Tractable Closure Stability and Polynomial Chain Blowup: If Every Step Is Easy, Is the Whole Path Necessarily Easy?

The previous twelve rounds kept asking, “is this step tractable?” — this round proves that question alone isn't enough. If the representation size satisfies s_(t+1)=s_t^d(d≥2), every step is a polynomial transformation relative to the current representation size, yet starting from s_0=n, after m steps s_m=n^(d^m) — as soon as the step count m grows with the input, the whole thing quickly exceeds any fixed polynomial bound, with no need to involve SAT at all; it's purely a combinatorial trap in algorithm design. Formally distinguishes Stepwise Polynomiality (local polynomiality relative to the current representation) from Pathwise Polynomiality (polynomiality over the entire run relative to the original input), and proposes Tractable Closure Stability (TCS), which requires the step count, peak representation size, cumulative cost, and answer-recovery cost to all be uniformly polynomially bounded in the original input. The Equals Team is forced to upgrade to an Amortized Tractability Certificate (ATC), while the Not-Equals Team proposes the Path-Chain Instability Conjecture (PCIC).