Fractalyx Vault: Cryptographic Primitives Unified by the Fractal-Stochastic Model (MFSU) with FractalShield Deniable Encryption

**[CORRIGENDUM — April 2026]**

Post-publication empirical validation revealed a discrepancy in Section 3.2 (Stream Cipher) and Remark 3.5. The statement "the fractal field alone fails the χ²-test (χ² = 1752, p ≈ 0)" was produced with an earlier version of the MFSU solver that lacked constant-time normalization. The current implementation (fractalshield.py, commit main) applies unconditional normalization `ψ / max(‖ψ‖∞, 1)` at every step, which stabilizes the field and yields χ² ≈ 245 (p = 0.66) for the raw fractal output alone.

Consequently, the corrected characterization of the defense-in-depth architecture is: the fractal field provides statistically acceptable distribution (χ² ≈ 245, p = 0.66); SHA3-256 whitening further improves uniformity to p ≈ 0.85 and provides security reducibility independent of the field's statistical properties. The conclusion that SHA3-256 is a correct and recommended layer is unchanged; only the empirical motivation is updated.

All other results, theorems, lemmas, and open problems stated in v3.0 remain valid. A formal v3.1 revision addressing this and completing the NIST STS battery is in preparation.

    This paper presents Fractalyx Vault v1.0, a complete symmetric cryptographic system derived from the Unified Fractal-Stochastic Model (MFSU): ∂ψ/∂t = −δF(−Δ)^(β/2)ψ + γ|ψ|²ψ + ση(x,t). All cryptographic primitives — a memory-hard key derivation function (8 MB scratchpad), a dual-layer stream cipher (fractal field + SHA3-256 whitening), a Merkle-Damgård fractal hash, and a TOTP 2FA module — emerge from the same physical equation with parameters δF=0.921, β=1.079, H=0.541. The centerpiece innovation is FractalShield: an oracle-free, multi-layer deniable encryption scheme where attacker cost escalates geometrically (3.5× / 7.5× / 15.5× per protection level) and all layers are statistically indistinguishable. Empirical validation includes chi-square uniformity tests (p=0.49), Pearson autocorrelation < 0.003, and avalanche effect 49–51% for all primitives. The reference implementation is open-source (Apache 2.0) and distributed as a Python package with CLI and Streamlit web interface.