Digit Genesis Matrix

Let $N$ be the set of base-10 digits. Define subset $A = \{0, 1, 2, 3, 4\}$ as the Numerical Axiomatic Foundation, from which all elements of $N$ emerge through reflective and inverse transformations.

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Transformation Rules

The entire set of digits $D = \{0, ..., 9\}$ is generated from the core set $A = \{0, 1, 2, 3, 4\}$ and the anchor $5$ through a simple involutional map.

1. Reflection

The primary transformation is reflection, which maps a digit $d$ to its additive inverse modulo 10.

Definition (Reflection): $m(d) = (10 - d) \pmod{10}$

• This map pairs digits: $\{1,9\}, \{2,8\}, \{3,7\}, \{4,6\}$.
• Two digits, $0$ and $5$, are anchors, as they map to themselves ($m(0)=0, m(5)=5$).
• Applying the reflection twice returns the original digit: $m(m(d)) = d$.

2. Negative Analogs (Symmetric Representation)

To reveal the underlying symmetry, we can represent the digits using the symmetric set $R = \{-4, -3, -2, -1, 0, 1, 2, 3, 4, 5\}$. The reflection $m(d)$ is now equivalent to simple negation, $m(r) = -r$.

• $9 \equiv -1 \pmod{10}$
• $8 \equiv -2 \pmod{10}$
• $7 \equiv -3 \pmod{10}$
• $6 \equiv -4 \pmod{10}$

In the diagram, these negative counterparts are shown on the "shadow lattice", illustrating how the reflected digits are also the negative images of the core set.