Architecture and computation boundaries

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The theory is tied to homology-operator-lab @ 6143729669902ee875b211b58085e954c76cdf88. The formal definition, proofs, projection space, and six-edge example are in operator theory. This page describes implementation responsibilities and computation boundaries. See the API, result model, and solver contract.

Chain window and projection

For a fixed degree k, the input is a finite based window over F2:

\[ C_{k+1}\xrightarrow{D}C_k\xrightarrow{A}C_{k-1},\qquad AD=0,\qquad w_i>0. \]

The reference selects generalized inverses with AGA=A and DUD=D and constructs

\[ R=I+GA,\qquad Q=I+DU,\qquad P=QR,\qquad L=I+P. \]

Every candidate is independently checked for P²=P, AP=0, PD=0, and z+Pz∈im(D) on a complete cycle basis. The last condition ensures preservation of cycle homology. A zero projection can satisfy the first three conditions and still lose nonzero homology. Other solver parameterizations use the same boundary.

HomologyOperator uses one P for topology, representatives, weighted geometry, and stretch. Queries do not choose another representative or run independent PH to populate the result. ker(L) identifies the selected homology representation.

Geometry in original coordinates

For cycles z,y, define

\[ m_w(x)=\sum_i w_i x_i,\qquad \mathrm{selected\_mass}(z)=m_w(Pz),\qquad d_P([z],[y])=m_w(P(z+y)). \]

Support and its intersection/union are read from the same projected coordinates. A basis change may preserve homology while changing mass, support, and stretch. Weight semantics and units belong to the input, not to an inferred geometric name.

True minimum class mass and the current/optimal projection objectives are different:

\[ \mu_w([z])=\min\{m_w(x):Ax=0,\ [x]=[z]\},\qquad \Gamma_w(P)=\max_{0\ne z,\ Az=0}\frac{m_w(Pz)}{m_w(z)},\qquad \Gamma_*=\min_P\Gamma_w(P). \]

minimum_class_mass is currently unavailable. Feasibility, exact current Γ, and global optimality are separate facts. Empty cycle space has declared stretch 0 with EmptyDomain; a nonempty cycle space with zero Betti has a legitimate computed 0. This is distinct from a separately defined section-accounting convention. Exact F2 algebra does not certify floating-point geometry or optimality.

Filtration transport

OperatorFamily accepts finite ordered windows and matching operators in one degree. Basis identifiers define inclusions in all three degrees; chain-map, coordinate, and weight policies are validated. Repeated scales preserve stage order, and the terminal extension is constant.

\[ \mathcal H_i=\ker L_i,\qquad T_{ij}=P_jJ_{ij}|_{\mathcal H_i}. \]

Transports retain kernel-coordinate action and original target chain action. They must satisfy T_ii=I and T_jl T_ij=T_il, and agree with the induced homology map under the chosen representation. Stage Betti alone does not determine this map.

barcode() reads adjacent kernel maps while preserving the full rank invariant. barcode_basis() and rank_table() explicitly request larger historical/all-rank outputs. Death yields a valid zero chain, mass, and support. Failed stages remain missing rather than becoming empty spaces. The historical proof is retained in the filtration guide.

Existing modules

Module	Responsibility
algebra.py	Explicit-shape F2 matrices and the supported CyclicAction
chain.py	Window, coordinate, AD=0, weight and source validation
solver.py	Requests, capabilities, budgets, solver records and dispatch
validation.py	Independent projection and optimization certificate validation
operator.py	Same-P single-scale readouts, query history and results
family.py	Inclusions, transport, rank/barcode, tracking and family snapshots
result.py	Six identities, query state, canonical JSON and cache boundaries
native.py / native/src	Optional safe Rust adaptation, packed algebra and supported batch operations

There is no public LinearAction base class or universal sparse backend; callers construct based chain windows for the supported input domain.

Action and native boundaries

Matrix stores explicit P/L. CyclicAction represents the fixed supported family. CompactAction stores generalized-inverse or HC factors without materializing full P/L. Every representation preserves the same action on all chains, including noncycle extension, and undergoes independent validation and recovery.

NativeFeasibleSolver and explicit apply_batch remain limited to 64 dimensions; geometry_batch supports multiword Matrix and Factorized/HC. GeometryWorkspace binds all six identities and reuses process-local action/weight buffers. Checked u64 mass and explicit exact/floating Python fallback preserve the arithmetic policy. Different representations may have different content identities.

Resource and evidence boundaries

Solver budgets are cooperative state, wall-time, and logical-entry checkpoints. They are not hard RSS limits or a unified preemption budget for validation, certificate replay, serialization, and recovery. Those costs remain part of the complete call. Missing or uncomputed objective is never replaced by zero.

Independent PH reduction and GUDHI are test/measurement oracles. They do not populate production barcodes, replace representatives, or fill missing results. Performance comparisons match input, P, arithmetic, certificates and budgets, and retain conversion, validation, query, snapshot and recovery costs.