Central Algebras and Central Cohomology: Central Actions, Lifting Towers, and Filtered Derived Centres

We develop a centre-first framework for constructing and controlling ambient algebraic objects from a prescribed central object and additional peripheral data. For associative algebras the basic datum is a central map \(C\to A\), not merely the ordinary centre \(Z(A)\). We distinguish central actions, faithful central embeddings, exact realizations of \(C\) as the full centre, and filtered or derived enhancements. A finite free two-corner construction realizes a prescribed commutative algebra with a multiplication parameter and shows that the central algebra alone does not determine the ambient algebra. Dynamic central flags enlarge the acting centre while contracting its ambient action domain. Retained corner bimodules and composition maps reconstruct split finite flags, whereas iterating the centre of one fixed algebra produces no such dynamics. Candidate choices are compared through their embedded control hulls: commuting hulls admit a minimal common refinement, noncommuting hulls do not, and cofinally equivalent infinite flags have the same limiting central pair. Split refinements transport their complete corner recovery data. Central action towers, fibre-product lift algebras, and commutator defects organize quotient lifting. Compatible quotient centres recover the actual centre when the ambient algebra realizes the quotient tower; a derived-limit class detects failure of compatibility when only levelwise lifts are given. For finite filtered algebras, weighted central lifts identify the degree-zero pages of the Hochschild spectral sequence and their radical filtration. A filtered normalized bar model provides the ambient derived centre and its comparison with these lifting pages. Joint changes of the central product, ambient product, central map, and finite diagram arrows are governed in low degrees by explicit relation complexes. Their zeroth and first cohomology describe infinitesimal automorphisms and deformations, while conormal obstruction classes give necessary and sufficient lifting criteria across small Artin extensions. A finite typed enhancement permits candidate embeddings, commutator defects, precision maps, bimodule actions, composition, control comparisons, and recovery tensors to vary in the same relation problem. Its algebra-arrow core and Hopf specialization recover the earlier complexes, while core--periphery connecting maps record mixed lifting failures. Polynomial simplicial resolutions give presentation-independent higher additive control in arbitrary characteristic; explicit redundant-relation contractions explain why all syzygies must be retained, and Frobenius cycles delimit ordinary polynomial--exterior dg replacements in positive characteristic. Semifree resolutions remain available in characteristic zero. Layer, precision, relation, coefficient, and ambient filtrations yield compared spectral sequences with computed nonzero differentials. These additive and cotangent models are compared with, but not identified with, the multiplicative or brace structure on the ambient Hochschild complex. Strict brace maps and strict pullbacks give a multiplicative comparison range; matrix Morita equivalence is realized by an explicit filtered brace quasi-isomorphism, while general derived Morita equivalence preserves the \(B_\infty\) homotopy type rather than an arbitrary chosen chain map. The constructions are realized for discrete algebras, complete cofinite ideal-filtered algebras, and scalar-split Banach and Fréchet settings. A finite generator certificate gives an explicit continuous double-commutant recovery theorem. Native Lie and group versions retain their own central objects: Lie brackets, group components, periods, central extensions, and quotient actions are not replaced by centres of enveloping or group algebras. A native group control package retains the central parameter, peripheral group, actions, bisets, extensions, representations, topology, and recovery maps as separately typed data. Central factor normal forms give exact realization criteria, while finite examples separate extension kernels, group-algebra centres, commutants, and Hecke algebras from the native centre. A typed derived comparison separates group cochains, module Ext, mapping spaces, strict path algebras, native central homotopy fibres, group-algebra Hochschild cochains, and Hopf--central relation tangents. The comparison records which arrows are chain maps, spectral maps, low-degree agreements, or full additive mapping-space equivalences. Multilayer central transport is then defined from vertex centres, ambient algebras, bimodule edges, composition, defects, precision and recovery data; repeated centres of a commutative algebra are excluded as degenerate layers. For a finite bimodule diagram its secondary centre is the universal algebra of vertex scalars balanced on every edge. It acts on all transport functors, identifies with the diagonal corner of the associated path-algebra centre, obeys flat base change, and acquires explicit two-sided Tor corrections under nonflat base change. For native group diagrams, compatible vertex-centre tuples form a secondary centre group computed from orbitwise biset relations. Its elements act through set and permutation-module transport. Principal bitorsors recover full actions, invertible linear bimodules recover module categories, and arbitrary balanced bisets retain only the compatible central actions. Effective profinite biset systems recover the continuous secondary group, whereas finite compatible shadows need not have a discrete source. For finite strict transport categories, recursive secondary, tertiary and longer merges flatten canonically to one global equalizer. A defect insertion identity makes composite constraints redundant when the comparison maps generate their targets. Proper comparison images and unseen biset orbits show why strictness is necessary. A typed non-collapse profile separates missing connections, nonfaithful and noninvertible transports, homotopy markings, precision, closure, inverse-limit effectivity, and empty central lift fibres. Tertiary transport retains a comparison from a balanced two-edge composite to a direct edge; its insertion complex has a degree-two triangle obstruction, with explicit algebraic and native examples which vanish on every proper face but not on the full triangle. Finite ordered semisimplicial packets then retain simplex comparisons, partition refinement defects, tetrahedral associativity, precision, topology and recovery data through arbitrary finite order. Strict packets flatten at the level of their transferable centre, while marked and defective tetrahedra show that their complete higher data need not flatten. A typed multilayer total complex places vertex centres, edge endomorphisms, simplex comparison maps, shell relations, recovery and precision corrections in one additive model. Its low cohomology recovers transferable centres, transport variations and joint lifting obstructions, while higher classes retain path-coherence syzygies. Over a general commutative base ring, a finite-projective commutator complex measures both central directions lost and central directions created by nonflat base change through explicit \(\operatorname{Tor}_2\) and \(\operatorname{Tor}_1\) terms. Invertible corner modules give exact finite-projective central realizations compatible with arbitrary base change, while unitization and transient quotient centres exhibit the necessary boundaries. For perfect underlying complexes, an additive derived commutator fibre commutes with arbitrary derived base change; its hyper-Tor sequence isolates the correction contributed by nonprojective scalar-module directions. This additive fibre is kept distinct from the ambient Hochschild derived centre. Beyond scalar-split exact structures, strictly bar-projective topological algebras admit a strict continuous derived centre. Banach algebras with \(\ell^1\) bar coordinates give an explicit class, completed Hopf operations require strict tensor flatness, and a densely glued endomorphism algebra shows that completion can create a new central direction. For the arithmetic model based on \(\mathrm{SL}_2(\mathbb Z)\) and its dyadic congruence quotients, we construct native central parameters, peripheral representations, centralizer depth--precision grids, and explicit lifting obstructions. Finite native flags admit module, path-algebra, and derived mapping-space models whose comparison detects mixed classes invisible on proper coordinate faces. Congruence transgressions produce obstruction death and revival, while centre turnover and inverse limits show that depth, precision, and actual discrete lifting remain distinct. For odd primes the rank-two centre tower is instead stable, while rank--characteristic resonance creates a transient scalar line in \(\mathrm{SL}_3(\mathbb Z/3^n)\). Products of local centres contain tuples not realized by the diagonal \(S\)-integer centre. Finally, a finite free Hopf--central parameter algebra permits source products, coproducts, quotient arrows, split kernels, representations, central characters, and recovery markings to vary jointly. At the first dyadic edge, crossed source complexes and their centre-fixed relatives have explicit ranks. On the retained cyclic Hopf arrow \(k[C_4]\to k[C_2]\), the complete marked Jacobian is infinitesimally rigid. A five-term bialgebra coefficient identity obstructs the crossed carry when the central involution is required to remain group-like. For the complete first-edge group-Hopf core, separability of the dual diagonal coalgebras shows that the full 2302-dimensional Jacobian kernel is exactly gauge, even when the cyclic summand and all Hopf arrows move. Split sections and ordinary representations give the remaining marked directions, but they do not cancel the carry. Infinite compatible central diagrams and unrestricted continuous or multiplicative hi