Machine-Checked Cardinality Bounds for Masked Barrett Reduction: A 1-Bit Side-Channel Leakage Barrier in Post-Quantum Cryptographic Hardware

Barrett reduction is the nonlinear core of every practical NTT-based post-quantum cryptography implementation. Existing composition frameworks (ISW, t-SNI, PINI, DOM) address Boolean masking over GF(2); none provides a machine-checked characterization of Barrett's leakage under first-order arithmetic masking and the first-order probing model over prime fields. Building on our prior series, QANARY [15], partial-NTT-masking margins [14], algebraic foundations [16], and butterfly composition [18], we close this gap. We prove a trichotomy: for any $q > 0$ and shift $s$, the Barrett internal wire map $f_x(m) = ((x + 2^s - m) \bmod 2^s) \bmod q$ has preimage cardinality in $\{0, 1, 2\}$, never more. We call this the 1-Bit Barrier: max-multiplicity 2 implies at most 1 bit of min-entropy loss per internal wire, universal over all moduli. The count-zero cases, unreachable output values, reveal that actual leakage is often strictly less than 1 bit, making the bound conservative. We introduce PF-PINI (Prime-Field PINI): Barrett satisfies PF-PINI(2); the Cooley-Tukey butterfly satisfies PF-PINI(1). We observe (not yet proved) that with fresh inter-stage masking, the composed pipeline has max-multiplicity $\max(k_1, k_2)$, so the 1-Bit Barrier propagates. The trichotomy, the PF-PINI instantiations, and cardinality results are machine-checked in Lean 4 with Mathlib: 12 proved results, zero sorry, universal over all $q > 0$ (the min-entropy bound follows by standard definitions). Adams Bridge lacks fresh inter-stage masking, violating PF-PINI composition and explaining why Papers 1 [15] and 2 [14] found vulnerabilities. NIST IR 8547 recommends formal methods for PQC implementation validation. The 1-Bit Barrier provides the first universal machine-checked cardinality bound for masked Barrett reduction in ML-KEM (FIPS 203) and ML-DSA (FIPS 204), with a corresponding 1-bit leakage interpretation.