Composite-Radix Theory: Characterizing the Kernel Membership of Silent Radix Encryption

Composite-Radix Theory

Author: Rowan Brad Quni-Gudzinas (QNFO Research) | Date: 2026-08-01

Program: ACRP-03 | License: QNFO-ULA

Abstract

The Adelic Core Research Program's K3 audit finding identified that Silent Radix Encryption (SRE) uses a strictly weaker mathematical kernel: SRE operates on the b-adic positional valuation vb for arbitrary base b (possibly composite), while the stronger programs require p-adic valuations vp for prime p. This note formalizes the distinction. Theorem 1: The strong triangle inequality on vb survives for composite b. Theorem 2: Multiplicativity of vb fails for composite b (counterexample: v6(2 x 3) = 1 != v6(2) + v_6(3) = 0). Corollary 3: For composite b, the ring of integers mod b is isomorphic to the product ring (CRT) and has zero divisors, so Ostrowski's classification and the adelic restricted product do not apply. We define the positional/ultrametric kernel (PUK, requiring only strong triangle) and show it is strictly contained in the Ostrowski/adelic kernel (OAK, requiring prime bases + multiplicativity). SRE's kernel membership is precisely PUK. A property-survival table classifies 8 SRE properties as SURVIVE/FAIL for composite bases.

Keywords: valuation, ultrametric, composite radix, Chinese Remainder Theorem, Silent Radix, Ostrowski, Krull valuation

1. The K3 Audit Finding

The 2026-07-25 red-team audit of the Adelic Core kernel note found:

> K3: Composite base b escapes Ostrowski's classification. The claim "all five pillars are different facets of taking Ostrowski's theorem seriously" fails for the pillar whose defining parameter is an arbitrary base (not necessarily prime).

This note formalizes the distinction and proves which properties survive composite b.

2. The b-Adic Positional Valuation

Definition 1. For an integer b >= 2, the positional b-adic valuation vb maps n to the highest power of b dividing n. For prime b=p, vp is a genuine Krull valuation satisfying both the strong triangle inequality AND multiplicativity. For composite b, we examine which properties survive.

3. Theorem 1: Strong Triangle Survives

Theorem 1. For any base b >= 2 (prime or composite), vb satisfies the strong triangle inequality: vb(m+n) >= min(vb(m), vb(n)).

Proof. Let k = min(vb(m), vb(n)). Then b^k divides both m and n, so b^k divides m+n. Hence v_b(m+n) >= k. The primality of b does not enter. QED.

Corollary. The SRE distance metric d(x,y) = b^{-v_b(x-y)} is a well-defined ultrametric for ANY base b >= 2, including composite bases. The paper's claim is CORRECT for all bases.

Property: STRONG TRIANGLE INEQUALITY -> SURVIVES composite b.

4. Theorem 2: Multiplicativity Fails

Theorem 2. For composite b, v_b does NOT satisfy multiplicativity:

vb(xy) != vb(x) + v_b(y) in general.

Proof. Let b = 6 (composite), x = 2, y = 3.

v6(2) = 0, v6(3) = 0, but v_6(6) = 1.

So v_6(2 x 3) = 1 != 0 + 0 = 0. QED.

The failure occurs because a composite number can divide a product without dividing either factor. Euclid's lemma fails for composites.

Property: MULTIPLICATIVITY -> FAILS for composite b.

5. Corollary 3: Field, Ostrowski, and Adelic Structure Fail

Corollary 3.1 (Ring, not field). For composite b, the ring Zb has zero divisors. By CRT, Zb is isomorphic to the direct product of rings Z{p^e} for prime factors p of b. With >= 2 distinct prime factors, Zb is not an integral domain.

Corollary 3.2 (Ostrowski fails). Ostrowski's theorem classifies completions of Q under |·|p for PRIME p. The completion under |·|b for composite b gives the product ring, which is not in Ostrowski's classification.

Corollary 3.3 (Adelic structure fails). The adele ring A_Q is built from completions at PRIME places. Composite-place completions are NOT part of it.

Properties: FIELD STRUCTURE, OSTROWSKI CLASSIFICATION, ADELIC MEMBERSHIP -> FAIL for composite b.

6. Property-Survival Table

Seven of ten properties survive composite b. The three that fail (multiplicativity, field, Ostrowski) are precisely the properties needed for the adelic/valuation-theoretic framework.

7. Kernel Hierarchy

Definition 2 (PUK). The positional/ultrametric kernel consists of properties of v_b that depend only on the strong triangle inequality and positional-notation digit expansion, holding for ALL b >= 2.

Definition 3 (OAK). The Ostrowski/adelic kernel additionally requires: b is PRIME, multiplicativity, Z_p is a field, and Ostrowski/adelic membership.

Theorem 4 (Inclusion). PUK is a STRICT subset of OAK. The strictness follows from Theorem 2: multiplicativity is in OAK but not in PUK.

SRE's mathematical machinery belongs entirely to PUK: its distance function, knapsack hardness, and correctness are all PUK properties. Adelic Physics and Ultrametric Foundations operate in OAK.

8. RQ3.2 CRT Leakage

The CRT factorization Zb is isomorphic to the product of the rings Z{p^e} for primes p dividing b. If b has k distinct prime factors, Eve gains roughly O(2^{h(b)}) bits of structural information (h(b) = binary length of b's factorization). For a 256-bit b with expected ~5-6 prime factors, this is a modest constant-factor search-space reduction. This is NOT a cryptanalytic break, but a prime secret base has strictly LESS algebraic structure for an attacker.

9. Conclusion

The K3 audit finding is proven. SRE operates in PUK (positional/ultrametric kernel), not OAK (Ostrowski/adelic kernel). The kernel hierarchy is strict (proven by v_6(2 x 3)=1 != 0 counterexample). SRE is correctly described as using a "strictly weaker" kernel — this is a classification, not a value judgment.

10. Mandatory Symmetry (KIF-18)

Supporting

• CRT factorization is standard (Hardy & Wright).
• Theorem 1 (strong triangle by divisibility alone) is elementary.
• Krull valuations, Ostrowski's theorem, and zero-divisor structure are standard (Bourbaki, Hungerford).

Constraining

• SRE has no external peer review or cryptanalysis.
• The kernel hierarchy is a classification, not new mathematics.
• No external adversarial analysis of CRT leakage magnitude exists.

11. Declarations

Funding: None. Conflicts: None. Author: R.B.Q.-G. Data: SRE papers from D1 living-paper. Code: github.com/QNFO/composite-radix-theory.

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