silva_certified_equilibrium Reproduction Dossier

contractive equilibrium with interval and semialgebraic certificate routes. This dossier connects the source mechanism to its SILVA implementation, compact evidence, replaceable components, and source-scale route. Existing tests and notebooks remain the executable authority.

Evidence boundary

The mechanism is compact-verified in the package suite. The final source-scale stage remains planned until the cited data, complete optimization budget, checkpoints, and evaluation protocol have actually run.

Identity and Sources

Field	Value
Domain	optimization and certified equilibria
Task contract	bounded B,D input -> bounded equilibrium and certified task output
Source relation	paper-adaptation
References	[97], [98]
Repositories	https://openreview.net/forum?id=y1PXylgrXZ https://github.com/locuslab/monotone_op_net
Editable scale plan	experiments/reproduction/configs/silva_certified_equilibrium.json

Governing Equation

The domain-level state contract is

\[ 0\in\mathcal A_\theta(z^\star;x)+\mathcal B(z^\star),\qquad y=Q_\psi(z^\star). \]

The implementation registry specializes it operationally as

z_star=phi(W z_star+U x+b), ||W||_infinity<1; [z_lower,z_upper]=IBP_fixed_point([x_lower,x_upper])

Define the root residual

\[ R_\theta(z;x)=z-T_\theta(z;x). \]

At a regular equilibrium, differentiating \(R_\theta(z^\star;x)=0\) gives

\[ \frac{\partial z^\star}{\partial x} = \left(I-\frac{\partial T_\theta}{\partial z}\right)^{-1} \frac{\partial T_\theta}{\partial x}. \]

This identity explains why the forward residual, the conditioning derivative, and the adjoint linear solve must be diagnosed separately from the task metric.

What Is Preserved

• contractive affine equilibrium with a monotone activation
• coupled lower/upper interval fixed point and signed-affine output bounds
• exportable ReLU affine system for semialgebraic certificate programs

What Can Be Replaced

Each item below is an explicit control rather than an undocumented modification:

• replace the monotone activation, bounded affine operator, or readout
• export the affine ReLU system for external semialgebraic certificate programs
• state width
• contraction margin
• input radius
• bound-solver budget

Constructor and Shape Contract

silva_certified_equilibrium(input_dim: 'int', state_dim: 'int', output_dim: 'int', *, activation: "Literal['relu', 'tanh']" = 'relu', contraction: 'float' = 0.8, config: 'SolverConfig | None' = None) -> 'None'

The transition must preserve the declared equilibrium-state shape even when the encoder, branch operators, constraints, solver, and readout are replaced. Test the transition by itself before testing the complete root solve.

Progressive Experiment Ladder

1. Equation and tensor contract

Objective: Make the state, conditioning variables, operator, and readout explicit.

Procedure:

• Write and evaluate the family equation: z_star=phi(W z_star+U x+b), ||W||_infinity<1; [z_lower,z_upper]=IBP_fixed_point([x_lower,x_upper])
• Declare every tensor axis, boundary, mask, graph, or physical unit.
• Check the transition output has exactly the same shape as the equilibrium state.

Acceptance checks:

• finite transition values
• shape-preserving state update
• all conditioning variables affect the intended branch

Evidence target: contract-verified.

2. Primitive mechanism reconstruction

Objective: Build the retained source mechanism from replaceable modules.

Procedure:

• contractive affine equilibrium with a monotone activation
• coupled lower/upper interval fixed point and signed-affine output bounds
• exportable ReLU affine system for semialgebraic certificate programs

Acceptance checks:

• primitive modules expose trainable parameters and gradients
• mechanism-specific invariance or constraint check passes
• direct transition evaluation is deterministic under a fixed seed

Evidence target: compact-verified.

3. Public abstraction equivalence

Objective: Verify that the assembled family evaluates the same transition as its primitives.

Procedure:

• Copy the primitive module parameters into the public family constructor.
• Evaluate one transition and one complete equilibrium with identical inputs.
• Compare outputs, residuals, and parameter gradients with declared tolerances.

Acceptance checks:

• transition outputs agree
• equilibrium residual is finite and decreases
• primitive and assembled gradients agree on the compact case

Evidence target: compact-verified.

4. Compact real or analytic task

Objective: Exercise training, evaluation, diagnostics, and serialization end to end.

Procedure:

• Use the cited notebook's deterministic compact fixture before replacing it with source-scale data.

Acceptance checks:

• record natural accuracy
• record certified accuracy by radius
• record certificate margin
• record bound residual and iterations
• checkpoint reload reproduces the recorded prediction
• result record contains data and configuration fingerprints

Evidence target: compact-verified.

5. Official-data subset

Objective: Validate the complete source data path before spending the full budget.

Procedure:

• Acquire the cited data and preserve its official split, preprocessing, units, and metric.
• Freeze preprocessing, split logic, metric code, and checkpoint format.
• Run a deterministic subset large enough to expose batching and memory failures.

Acceptance checks:

• dataset receipt and checksum are stored
• resume and evaluation paths reproduce the same subset metric
• memory and runtime are measured rather than estimated

Evidence target: subset-verified.

6. Source-scale reproduction or declared extension

Objective: Run the cited protocol, or change it explicitly as a SILVA extension.

Procedure:

• Acquire the cited data and preserve its official split, preprocessing, units, and metric.
• Build the same SILVA family with source-aligned task modules and scale controls.
• Run forward, loss, backward, checkpoint resume, and metric validation on a small shard before the complete experiment.
• source dataset, perturbation norm/radius, contraction parameterization, bound solver, training schedule, seeds, natural accuracy, certified accuracy, and certificate runtime

Acceptance checks:

• all required artifacts are archived
• reported metrics use the cited evaluation protocol
• every architectural or training deviation is listed
• claims match the achieved evidence status

Evidence target: planned.

Data, Access, and Storage

Candidate datasets:

• MNIST
• CIFAR-10
• SVHN
• robustness certification

Authoritative routes:

• https://openreview.net/forum?id=y1PXylgrXZ
• https://github.com/locuslab/monotone_op_net

Access obligations:

• Follow the cited repository and dataset terms, then record source revisions and archive checksums.

Storage planning:

• Measure one processed sample, estimate the complete split, and budget raw data, processed shards, checkpoints, optimizer state, and diagnostics separately.

Preprocessing record:

• record dataset version, split, normalization, shape convention, and seed
• preserve masks, graph indices, boundaries, or physical units required by the domain

Metrics and Current Evidence

Required metrics:

• natural accuracy
• certified accuracy by radius
• certificate margin
• bound residual and iterations
• certificate runtime and memory

This family is verified through its listed mechanism tests and executed notebook. It is not included in a same-task comparison when another family does not share its input, state, output, and loss contract. The absence of a comparison row is therefore a scope decision, not missing implementation evidence.

Executed notebook paths:

• notebooks/package_api/58_silva_certified_equilibrium.ipynb
• notebooks/package_api/54_statistical_benchmarking.ipynb

Mechanism tests:

• tests/test_advanced_expansions.py

Compact Defaults

Option	Value
tier	'smoke'
config	SolverConfig(solver='anderson', max_iter=12, tol=1e-05, alpha=1.0, history=3, ridge=0.0001, beta=1.0, stop_mode='relative', relative_eps=1e-08, anderson_batch_dims=1, track_residuals=True, reengage=True, backward_mode='implicit', backward_solver='gmres', backward_max_iter=20, backward_tol=1e-05, backward_stop_mode='relative', backward_relative_eps=1e-08, phantom_steps=1, phantom_tau=1.0, neumann_terms=5, shine_refine_steps=0, indexing=(), return_best=True)

Full Defaults

Option	Value
tier	'full'
config	SolverConfig(solver='anderson', max_iter=60, tol=1e-05, alpha=1.0, history=6, ridge=0.0001, beta=1.0, stop_mode='relative', relative_eps=1e-08, anderson_batch_dims=1, track_residuals=True, reengage=True, backward_mode='implicit', backward_solver='gmres', backward_max_iter=80, backward_tol=1e-05, backward_stop_mode='relative', backward_relative_eps=1e-08, phantom_steps=1, phantom_tau=1.0, neumann_terms=5, shine_refine_steps=0, indexing=(), return_best=True)

Defaults establish a starting budget; the cited source protocol takes precedence whenever reproduction is the claim.

Source-Scale Checklist

• Acquire the cited data and preserve its official split, preprocessing, units, and metric.
• Build the same SILVA family with source-aligned task modules and scale controls.
• Run forward, loss, backward, checkpoint resume, and metric validation on a small shard before the complete experiment.

Benchmark-specific requirements:

• source dataset, perturbation norm/radius, contraction parameterization, bound solver, training schedule, seeds, natural accuracy, certified accuracy, and certificate runtime

Required archived artifacts:

• machine-readable model and solver configuration
• dataset receipt with source revision, split, license, and checksum
• preprocessing and normalization record
• seeded training and evaluation log
• checkpoint and optimizer-resume state for trained experiments
• task metrics and equilibrium diagnostics in a machine-readable result
• runtime, peak-memory, device, precision, and dependency record
• declared deviations from the cited protocol

Reporting Rule

Report the achieved evidence status, not the intended one. A compact or subset run may validate the implementation and data path, but only a completed cited protocol supports a source-scale reproduction statement. Modified operators are valuable SILVA extensions when every deviation is named and measured.

Where to Go Next

Question	Page
Where are all family dossiers?	Family Dossier Index
How is a custom family assembled?	Advanced Extension Handbook
How are experiment stages represented in the API?	Research-Depth API
Which lab inspects every dossier?	Family Dossier Lab