Paper II visual guide · Visual 17 of 24 · MODEL E / TWO DIRECTIONAL WEIGHTS

See every Smith fiber modulo eight

Traditional paper: Theorem 10.6 (p. 36)

Consider the actual map of finite sets

\[ ({\mathbb Z}/8{\mathbb Z})^2\longrightarrow({\mathbb Z}/8{\mathbb Z})^2, \qquad (x,y)\longmapsto(2x,4y). \]

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.

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All inputs and targets for a finite diagonal map — Original drawing. Read the description below; expand the labels and data for selectable text.

Reading the drawing

Both grids show all sixty-four residue pairs. Only eight target points are attained; each attained target has eight preimages.

Drawing labels and data 37 labels

Read the original labels as selectable text. Tables preserve grid relationships; other short labels are grouped compactly.

64 inputs \((x,y)\)
\(0\)
\(1\)
\(2\)
\(3\)
\(4\)
\(5\)
\(6\)
\(7\)
\(0\)
\(1\)
\(2\)
\(3\)
\(4\)
\(5\)
\(6\)
\(7\)
8 attainable targets \((2x,4y)\)
\(0\)
\(1\)
\(2\)
\(3\)
\(4\)
\(5\)
\(6\)
\(7\)
\(0\)
\(1\)
\(2\)
\(3\)
\(4\)
\(5\)
\(6\)
\(7\)
\({\operatorname{diag}}(2,4)\)
\(\bmod 8\)

Every filled target has exactly \(2\cdot4=8\) preimages.

Open the full-size drawing

Drawing source with line numbers

Evidence note from the source edition

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.

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For a specific attainable target \((2,4)\),

\[ 2x\equiv2\pmod8\iff x\in\{1,5\},\qquad 4y\equiv4\pmod8\iff y\in\{1,3,5,7\}. \]

The coordinate choices are independent, giving \(2\cdot4=8\) solutions. The entire carrier is partitioned consistently:

\[ 8\text{ attainable targets}\ \times\ 8\text{ inputs per target}=64. \]

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.

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The arithmetic behind the picture

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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 \((1,2)\).

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Read this as

A directional weight has two consequences: a divisibility condition on the target and a conditional multiplicity of inputs. Solvability comes before counting.

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