Abstract

We adapt the sequential IX-C–IX-Free–IX-Free-2 programme to a causally saturated regular laboratory on the prescribed I-B conical spacetime. The three gates are retained in order: first-stage products and exact descent criteria; canonical two-edge outputs and degree-two second-stage maps; and degree-four descent with simultaneous overlap agreement. The carrier is the already adapted IX-A-ST space of actual off-shell polynomial functionals. The covariance and the VII-D triangle finite part are supplied chart data.

For the selected word $B_f,W_g,B_{f^{\prime }}$, we prove injectivity of both degree-two and both degree-four presentation maps. A chart-local advanced-trace witness separates every nonzero sharp generator from the partial contractions and the old microcausal algebra. An anchored tensor-domain argument then eliminates the old overlap of the entire new-domain sum, not merely its pairwise intersections. Consequently every fixed real common parent parameter passes both scalar descent and gluing: ${\mathcal{C}}_{\mathrm{desc}}={\mathcal{C}}_{\mathrm{glue}}=\mathbb{R}$. The prescribed coefficients define one truncated partial product modulo ${\hslash }^4$, with presentation independence, reversal, and coherence of both associations of this word through order ${\hslash }^3$.

The potential value of the result is a controlled example in which actual-functional factorization is achieved without assuming continuity of unchanged sharp approximants or altering the old product. The algebraic findings are adapted from the cited Cheung manuscripts; the added construction is their supported conical transfer and a complete finite certificate interface, including destructive-clock readout and reference-preserving causal transport. No global product, actual-stage continuity, causal lift, interacting Ward identity, or spacetime emergence follows. The inherited ultraviolet analysis remains an explicit premise and is not re-proved.