 \takeaway{A small local defect is meaningful only together with control of the surrounding operators or inverses. Near a pole, the domain and amplification factors are part of the theorem, not technical decoration.}
576 
577 \vpage{24}{Read the certificates with the pictures}{PROOF BOUNDARY / HOW TO USE THIS GUIDE}{Formal evidence: chapter-wide correspondence and the 25-row foundational contract}
578 The foundational formal contract has twenty-five items: twenty-one numbered definitions and results in the main text, one Smith-phase identity, and three algorithm specifications. The source archive preserves its evidence. The broader constructions have a separate correspondence and replay record. The table below describes the foundational interfaces. The chapter-wide map identifies the broader constructions. The models illustrate the mathematics, and their finite checks have recorded ranges.
579 
580 \input{generated/current_visual_scope.tex}
581 
582 {\small
583 \begin{tabularx}{\linewidth}{@{}p{.28\linewidth}YY@{}}\toprule
584 \textbf{Inherited interface} & \textbf{Inputs and checked scope} & \textbf{Boundary of that interface}\\\midrule
585 Exact refinement & Complete projector families and a parent map when formulas relating parents to children are used. & A canonical refinement for every arbitrary pair of decompositions.\\
586 Fourier realization & A finite orthogonality table, or field/root hypotheses that provide it. & Orthogonality from multiplicative order alone over every ring.\\
587 Weighted inversion & A two-sided inverse of $D$, a multiplicative filtration, and positive certified gains. & Convergence in every possible complete filtered algebra.\\
588 Graph rationality & A finite multigraph with positive weights, together with its actual determinant and adjugate. & Simple-path counts, analytic convergence everywhere, or operator noncancellation.\\
589 Spectral separation & Evaluated B\'ezout identities and explicit lattice-preservation data. & Automatic construction or minimality of every resultant certificate.\\
590 Smith congruences & Concrete diagonal data modulo $p^L$, with $p>1$. & An automatically constructed general-DVR Smith normal form.\\
591 Linear defects & Actual typed maps and the stated inverse or norm witnesses. & Small propagated errors independent of time or distance from a pole.\\\bottomrule
592 \end{tabularx}}
593 
594 \head{Reproduce the explanatory calculations}
595 The preserved model source contains \textbf{118 named exact checks}. Each finite range is recorded in the file \texttt{EXACT\_CHECKS.json}. They cover masks, CRT labels, modular inverse products, graph enumeration, rational recurrences, lattice denominators, complete finite fibers, phase identities, and defect formulas. Graphics are generated from those same exact model definitions. Numerical placement of geometric points is only a rendering step.
596 
597 The computational report records 48 passing Python tests, 118 exact visual checks, 22 byte-matching regenerated vector plates, 13 regenerated PDF/PNG figure pairs, and nine matching numerical summary rows. Each record retains its tested range and source identity. Lean replay and source coverage are documented separately.
598 
599 \takeaway{Use the guide to understand what a theorem says and why its hypotheses matter. Use the theorem map and preserved proof sources to identify the precise formal result. Neither a picture nor a finite test expands that result's scope.}

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