Unified executable language

Read side by side

Komorebi Math uses one sequence of operations across its mathematical domains:

seed
  -> address
  -> operate
  -> project
  -> transfer
  -> refine
  -> constrain
  -> complete
  -> reconstruct

This is an execution language, not an assertion that every domain contains the same objects. A signal packet, a module summand, a symmetry sector, and a residual deformation class have different meanings. They share an interface only when the corresponding operations and identities are preserved.

1. Seed

A seed is finite generating data for an operator action.

SeedSystem(operators, seeds)

Implemented operations include orbit hulls, generation certificates, finite free presentations, semisimple seed rank, rational single operator seed rank, and finite seed searches.

2. Address

An address names a legal operator path or packet coordinate.

OperatorGrammar
residue_analyze(values, modulus)
crt_many(residues, moduli)
polyphase_analyze(signal, order)

Typed paths, residue addresses, Chinese remainder addresses, mixed radix signal addresses, and packet tree paths are implemented.

3. Operate

OperatorRealization(grammar, spaces, operators)
OperatorAlgebra(generators)
StructuredOperator
MatrixSemigroup(generator)

Relations are checked exactly with an exact backend or numerically with a residual and tolerance. Structured operators and finite semigroups provide executable domain realizations.

4. Project

PacketSystem(projectors)
PacketIR.from_dense(matrix, layout)
ConditionalExpectation(weights, partition)

Projection covers algebraic packets, structured computational blocks, signal phases, and finite conditional expectation packets.

5. Transfer

PacketOperator
TransferGraph
PolyphaseConvolution
PacketSemigroup

Transfer blocks reconstruct operators, compose by packet matrix multiplication, propagate through paths, and specialize to convolution, semigroup, Markov, and memory systems.

Computational choice

The computational namespace implements packet intermediate representations, structured operators, low rank recompression, support propagation, Krylov methods, precision allocation, exact finite tree compilation, scheduling, and optional CuPy execution.

The compiler is exact for the supplied finite additive tree. A precision assignment is exact for the supplied integer budget. Threshold structure discovery remains heuristic. Cost and roofline results are declared models rather than hardware guarantees.

Signal realization

Signals realize the address, project, transfer, refine, and reconstruct stages through finite polyphase coordinates and packet trees.

coordinates = polyphase_analyze(signal, order)
filtered = polyphase_convolution(signal, kernel, order)
tree = HaarPacketTree.analyze(signal, depth)
reconstruction = retained_packet_reconstruct(signal, packet, approximation)

The signal implementation includes exact finite coordinate identities, filter bank reconstruction, additive best basis selection, truncated power singular packets, stable separated jump recovery under declared assumptions, nonlinear residue products, and Smith chart coordinates.

A recovery certificate does not infer unknown smoothness or noise assumptions from the data. It records which finite threshold conditions were checked and which model hypotheses remain external.

Dynamics realization

Dynamics realizes operate, transfer, refine, constrain, complete, and reconstruct through finite matrix evolutions and finite coarse models.

semigroup = MatrixSemigroup(generator)
lumping = MarkovLumping(generator, partition)
posterior = BayesianUpdate(prior, likelihood)
fractional = fractional_power(operator, exponent)
reduction = FeshbachReduction(operator, retained)
kernel = reduction.memory_kernel()

The implementation includes semigroup and resolvent defects, compatible finite towers, exact Markov lumping, conditional expectation packets, Bayesian sufficiency, fractional spectral calculus, Green operators, finite Feynman Kac transfer, Schur Feshbach elimination, rational memory kernels, and stable phase type survival models.

Finite matrix theorems do not claim an unbounded continuum construction. Completion certificates state the topology, compatibility, and stability assumptions that were actually checked.

Analytic realization

The analytic realization uses residue projection to turn one entire function into a finite cyclic differential system. Reciprocal, logarithmic, Appell, and theta families then operate on the resulting coefficient coordinates.

packets = cyclic_packets(coefficients, order)
section = signed_section(order, residue, z)
coefficients = reciprocal_coefficients(order, parameter, degree)
zeros = positive_zero_string(order, count)

Exact coefficient operations and numerical zero operations remain distinct. A zero certificate checks only the returned residuals, ordering, and requested finite computation.

Quantum realization

Quantum systems realize seed, project, transfer, constrain, and reconstruct through observable algebras, symmetry sectors, circuit paths, moving packets, channels, error correction, and effective dynamics.

packets = cyclic_symmetry_projectors(symmetry, order)
circuit = CircuitPacketExpansion(gates, packet_systems)
channel = QuantumChannel(kraus_operators)
code = QuantumCode(projector, errors)
transport = KatoTransport(moving_packet_system)

Every operator acts on a declared finite Hilbert space. Continuum limits and gravitational endpoints require additional mathematics and are not represented as executable conclusions.

6. Refine

PacketRefinement
RefinementTower
WeightedTransferGraph
LatticePacketData
SmithCongruenceProblem
SemigroupRefinement

Implemented refinement includes coarse and fine packet relations, residue and CRT refinement, weighted transfer paths, filtered inverses, integral separation, Smith congruences, operator defects, semigroup Duhamel defects, and finite compatible towers.

7. Constrain

LinearObstructionProblem
LinearLiftProblem
PolynomialLiftProblem
TransportGraph
recover_separated_jumps
MarkovLumping
BayesianPartition

The finite constraint layer contains cokernel obstructions, prime power correction spaces, lift trees, finite holonomy, signal recovery under declared threshold hypotheses, exact Markov lumping, and fixed Bayesian sufficiency.

The nonlinear namespace implements packet jets, Faà di Bruno composition, rooted nonlinear operator trees, graded operations, nonlinear Schur and Lyapunov-Schmidt elimination, finite filtered lifting, and obstruction spectral sequences. Smooth Kato transport is shared by the nonlinear and quantum namespaces through one implementation.

8. Complete

There is no single universal completion object. A Banach inverse limit, a finite lift forest, a compatible semigroup tower, and a global arithmetic system require different runtime representations.

The library implements finite semigroup towers, finite hidden state realizations, finite survival systems, finite Kato transport, Euclidean transfer, and bounded Hamiltonian tower diagnostics. These finite representations do not construct a general continuum semigroup.

9. Reconstruct

Implemented reconstruction includes:

PacketCoordinates.reconstruct()
PacketOperator.reconstruct()
ResidueCoordinates.reconstruct()
PolyphaseCoordinates.reconstruct()
PeriodicFilterBank.synthesize()
PrunedPacketCoefficients.reconstruct()
retained_packet_reconstruct(...)
FeshbachReduction.memory_kernel()

Signal reconstruction separates smooth approximation error from packet estimation error. Memory reconstruction separates retained dynamics from an auxiliary hidden realization.

Execution classes

Examples:

• a SymPy Smith solve is exact
• a NumPy cokernel solve is numerical
• a jump recovery threshold can be checked numerically while regularity and noise hypotheses remain conditional
• exact finite Markov lumping is certified from the supplied matrix and partition
• a Feshbach resolvent identity is numerical unless exact matrix arithmetic is supplied
• future automatic packet discovery remains heuristic

Arithmetic and sequence realization

For arithmetic and sequence realizations, a seed is an initial coefficient family, address means a residue or Mahler coordinate, operate means convolution or recurrence evolution, project means local or residue selection, transfer means triangular or affine coordinate transport, refine means prime power or polyphase refinement, constrain means a congruence or recurrence relation, complete means a certified finite model, and reconstruct means exact coefficient or term recovery.

Deformation and Langlands realization

In deformation and arithmetic models, the unified stages acquire additional finite meanings. A residual class is a seed, a lift level is an address, reduction and transport are operators, packet cohomology and finite Hecke idempotents are projections, stable cores and support impose constraints, and a completion report records whether every required gate has been supplied.

forest = residue_forest(levels, prime=p)
report = forest.report()

packet = TameWeilPacket(FrobeniusOrbit(q, degree, exponent))
packet.relation_certificate().require()

completion = AutomorphicCompletionReport(gates)
result = completion.result()

The same words do not identify deformation support with a finite spectral packet. Fitting support is computed from a module presentation. Central idempotent packets are used only where a finite algebra actually decomposes as a product.

Finite field, tame, Heisenberg, orbit, Mackey, and Clifford computations may be exact. Ramification comparisons, automorphic induction plans, patching support, compatible propagation, and final completion remain conditional whenever they rely on external arithmetic existence or comparison theorems.

Portable output realization

Portable output supplies the stage after reconstruction. A mathematical value may be normalized, rendered, written, and compiled without changing its execution classification.

reconstruct
  -> normalize
  -> render
  -> write
  -> optionally compile LaTeX
  -> validate the written output

MathDocument is renderer independent. Ordinary prose is escaped. Equations contain trusted mathematical LaTeX. RawLatex is an explicit trusted escape hatch. export_data supports data formats and report formats. compile_latex is a local tool with shell escape disabled by default.

Export success certifies that the requested file was written and, for PDF, that the local compiler returned success and produced a nonempty PDF. It does not add a mathematical proof to the underlying value.

Advanced nonlinear realization

Nonlinear realization gives the constrain and complete stages a higher-order meaning. A derivative provider supplies the local multilinear data. Packet jets project each derivative slot. Faà di Bruno composes those jets. Rooted trees encode finite nonlinear iteration. Schur and Lyapunov-Schmidt reductions eliminate hidden or range variables. Filtered correction spaces organize successive obstructions. Finite spectral-sequence pages record which classes survive each filtration depth.

derivative data
  -> packet jets
  -> higher composition or rooted trees
  -> obstruction projection
  -> correction or elimination
  -> filtered lift
  -> finite spectral sequence
  -> conditional completion record

The final arrow is conditional whenever it refers to an infinite inverse limit. The runtime certifies finite compatibility but does not infer completeness or the existence of an infinite branch.