Consolidate numerical foundations and conditional formation research

Unify mathematical admission and reuse scoped algebra across mathematics,
physics and SDK consumers. Preserve native backend paths and public adapters,
replace eager physics exports with a checked lazy facade, and retire redundant
helpers with immutable recovery manifests.

Consolidate pair-state response and prepared positive-loss acquisition under
their declared complete laws. Share preparation and independent test kernels,
retain frozen protocols and counterexamples, and reduce the execution plan to
one queue with explicit resumption boundaries. Physical identification,
autonomous preparation and unique law selection remain open.

Fix overflow-based false verdicts in regularity, contractivity and dissipative
diagnostics. Reject unrepresentable arithmetic before comparisons and keep
tracker histories unchanged on rejected observations. Correct unsupported
universal-scale and threshold claims in the arithmetic documentation.

Validation:
- Routine engine/API gate: 9761 passed, 13 skipped.
- Full mathematics and dissipative consumer: 2005 passed.
- Broad physics integration: 1399 passed; final affected rerun: 168 passed.
- Final facade and module compatibility: 45 passed.
- CI-pinned Black/isort, flake8, physics stub, documentation integrity,
  6678 internal references, strict MkDocs, and isolated wheel build passed.
- Contributor mirror and frozen pair/formation proof owners preserved.

Validation is scoped to the selected contracts and local Python 3.13;
not every retained research campaign or CI platform was replayed.

Author: TNFR AI Agent

ARCHITECTURE.md

@@ -18,6 +18,11 @@ boundary, not a measured speedup. Other state/configuration dependencies remain
 part of each consumer's key or explicit invalidation contract. Graph views use
 fresh computations; concurrent mutation during a read is unsupported.
 
+The structural random-walk verifier reuses one resistance geometry within a
+call for resistance and commute-time projections. The comparison still uses
+the separate component-volume multiplication of materialized resistance.
+There is no persistent geometry cache; a later call re-admits the graph.
+
 LRU storage and callback handling have one implementation in
 [`utils/unified_cache.py`](src/tnfr/utils/unified_cache.py). The compatibility
 `InstrumentedLRUCache` and `ManagedLRUCache` names remain aliases in `utils.cache`;
@@ -61,6 +66,23 @@ graph invalidation and persistence retain their separate responsibilities.
 These are implementation responsibilities. The [documentation ownership map](docs/README.md)
 identifies the single maintained guide for each responsibility.
 
+The `tnfr.physics` public namespace resolves its exports on first access.
+One ordered owner map supplies `__all__`, discovery through `dir()` and lazy
+dispatch; resolved names are the actual owner objects, not proxies. Importing
+one field or observer therefore does not eagerly load unrelated research and
+runtime certificates. Direct module imports retain their own dependencies.
+The [facade controls](tests/physics/test_physics_facade_imports.py) check cold
+imports, public object identity and discovery independently of timing.
+The generated [typing facade](src/tnfr/physics/__init__.pyi) re-exports those
+same owners for static analyzers. Its
+[generator](scripts/generate_physics_stub.py) reads the runtime map without
+importing the package; the [testing guide](TESTING.md#code-quality-and-documentation-checks)
+owns regeneration and drift checks.
+
+Physics imports NumPy and NetworkX as core dependencies declared in
+`pyproject.toml`. Optional SciPy and acceleration paths retain their own
+availability checks and fallback behavior.
+
 ## Nodal execution flow
 
 The ordinary runtime composes the following configured operations. The diagram
@@ -203,7 +225,8 @@ lost in that conversion.
 | [`physics/source_relative_form.py`](src/tnfr/physics/source_relative_form.py) | Composes the form observer with an independently supplied held rate source; `Network.source_relative_form` is the adapter. |
 | [`physics/relational_observations.py`](src/tnfr/physics/relational_observations.py) | Native field, region and hypothetical support-change reports; separately, graph-independent coefficient/rate/sample-jet bounds. Sample adapters share stencils and outward error propagation. |
 | [`mathematics/linear_observation.py`](src/tnfr/mathematics/linear_observation.py) | Exact row-space realization and visible/hidden memory for a supplied rational generator; scoped lower bounds distinguish conservative observation from a supplied form-loss model. `physics/epi_memory.py` retains its own diffusion admission. |
-| [`mathematics/_exact_linear_algebra.py`](src/tnfr/mathematics/_exact_linear_algebra.py) | Shared exact products, inverses and rank algebra; compatibility physics imports delegate here. |
+| [`mathematics/_exact_linear_algebra.py`](src/tnfr/mathematics/_exact_linear_algebra.py) | Shared exact products, powers, inverses and semidefinite tests; compatibility physics imports, fixed-map network-stage powers and p-adic tower products delegate here. The tower adapter retains its mutable list rows. Products skip exact zero factors while retaining full rational output shapes and the consumers' product-call accounting. |
+| [`mathematics/krylov.py`](src/tnfr/mathematics/krylov.py) | Exact rational rank, Krylov reachability and Hankel moment calculations under their declared input/output premises. |
 | [`mathematics/_phase_resultant_chamber.py`](src/tnfr/mathematics/_phase_resultant_chamber.py) | Rational trigonometric/resultant bounds, principal-argument charts and supplied-rate kinematics. Reused geometry does not transfer a law. |
 | [`mathematics/_validated_taylor.py`](src/tnfr/mathematics/_validated_taylor.py) | Strict Picard tubes, Taylor remainders and initial-box propagation; the comparison kernel owns the shared 1–24-coordinate work limit. Each flow adapter retains its layout and other admission budgets. |
 | [`mathematics/_validated_metric.py`](src/tnfr/mathematics/_validated_metric.py) | Reuses Picard/Taylor machinery while retaining an SPD-metric radius between steps. The caller proves its whole-tube logarithmic bound; local truncation/rounding errors enter the same norm. Coordinate projections do not replace the retained uncertainty. |
@@ -239,11 +262,34 @@ The sine layer shares four boundaries:
    work and resultant kinematics. Mobility and regional/mediated readers reuse
    these primitives while retaining their own complete-law identifiers.
 3. `_sine_preparation.py` supplies weighted preparation and uncertainty bounds.
+   Its report reader re-admits original source primitives before the shared
+   row calculation. Fixed-source consumers can use that calculation after
+   admitting their law, support, capacities and exact rows, without constructing
+   an observation report whose derived rates they do not consume.
    Forecasts share validated Taylor/metric kernels and retain environmental
    coordinates, hidden initialization and original error associations.
 4. `sdk/relational_reports.py` delegates reports and projects exact JSON values;
    it neither installs a solver nor authenticates an evaluated response.
 
+`relational_sine_pair.py` owns exact global pair state, cancellation
+observations and finite exchange/receiver certificates. It depends directly on
+shared scalar, phasor and rational-interval admission. The larger
+`relational_sine_scale.py` retains replica geometry, grouping and emission.
+`relational_sine_replica_pulse.py` owns prepared internal pulses, their
+variations, splitting and work-response assessments, including the validated
+Taylor solver dependency. The scale module re-exports the established pair
+and pulse APIs for compatibility. Their defining classes/functions live in
+the corresponding owners; old imports resolve to those same objects,
+including SDK dispatch and older pickle class lookups. These boundaries
+change neither the complete laws nor the report schemas.
+
+`relational_sine_formation_response.py` combines the shared preparation kernel
+with exact Poisson and interval bounds for its fixed positive-loss sources.
+One fresh domain admission serves both source rows in a call; all public
+budgets are admitted anew. It keeps its own complete-law and clock premises,
+independent of the conservative pair certificates. No prepared-state or
+verdict cache substitutes for source admission.
+
 Downstream calculations re-admit the producer's primitive source and rebuild
 consumed gradients, rates, work and bounds. Forecast readers also check the
 source association at the actual validated time. The
@@ -321,7 +367,14 @@ or second transport law is introduced.
   classifications.
 - `tnfr.errors` provides contextual public exceptions.
 - `tnfr.mathematics` owns numerical backends and domain-neutral mathematical
-  structures.
+  structures. Its private `_complex_arrays` reader shares original-component
+  admission and range-safe normalization across spectral consumers.
+  `_integer_admission` owns exact integer/index admission for arithmetic
+  modules; primality dispatch and optional SymPy selection remain with
+  `number_theory`.
+- `tnfr.math.symbolic` owns supplied-law symbolic calculus when SymPy is
+  installed. Its public helpers are re-exported by `tnfr.mathematics` and
+  `tnfr`; those import paths share the same implementations.
 
 ### Structural dynamics
 

TESTING.md

@@ -274,6 +274,16 @@ For examples with `run_protocol` and `build_report`, reuse the real report throu
 a module fixture for numerical and scope checks. The shared
 [example helper](tests/example_protocol_helpers.py) checks `main` separately with
 a sentinel protocol and synthetic report, without rerunning a scientific producer.
+
+Reusable independent oracles belong in test support modules rather than in
+another collected test suite. The [pair oracle helper](tests/physics/_sine_pair_oracles.py)
+shares fine support, exact jets and high-precision reductions without importing
+production certificates or pytest fixtures. Each consuming suite retains its
+own complete-law assumptions, preparation budgets and module-scoped reports.
+When optimizing a shared certificate kernel, compare complete exact projections
+on representative available and unavailable cases as well as these independent
+controls; timing differences alone cannot establish unchanged evidence.
+
 [Runtime facade tests](tests/physics/test_runtime_facade_imports.py) own the two
 cold import orders for the P2/REMESH example families; those checks still use
 fresh processes and resolve the actual public APIs.
@@ -354,6 +364,14 @@ Pass affected files for a focused edit where supported. Black and isort use
 in [the workflow guide](.github/WORKFLOWS.md). `make format` modifies files;
 it is not a read-only validation step.
 
+The lazy `physics` namespace has one maintained export map and a generated
+typing facade. After changing that map, run
+`python scripts/generate_physics_stub.py --write`; validate it with
+`python scripts/generate_physics_stub.py --check`. The generator reads the
+source AST without importing TNFR, and the
+[facade tests](tests/physics/test_physics_facade_imports.py) check typed exports,
+runtime object identity and cold import boundaries.
+
 Optional pre-commit setup requires installing `pre-commit` separately before
 `pre-commit install`. [.pre-commit-config.yaml](.pre-commit-config.yaml)
 configures Black, isort, pydocstyle and a local Bandit-command guard. That guard