The TCAP Transformation (Ξ) VI — Self-Atomized Odd Hybrids: Prime Fixed-Point Convergence, Deterministic Prime Generation, and One-Sided Primality Certification

This sixth paper in the TCAP series develops an exact convergent branch of the Transformation by Combination of Prime Adjacencies (TCAP), denoted Ξ. For every odd input n ≥ 3, the same integer n is selected as a fixed opaque atom and inserted into the ordered odd hybrid Hₙ(n) = (3, 5, 7, …, n − 2, n, n). The associated odd-double-factorial product is never computed. The first Ξ image has the exact closed form X₁(n) = (n³ + 6n² − 4n + 3) / 6. At later ranks, the raw transformation uses only the two-atom identity Ξₐₜ(a,b) = a(b + 1), for a ≤ b. Between ranks, a canonical exact reduction uses only integer divisibility tests, exact divisions, comparisons and the square-root certificate 1 < R ≤ B². No GCD, factorization routine, classical sieve, prime table, modular-exponentiation primality test, or external primality test is part of the mechanism. The central theorem is based on a uniform half-descent. If S > n is any reduced state and the next reduced state T also remains above n, then T − 1 ≤ (S − 1) / 2. Thus states above n descend geometrically and cannot form a non-trivial cycle. Once the orbit reaches a reduced state at or below n, that state is necessarily a certified prime. The state n is fixed when reached; if a certified prime p < n is reached, the terminal prime fixed point is obtained after at most one further transition. Consequently, for every odd integer n ≥ 3, the self-atomized Ξ dynamics converges in finitely many ranks to a prime fixed point. The number of transitions that can remain above n is bounded by ceil(log₂((X₁(n) − 1)/(n − 1))), so the outer Ξ-rank depth is O(log n). This is a bound on the number of outer transitions only and is not a claim that the complete bit complexity, including canonical reduction, is O(log n). If F(n) denotes the terminal fixed point, then G(n)=F(n) is a deterministic prime-output map on every odd input n ≥ 3. The word “generator” is used in this operational sense only; G is not claimed to be injective, ordered, or surjective onto the primes. For composite n, the terminal fixed point is necessarily a prime strictly below n. Hence F(n) = n ⇒ n is prime. Self-fixation is therefore a deterministic one-sided primality certificate with no composite false positives. The converse is not universal; for example, n=11 gives 11 → 7 → 7. The paper also derives an exact pure-TCAP optimization. For every state S>n, the literal image n(S+1) need not be formed: S+1 can be reduced directly because the fixed factor n is predictably eliminated by the canonical sweep. The auxiliary reductions of n and n+1 are computed only when needed, the terminal tail below n is closed algebraically, and after saturation of 2 and 3 the reducer skips control values divisible by 2 or 3 while retaining composite values of the form 6k±1. No sieve or external primality machinery is introduced. The literal canonical and optimized implementations were exhaustively cross-checked on all 4,999 odd inputs 3≤n≤10,000. No divergence was found in the terminal prime fixed point, the logical rank of first arrival, or the logical rank of fixed-point confirmation. The work separates proved statements from finite computational checks and does not claim a complete primality test or an established asymptotic advantage over classical primality algorithms.