Causal Factorization Algebras and the BV Pushforward: Deriving the $L_{\infty}$ Quasi-Isomorphism from Cohesive Topology

The Spectral Realization program establishes a constructive traversal of the differential cohomology hexagon, deriving the Batalin-Vilkovisky (BV) BRST complex from the Lie differentiation of the automorphism groupoid of the synthesized physical state. The present paper proves the foundational theorem that the dynamical chaining mechanism (the cohesive regulator map $\mathrm{reg}\colon\Gamma({\flat\hat{\mathbb{E}}_{\omega}})\to({\Pi\hat{\mathbb{E}}_{\omega}})$ is isomorphic, upon Lie differentiation, to the BV pushforward $p_{\mathrm{int}}$ of Cattaneo and Mnev as an $L_{\infty}$-quasi-isomorphism.
        
        The proof proceeds in four stages. We first formally define the physical slice as a Causal Factorization Algebra $\mathcal{F}$, with local observables $\mathcal{F}(I) = \Gamma({\flat\hat{\mathbb{E}}_{\omega}}\vert_{I})$ over each directed causal interval $I$, and prove that the regulator satisfies the sewing axioms. We then apply the cubical Dold-Kan correspondence to show that the topological Higher-Dimensional Automaton (HDA) of the slice linearizes, under $\mathfrak{Lie}(-)$, into a tangent $L_{\infty}$-algebra, where commuting 2-cells map to vanishing brackets $\ell_2(a,b)=0$ and obstructed 2-cells map to curvature brackets $\ell_2(a,b)=F$. We invoke Lurie's directed $\infty$-operads to identify the resulting locally constant factorization algebra with a forward-directed $E_1$-algebra (an $A_\infty$-algebra), and apply the Costello-Gwilliam theorem to construct its BV quantization. Finally, we construct the strong deformation retract data $(i, p, K)$ identifying the Synthesis lift with the Homological Perturbation Lemma (HPL) inclusion $i_{\mathrm{int}}$, the global sections functor $\Gamma$ with the HPL projection $p_{\mathrm{int}}$, and the gauge-fixing propagator with the chain homotopy operator $K$, establishing that Spectral Realization $S\to A$ is the geometric HPL.