UV renormalization architecture

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current

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Ownership decision required

Review

2026-08-17 · c9f4e32acd2c

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reviewed

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• gammalooprs::uv::UVOrchestrator
  crates/gammalooprs/src/uv/orchestrator.rs · digest b57273870e63

Reviewed: 2026-08-17 against c9f4e32acd2c

Lifecycle: Current implementation architecture. Unsupported prescription paths are recorded explicitly under Current Boundaries.

Scope

This document records the implemented UV-renormalization invariants shared by the legacy DAG-forest and hedge-poset orchestrators. Historical revision comparisons, temporary log locations, and external-reference convention tables belong in tests or investigation records rather than in the current architecture.

Orchestrators

UVgenerationSettings::orchestrator selects one of three execution modes:

• legacy_dag_forest computes the established Approximation DAG and remains the reference implementation for connected graphs.
• hedge_poset is the default. It unfolds commuting UV operations into trace levels and stores their results in a compute store. It supports disconnected unions.
• compare computes both connected-graph backends, compares normalized expressions, and returns the legacy result. It is a validation mode, not a request to execute the returned integrand with the hedge-poset backend.

The backends share analytic operations and result types. Scheduling, dependency lookup, caching, and disconnected-component composition belong to the orchestrator because their traversal models differ.

Scheme and Computation Ownership

The renormalization prescription belongs to the Spinney. A compute node does not copy that policy; it stores computed values:

• the signed local four-dimensional counterterm;
• the integrated counterterm projections;
• cut-dependent local three-dimensional and final integrands.

This separation lets the same integrated result be consumed differently without storing scheme-specific copies.

Signs and Integrated Projections

Each UV operation supplies its own subtraction sign. In particular, the local four-dimensional limit constructs the next counterterm as a signed -T(...) operation. Nested signs therefore arise from recursive operation composition, not from terminal depth or parity.

For a connected counterterm, integration stores a Laurent series from which two physical projections are obtained:

pole projection:               negative powers of integral(local)
finite-counterterm projection: -nonnegative powers of integral(local)

The second projection includes ε⁰ and the retained positive ε powers, not only the strictly finite coefficient. Those positive powers are required when factorized component series multiply into the finite part of a disconnected counterterm.

No extra terminal parity is applied. Adding a sign derived from operation count would duplicate signs already supplied by the local operations.

The projections are used as follows:

• a non-root PolePart dependency recurses through the completed integrated pole only;
• a root PolePart result combines its local term with the integrated pole;
• an MUV dependency combines its local term with the signed finite counterterm;
• terminal renormalization output selects the pole or finite projection from the source Spinney's prescription;
• final three-dimensional integrand assembly always localizes the signed finite counterterm, regardless of the terminal prescription.

The last distinction is essential: changing terminal output policy must not remove the finite addback required when assembling an integrable 3D expression.

Disconnected Composition

Disconnected composition depends on which object is being combined.

Four-dimensional local terms

Each connected component first obtains its complete recursive counterterm, including the projection selected by that component's Spinney. The union's local counterterm is the product of those full component counterterms.

Integrated terms

Integrated counterterms are scalars and factorize over connected components. The stored disconnected value provides the product of component pole projections and the product of component finite-counterterm projections.

The aggregate is sufficient when every component uses the same terminal prescription. At terminal use, the hedge-poset backend nevertheless returns to the component nodes, selects each component's own pole or finite projection, and only then multiplies them. This also handles mixed prescriptions without storing scheme-specific variants of the aggregate.

Three-dimensional local terms

Cut CFF structures need not factorize over disconnected components. Multiplying complete per-component local results would also multiply the common root and can duplicate or incorrectly separate intertwined CFF structure.

The hedge-poset backend therefore replays component-local operation paths from their common root. At a union it:

1. obtains the replay states for every component;
2. forms compatible combinations of their integrated prefixes;
3. joins those prefixes in the trace/Foata representation;
4. applies each component's local operations to the shared incoming state.

The construction operates over an arbitrary number of components and is not special-cased for a two-component spectacles graph.

Marker Representation

UV markers are diagnostic symbolic structure. Approximation, integration, series, and truncation are distinct canonical function heads applied to the history of current/given subgraphs. Spenso printing supplies the compact K[...], angled-bracket, Σ(...), and truncation notation while structured DOT uses the same underlying atoms. Typst presentation combines Symbolica's Typst arithmetic mode with those Spenso shorthands; truncation is emitted as op("Tr")(…), which is valid Typst math, while compact terminal output remains Tr(…).

DOT full_num fields contain Typst math fragments because the drawing template evaluates them in math mode. Per-graph renormalization .typ files wrap the same fragment in a display-math document, while the accompanying .txt files remain the round-trippable Symbolica representation.

Subgraph labels must be created through the subgraph symbol factory so every consumer receives the same symbol metadata and print behavior.

Current Boundaries

• Local four-dimensional and integrated counterterm generation currently support only MUV and PolePart. The local three-dimensional kernel has an IR branch, but integrated IR generation is not implemented; VaccuumLimit and OS are also unsupported, while Unsubtracted is expected to be filtered out before these operations.
• Parametric integrand generation currently supports final 3D output only. FourD is used by integrated renormalization internally but is rejected by the parametric orchestrator.
• The legacy DAG executor constructs union-shaped nodes but does not execute multi-parent unions. Disconnected generation therefore requires the hedge-poset backend.
• compare cannot validate disconnected unions until the legacy backend has a corresponding execution path.
• External systems such as RQFT may use different propagator or vertex sign conventions. Those graph-specific conversion factors are recorded next to the reference expressions in renormalization tests; they are not part of the internal recursive sign convention.