Dual-Mode Arithmetic Representation

In the paper's words

This presents a framework that resolves the apparent incompatibility between algebraic and geometric representations of natural numbers. We demonstrate that while static two-dimensional coordinate systems are informationally incomplete for capturing the full arithmetic structure of prime factorization, a dynamic spectral representation—constructed as a holographic boundary dual—admits a functorial equivalence with the algebraic structure. Through categorical formalization, analytic number theory, and network embedding theory, we establish that prime factorization and enriched ordinal–rotational addressing are complementary epistemic interfaces to a single arithmetic reality.

The abstract, verbatim, and its opening sentence is the claim. Every line below is quoted from the text.

Numbers it reports

No sentence of this paper reports a figure with a unit or a quantity word, so none is shown.

How it is done

1. The algebraic representation through prime factorization provides a deep structural description grounded in the Fundamental Theorem of Arithmetic, where each integer greater than one is uniquely expressible as a product of prime powers.

   from 1.0 Arithmetic Representation

2. By applying the holographic principle, we construct a boundary encoding of the bulk prime exponent data through spectral enrichment.

   from 3.0 Holographic

3. This categorical framework reveals deeper structural connections: We define two functors: $F_{\text{prime}}$ maps $\mathcal{N}$ to a category of vector translations preserving multiplicative structure as vector addition, while $F_{\text{spectral}}$ maps $\mathcal{N}$ to a category of signal additions preserving multiplication as signal superposition.

   from 4.0 Categorical

4. The linear independence of $\{\log p_i\}$ over $\mathbb{Q}$, established through the Lindemann–Weierstrass theorem, guarantees the injectivity of the map $n \mapsto S_n(t)$—different numbers produce distinct signals (Lindemann, 1885; Weierstrass, 1885).

   from 5.0

What would falsify it

The paper's text states no explicit falsification criterion; the limits below are what it says about where the claim stops.

Where the claim stops

1. The resonance conditions inherent in the spectral representation do not yield efficient factorization algorithms, as the inverse problem (phase retrieval) maintains equivalent computational complexity to traditional factorization.

   from Dual-Mode

2. Critically, this unification does not reduce the computational complexity of integer factorization; rather, it demonstrates how the same hardness manifests across representational domains.

   from 1.0 Arithmetic Representation

3. In contrast, static two-dimensional encodings lack native multiplicative operations and cannot replicate the algebraic structure of prime factorization.

   from 2.0

Where to read

1. Dual-Mode220 words · 1 min
2. 1.0 Arithmetic Representation300 words · 1 min
3. 2.0210 words · 1 min
4. 3.0 Holographic335 words · 1 min
5. 4.0 Categorical396 words · 2 min
6. 5.0357 words · 2 min
7. 6.0258 words · 1 min
8. 7.0 Geometric315 words · 1 min
9. 8.0 Generalization205 words · 1 min
10. 9.0240 words · 1 min
11. 10.0 Gauge Theory197 words · 1 min
12. 11.0145 words · 1 min