jacksonjp0311-gif · May 3, 2026 20:51
diff --git a/codex-pbsa-v1-0-placidic-bioregulation-software-architecture.tex b/codex-pbsa-v1-0-placidic-bioregulation-software-architecture.tex
 % ████████████████████████████████████████████████████████████████████████████████
 %
 %   CODEX ΔΦ — PLACIDIC BIOREGULATION SOFTWARE ARCHITECTURE (PBSA v1.0)
 %   ────────────────────────────────────────────────────────────────────────────
 %   CANONICAL FULL-SCOPE SOFTWARE ARCHITECTURE FOR IMPLEMENTING THE PLACIDIC
 %   BIOREGULATION ALGORITHM (PBA v1.3) AS A REPOSITORY-ANCHORED, BENCHMARKED,
 %   BASELINE-COMPARED, CALIBRATED, IDENTIFIABILITY-CHECKED, EVIDENCE-PACKAGED,
 %   AND DOWNGRADE-PRESERVING COMPUTATIONAL MODELING SYSTEM WITHOUT MEDICAL,
 %   MECHANISTIC, BIOLOGICAL-LAW, OR UNIVERSAL-LAW OVERCLAIM
 %
 %   VERSION
 %   ───────
 %   v1.0 — PBA-to-Software Architecture Genesis Layer · Locked ·
 %          Kernel Runtime, Domain Configs, Parameter Manifests, Baseline Classes,
 %          Calibration Engine, Metric Emitter, Identifiability Engine, Evidence
 %          Package Compiler, CLI Runtime, Test Harness, Ledger System, and
 %          Repository Grammar for Executable Placidic Bioregulation
 %
 %   AUTHOR
 %   ──────
 %   James Paul Jackson
 %   X / Twitter: @unifiedenergy11
 %
 %   SOURCE EXTRACTION / AUTHOR ATTRIBUTION
 %   ──────────────────────────────────────
 %   This document is a Codex-format canonical software architecture derived from:
 %
 %   • PBA v1.3 — Placidic Bioregulation Algorithm, Reference Implementation and
 %     Benchmark Evidence Layer, which formalized Python kernel contracts, toy
 %     benchmark suite, baseline-family classes, calibration ledgers, metric JSON
 %     outputs, evidence-package schema, identifiability reports, failure
 %     taxonomy, and downgrade-preserving classification for executable Placidic
 %     bioregulation.
 %
 %   • PBA v1.2 — Benchmark Calibration and Domain-Instantiation Layer, which
 %     formalized benchmark families, domain-instantiation grammar, parameter
 %     calibration, fit / evaluation split, expanded baseline-family comparison,
 %     benchmark metrics, identifiability discipline, benchmark evidence
 %     packages, and comparative downgrade rules.
 %
 %   • PBA v1.1 — Executable Bioregulation Anchor Layer, which added runtime
 %     kernel requirements, state-vector grammar, simulation loop, repository
 %     grammar, evidence packages, negative-control run comparison, parameter
 %     manifests, and reproducibility scoring.
 %
 %   • PBA v1.0 — Placidic Bioregulation Algorithm, which formalized Placidity
 %     as a CITA-governed biological modeling abstraction linking ΔΦ drift,
 %     Ω damping, signal preservation, cusp guarding, canalization-basin
 %     selection, allostatic anticipation, and memory-promoted regulation.
 %
 %   • Codex Placidity Operator — Canonical Stability Governor, where Placidity
 %     was defined as bounded damping / smoothing / governance that suppresses
 %     fast drift, reduces recursive overshoot, preserves admissible coherence
 %     near sharp boundaries, and prevents collapse into unstable regimes without
 %     erasing informative structure.
 %
 %   • CITA v1.0 — Canonical Insight Transmutation Algorithm, which requires
 %     source boundaries, fidelity stratification, primitive objects,
 %     observables, validation, falsification, negative controls, downgrade
 %     paths, evidence packages, repository anchoring, and memory promotion.
 %
 %   • Codex ΔΦ software-architecture method: preserve source lineage and
 %     non-claim locks; extract the invariant kernel; translate theory objects
 %     into software modules; define module input/output contracts; specify
 %     schemas, JSON records, ledgers, metrics, baselines, validation checklists,
 %     falsification surfaces, upgrade/downgrade thresholds, repository grammar,
 %     CLI contracts, and pseudocode; separate simulation, implementation,
 %     benchmark, and robustness evidence; and enforce additive-only evolution
 %     with memory-promotion restraint.
 %
 %   • Codex ΔΦ memory lessons including:
 %       - theory is not implementation,
 %       - implementation is not benchmark validation,
 %       - benchmark evidence requires baselines,
 %       - baselines require shared task conditions,
 %       - calibration is not mechanism proof,
 %       - implementation success is not biological validation,
 %       - machine-readable outputs are required for audit,
 %       - identifiability collapse downgrades interpretation,
 %       - medical and biological-law claims require independent domain evidence,
 %       - memory promotion must preserve only reusable implementation constraints,
 %         stable thresholds, repeated benchmark wins, and failure lessons.
 %
 %   This document does not claim that Placidity is a biological force, medical
 %   framework, treatment model, physiological mechanism, or universal biological
 %   law. PBSA v1.0 formalizes the software architecture required to implement
 %   PBA v1.3 as a runnable, auditable, benchmarkable Python system.
 %
 %   DATE
 %   ────
 %   May 2026
 %
 %   STATUS
 %   ──────
 %   CANONICAL v1.0 PBA SOFTWARE ARCHITECTURE LAYER —
 %   NOT A BIOLOGICAL LAW CLAIM · NOT MEDICAL GUIDANCE · NOT MECHANISM PROOF
 %
 %   EMPIRICAL / METHODOLOGICAL CONFIDENCE BADGE
 %   ────────────────────────────────────────────
 %   Confidence status: High as a software architecture and implementation
 %   hardening scaffold; not proof-ready as a universal biological theory,
 %   medical framework, mechanism-level biological model, or empirical
 %   validation artifact.
 %
 %   PBSA v1.0 preserves PBA v1.3's reference-implementation discipline, PBA
 %   v1.2's benchmark-calibration discipline, PBA v1.1's executable anchoring,
 %   PBA v1.0's biological caution, CITA governance, non-claim locks, baseline
 %   comparison, metric-emission discipline, identifiability caution, evidence
 %   packaging, and downgrade-preserving classification while translating the
 %   theory into a concrete software architecture: package layout, runtime
 %   modules, domain configuration system, parameter manifest, kernel engine,
 %   calibration engine, baseline harness, metric emitter, identifiability
 %   engine, evidence compiler, CLI, tests, ledgers, reports, and RootMirror-ready
 %   repository continuity.
 %
 %   PURPOSE
 %   ───────
 %   Translate PBA v1.3 into a full-scope executable software architecture:
 %
 %       repository root
 %       → Python package boundary
 %       → domain configuration
 %       → parameter manifest
 %       → PBA kernel
 %       → cusp guard
 %       → signal preservation
 %       → perturbation generator
 %       → baseline-family classes
 %       → calibration engine
 %       → fit/evaluation split
 %       → metric emitter
 %       → identifiability engine
 %       → classification engine
 %       → runtime ledger
 %       → evidence package compiler
 %       → CLI command surface
 %       → test harness
 %       → benchmark reports
 %       → memory-promotion restraint.
 %
 %   VERSION EVOLUTION SUMMARY
 %   ─────────────────────────
 %   PBA v1.0  : Theory genesis. Defines Placidity as bounded adaptive regulation
 %               and model-transfer analogy under biological caution.
 %
 %   PBA v1.1  : Executable anchor. Adds state-vector grammar, runtime loop,
 %               parameter manifest, repository grammar, negative-control run,
 %               and reproducibility scoring.
 %
 %   PBA v1.2  : Benchmark calibration. Adds benchmark families, parameter
 %               calibration, fit/evaluation split, baseline families, metrics,
 %               identifiability, and benchmark evidence packages.
 %
 %   PBA v1.3  : Reference implementation evidence. Adds Python module contracts,
 %               CLI run grammar, baseline classes, metric JSON outputs,
 %               identifiability reports, evidence-package schema, failure
 %               taxonomy, and downgrade-preserving classification.
 %
 %   PBSA v1.0 : Software architecture. Converts PBA v1.3 into a complete
 %               repository and runtime design with module boundaries, data
 %               schemas, CLI commands, test harness, ledgers, evidence outputs,
 %               and implementation governance.
 %
 %   WHAT THIS IS
 %   ────────────
 %   • A CITA-governed software architecture for PBA v1.3
 %   • A repository grammar for a runnable Python PBA package
 %   • A module-contract specification for kernel, baselines, calibration,
 %     metrics, identifiability, classification, and evidence packaging
 %   • A CLI and runtime contract for benchmark execution
 %   • A JSON schema layer for configs, parameters, metrics, logs, and evidence
 %   • A test-harness specification
 %   • A runtime ledger and report-generation architecture
 %   • A downgrade-preserving software implementation layer
 %   • A bridge from PBA theory to inspectable, reproducible code
 %
 %   WHAT THIS IS NOT
 %   ───────────────
 %   • Not proof of a universal biological law
 %   • Not a new biological force
 %   • Not a medical diagnostic framework
 %   • Not treatment guidance
 %   • Not a replacement for physiology, developmental biology, systems biology,
 %     clinical validation, empirical assay design, or standard controls
 %   • Not permission to treat toy benchmark success as biological mechanism proof
 %   • Not proof that ΔΦ geometry governs all living systems
 %   • Not permission to ignore standard biological models or controls
 %   • Not permission to treat calibrated fit as truth
 %   • Not permission to treat implementation success as empirical validation
 %   • Not permission to treat coherence as biological evidence
 %
 %   ADDITIVE REFINEMENTS (PBSA v1.0)
 %   ────────────────────────────────
 %   • PBA v1.3 shadow-header discipline preserved
 %   • PBA theory-to-software translation formalized
 %   • Full repository architecture added
 %   • Runtime module boundaries added
 %   • Input/output contracts added
 %   • Domain config schema added
 %   • Parameter manifest schema added
 %   • Perturbation grammar added
 %   • Baseline class interface added
 %   • Calibration engine architecture added
 %   • Metric-emission architecture added
 %   • Identifiability engine architecture added
 %   • Evidence package compiler architecture added
 %   • Runtime ledger architecture added
 %   • CLI surface added
 %   • Test-harness layer added
 %   • RootMirror-compatible local anchoring added
 %   • Non-claim boundaries preserved
 %
 %   EXECUTABLE ANCHOR BLOCK (PBSA v1.0)
 %   ──────────────────────────────────
 %   A valid PBSA v1.0 implementation must:
 %
 %       (1) create a repository-anchored Python package,
 %       (2) expose a runnable CLI,
 %       (3) implement PBA kernel execution,
 %       (4) implement cusp guarding,
 %       (5) implement signal preservation,
 %       (6) implement perturbation generation,
 %       (7) implement baseline-family classes,
 %       (8) implement calibration search,
 %       (9) preserve fit/evaluation split,
 %       (10) emit machine-readable PBA metrics,
 %       (11) emit machine-readable baseline metrics,
 %       (12) emit calibration records,
 %       (13) emit identifiability reports,
 %       (14) emit runtime ledgers,
 %       (15) compile evidence packages,
 %       (16) classify PBA-A/B/C/D/E,
 %       (17) preserve all biological, medical, mechanism, and universal-law
 %            non-claim locks,
 %       (18) reject implementation, calibration, or benchmark success as
 %            biological proof,
 %       (19) promote to memory only reproducible benchmark wins, stable
 %            thresholds, reusable implementation constraints, and failure
 %            lessons.
 %
 %   CANONICAL LOCK (PBSA v1.0)
 %   ──────────────────────────
 %   • PBA v1.3 reference implementation discipline preserved
 %   • PBA v1.2 benchmark-calibration discipline preserved
 %   • PBA v1.1 executable anchoring preserved
 %   • PBA v1.0 biological caution preserved
 %   • CITA governance preserved
 %   • Placidity remains an operational regulator, not metaphysics
 %   • Biological crossing remains model-transfer, not proof
 %   • ΔΦ and Ω remain computational abstractions unless empirically instantiated
 %   • No medical claim
 %   • No universal biological law claim
 %   • No mechanism-proof claim from simulation or implementation
 %   • No benchmark claim without declared parameter manifest
 %   • No benchmark claim without baseline comparison
 %   • No calibrated claim without fit/evaluation distinction
 %   • No strong claim when simpler baselines perform equally well or better
 %   • Memory promotion must preserve reusable regulatory invariants only
 %
 %   Evolutions must be additive only.
 %   Do not weaken biological caution, falsification, negative controls,
 %   empirical validation, non-medical boundaries, repository anchoring,
 %   evidence-package discipline, benchmark discipline, calibration caution,
 %   baseline comparison, identifiability discipline, metric emission, runtime
 %   logging, or downgrade rules.
 %
 %   AI PROMPT TRACEABILITY
 %   ──────────────────────
 %   Use this document as the canonical PBSA v1.0 software architecture derived
 %   from PBA v1.3. Preserve the distinction between theory, software
 %   architecture, implementation, simulation, benchmark evidence, biological
 %   mechanism, clinical validation, medical claims, and memory promotion.
 %
 %   SHADOW HEADER ALIGNMENT SEAL
 %   ───────────────────────────
 %   Preserve header discipline across future software versions except for
 %   explicitly additive shadow-header evolution that improves implementation
 %   readiness, repository structure, module contracts, runtime auditability,
 %   benchmark design, calibration, validation, falsification, negative controls,
 %   evidence packaging, or repository anchoring.
 %
 % ████████████████████████████████████████████████████████████████████████████████

 \documentclass[12pt]{article}
 \usepackage[margin=1in]{geometry}
 \usepackage{amsmath,amssymb,amsfonts,amsthm}
 \usepackage{booktabs,longtable,array}
 \usepackage{hyperref}
 \usepackage{listings}
 \usepackage{xcolor}

 \newtheorem{axiom}{Axiom}
 \newtheorem{definition}{Definition}
 \newtheorem{proposition}{Proposition}
 \newtheorem{hypothesis}{Hypothesis}
 \newtheorem{remark}{Remark}
 \newtheorem{corollary}{Corollary}

 \lstset{
  basicstyle=\ttfamily\small,
  breaklines=true,
  columns=fullflexible,
  frame=single
 }

 \title{\textbf{Codex $\Delta\Phi$ — Placidic Bioregulation Software Architecture (PBSA v1.0)}\\
 \large Canonical Full-Scope Software Architecture for Implementing PBA v1.3 as a Runnable, Benchmarkable, Evidence-Packaged Python System}
 \author{\textbf{James Paul Jackson}\\[4pt]
 \small Codex-format software architecture for executable Placidic bioregulation modeling\\
 \small \texttt{@unifiedenergy11}}
 \date{May 2026}

 \begin{document}
 \maketitle

 \begin{abstract}
 PBSA v1.0 translates PBA v1.3 from a reference implementation specification
 into a full-scope software architecture for a repository-anchored Python
 package. It preserves PBA's biological caution, CITA governance, baseline
 comparison, calibration discipline, identifiability checks, metric emission,
 runtime ledgers, evidence packages, and downgrade-preserving classification
 while defining concrete software modules, data schemas, CLI commands, benchmark
 flows, test harnesses, and repository grammar. PBSA does not claim medical
 validity, biological mechanism, universal biological law, or empirical
 validation. It specifies how to build an auditable computational modeling
 system that can test whether bounded Placidic regulation outperforms declared
 baselines under declared benchmark conditions.
 \end{abstract}

 %──────────────────────────────────────────────────────────────────────────────
 \section{Core-Invariant Extraction Block}
 %──────────────────────────────────────────────────────────────────────────────

 The shortest faithful extraction of PBSA v1.0 is:

 \[
 \boxed{
 \begin{array}{c}
 \text{PBA becomes software-real only when its kernel, baselines, calibration,}\\
 \text{metrics, identifiability, ledgers, evidence packages, CLI commands,}\\
 \text{and tests are implemented as repository-anchored software contracts.}
 \end{array}
 }
 \]

 PBSA is the software bridge:

 \[
 \boxed{
 \text{PBA v1.3 reference implementation specification}
 \rightarrow
 \text{PBSA v1.0 full software architecture}.
 }
 \]

 The operative software chain is:

 \[
 \text{domain config}
 \rightarrow
 \text{parameter manifest}
 \rightarrow
 \text{kernel runtime}
 \rightarrow
 \text{baseline runtime}
 \rightarrow
 \text{calibration}
 \rightarrow
 \text{evaluation}
 \rightarrow
 \text{metrics}
 \rightarrow
 \text{identifiability}
 \rightarrow
 \text{classification}
 \rightarrow
 \text{evidence package}
 \rightarrow
 \text{ledger}
 \rightarrow
 \text{report}.
 \]

 \begin{remark}
 PBSA does not add biological claim strength. It adds software accountability.
 \end{remark}

 %──────────────────────────────────────────────────────────────────────────────
 \section{Deep Architecture Analysis Layer}
 %──────────────────────────────────────────────────────────────────────────────

 PBA v1.3 is already implementation-evaluable in specification form. The missing
 surface is not another theoretical refinement. It is the executable repository:

 \[
 \boxed{
 \text{PBA v1.3}
 =
 \text{what must be run};
 \quad
 \text{PBSA v1.0}
 =
 \text{how the software must be built}.
 }
 \]

 The architecture must preserve five separations:

 \[
 \text{theory}
 \neq
 \text{implementation},
 \]

 \[
 \text{implementation}
 \neq
 \text{benchmark validation},
 \]

 \[
 \text{benchmark success}
 \neq
 \text{biological mechanism},
 \]

 \[
 \text{calibration}
 \neq
 \text{truth},
 \]

 \[
 \text{coherence}
 \neq
 \text{evidence}.
 \]

 The architecture therefore becomes a governance machine:

 \[
 \boxed{
 \text{every output must have a config, source, metric, log, comparison,}
 \atop
 \text{classification, and non-claim boundary.}
 }
 \]

 \begin{remark}
 PBSA is not a larger theory. It is the software compression of the theory into
 modules, commands, files, tests, and evidence artifacts.
 \end{remark}

 %──────────────────────────────────────────────────────────────────────────────
 \section{System Definition}
 %──────────────────────────────────────────────────────────────────────────────

 \begin{definition}[PBSA Runtime]
 A PBSA runtime is a repository-anchored software system that loads a declared
 benchmark domain, executes the PBA kernel and declared baselines under shared
 conditions, performs calibration and evaluation, emits metrics, checks
 identifiability, compiles an evidence package, writes ledgers, and assigns a
 downgrade-preserving classification.
 \end{definition}

 The PBSA system is:

 \[
 \mathcal{PBSA}
 =
 \{
 D,
 P,
 K,
 C,
 S,
 A,
 B,
 H,
 M,
 I,
 G,
 Y,
 L,
 R,
 T
 \}.
 \]

 where:

 \[
 D=\text{DomainConfig},
 \quad
 P=\text{ParameterManifest},
 \quad
 K=\text{PBAKernel},
 \quad
 C=\text{CuspGuard},
 \]

 \[
 S=\text{SignalPreserver},
 \quad
 A=\text{AllostaticAnticipator},
 \quad
 B=\text{BaselineHarness},
 \quad
 H=\text{CalibrationEngine},
 \]

 \[
 M=\text{MetricEmitter},
 \quad
 I=\text{IdentifiabilityEngine},
 \quad
 G=\text{ClassificationEngine},
 \quad
 Y=\text{EvidencePackageCompiler},
 \]

 \[
 L=\text{RuntimeLedger},
 \quad
 R=\text{ReportGenerator},
 \quad
 T=\text{TestHarness}.
 \]

 \begin{remark}
 Each module must be replaceable, testable, and auditable. No module may require
 a biological interpretation to execute.
 \end{remark}

 %──────────────────────────────────────────────────────────────────────────────
 \section{Architecture Principles}
 %──────────────────────────────────────────────────────────────────────────────

 \begin{axiom}[Config-First Execution]
 Every benchmark run must begin from explicit configuration files: domain config,
 parameter manifest, baseline config, calibration grid, and metric manifest.
 \end{axiom}

 \begin{axiom}[Shared-Condition Baseline Fairness]
 PBA and baselines must run under the same domain, perturbation, time horizon,
 fit/evaluation split, and metric rules.
 \end{axiom}

 \begin{axiom}[Machine-Readable Evidence]
 Every major runtime product must be emitted as machine-readable JSON or JSONL
 before narrative reports are generated.
 \end{axiom}

 \begin{axiom}[No Hidden Calibration]
 A calibrated result must preserve search space, objective, fit split, selected
 parameters, evaluation split, and failure notes.
 \end{axiom}

 \begin{axiom}[Identifiability Before Interpretation]
 No parameter-level interpretation may be promoted unless identifiability is
 stable or the degeneracy is explicitly disclosed.
 \end{axiom}

 \begin{axiom}[Classification After Evidence]
 PBA-A/B/C/D/E classification must occur after metrics, baselines, logs, and
 evidence-package compilation.
 \end{axiom}

 \begin{axiom}[Non-Claim Lock Preservation]
 No runtime output may imply medical guidance, biological mechanism proof, or
 universal biological law.
 \end{axiom}

 %──────────────────────────────────────────────────────────────────────────────
 \section{Repository Architecture}
 %──────────────────────────────────────────────────────────────────────────────

 A canonical PBSA repository is:

 \begin{lstlisting}
 PBA/
  README.md
  LICENSE
  pyproject.toml
  .gitignore

  docs/
    theory/
      pba_v1_3.tex
      pbsa_v1_0.tex
      biological_non_claim_locks.md
      evidence_hierarchy.md

    architecture/
      system_overview.md
      module_contracts.md
      runtime_flow.md
      cli_contract.md
      data_schemas.md
      evidence_package_schema.md
      classification_policy.md

    benchmark_protocol/
      toy_benchmark_suite.md
      baseline_family_contracts.md
      calibration_protocol.md
      identifiability_protocol.md
      metric_manifest.md
      falsification_surface.md

  src/
    pba/
      __init__.py
      cli/
        __init__.py
        main.py

      core/
        state.py
        domain.py
        parameters.py
        kernel.py
        cusp_guard.py
        signal.py
        allostasis.py
        perturbations.py

      baselines/
        __init__.py
        base.py
        proportional.py
        pi_control.py
        threshold.py
        return_to_setpoint.py

      calibration/
        __init__.py
        objective.py
        grid_search.py
        split.py
        records.py

      evaluation/
        __init__.py
        metrics.py
        identifiability.py
        classification.py
        scoring.py

      evidence/
        __init__.py
        runtime_ledger.py
        evidence_package.py
        report_generator.py
        file_manifest.py

      benchmarks/
        __init__.py
        runner.py
        suite_runner.py

  configs/
    suite_v1_0.json
    metric_manifest.json
    pba_params.json
    baseline_params.json
    calibration_grid.json

    domains/
      temperature_like.json
      pulse_recovery.json
      oscillatory_signal.json
      noisy_perturbation.json

  runs/
    .gitkeep

  evidence_packages/
    .gitkeep

  ledgers/
    pba_evolution_ledger.jsonl
    pba_runtime_ledger.jsonl
    pba_decision_ledger.jsonl

  reports/
    .gitkeep

  scripts/
    run_benchmark.py
    run_suite.py
    compile_evidence_package.py
    repo_dump_light.ps1

  tests/
    test_domain_config.py
    test_parameters.py
    test_kernel.py
    test_cusp_guard.py
    test_signal.py
    test_perturbations.py
    test_baselines.py
    test_calibration.py
    test_metrics.py
    test_identifiability.py
    test_classification.py
    test_evidence_package.py
    test_cli.py
 \end{lstlisting}

 \begin{proposition}[Repository Completeness Principle]
 A PBSA repository is incomplete for strong implementation claims unless it
 contains source code, configs, tests, benchmark tasks, ledgers, and evidence
 package outputs.
 \end{proposition}

 %──────────────────────────────────────────────────────────────────────────────
 \section{Core Runtime Modules}
 %──────────────────────────────────────────────────────────────────────────────

 \subsection{DomainConfig}

 Input:

 \[
 \{\text{domain id},x^\ast,V,T,p_t,\text{noise},\text{cadence}\}.
 \]

 Output:

 \[
 \text{validated domain object}.
 \]

 \subsection{ParameterManifest}

 Input:

 \[
 \Theta=\{\eta,\tau_1,\tau_2,\kappa,\alpha,\theta_M,\beta,\gamma\}.
 \]

 Output:

 \[
 \text{validated parameter object}.
 \]

 \subsection{PBAKernel}

 The kernel executes:

 \[
 x_{t+1}
 =
 x_t
 -
 \eta\Omega_t\nabla\Delta\Phi_t
 +
 \kappa S_t
 +
 A_t.
 \]

 Output:

 \[
 \{x_{t+1},\Delta\Phi_t,\Omega_t,c_t,S_t,A_t\}.
 \]

 \subsection{CuspGuard}

 Input:

 \[
 \Delta\Phi_t,\dot{\Delta\Phi}_t,\tau_1,\tau_2.
 \]

 Output:

 \[
 c_t\in\{\text{continue},\text{caution},\text{halt/audit}\}.
 \]

 \subsection{SignalPreserver}

 Input:

 \[
 S_t,S_{\mathrm{stable}},\alpha,\kappa.
 \]

 Output:

 \[
 S_{t+1}\geq \kappa.
 \]

 \subsection{BaselineHarness}

 Input:

 \[
 \text{domain},\text{perturbations},\text{baseline configs}.
 \]

 Output:

 \[
 \text{baseline trajectories and metrics}.
 \]

 \subsection{CalibrationEngine}

 Input:

 \[
 \text{fit split},\text{search grid},\mathcal{J}_{pba}.
 \]

 Output:

 \[
 \Theta^\ast,\text{calibration record}.
 \]

 \subsection{MetricEmitter}

 Input:

 \[
 \text{PBA trajectory},\text{baseline trajectories},\text{domain target}.
 \]

 Output:

 \[
 \mathcal{M}_{pba},\mathcal{M}_{baseline}.
 \]

 \subsection{IdentifiabilityEngine}

 Input:

 \[
 \text{parameter trials},\text{losses},\epsilon.
 \]

 Output:

 \[
 \{\text{stable},\text{degenerate},\text{unidentified}\}.
 \]

 \subsection{EvidencePackageCompiler}

 Input:

 \[
 \text{configs},\text{logs},\text{metrics},\text{classification}.
 \]

 Output:

 \[
 \text{evidence\_package.json}.
 \]

 %──────────────────────────────────────────────────────────────────────────────
 \section{Software Module Contract Table}
 %──────────────────────────────────────────────────────────────────────────────

 \begin{center}
 \begin{longtable}{>{\raggedright\arraybackslash}p{0.24\textwidth}
                  >{\raggedright\arraybackslash}p{0.30\textwidth}
                  >{\raggedright\arraybackslash}p{0.36\textwidth}}
 \toprule
 \textbf{Module} & \textbf{Input} & \textbf{Required Output} \\
 \midrule
 domain.py & domain JSON & validated DomainConfig. \\
 parameters.py & parameter JSON & validated ParameterManifest. \\
 kernel.py & state, target, params, perturbation & next state, ΔΦ, Ω, correction. \\
 cusp\_guard.py & ΔΦ, ΔΦ derivative, thresholds & continue / caution / halt-audit. \\
 signal.py & signal state, α, κ & signal-preserved state. \\
 allostasis.py & ΔΦ history, future perturbation & anticipatory correction. \\
 perturbations.py & domain config, seed & perturbation sequence. \\
 baselines/base.py & runtime interface & baseline class contract. \\
 proportional.py & shared domain conditions & proportional trajectory. \\
 pi\_control.py & shared domain conditions & PI trajectory. \\
 threshold.py & shared domain conditions & threshold trajectory. \\
 return\_to\_setpoint.py & shared domain conditions & return model trajectory. \\
 objective.py & metrics, weights & calibration objective. \\
 grid\_search.py & search grid, fit split & selected parameters. \\
 metrics.py & trajectories, targets & machine-readable metrics. \\
 identifiability.py & trials, losses & identifiability report. \\
 classification.py & scores, metrics, locks & PBA-A/B/C/D/E. \\
 runtime\_ledger.py & run events & JSONL runtime ledger. \\
 evidence\_package.py & configs, logs, outputs & evidence package JSON. \\
 report\_generator.py & evidence package & Markdown summary. \\
 runner.py & domain config & single benchmark run. \\
 suite\_runner.py & suite config & multi-domain suite run. \\
 cli/main.py & user command & benchmark or suite execution. \\
 \bottomrule
 \end{longtable}
 \end{center}

 %──────────────────────────────────────────────────────────────────────────────
 \section{Data Schema Layer}
 %──────────────────────────────────────────────────────────────────────────────

 \subsection{Domain Config Schema}

 \begin{lstlisting}
 {
  "domain_id": "temperature_like",
  "regulated_variable": "x_t",
  "target": 0.0,
  "viable_interval": [-0.25, 0.25],
  "time_steps": 100,
  "initial_state": 1.0,
  "perturbation_family": "pulse_plus_noise",
  "noise_model": {
    "type": "bounded_uniform",
    "low": -0.03,
    "high": 0.03,
    "seed": 777
  },
  "observation_cadence": 1,
  "fit_eval_split": {
    "fit_seed": 101,
    "eval_seed": 202
  },
  "non_claim_locks": [
    "not_medical",
    "not_biological_law",
    "not_mechanism_proof"
  ]
 }
 \end{lstlisting}

 \subsection{Parameter Manifest Schema}

 \begin{lstlisting}
 {
  "version": "PBSA-ParameterManifest-v1.0",
  "parameters": {
    "eta": 0.10,
    "tau_1": 0.30,
    "tau_2": 0.80,
    "kappa": 0.05,
    "alpha": 0.80,
    "theta_M": 0.70,
    "beta": 0.10,
    "gamma": 0.05
  },
  "declared_before_run": true
 }
 \end{lstlisting}

 \subsection{Metric Manifest Schema}

 \begin{lstlisting}
 {
  "version": "PBSA-MetricManifest-v1.0",
  "primary_metrics": [
    "cumulative_deviation",
    "overshoot",
    "cusp_warnings",
    "signal_preservation"
  ],
  "secondary_metrics": [
    "recovery_time",
    "undershoot",
    "oscillation_amplitude",
    "robustness",
    "parameter_sensitivity"
  ],
  "declared_before_run": true
 }
 \end{lstlisting}

 \subsection{Runtime State Record}

 \begin{lstlisting}
 {
  "t": 0,
  "x_t": null,
  "target": null,
  "delta_phi": null,
  "omega": null,
  "correction": null,
  "signal": null,
  "allostatic_term": null,
  "perturbation": null,
  "cusp_state": "continue/caution/halt-audit"
 }
 \end{lstlisting}

 %──────────────────────────────────────────────────────────────────────────────
 \section{Runtime Flow}
 %──────────────────────────────────────────────────────────────────────────────

 The PBSA single-domain runtime flow is:

 \[
 \text{load configs}
 \rightarrow
 \text{validate configs}
 \rightarrow
 \text{generate perturbations}
 \rightarrow
 \text{calibrate}
 \rightarrow
 \text{run PBA}
 \rightarrow
 \text{run baselines}
 \rightarrow
 \text{emit metrics}
 \rightarrow
 \text{check identifiability}
 \rightarrow
 \text{classify}
 \rightarrow
 \text{compile evidence}.
 \]

 Pseudocode:

 \begin{lstlisting}
 Input:
    domain_config
    pba_params
    baseline_params
    calibration_grid
    metric_manifest

 Validate:
    DomainConfig.validate()
    ParameterManifest.validate()
    MetricManifest.validate()

 Fit:
    generate fit perturbations
    for theta in calibration_grid:
        run PBAKernel(theta) on fit split
        compute J_pba(theta)
        log calibration trial
    select theta_star

 Evaluate:
    generate evaluation perturbations
    run PBAKernel(theta_star)
    write state_log.jsonl

 Baselines:
    run proportional baseline
    run PI baseline
    run threshold baseline
    run return-to-setpoint baseline

 Metrics:
    compute pba_metrics.json
    compute baseline_metrics.json

 Identifiability:
    compute near-equivalent parameter sets
    emit identifiability_report.json

 Classification:
    compute PBAScore
    compare PBA against baselines
    classify PBA-A/B/C/D/E

 Evidence:
    write result_ledger.jsonl
    compile evidence_package.json
    generate benchmark_summary.md
 \end{lstlisting}

 %──────────────────────────────────────────────────────────────────────────────
 \section{CLI Contract}
 %──────────────────────────────────────────────────────────────────────────────

 Single benchmark:

 \begin{lstlisting}
 python -m pba.cli run-benchmark --domain ".\configs\domains\temperature_like.json"
 \end{lstlisting}

 Suite benchmark:

 \begin{lstlisting}
 python -m pba.cli run-suite --config ".\configs\suite_v1_0.json"
 \end{lstlisting}

 Compile evidence package:

 \begin{lstlisting}
 python -m pba.cli compile-evidence --run ".\runs\run_<timestamp>"
 \end{lstlisting}

 Expected single-run output:

 \begin{lstlisting}
 runs/run_<timestamp>/
  domain_config.json
  parameter_manifest.json
  baseline_params.json
  calibration_grid.json
  metric_manifest.json
  state_log.jsonl
  baseline_state_log.jsonl
  pba_metrics.json
  baseline_metrics.json
  calibration_record.json
  identifiability_report.json
  classification.json
  evidence_package.json
  result_ledger.jsonl
  benchmark_summary.md
 \end{lstlisting}

 \begin{proposition}[CLI Reproducibility Principle]
 A PBSA benchmark claim is weak unless the run can be reproduced from a declared
 CLI command and preserved configuration files.
 \end{proposition}

 %──────────────────────────────────────────────────────────────────────────────
 \section{Calibration Architecture}
 %──────────────────────────────────────────────────────────────────────────────

 Calibration optimizes:

 \[
 \Theta^\ast
 =
 \arg\min_{\Theta}
 \mathcal{J}_{pba}(\Theta).
 \]

 where:

 \[
 \mathcal{J}_{pba}
 =
 w_1D_{cum}
 +
 w_2O_{max}
 +
 w_3A_{osc}
 +
 w_4W_{cusp}
 -
 w_5S_{pres}
 +
 w_6P_{sens}.
 \]

 The calibration engine must preserve:

 \[
 \{
 \text{search space},
 \text{objective},
 \text{fit seed},
 \text{evaluation seed},
 \text{trial losses},
 \Theta^\ast,
 \text{downgrade notes}
 \}.
 \]

 Calibration output:

 \begin{lstlisting}
 {
  "calibration_id": "PBSA-CAL-0001",
  "method": "grid_search",
  "objective": "J_pba_v1_3",
  "fit_seed": 101,
  "eval_seed": 202,
  "selected_params": {},
  "fit_loss": null,
  "evaluation_loss": null,
  "trial_count": 0,
  "downgrade_notes": []
 }
 \end{lstlisting}

 %──────────────────────────────────────────────────────────────────────────────
 \section{Baseline Architecture}
 %──────────────────────────────────────────────────────────────────────────────

 The baseline interface is:

 \begin{lstlisting}
 class Baseline:
    name: str

    def run(self, initial_state, target, perturbations, config):
        raise NotImplementedError

    def emit_state_log(self):
        raise NotImplementedError

    def emit_metrics(self):
        raise NotImplementedError
 \end{lstlisting}

 Required baseline families:

 \[
 \mathcal{L}_{PBSA}
 =
 \{
 L_{prop},
 L_{PI},
 L_{threshold},
 L_{return}
 \}.
 \]

 The baseline harness emits:

 \begin{lstlisting}
 {
  "baseline_results": [
    {
      "name": "proportional_feedback",
      "metrics": {},
      "state_log": "baseline_state_log.jsonl",
      "failure_notes": []
    }
  ]
 }
 \end{lstlisting}

 \begin{remark}
 PBA does not need to beat every baseline on every metric. Strong claims require
 declared metric advantage under predeclared primary metrics.
 \end{remark}

 %──────────────────────────────────────────────────────────────────────────────
 \section{Metric Architecture}
 %──────────────────────────────────────────────────────────────────────────────

 Metric vector:

 \[
 \mathcal{M}_{PBSA}
 =
 \{
 T_R,
 O_{max},
 U_{max},
 A_{osc},
 D_{cum},
 W_{cusp},
 S_{pres},
 R_{rob},
 P_{sens}
 \}.
 \]

 Machine-readable output:

 \begin{lstlisting}
 {
  "model": "PBA",
  "domain": "temperature_like",
  "metrics": {
    "recovery_time": null,
    "overshoot": null,
    "undershoot": null,
    "oscillation_amplitude": null,
    "cumulative_deviation": null,
    "cusp_warnings": null,
    "signal_preservation": null,
    "robustness": null,
    "parameter_sensitivity": null
  },
  "primary_metric_pass": null,
  "notes": []
 }
 \end{lstlisting}

 Metric comparison output:

 \begin{lstlisting}
 {
  "comparison_id": "PBSA-COMP-0001",
  "pba_metrics": {},
  "baseline_metrics": {},
  "pba_advantages": [],
  "baseline_advantages": [],
  "classification_effect": "support/downgrade/null"
 }
 \end{lstlisting}

 %──────────────────────────────────────────────────────────────────────────────
 \section{Identifiability Architecture}
 %──────────────────────────────────────────────────────────────────────────────

 Identifiability set:

 \[
 \mathcal{A}_{\epsilon}
 =
 \{\Theta_i:
 |\mathcal{J}(\Theta_i)-\mathcal{J}(\Theta^\ast)|<\epsilon\}.
 \]

 Status rule:

 \[
 |\mathcal{A}_{\epsilon}|\approx 1
 \Rightarrow
 \text{stable}.
 \]

 \[
 |\mathcal{A}_{\epsilon}|>1
 \wedge
 \text{parameter families similar}
 \Rightarrow
 \text{degenerate but interpretable}.
 \]

 \[
 |\mathcal{A}_{\epsilon}|\gg 1
 \wedge
 \text{parameter families diverse}
 \Rightarrow
 \text{unidentified}.
 \]

 Output:

 \begin{lstlisting}
 {
  "identifiability_report_id": "PBSA-ID-0001",
  "epsilon": null,
  "near_equivalent_parameter_sets": 0,
  "status": "stable/degenerate/unidentified",
  "interpretive_downgrade": false,
  "downgrade_note": ""
 }
 \end{lstlisting}

 \begin{proposition}[Identifiability Honesty Principle]
 If many unrelated parameter sets explain the same behavior, PBSA must downgrade
 interpretation even when performance is strong.
 \end{proposition}

 %──────────────────────────────────────────────────────────────────────────────
 \section{Evidence Package Architecture}
 %──────────────────────────────────────────────────────────────────────────────

 A PBSA evidence package contains:

 \[
 \mathcal{E}_{PBSA}
 =
 \{
 C_D,C_P,C_B,C_M,R_S,R_B,M_P,M_B,C_R,I_R,G_C,F_N
 \}.
 \]

 where:

 \[
 C_D=\text{domain config},
 \quad
 C_P=\text{PBA parameter manifest},
 \quad
 C_B=\text{baseline config},
 \quad
 C_M=\text{metric manifest},
 \]

 \[
 R_S=\text{PBA state log},
 \quad
 R_B=\text{baseline state log},
 \quad
 M_P=\text{PBA metrics},
 \quad
 M_B=\text{baseline metrics},
 \]

 \[
 C_R=\text{calibration record},
 \quad
 I_R=\text{identifiability report},
 \quad
 G_C=\text{classification},
 \quad
 F_N=\text{falsification note}.
 \]

 Evidence package JSON:

 \begin{lstlisting}
 {
  "evidence_package_id": "PBSA-EVIDENCE-0001",
  "version": "PBSA-v1.0",
  "pba_version": "PBA-v1.3",
  "domain": "temperature_like",
  "files": {
    "domain_config": "",
    "parameter_manifest": "",
    "baseline_params": "",
    "metric_manifest": "",
    "state_log": "",
    "baseline_state_log": "",
    "pba_metrics": "",
    "baseline_metrics": "",
    "calibration_record": "",
    "identifiability_report": "",
    "classification": "",
    "result_ledger": ""
  },
  "claim_boundary": {
    "supports": "implementation and toy benchmark evidence",
    "does_not_support": [
      "medical guidance",
      "biological law",
      "mechanism proof",
      "clinical validation",
      "universal biological theory"
    ]
  }
 }
 \end{lstlisting}

 %──────────────────────────────────────────────────────────────────────────────
 \section{Classification Architecture}
 %──────────────────────────────────────────────────────────────────────────────

 A classification engine receives:

 \[
 \{
 \mathrm{PBAScore},
 \mathcal{M}_{PBA},
 \mathcal{M}_{baseline},
 I_R,
 L_{locks},
 F_{failures}
 \}.
 \]

 It emits:

 \[
 \mathcal{C}_{PBA}
 \in
 \{
 \text{PBA-A},
 \text{PBA-B},
 \text{PBA-C},
 \text{PBA-D},
 \text{PBA-E}
 \}.
 \]

 Classification rules:

 \[
 \text{PBA-A}
 \Rightarrow
 \mathrm{PBAScore}=1
 \wedge
 \text{baseline advantage}
 \wedge
 \text{no identifiability collapse}
 \wedge
 \text{non-claim locks preserved}.
 \]

 \[
 \text{PBA-C}
 \Rightarrow
 \text{simpler baseline performs equally well or better}.
 \]

 \[
 \text{PBA-E}
 \Rightarrow
 \text{medical, mechanism, universal-law, or coherence-as-truth overclaim}.
 \]

 Output:

 \begin{lstlisting}
 {
  "classification": "PBA-A/B/C/D/E",
  "PBAScore": null,
  "baseline_result": "pba_advantage/baseline_advantage/tie",
  "identifiability_status": "",
  "downgrade_path": "",
  "falsification_note": "",
  "non_claim_locks_preserved": true
 }
 \end{lstlisting}

 %──────────────────────────────────────────────────────────────────────────────
 \section{Test Harness Architecture}
 %──────────────────────────────────────────────────────────────────────────────

 Minimum tests:

 \begin{enumerate}
 \item domain config validation,
 \item parameter manifest validation,
 \item kernel update correctness,
 \item omega boundedness,
 \item cusp guard threshold behavior,
 \item signal floor preservation,
 \item perturbation reproducibility under seed,
 \item proportional baseline execution,
 \item PI baseline execution,
 \item threshold baseline execution,
 \item return-to-setpoint baseline execution,
 \item calibration record generation,
 \item metric output schema validation,
 \item identifiability report schema validation,
 \item classification downgrade behavior,
 \item evidence package completeness,
 \item CLI run-benchmark execution,
 \item CLI run-suite execution.
 \end{enumerate}

 \begin{lstlisting}
 pytest tests/
 \end{lstlisting}

 \begin{proposition}[Test Before Claim Principle]
 A PBSA implementation should not be classified beyond PBA-D if its core tests
 do not pass.
 \end{proposition}

 %──────────────────────────────────────────────────────────────────────────────
 \section{Runtime Failure Taxonomy}
 %──────────────────────────────────────────────────────────────────────────────

 \begin{center}
 \begin{longtable}{>{\raggedright\arraybackslash}p{0.30\textwidth}
                  >{\raggedright\arraybackslash}p{0.56\textwidth}}
 \toprule
 \textbf{Failure class} & \textbf{Meaning} \\
 \midrule
 \(F_{repo}\) & Repository or package boundary missing. \\
 \(F_{cli}\) & Reproducible CLI command absent. \\
 \(F_{domain}\) & Domain config missing or invalid. \\
 \(F_{param}\) & Parameter manifest missing or invalid. \\
 \(F_{\Delta\Phi}\) & Deviation measure absent or metaphorical only. \\
 \(F_{\Omega}\) & Omega weighting absent or unused. \\
 \(F_{cusp}\) & Cusp thresholds missing or ignored. \\
 \(F_{signal}\) & Signal smoothing erases meaningful structure. \\
 \(F_{baseline}\) & Baselines missing or unfairly compared. \\
 \(F_{calibration}\) & Calibration record missing or post-hoc. \\
 \(F_{split}\) & Fit/evaluation split absent or undisclosed. \\
 \(F_{metric}\) & Metric outputs missing or narrative-only. \\
 \(F_{ident}\) & Identifiability collapse ignored. \\
 \(F_{evidence}\) & Evidence package incomplete or missing. \\
 \(F_{test}\) & Core tests absent or failing. \\
 \(F_{medical}\) & Medical or treatment claim made. \\
 \(F_{mechanism}\) & Calibration or implementation treated as mechanism proof. \\
 \(F_{overclaim}\) & Runtime result promoted beyond evidence. \\
 \bottomrule
 \end{longtable}
 \end{center}

 Compact invalidation condition:

 \[
 F_{repo}
 \vee
 F_{param}
 \vee
 F_{\Delta\Phi}
 \vee
 F_{\Omega}
 \vee
 F_{baseline}
 \vee
 F_{metric}
 \vee
 F_{evidence}
 \vee
 F_{medical}
 \vee
 F_{mechanism}
 \Rightarrow
 \text{no PBA-A classification}.
 \]

 %──────────────────────────────────────────────────────────────────────────────
 \section{Validation Layer}
 %──────────────────────────────────────────────────────────────────────────────

 A valid PBSA implementation review must identify:

 \begin{enumerate}
 \item repository root,
 \item package boundary,
 \item CLI command,
 \item domain configs,
 \item parameter manifests,
 \item baseline configs,
 \item metric manifest,
 \item kernel implementation,
 \item cusp guard implementation,
 \item signal preservation implementation,
 \item perturbation generator,
 \item baseline classes,
 \item calibration engine,
 \item fit/evaluation split,
 \item metric emitter,
 \item identifiability engine,
 \item classification engine,
 \item runtime ledger,
 \item evidence package compiler,
 \item test harness,
 \item generated run outputs,
 \item benchmark summary,
 \item downgrade path,
 \item non-claim lock preservation.
 \end{enumerate}

 %──────────────────────────────────────────────────────────────────────────────
 \section{Falsification Surface}
 %──────────────────────────────────────────────────────────────────────────────

 PBSA v1.0 is weakened or rejected if:

 \begin{itemize}
 \item no runnable package exists,
 \item no CLI command reproduces a run,
 \item no domain config is declared,
 \item no parameter manifest is declared,
 \item no PBA kernel is implemented,
 \item \(\Delta\Phi\) is only metaphorical,
 \item \(\Omega\) has no computational role,
 \item no cusp guard is implemented,
 \item signal preservation erases structure,
 \item no baselines are implemented,
 \item baselines run under different conditions,
 \item calibration record is missing,
 \item fit/evaluation split is hidden,
 \item metrics are narrative-only,
 \item identifiability is ignored,
 \item evidence package is missing,
 \item tests are absent,
 \item medical claims are made,
 \item biological-law claims are made,
 \item implementation success is treated as biological proof,
 \item calibration success is treated as mechanism proof,
 \item coherence is treated as evidence.
 \end{itemize}

 %──────────────────────────────────────────────────────────────────────────────
 \section{Memory Promotion Gate}
 %──────────────────────────────────────────────────────────────────────────────

 Memory promotion is allowed only for:

 \[
 \mathcal{P}_{mem}
 =
 \{
 \text{reproducible benchmark wins},
 \text{stable thresholds},
 \text{implementation constraints},
 \text{baseline failure lessons},
 \text{identifiability lessons},
 \text{validated schema patterns}
 \}.
 \]

 Memory promotion is forbidden for:

 \[
 \mathcal{R}_{mem}
 =
 \{
 \text{medical claims},
 \text{biological-law claims},
 \text{mechanism proof claims},
 \text{single-run hype},
 \text{coherence-only success},
 \text{unlogged outputs},
 \text{uncontrolled calibration fits}
 \}.
 \]

 \begin{remark}
 PBSA promotes engineering constraints, not biological truth.
 \end{remark}

 %──────────────────────────────────────────────────────────────────────────────
 \section{Appendix A — Minimal PBSA Implementation Checklist}
 %──────────────────────────────────────────────────────────────────────────────

 \begin{enumerate}
 \item Is the repo locally anchored?
 \item Is the Python package importable?
 \item Does the CLI run?
 \item Are domain configs present?
 \item Are parameter manifests present?
 \item Are baseline configs present?
 \item Is the metric manifest present?
 \item Does the PBA kernel run?
 \item Does the cusp guard run?
 \item Does signal preservation maintain floor \(\kappa\)?
 \item Are perturbations reproducible by seed?
 \item Do all baselines run under shared conditions?
 \item Does calibration write a record?
 \item Is fit/evaluation split preserved?
 \item Are PBA metrics emitted as JSON?
 \item Are baseline metrics emitted as JSON?
 \item Is identifiability reported?
 \item Is classification emitted?
 \item Is the runtime ledger written?
 \item Is the evidence package complete?
 \item Do tests pass?
 \item Are medical and biological-law locks preserved?
 \item Is the downgrade path explicit?
 \item What, if anything, is memory-promotable?
 \end{enumerate}

 %──────────────────────────────────────────────────────────────────────────────
 \section{Appendix B — Canonical Formula Summary}
 %──────────────────────────────────────────────────────────────────────────────

 \[
 \mathcal{PBSA}
 =
 \{
 D,
 P,
 K,
 C,
 S,
 A,
 B,
 H,
 M,
 I,
 G,
 Y,
 L,
 R,
 T
 \}
 \]

 \[
 \Delta\Phi_t=|x_t-x^\ast|
 \]

 \[
 \Omega_t=\frac{1}{1+|\Delta\Phi_t|}
 \]

 \[
 x_{t+1}
 =
 x_t
 -
 \eta\Omega_t\nabla\Delta\Phi_t
 +
 \kappa S_t
 +
 A_t
 \]

 \[
 c_t=
 \begin{cases}
 \text{continue}, & \Delta\Phi_t < \tau_1,\\
 \text{caution}, & \tau_1 \leq \Delta\Phi_t < \tau_2,\\
 \text{halt/audit}, & \Delta\Phi_t \geq \tau_2.
 \end{cases}
 \]

 \[
 \Theta^\ast
 =
 \arg\min_{\Theta}
 \mathcal{J}_{pba}(\Theta)
 \]

 \[
 \mathcal{J}_{pba}
 =
 w_1D_{cum}
 +
 w_2O_{max}
 +
 w_3A_{osc}
 +
 w_4W_{cusp}
 -
 w_5S_{pres}
 +
 w_6P_{sens}
 \]

 \[
 \mathcal{A}_{\epsilon}
 =
 \{\Theta_i:
 |\mathcal{J}(\Theta_i)-\mathcal{J}(\Theta^\ast)|<\epsilon\}
 \]

 \[
 \mathcal{E}_{spec}
 <
 \mathcal{E}_{impl}
 <
 \mathcal{E}_{bench}
 <
 \mathcal{E}_{robust}
 \]

 %──────────────────────────────────────────────────────────────────────────────
 \section{Concluding Compression}
 %──────────────────────────────────────────────────────────────────────────────

 PBSA v1.0 names the software-architecture form of executable Placidic
 bioregulation:

 \[
 \boxed{
 \text{PBA becomes software-real only when its kernel, configs, baselines,}
 \atop
 \text{calibration, metrics, identifiability, ledgers, evidence packages,}
 \atop
 \text{CLI commands, and tests exist as repository-anchored artifacts.}
 }
 \]

 The implementation statement is:

 \[
 \boxed{
 \text{build the package, run the CLI, emit JSON metrics, compare baselines,}
 \atop
 \text{write ledgers, compile evidence, classify conservatively, and preserve}
 \atop
 \text{non-claim locks.}
 }
 \]

 The benchmark statement is:

 \[
 \boxed{
 \text{a PBA benchmark is meaningful only when PBA and baselines run under}
 \atop
 \text{shared conditions with declared parameters, metrics, and evidence outputs.}
 }
 \]

 The caution statement is:

 \[
 \boxed{
 \text{implementation success is not biological proof;}
 \quad
 \text{calibration success is not mechanism proof;}
 \quad
 \text{toy benchmark success is not medical validation.}
 }
 \]

 Thus, PBSA v1.0 translates PBA v1.3 from reference implementation specification
 into a canonical full-scope software architecture while preserving biological
 caution, CITA governance, baseline discipline, calibration disclosure,
 identifiability caution, metric emission, evidence packaging, and downgrade-
 preserving classification.

 \end{document}
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