Generalize database-derived empirical scaling laws into a machine-independent VAFT API

[HongSik-Yun-Fusion]

Summary

Generalize VAFT's current confinement-scaling support into a machine-independent architecture for database-derived empirical scaling laws, operational boundaries, and reduced predictive relations used across tokamak physics.

The scope is broader than energy confinement. Representative targets include:

• energy confinement time;
• L–H transition power threshold;
• minimum L–H transition density;
• Greenwald and modern density-limit relations;
• separatrix density;
• SOL heat-flux width;
• divertor / ELM heat-load scalings;
• pedestal empirical relations;
• reduced operational-space boundaries;
• detachment-related empirical relations;
• other multi-machine database-derived physics relations.

The main goal is not to add many formulas at once. It is to establish a scientific API contract that allows physically different empirical relations to be represented, discovered, evaluated, and later applied to ODS/IDS data without embedding machine-specific assumptions in the formulas themselves.

---

1. Motivation: empirical scaling is not only a confinement problem

VAFT currently represents confinement scalings through a shared coefficient table and a common evaluator, conceptually:

_SCALING_COEFS
    -> confinement_time_from_engineering_parameters(...)
    -> OMAS wrapper extracts engineering parameters

This works reasonably well for a small family of confinement laws such as ITER89P, IPB98(y,2), NSTX scalings, and ST multi-machine scalings.

However, tokamak empirical physics spans a much wider set of quantities:

global confinement
    tau_E

regime transition
    P_LH
    n_e,min for the L-H transition

operational limits
    n_G
    modern density-limit boundaries
    beta_N-related reference limits

edge / pedestal
    n_e,sep
    pedestal width / height relations

SOL / exhaust
    lambda_q
    divertor heat load

transients
    ELM energy / parallel fluence

operational-space classification
    SepOS-type boundaries

These relations are all empirical or reduced-order in a broad sense, but their predictor semantics differ substantially.

For example, these are not interchangeable quantities:

n_e_line_avg
n_e_volume_avg
n_e_ped
n_e_sep

p_heat
p_abs
p_loss
p_net
p_transport
p_sol

b_t_axis
b_t_ref
b_pol_omp

kappa_boundary
kappa_area
plasma_surface_area

Therefore, extending the current confinement coefficient dictionary into a universal empirical-scaling registry would eventually obscure rather than clarify the physical contract.

The new architecture should preserve VAFT's established layer boundaries:

vaft.machine_mapping
    machine-specific source, hardware, calibration, geometry and shot-era policy

vaft.formula
    pure machine-independent physical / numerical formulas

vaft.process
    reusable scientific processing, fitting and inference

vaft.omas / vaft.imas
    ODS/IDS-aware extraction and schema-facing adapters

vaft.validation
    scientific applicability / validity assessment

Do not introduce a new top-level vaft.physics package unless a later architecture decision independently justifies it.

---

2. Representative scaling laws

The following examples are not all required in the first PR. They define the design envelope: the API should be able to represent these different physical signatures without forcing them into one universal vector of generic engineering parameters.

2.1 L–H transition power — Martin 2008

A widely used ITPA multi-machine L–H transition threshold scaling is

$$
P_{L-H}^{2008},[\mathrm{MW}]

0.0488,
\bar n_e^{0.717}
B_t^{0.803}
S^{0.941},
$$

with approximately

n_e_bar : line-averaged electron density [10^20 m^-3]
B_t     : toroidal magnetic field [T]
S       : plasma surface area [m^2]

A natural Formula API is therefore closer to

lh_transition_power_martin_2008(
    n_e_line_avg,
    b_t,
    plasma_surface_area,
)

than to

evaluate_scaling(n_e=..., B=..., area=..., model="Martin2008")

because the signature itself documents which density, magnetic field and geometric quantity the model expects.

The Martin scaling also has regime/configuration applicability assumptions and is normally interpreted on the high-density branch. This already motivates separating:

numerical formula evaluation

from

validity / applicability assessment

Reference:

• Y. R. Martin et al., Power requirement for accessing the H-mode in ITER, Journal of Physics: Conference Series 123 (2008).

---

2.2 Updated metallic-wall L–H threshold — ITPA TC-26 / Delabie 2026

A recent metallic-wall multi-machine regression gives a relation of the form

$$
\frac{P_{L-H}}{S}

0.0441,
B_t^{0.580}
\bar n_e^{1.08}
\frac{2}{M_{\mathrm{eff}}^{0.975}}
D,
$$

where, schematically,

P_LH / S : transition power per plasma surface area
B_t      : toroidal magnetic field
n_e_bar  : line-averaged electron density
M_eff    : effective ion mass
D        : divertor-configuration factor

This predicts the same broad target as Martin 2008 but requires a different physical signature, including isotope mass and a categorical/configuration factor.

The API should therefore naturally allow distinct public functions such as

lh_transition_power_martin_2008(...)
lh_transition_power_delabie_2026(...)

rather than hiding materially different physical models behind one undifferentiated argument set.

Reference:

• E. Delabie et al., updated ITPA metallic-wall L–H transition threshold scaling, Nuclear Fusion 66 (2026).

---

2.3 Minimum density for the L–H transition

The L–H threshold is non-monotonic with density, with a low-density branch where the threshold rises again.

One semi-empirical minimum-density relation has the form

$$ \langle n_{e,\min}\rangle \propto A_{\mathrm{eff}}^{0.07} I_p^{0.32} B_0^{0.62} a^{-1.32} R_0^{0.35} \frac{(1+\kappa^2)^{0.36}}{\kappa^{0.8}}. $$

This is useful not merely as another density predictor, but as a guard on which density branch is being used for a threshold law.

A future analysis should be able to compose independent empirical relations explicitly:

p_lh = lh_transition_power_martin_2008(...)
n_lh_min = lh_transition_minimum_density(...)
branch_ratio = n_e_line_avg / n_lh_min

Do not hide one empirical relation inside another unless that composition is itself the published model.

---

2.4 Energy confinement — IPB98(y,2), ITPA20, and ST scalings

Confinement remains an important reference family.

IPB98(y,2) has the familiar structure

$$
\tau_E

C,
I_p^{0.93}
B_t^{0.15}
\bar n_e^{0.41}
P^{-0.69}
R^{1.97}
\epsilon^{0.58}
\kappa^{0.78}
M^{0.19}.
$$

A more recent ITPA20 engineering scaling is approximately

$$
\tau_{E,\mathrm{th}}

0.053,
I_p^{0.98}
B_t^{0.22}
\bar n_e^{0.24}
P_{l,\mathrm{th}}^{-0.669}
R_{\mathrm{geo}}^{1.71}
(1+\delta)^{0.36}
\kappa_a^{0.80}
\epsilon^{0.35}
M_{\mathrm{eff}}^{0.20}.
$$

These models demonstrate why semantic input names matter even within confinement physics:

p_loss
p_l_th
p_abs
p_heat

are not automatically equivalent, and

kappa_boundary
kappa_area

need not represent the same shape definition.

Long term, confinement relations should follow the same explicit semantic contract as the new empirical-model families rather than relying only on a shared coefficient dictionary.

References:

• ITER Physics Basis / IPB98(y,2).
• G. Verdoolaege et al., updated ITPA H-mode energy-confinement scaling, 2021.

---

2.5 Greenwald density limit

The Greenwald relation is a simple but important operational reference:

$$
n_G[10^{20},\mathrm{m^{-3}}]

\frac{I_p[\mathrm{MA}]}{\pi a^2[\mathrm{m^2}]},
$$

with

$$ f_G = \frac{\bar n_e}{n_G}. $$

VAFT already provides Greenwald-related Formula functions, making this a useful existing reference for the generalized architecture.

The API must preserve the physical definition of the comparison density. n_e_line_avg should not silently become n_e_volume_avg merely because both have density units.

---

2.6 Modern density-limit boundaries

Modern multi-machine work has proposed operational boundaries that go beyond a simple Greenwald fraction.

For example, an L-mode multi-machine analysis reports an edge-collisionality / beta boundary of the approximate form

$$ \nu_{\mathrm{limit}} = 3.5,\beta^{-0.40}. $$

This kind of model is better interpreted as an operational boundary or classifier than as a scalar engineering prediction.

The architecture should therefore eventually support both:

predicted = model(...)

and higher-level constructions such as

margin = observed - boundary(...)

without forcing every empirical model into one identical user-facing abstraction.

---

2.7 SOL heat-flux width — Eich 2013

One of the most important multi-machine H-mode SOL relations is

$$
\lambda_q^{H\text{-mode}}[\mathrm{mm}]

0.63,
B_{\mathrm{pol,OMP}}^{-1.19}.
$$

The predictor is specifically the outboard-midplane poloidal magnetic field.

This is a strong example of why a generic B argument is insufficient:

b_t_axis
b_t_ref
b_pol_omp

are physically different quantities.

A natural Formula API is

heat_flux_width_eich_2013(
    b_pol_omp,
)

This model is particularly attractive for VAFT because the workflow can eventually be

equilibrium
    -> B_pol at OMP
    -> Eich lambda_q

The underlying database also included spherical tokamak data such as NSTX and MAST, making it a useful architecture test for both conventional and low-aspect-ratio machines.

Reference:

• T. Eich et al., Scaling of the tokamak near the scrape-off layer H-mode power width and implications for ITER, Nuclear Fusion 53 (2013) 093031.

---

2.8 H-mode separatrix density

Separatrix electron density connects pedestal, SOL, detachment, and confinement physics.

Recent multi-machine work across C-Mod, ASDEX Upgrade, and JET considers relations of the form

$$
n_{e,\mathrm{sep}}

C_{\mathrm{dev}}
p_{0,\mathrm{div}}^{\alpha_p}
I_p^{\alpha_I}
B_t^{\alpha_B}
\left(\frac{P_{\mathrm{SOL}}}{R_{\mathrm{geo}}}\right)^{\alpha_{P/R}}
a_{\mathrm{geo}}^{\alpha_a}.
$$

Representative input semantics include

divertor_neutral_pressure
p_sol
r_geo
a_geo
b_t

which are very different from confinement or L–H threshold predictors.

This is also an important applicability example: a relation trained mainly on conventional-aspect-ratio C-Mod/AUG/JET data may be mathematically evaluable for MAST-U or VEST while still being out of the original regression domain.

Therefore the architecture must distinguish

numerically evaluable

from

supported by the original database domain

Reference:

• Silvagni et al., A predictive formula for the H-mode electron separatrix density: Bridging regression and physics-based models across C-Mod, AUG and JET tokamaks, 2026 preprint / submitted work.

Before implementation, confirm the final publication status and coefficients.

---

2.9 ELM parallel energy fluence

Transient ELM heat loading also has multi-machine empirical relations.

A representative JET / MAST / ASDEX Upgrade relation is

$$
\epsilon_{\parallel,\mathrm{ELM}}

0.28,
n_{e,\mathrm{ped}}^{0.75}
T_{e,\mathrm{ped}}^{0.98}
(\Delta E_{\mathrm{ELM}}[%])^{0.52}
R_{\mathrm{geo}},
$$

with the published engineering-unit convention documented by the source.

This requires predictor semantics such as

n_e_ped
t_e_ped
relative_elm_energy_loss
r_geo

By this point, density alone already appears in at least four distinct scientific forms:

n_e_line_avg
n_e_volume_avg
n_e_sep
n_e_ped

This is one of the clearest reasons to make semantic naming part of the API contract.

Reference:

• T. Eich et al., multi-machine ELM divertor energy-fluence analysis, Nuclear Materials and Energy 12 (2017).

---

2.10 Pedestal width — EPED-type reduced relation

Not every reduced relation is a pure database power law.

The EPED pedestal-width relation is commonly written approximately as

$$ \Delta_\psi = 0.076\sqrt{\beta_{p,\mathrm{ped}}}. $$

This is better regarded as a hybrid empirical / physics-informed reduced relation than as a generic engineering regression.

Long term, VAFT should be able to distinguish concepts such as

pure empirical regression
semi-empirical relation
physics-informed reduced model
operational boundary

without requiring a new top-level architecture for each category.

Reference:

• P. B. Snyder et al., EPED pedestal-model literature.

---

2.11 Separatrix Operational Space — SepOS

SepOS is not one scalar power law. It is a reduced operational-space framework involving quantities such as

n_e_sep
T_e_sep
alpha_t
collisionality
pressure-related edge variables

and is used to describe boundaries between operating regimes such as H-mode, QCE-like operation, and density-limit-related regions.

This shows that future empirical/reduced-model infrastructure may need to support

scalar prediction
boundary
classifier
operational-space coordinate

as related but distinct concepts.

Implementing SepOS itself is not required in the first PR. The architecture should simply avoid making it impossible later.

---

3. What these examples imply for the API

The examples above show that the central abstraction should not be one universal vector of engineering parameters.

Model	Representative inputs
Martin L–H	n_e_line_avg, b_t, plasma_surface_area
TC26 L–H	n_e_line_avg, b_t, m_eff, divertor configuration
minimum L–H density	i_p, b_t, a_minor, r_geo, kappa, m_eff
IPB98 / ITPA20	i_p, b_t, density, power, geometry, shaping, mass
Greenwald	i_p, a_minor
modern density limit	edge collisionality, beta, related boundary variables
Eich lambda_q	b_pol_omp
separatrix density	divertor_neutral_pressure, p_sol, geometry, b_t
ELM fluence	n_e_ped, t_e_ped, relative ELM loss, r_geo
EPED width	beta_p_ped

The target abstraction is therefore:

$$ \boxed{ \text{model-specific physical signature} + \text{canonical semantic variable names} + \text{shared discovery / provenance / validation infrastructure} } $$

---

4. Pure empirical laws belong to vaft.formula

Published empirical relations should be implemented as machine-independent pure functions.

Conceptually:

lh_transition_power_martin_2008(
    n_e_line_avg,
    b_t,
    plasma_surface_area,
)

heat_flux_width_eich_2013(
    b_pol_omp,
)

separatrix_density_silvagni_2026(
    divertor_neutral_pressure,
    b_t,
    p_sol,
    r_geo,
    a_geo,
)

Formula functions must:

• accept scalar or NumPy-compatible physical inputs;
• return physical numerical outputs;
• not read ODS/IDS;
• not depend on shot number;
• not contain VEST, MAST-U, KSTAR, ITER, or other machine-specific branches;
• not perform database loading;
• not decide whether a discharge lies inside the original regression domain.

The function signature itself should communicate the physical definition of each predictor.

Avoid a primary Formula API such as

evaluate_scaling(
    n_e=...,
    P=...,
    B=...,
    model="...",
)

when n_e, P, and B mean different quantities for different published laws.

---

5. Canonical semantic variable names

Define and use consistent names for common empirical-model inputs.

Examples:

i_p

n_e_line_avg
n_e_volume_avg
n_e_sep
n_e_ped

t_e_sep
t_e_ped

p_heat
p_abs
p_loss
p_net
p_transport
p_sol

b_t_axis
b_t_ref
b_pol_omp

r_geo
a_minor

kappa_boundary
kappa_area
triangularity

plasma_surface_area
divertor_neutral_pressure

m_eff

Do not use a shorter generic name when multiple physically distinct definitions are common in the literature.

The public signature is part of the scientific contract.

---

6. Unit convention

Where practical, new Formula API inputs and outputs should follow a consistent canonical convention, preferably SI.

Published equations should remain documented in their original engineering units.

Examples:

published:
    lambda_q [mm]

VAFT public result:
    lambda_q [m]

and

published:
    n_e [10^20 m^-3]

VAFT public input:
    n_e [m^-3]

For each implemented law, document:

published unit convention
internal conversion
returned unit

and add regression tests that reproduce the published expression numerically.

Do not detach a fitted prefactor from the unit convention in which it was derived.

---

7. Reuse the existing Formula catalog

VAFT already provides a docstring-derived discovery layer:

vaft.formula.describe(...)
vaft.formula.search(...)
vaft.formula.list_formulas(...)
vaft.formula.categories()

Do not create a parallel empirical-scaling registry solely for discovery or documentation.

New models should be discoverable through the existing catalog, for example:

vaft.formula.describe(
    "transition.lh_transition_power_martin_2008"
)

vaft.formula.search("Eich")

Potential future catalog metadata may include distinctions such as

empirical
semi_empirical
reduced_model
operational_boundary

and database / regime information, but the implementation should avoid creating a second source of truth unless the existing Formula contract genuinely cannot express the required metadata.

---

8. Organize Formula modules by physics domain

Do not create one large scaling.py containing unrelated empirical models.

Prefer flat physics-domain modules consistent with the current Formula architecture:

vaft/formula/
    equilibrium.py
    stability.py
    transition.py
    edge.py
    exhaust.py

Conceptual ownership:

transition.py
    L-H / H-L transition relations
    minimum transition density
    related regime-transition empirical laws

edge.py
    separatrix quantities
    pedestal-related empirical relations
    edge operational relations

exhaust.py
    SOL heat-flux width
    divertor heat-load relations
    ELM energy / fluence relations
    detachment-related reduced relations where appropriate

Existing confinement functions may initially remain in equilibrium.py for compatibility.

A later issue may decide whether confinement/transport relations justify a dedicated transport.py.

Avoid nested packages solely for organizational symmetry unless the public responsibilities later justify them.

---

9. Separate Formula evaluation from ODS/IDS input resolution

Formula functions must not know how ODS/IDS stores the required physical quantity.

A later schema-aware adapter may live conceptually under

vaft.omas.scaling
vaft.imas.scaling

with APIs such as

inputs = vaft.omas.scaling.resolve_inputs(
    ods,
    model="transition.lh_transition_power_martin_2008",
    time_slice=...,
)

result = vaft.omas.scaling.evaluate(
    ods,
    model="transition.lh_transition_power_martin_2008",
    time_slice=...,
)

The intended flow is

published formula signature
        ↓
required canonical physical quantities
        ↓
ODS/IDS resolvers
        ↓
pure Formula API
        ↓
prediction

The first implementation does not need to complete this adapter, but the Formula API must be designed so that the adapter can later be added without changing model semantics.

---

10. ODS resolution should be based on physical semantics

A future resolver may map canonical semantic names to ODS extraction logic, conceptually:

_INPUT_RESOLVERS = {
    "i_p": resolve_i_p,
    "b_t_axis": resolve_b_t_axis,
    "n_e_line_avg": resolve_n_e_line_avg,
    "n_e_volume_avg": resolve_n_e_volume_avg,
    "n_e_sep": resolve_n_e_sep,
    "p_loss": resolve_p_loss,
    "p_sol": resolve_p_sol,
    "plasma_surface_area": resolve_plasma_surface_area,
    "b_pol_omp": resolve_b_pol_omp,
}

Do not create per-machine wrappers such as

compute_vest_lh_threshold(...)
compute_mastu_lh_threshold(...)

unless the underlying empirical relation itself is genuinely machine-specific.

Machine-specific acquisition, calibration, geometry and shot-era knowledge should have been resolved before the generic scaling evaluator is called.

---

11. Remove machine-specific assumptions from generic scaling extraction

Generic scaling evaluation must not contain assumptions such as

R_ref = 0.4  # VEST reference

or conditionals such as

if machine == "VEST":
    ...

The intended data flow is

machine-specific raw/source knowledge
        ↓
machine_mapping
        ↓
canonical ODS / IDS physical quantities
        ↓
generic scaling input resolution
        ↓
machine-independent Formula API

If a published model requires a physical quantity that is unavailable, the resolver should report it as missing rather than inventing a machine-specific approximation.

---

12. Do not silently substitute physically different predictors

The resolver must distinguish physically different measurements and derived quantities.

Examples:

measured interferometer line-average density
profile-derived midplane line-average density
volume-averaged density
pedestal density
separatrix density

Likewise:

p_abs
p_loss
p_net
p_transport
p_sol

Recommended policy:

1. exact requested physical definition exists
      -> use it

2. formally equivalent derivation exists
      -> derive it and retain provenance

3. only a physically different proxy exists
      -> do not silently substitute

Explicit caller overrides may later be supported, but they must remain explicit and provenance-aware.

---

13. Preserve input provenance

ODS-aware scaling evaluation should eventually retain not only predictor values but also their provenance.

Conceptually:

ResolvedInput(
    name="n_e_line_avg",
    value=...,
    unit="m^-3",
    source="interferometer",
    representation="measured",
    time=...,
)

versus

ResolvedInput(
    name="n_e_line_avg",
    value=...,
    unit="m^-3",
    source="core_profiles",
    representation="derived",
    method="midplane_profile_integral",
)

Pure Formula functions should continue to return numerical values only.

Structured provenance belongs to the ODS/IDS-aware evaluation layer.

---

14. Separate numerical evaluation from applicability assessment

A published empirical law may be mathematically evaluable outside the parameter domain of its original regression database.

These are different questions:

Can the formula be evaluated?

and

Is this application supported by the original database?

Formula functions answer only the first.

A later validation layer should assess the second, conceptually under

vaft.validation.scaling

using evidence such as

aspect ratio
magnetic-field range
density range
power range
plasma regime
divertor configuration
species / isotope
machine class
database coverage

Possible states may include

in_domain
extrapolated
unknown
invalid

Do not embed machine-specific applicability warnings directly in pure Formula functions.

---

15. Separate published-law evaluation from new database inference

There are two distinct workflows.

A. Evaluate an existing published relation

vaft.formula
    published empirical / reduced relation

B. Infer a new scaling from experimental data

vaft.process
    regression
    bootstrap
    model comparison
    uncertainty / identifiability analysis

For example, the VEST confinement analysis in #548 belongs primarily to the second category.

Conceptual future APIs may include

fit_power_law_scaling(...)
bootstrap_scaling(...)
compare_scaling_models(...)

under the appropriate processing domain.

Do not mix newly fitted VEST-specific coefficients into the generic published-model registry unless they are intentionally promoted into a separately documented empirical model with explicit provenance and applicability.

---

16. Support the same model across machines and datasets

The long-term target is that the same pure Formula function can be evaluated from

VEST ODS
MAST-U ODS
KSTAR ODS
ITER / reactor design point
multi-machine DataFrame
published database row
user-supplied scalar inputs

without changing the model implementation.

The central interface should therefore be

canonical semantic physical variables
               ↓
          Formula model

rather than machine-specific wrappers.

---

17. Initial implementation set

The first implementation should deliberately use models with different physical signatures.

Recommended minimum:

Transition

Martin 2008 L-H threshold

Exhaust

Eich 2013 H-mode lambda_q

Edge

one qualified H-mode n_e,sep relation

Existing:

Greenwald density

can serve as an operational-limit reference.

Optional additional model:

ITPA20 confinement

This set exercises, at minimum:

line-average density
surface area
toroidal field
OMP poloidal field
SOL power
divertor pressure
machine geometry

and is therefore a meaningful test of whether the abstraction is genuinely general.

---

Proposed implementation phases

Phase 1 — Formula API contract

• define canonical semantic predictor naming;
• add formula.transition;
• add formula.edge;
• add formula.exhaust;
• implement a small representative literature-model set;
• integrate the new functions with the existing Formula catalog;
• document original unit conventions, databases, validity and references;
• add published-value regression tests.

Do not add generic ODS auto-resolution in this phase.

Phase 2 — ODS/IDS model evaluation

Add generic input resolution and provenance-aware evaluation, conceptually:

vaft.omas.scaling
vaft.imas.scaling

resolve_inputs(...)
evaluate(...)
ResolvedInput
ScalingEvaluation

Phase 3 — applicability / validation

Add regression-domain and regime applicability assessment.

Phase 4 — database inference

Add reusable scaling fitting / model-comparison operations under the appropriate vaft.process domain, reusing generic statistics and validation infrastructure.

---

Tests

Formula-level reference tests

For every implemented empirical law:

• reproduce at least one numerical value independently calculated from the published expression;
• verify the published engineering-unit expression and the VAFT canonical-unit implementation are equivalent;
• verify scalar and NumPy-array evaluation where applicable;
• verify invalid non-physical inputs are handled consistently;
• verify no ODS/OMAS/IMAS dependency exists.

Semantic-definition tests

Explicitly protect distinctions such as

n_e_line_avg != n_e_volume_avg
n_e_line_avg != n_e_sep
n_e_ped != n_e_sep
p_sol != p_loss
b_pol_omp != b_t_axis
kappa_area != kappa_boundary

Catalog tests

Verify new models are discoverable through

vaft.formula.describe(...)
vaft.formula.search(...)
vaft.formula.list_formulas(...)

and retain their empirical/reference metadata.

Import tests

Preserve Formula lazy-loading behavior. Adding new physics modules must not make

import vaft.formula

eagerly load unrelated heavy modules.

---

Relationship to existing work

Formula catalog

Reuse the existing Formula catalog and docstring parser rather than introducing a second model-discovery framework.

#103 — architectural layer boundaries

Follow the established separation between

machine_mapping
formula
process
omas / imas

and avoid reintroducing machine-specific behavior into generic physics functions.

#252 — processing contracts and provenance

Any future ODS-aware scaling extraction/evaluation should follow the established provenance and input/output semantic contracts.

#253 — validation architecture

Applicability evidence, scientific assessment, and downstream acceptance policy should remain separate concepts.

#263 — process ownership rationalization

Any later scaling regression/inference implementation should follow the final vaft.process ownership rules rather than expanding miscellaneous statistical modules.

#548 — VEST confinement scaling

#548 owns the VEST-specific scientific inference workflow for deriving and assessing confinement scaling from VEST observations.

This issue owns the generic architecture for representing and evaluating published empirical / reduced scaling laws.

The two should share canonical physical-variable semantics while remaining separate workflows.

---

Acceptance criteria

• The architecture explicitly covers representative empirical relations from transition, confinement, operational-limit, edge and exhaust physics.
• New empirical models are machine-independent Formula functions.
• No new generic scaling Formula API requires an ODS, IDS, shot number, or machine name.
• Physical predictor definitions are explicit in public signatures.
• Line-average, volume-average, pedestal and separatrix densities remain semantically distinct.
• p_loss, p_sol, p_abs, p_net, and related power definitions remain semantically distinct.
• Toroidal field and OMP poloidal field are not represented by one ambiguous generic field variable.
• Published unit conventions and VAFT canonical-unit conventions are documented.
• At least Martin 2008, Eich 2013, and one qualified separatrix-density model are implemented as architecture tests.
• Existing Greenwald functionality remains a compatible operational-limit reference.
• New formulas integrate with the existing Formula catalog.
• No redundant model-discovery registry is introduced.
• Formula modules are organized by physics domain rather than one monolithic scaling.py.
• Generic Formula functions contain no VEST-specific constants or branches.
• Physically different proxy quantities are not silently substituted.
• Numerical evaluation and applicability assessment remain separate.
• Published-law evaluation and experimental regression/inference remain separate.
• Published numerical values are reproduced by regression tests.
• Formula lazy-import behavior is preserved.

---

Non-goals

This issue does not require:

• implementing every scaling listed above;
• treating every reduced model as the same mathematical object;
• rebuilding the VEST confinement regression workflow in Rebuild VEST confinement-time scaling around validated power balance and physics-constrained inference #548;
• fitting new empirical coefficients;
• implementing EPED itself;
• implementing SepOS as a full classifier;
• implementing a detachment solver;
• implementing predictive transport solvers;
• creating a new vaft.physics top-level package;
• creating a second literature/model registry independent of vaft.formula.catalog;
• automatically converting every possible density or power definition;
• silently inferring unavailable scaling inputs;
• considering an extrapolated result scientifically valid merely because the numerical formula can be evaluated;
• replacing stability solvers with empirical operational boundaries.

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Completion condition

Closing this issue means VAFT can represent physically different relations such as

Martin P_LH
Eich lambda_q
Greenwald n_G
H-mode n_e,sep
ITPA confinement
ELM fluence

within one consistent architectural philosophy without pretending that they share one universal engineering-parameter vector.

The target abstraction is

$$ \boxed{ \text{model-specific physical signature} + \text{canonical semantic variables} + \text{shared discovery} + \text{generic data resolution} + \text{separate applicability assessment} } $$

Adding a new published relation should primarily require:

1. identify the exact physical predictor definitions;
2. implement the pure numerical relation;
3. document the original database, units, assumptions and validity;
4. add published-reference tests;
5. optionally connect the semantic predictors to generic ODS/IDS resolvers.

It should not require a new machine-specific scaling workflow.