Conditional balances from scalar reactions

reaction_balances(compiled) derives exact integer coordinate-sum identities from named scalar unit-transfer reactions. It does not evaluate propensities or change the compiled equations. The report distinguishes represented closed transfers, explicit external sources, unsupported contributions and scientific applicability conditions.

import numpy as np
from op_system import compile_spec, reaction_balances

compiled = compile_spec({
    "kind": "transitions",
    "state": ["a", "b"],
    "transitions": [
        {"name": "transfer", "from": "a", "to": "b", "rate": "k"},
        {"name": "inflow", "to": "a", "rate": "q"},
    ],
})
report = reaction_balances(compiled)
law = report.laws[0]
assert report.fully_covered
assert law.coefficients == (1, 1)
assert not law.closed
assert law.sources[0].reaction_name == "inflow"
assert law.sources[0].coefficient == 1

derivative = compiled.eval_fn(0.0, np.asarray([2.0, 3.0]), k=0.2, q=0.5)
np.testing.assert_allclose(np.dot(law.coefficients, derivative), 0.5)
# The identity is d(a + b)/dt = q, not a closed conservation law.

Removing the inflow transition gives a closed coordinate total. Changing its rate, its time dependence or the state dependence of the internal transfer does not change the incidence identity. An isolated state receives its own constant-coordinate law when it has no represented source or transfer.

Supplied model	fully_covered	Issued law
Complete scalar unit transfers	True	Closed component totals
Those transfers plus source-only reactions	True	Explicit open balances for affected components
Omitted, unsupported or inconsistent dynamics	False	No laws; inspect coverage_gaps

Exact derivation and recorded meaning

An internal column has -1 at its consumed state and +1 at its target. The condition c.T @ S_internal = 0 therefore requires equal coefficients on states joined by a transfer. Indicators of the undirected connected components form an exact integer basis. The implementation traverses this graph without adding a symbolic/SVD dependency or selecting a numerical rank tolerance. This conclusion depends on the unit-transfer contract; it does not generalize to arbitrary molecular stoichiometry or all nonlinear first integrals.

Every coefficient vector follows report.state_names. Component names follow compiled first-state order and are stable within that report; they are not persistent identities across revised models. reaction_names records every covered channel. Each ReactionBalanceSource names a source-only channel, its target and its integer propensity multiplier. Resolve its numerical flux using the matching compiled reaction/parameters. The records contain no callbacks or parameter values and can be inspected, hashed and pickled.

origin="compiled_unit_transfers" records the derivation method. It differs from a user-authored InvariantCheck and from a numerical residual check. The report is a conditional structural artifact, without authenticated provenance or a claim about an arbitrary replacement runtime callback.

Scientific metadata and coverage

state_meta and axis_meta are detached tuples of immutable declaration records. Missing units or representation labels remain unknown. Each law and report includes applicability assumptions: summed coordinates need compatible units/representations, and execution must faithfully realize these channels without additional state updates, fluxes, clipping, projections, resets or impulses. Unit labels are compared for known component conflicts; no dimensional analysis or unit conversion is performed. Disconnected components can carry different unit labels.

A numeric coordinate sum is not automatically a physical mass or spatial integral. Intensive quantities need their physical measures before making that interpretation; no mesh weights are inferred here. Use state representations and numerical qualification for declared measures and independent runtime checks.

This first analysis requires scalar templates and complete, uniquely named reaction artifacts matching normalized transition identities. Original ReactionGap records retain their origin and selectors inside BalanceCoverageGap. Templated/routed/shifted/spatial layouts, operators, unknown or duplicate identities, missing artifacts, known conflicting scientific labels, and declared sources/couplings/constraints blocks (including empty reserved blocks) prevent all issued laws. Unsupported contributions are never assigned zero flux. Reaction reactant/dependency completeness is a separate contract: it constrains some stochastic methods, while this analysis uses the represented net unit-transfer columns.

Weighted spatial balances, boundary/operator fluxes and more general stoichiometric derivations are not implemented by this analysis. The analysis leaves normalized/compiled storage and numerical evaluation unchanged, and adds no engine, framework or statistical dependency.