Prime-Axis Normalized Addition

Abstract

Rowwise W normalization of prime-axis table sums decodes as multiplication.

Theorem 1.1 (Rowwise normalized addition and decoder multiplication).

$$\forall z,w\in \mathrm{PrimeAxisTable},\mathrm{Bijective}(\mathrm{primeAxisEncoding})\wedge \mathrm{decodePrimeAxisTable}(\mathrm{normalizedPrimeAxisAdd}(z,w))=\mathrm{decodePrimeAxisTable}(z)\mathrm{decodePrimeAxisTable}(w)$$

Proof. Machine-checked in Lean as D5/S1/Digit/PrimeAxisAddition.prime_axis_addition_spec (✓ std3). ∎

Source. Repository-derived.

Commentary.

Prime-indexed canonical W rows are equivalent to positive naturals. Adding raw rows and applying the existing local W normalizer preserves exponent sums, so the finite prime-power decoder turns the normalized table sum into multiplication.

References

• Truth anchor: D5/S1/Digit/PrimeAxisAddition.prime_axis_addition_spec
• Dependency: D5/S1/Digit/PrimeAxisEncoding