silva_positive_concave_equilibrium Reproduction Dossier

positive-concave fixed point with existence and uniqueness structure. 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	positive vector or image input -> positive equilibrium and task readout
Source relation	paper-adaptation
References	[76]
Repositories	https://github.com/mateuszgabor/pcdeq
Editable scale plan	experiments/reproduction/configs/silva_positive_concave_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_positive z_star+s_positive(x)); W_positive>=0

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

• entrywise nonnegative recurrent operators and nonnegative source injection
• published variant-one tanh/softsign/ReLU6 and variant-two sigmoid maps
• fixed-point iteration over vector or convolutional positive-concave states

What Can Be Replaced

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

• replace the nonnegative linear or convolutional transition, source, or readout
• select either published activation/injection variant under the positive-concave contract
• state width
• positive operator kernel
• activation variant
• solver budget

Constructor and Shape Contract

silva_positive_concave_equilibrium(in_dim: 'int', state_dim: 'int', out_dim: 'int', *, variant: 'PositiveVariant' = 1, operator: 'PositiveOperator' = 'linear', activation: 'str | None' = None, kernel_size: 'int' = 3, weight_parameterization: 'PositiveWeightParameterization' = 'softplus', transition: 'nn.Module | None' = None, source: 'nn.Module | None' = None, readout: 'nn.Module | None' = None, 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_positive z_star+s_positive(x)); W_positive>=0
• 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:

• entrywise nonnegative recurrent operators and nonnegative source injection
• published variant-one tanh/softsign/ReLU6 and variant-two sigmoid maps
• fixed-point iteration over vector or convolutional positive-concave states

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:

• make_positive_concave_dataset gives a seeded nonnegative map and bounded positive equilibrium.

Acceptance checks:

• record task accuracy
• record minimum state and weight
• record fixed-point residual
• record runtime and memory
• 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 one source vision task and reproduce its image preprocessing, split, and classifier head.
• 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 one source vision task and reproduce its image preprocessing, split, and classifier head.
• Match published variant 1 or 2, nonnegative parameterization, activation, convolutional width, and fixed-point budget.
• Verify positivity and compact convergence first, then report task accuracy, residual, runtime, and memory with all source hyperparameters.
• source data split, preprocessing, positive parameterization, widths, kernels, and activations
• solver iterations, optimizer, learning-rate schedule, regularization, and seeds
• task accuracy, fixed-point residual, positivity minimum, runtime, and source baselines

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
• compact positive-concave vector and image equilibria

Authoritative routes:

• https://proceedings.mlr.press/v235/gabor24a.html
• https://github.com/mateuszgabor/pcdeq

Access obligations:

• Acquire the declared image benchmark through its official or framework-provided route and preserve the source split.
• Record preprocessing, positivity parameterization, activation variant, and source revision before comparing results.

Storage planning:

• Vector tasks are small; convolutional tasks are dominated by equilibrium feature maps times solver history and precision.
• Store raw positive parameters and transformed nonnegative weights only when diagnostics cannot be regenerated from the checkpoint.

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:

• task accuracy
• minimum state and weight
• fixed-point residual
• runtime and memory

The vector compact suite ran this family on the same task and data as the other compatible families in that suite.

Measure	Recorded value
Initial loss	0.208127
Final loss	0.0375294
Fractional loss reduction	0.820
Residual or final increment norm	9.57392e-07
Iterations or tied increments	17
Parameter count	67
Final gradient norm	0.488018

These values are compact-verified evidence. They establish finite optimization, gradient flow, and diagnostic reporting; they are not a publication ranking.

Executed notebook paths:

• notebooks/package_api/37_silva_positive_concave_equilibrium.ipynb

Mechanism tests:

• tests/test_structured_equilibria.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 one source vision task and reproduce its image preprocessing, split, and classifier head.
• Match published variant 1 or 2, nonnegative parameterization, activation, convolutional width, and fixed-point budget.
• Verify positivity and compact convergence first, then report task accuracy, residual, runtime, and memory with all source hyperparameters.

Benchmark-specific requirements:

• source data split, preprocessing, positive parameterization, widths, kernels, and activations
• solver iterations, optimizer, learning-rate schedule, regularization, and seeds
• task accuracy, fixed-point residual, positivity minimum, runtime, and source baselines

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