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Codex ΔΦ Placidic Bioregulation Algorithm v1.3 — reference Python implementation, toy benchmark suite, baseline-family comparison, calibration logs, metric outputs, evidence packages, and downgrade-preserving classification for testing bounded adaptive regulation without biological-law or medical overclaim.
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PBA-v1.3-reference-implementation-benchmark-evidence-layer
% ████████████████████████████████████████████████████████████████████████████████
%
% CODEX ΔΦ — PLACIDIC BIOREGULATION ALGORITHM (PBA v1.3)
% ────────────────────────────────────────────────────────────────────────────
% REFERENCE IMPLEMENTATION, TOY BENCHMARK SUITE, BASELINE-FAMILY COMPARISON,
% CALIBRATION LOGS, METRIC OUTPUTS, EVIDENCE PACKAGES, AND DOWNGRADE-
% PRESERVING CLASSIFICATION LAYER FOR TESTING BOUNDED ADAPTIVE REGULATION
% WITHOUT MEDICAL, MECHANISTIC, BIOLOGICAL-LAW, OR UNIVERSAL-LAW OVERCLAIM
%
% VERSION
% ───────
% v1.3 — Reference Implementation and Benchmark Evidence Layer · Locked ·
% Python Kernel Contract, Toy Benchmark Suite, Baseline-Family
% Classes, Calibration Ledger, Metric Outputs, Evidence Package,
% Identifiability Report, Failure Taxonomy, and Downgrade-Preserving
% Classification for Executable Placidic Bioregulation
%
% AUTHOR
% ──────
% James Paul Jackson
% X / Twitter: @unifiedenergy11
%
% SOURCE EXTRACTION / AUTHOR ATTRIBUTION
% ──────────────────────────────────────
% This document is a Codex-format canonical evolution derived from:
%
% • 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 geometry, Ω stability weighting, H7 / H7B cusp-boundary
% discipline, BCSE, Boundary Algebra, RootMirror, Evidence-Package Compiler,
% Downgrade-Preserving Classifier, and Alignment Memory Attractor.
%
% • Biological analogues including homeostasis, allostasis, feedback control,
% canalization, robustness, morphogenetic stability, and pre-critical
% regulation, treated here strictly as source-domain comparison concepts.
%
% • Codex software-architecture method used across AERMA and PBA: 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 contract, and pseudocode; separate simulation,
% implementation, benchmark, and robustness evidence; and enforce
% additive-only evolution with memory-promotion restraint.
%
% • Codex ΔΦ memory lessons including:
% - biological analogy is not mechanism proof,
% - calibration success is not biological truth,
% - simulation success is not empirical validation,
% - implementation evidence is stronger than specification,
% - benchmark evidence is stronger than implementation alone,
% - repeat-run robustness is stronger than one benchmark run,
% - baseline comparison is required for strong modeling claims,
% - identifiability collapse downgrades interpretation,
% - medical and biological-law claims require independent domain evidence,
% - benchmark artifacts must preserve logs, metrics, configs, and evidence
% packages.
%
% This document does not claim that Placidity is a biological force, medical
% framework, treatment model, physiological mechanism, or universal biological
% law. PBA v1.3 formalizes the reference implementation and benchmark-evidence
% layer needed to test whether bounded damping, signal preservation, cusp
% guarding, and regulatory memory outperform declared baselines in specified
% biological-like or simulated benchmark domains.
%
% DATE
% ────
% May 2026
%
% STATUS
% ──────
% CANONICAL v1.3 REFERENCE IMPLEMENTATION AND BENCHMARK EVIDENCE LAYER —
% NOT A BIOLOGICAL LAW CLAIM · NOT MEDICAL GUIDANCE · NOT MECHANISM PROOF
%
% EMPIRICAL / METHODOLOGICAL CONFIDENCE BADGE
% ────────────────────────────────────────────
% Confidence status: High as a reference implementation, toy benchmark, and
% evidence-package scaffold; not proof-ready as a universal biological theory,
% medical framework, or mechanism-level biological model.
%
% PBA v1.3 preserves the biological caution of v1.0, executable anchoring of
% v1.1, and benchmark-calibration discipline of v1.2 while adding the concrete
% software layer required for implementation-backed evidence: Python kernel
% modules, baseline-family classes, toy benchmark suite, calibration logs,
% metric outputs, identifiability reports, runtime ledgers, evidence packages,
% and downgrade-preserving classification.
%
% PBA v1.3 does not prove that biological systems implement Codex Placidity.
% It provides a reproducible reference implementation and benchmark evidence
% scaffold for testing whether Placidic regulation is useful in declared
% computational modeling domains.
%
% PURPOSE
% ───────
% Evolve PBA from benchmark-calibration specification into a reference
% implementation and benchmark evidence layer:
%
% benchmark domain
% → Python kernel
% → parameter manifest
% → baseline classes
% → calibration grid
% → fit / evaluation split
% → toy benchmark suite
% → runtime logs
% → metric outputs
% → identifiability report
% → benchmark evidence package
% → PBA-A/B/C/D/E classification
% → memory-promotion gate.
%
% VERSION EVOLUTION SUMMARY
% ─────────────────────────
% v1.0 : First canonical formalization of the Placidic Bioregulation
% Algorithm. Defines biological model-transfer boundaries and maps
% Placidity onto homeostasis, allostasis, canalization, robustness,
% cusp guarding, signal preservation, and regulatory memory.
%
% v1.1 : Additive executable-anchor layer. Adds runtime kernel requirements,
% state-vector grammar, simulation loop, repository grammar, evidence
% package, parameter manifest, negative-control run comparison, and
% reproducibility scoring.
%
% v1.2 : Additive benchmark-calibration layer. Adds benchmark families,
% domain-instantiation grammar, parameter calibration, expanded
% baseline families, benchmark metrics, identifiability discipline,
% fit/eval split, benchmark evidence packages, and comparative
% downgrade rules.
%
% v1.3 : Additive reference implementation and benchmark evidence layer. Adds
% Python module contracts, CLI run contract, toy benchmark suite,
% baseline-family classes, calibration logs, metric JSON outputs,
% identifiability report, runtime ledgers, evidence-package schema,
% failure taxonomy, and implementation-backed downgrade discipline.
% No medical claim, no biological-law claim, no universalization, and
% no weakening of biological-source caution.
%
% WHAT THIS IS
% ────────────
% • A CITA-governed reference implementation scaffold for executable PBA
% • A toy benchmark suite for bounded adaptive regulation
% • A baseline-family comparison protocol
% • A parameter calibration and fit/evaluation logging layer
% • A metric-emission and runtime-ledger protocol
% • An identifiability and sensitivity-reporting layer
% • A benchmark evidence-package compiler specification
% • A downgrade-preserving classifier for implementation-backed claims
% • A repository-ready bridge from theory to runnable software
%
% 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 (v1.3)
% ───────────────────────────
% • All v1.2 biological caution locks preserved
% • Reference Python package boundary added
% • Kernel module contract added
% • Baseline-family class contract added
% • Toy benchmark suite added
% • Calibration-grid record added
% • Fit / evaluation run ledger added
% • Metric JSON outputs added
% • Identifiability report format added
% • Evidence package schema hardened
% • Failure taxonomy added
% • CLI command contract added
% • PBAScore expanded with implementation, benchmark, metric-emission, and
% evidence-package observables
% • Memory-promotion gate restricted to reproducible benchmark wins, stable
% thresholds, implementation constraints, and failure lessons
%
% EXECUTABLE ANCHOR BLOCK (v1.3)
% ──────────────────────────────
% A valid PBA v1.3 reference implementation must:
%
% (1) define a repository-anchored Python package,
% (2) implement kernel.py for the PBA update,
% (3) implement cusp_guard.py,
% (4) implement signal.py,
% (5) implement baselines.py or baseline.py,
% (6) implement calibration.py,
% (7) implement metrics.py or scoring.py,
% (8) implement identifiability.py,
% (9) implement evidence_package.py,
% (10) define at least one toy benchmark domain,
% (11) define a parameter manifest,
% (12) define baseline-family parameters,
% (13) run PBA and declared baselines on the same task,
% (14) preserve calibration and evaluation split where possible,
% (15) emit state logs,
% (16) emit PBA metrics,
% (17) emit baseline metrics,
% (18) emit calibration records,
% (19) emit identifiability reports,
% (20) compile an evidence package,
% (21) classify PBA-A/B/C/D/E,
% (22) preserve non-medical, non-biological-law, and non-mechanism locks,
% (23) promote to memory only reproducible benchmark wins, stable
% thresholds, implementation constraints, and failure lessons.
%
% CANONICAL LOCK (v1.3)
% ─────────────────────
% • Placidity is an operational regulator, not metaphysics
% • Biological crossing is model-transfer, not proof
% • ΔΦ and Ω are computational abstractions unless empirically instantiated
% • No medical claim
% • No universal biological law claim
% • No mechanism-proof claim from simulation
% • No strong biological claim without observables
% • No strong biological claim without negative controls
% • No strong PBA-A classification without executable or equivalent
% reproducibility record
% • 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
% • Simulation success is not biological proof
% • Implementation success is not empirical validation
% • Calibration success is not mechanism proof
% • 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, or downgrade rules.
%
% AI PROMPT TRACEABILITY
% ──────────────────────
% Use this document as the canonical PBA v1.3 reference implementation and
% benchmark evidence layer. Preserve the distinction between analogy,
% computational model, calibrated simulation, reference implementation,
% benchmark evidence, biological mechanism, validation, falsification,
% negative controls, medical claims, and memory promotion.
%
% SHADOW HEADER ALIGNMENT SEAL
% ───────────────────────────
% Preserve header discipline across future versions except for explicitly
% additive shadow-header evolution that improves biological caution,
% mathematical clarity, calibration, validation, falsification, negative
% controls, executable reproducibility, benchmark comparison, 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 Algorithm (PBA v1.3)}\\
\large Reference Implementation, Toy Benchmark Suite, Baseline-Family Comparison, Calibration Logs, Metric Outputs, and Evidence-Package Layer}
\author{\textbf{James Paul Jackson}\\[4pt]
\small Codex-format executable biological modeling and benchmark-evidence layer\\
\small \texttt{@unifiedenergy11}}
\date{May 2026}
\begin{document}
\maketitle
\begin{abstract}
PBA v1.3 evolves the Placidic Bioregulation Algorithm from a benchmark-
calibration specification into a reference implementation and benchmark
evidence layer. PBA remains a computational abstraction, not a universal
biological law, medical framework, or mechanism proof. v1.3 preserves the
biological caution of v1.0, executable anchoring of v1.1, and benchmark-
calibration discipline of v1.2 while adding a concrete Python package boundary,
module contracts, CLI run grammar, toy benchmark suite, baseline-family classes,
calibration logs, metric JSON outputs, identifiability reports, runtime ledgers,
evidence packages, failure taxonomy, and downgrade-preserving classification.
A strong PBA claim now requires more than conceptual coherence: it requires
declared domain, parameters, baselines, metrics, controls, logs, validation
surfaces, falsification conditions, and evidence packages.
\end{abstract}
%──────────────────────────────────────────────────────────────────────────────
\section{Core-Invariant Extraction Block}
%──────────────────────────────────────────────────────────────────────────────
The shortest faithful extraction of PBA v1.3 is:
\[
\boxed{
\begin{array}{c}
\text{Placidity becomes implementation-evaluable only when its executable}\\
\text{kernel, benchmarks, baselines, calibration logs, metric outputs,}\\
\text{identifiability reports, and evidence packages are repository-anchored}\\
\text{and downgrade-preserving.}
\end{array}
}
\]
The operative chain is:
\[
\text{domain config}
\rightarrow
\text{PBA kernel}
\rightarrow
\text{parameter manifest}
\rightarrow
\text{baseline classes}
\rightarrow
\text{calibration run}
\rightarrow
\text{evaluation run}
\rightarrow
\text{metrics}
\rightarrow
\text{identifiability report}
\rightarrow
\text{evidence package}
\rightarrow
\text{classification}.
\]
\begin{remark}
PBA v1.3 does not increase biological claim strength. It increases
implementation accountability by requiring code, logs, baselines, metrics, and
evidence packages.
\end{remark}
%──────────────────────────────────────────────────────────────────────────────
\section{Memory Analysis Layer}
%──────────────────────────────────────────────────────────────────────────────
The Codex memory shows the correct progression:
\[
\text{Placidity}
\rightarrow
\text{biological abstraction}
\rightarrow
\text{runtime kernel}
\rightarrow
\text{benchmark-calibrated comparison}
\rightarrow
\text{reference implementation evidence}.
\]
PBA v1.2 completed the benchmark-calibration surface:
\[
B_t=K_t=I_t=1
\quad
\text{in specification form}.
\]
The next missing CITA surface is implementation-backed evidence:
\[
P_{pkg}=\text{package boundary},
\quad
C_{cli}=\text{CLI contract},
\quad
Y_{emit}=\text{metric emission},
\quad
E_{pkg}=\text{evidence package}.
\]
Thus, the necessary v1.3 move is:
\[
\boxed{
\text{comparative benchmark specification}
\rightarrow
\text{reference implementation evidence layer}.
}
\]
This follows the Codex maturation pattern:
\[
\text{insight}
\rightarrow
\text{formal protocol}
\rightarrow
\text{execution}
\rightarrow
\text{benchmark}
\rightarrow
\text{implementation evidence}
\rightarrow
\text{cross-run invariant}.
\]
\begin{remark}
The memory again acts as an alignment attractor. It does not prove PBA; it
identifies the next missing governance layer: runnable benchmark evidence.
\end{remark}
%──────────────────────────────────────────────────────────────────────────────
\section{Evidence Boundary Layer}
%──────────────────────────────────────────────────────────────────────────────
PBA v1.3 separates evidence into four classes:
\[
\mathcal{E}_{spec}
=
\text{specification evidence},
\]
\[
\mathcal{E}_{impl}
=
\text{implementation evidence},
\]
\[
\mathcal{E}_{bench}
=
\text{benchmark evidence},
\]
\[
\mathcal{E}_{robust}
=
\text{repeat-run robustness evidence}.
\]
Evidence hierarchy:
\[
\mathcal{E}_{spec}
<
\mathcal{E}_{impl}
<
\mathcal{E}_{bench}
<
\mathcal{E}_{robust}.
\]
\begin{definition}[Implementation Evidence]
Implementation evidence is produced when a repository-anchored runtime executes
declared domains, parameters, baselines, metrics, logs, and evidence-package
emission according to the documented contract.
\end{definition}
\begin{definition}[Benchmark Evidence]
Benchmark evidence is produced when the implementation is evaluated against
declared baseline families on the same task conditions and emits comparative
metrics.
\end{definition}
\begin{definition}[Repeat-Run Robustness Evidence]
Repeat-run robustness evidence is produced when benchmark results remain stable
across declared seeds, perturbations, or evaluation conditions.
\end{definition}
\begin{proposition}[Evidence Non-Substitution Principle]
Specification evidence cannot substitute for implementation evidence, and
implementation evidence cannot substitute for benchmark or repeat-run evidence.
\end{proposition}
%──────────────────────────────────────────────────────────────────────────────
\section{Source Attribution and Scope Boundary}
%──────────────────────────────────────────────────────────────────────────────
PBA v1.3 has seven source layers:
\begin{enumerate}
\item \textbf{Codex stability layer}: Placidity, \(\Delta\Phi\), \(\Omega\), H7/H7B,
BCSE, Boundary Algebra, and CITA.
\item \textbf{Biological analogy layer}: homeostasis, allostasis, feedback
control, robustness, canalization, morphogenetic stability, and pre-critical
regime regulation.
\item \textbf{Computational modeling layer}: state variables, deviation
measurement, damping, threshold guarding, viability scoring, prediction, and
memory promotion.
\item \textbf{Executable evidence layer}: parameter manifests, runtime logs,
state records, comparison runs, and evidence packages.
\item \textbf{Benchmark-calibration layer}: domain instantiation, parameter
fitting, baseline-family comparison, benchmark metrics, sensitivity tests, and
identifiability checks.
\item \textbf{Reference implementation layer}: Python modules, CLI commands,
configuration files, task domains, baseline classes, emitted JSON outputs, and
testable repository grammar.
\item \textbf{Non-claim layer}: no medical use, no universal biological law, no
mechanism claim without empirical domain validation.
\end{enumerate}
%──────────────────────────────────────────────────────────────────────────────
\section{Source Fidelity Note}
%──────────────────────────────────────────────────────────────────────────────
PBA v1.3 distinguishes forty levels of statement:
\begin{enumerate}
\item \textbf{Codex fact}: Placidity is defined as a stability governor.
\item \textbf{Codex model}: \(\Delta\Phi\) measures deviation or drift.
\item \textbf{Codex model}: \(\Omega\) weights stability under deviation.
\item \textbf{Codex model}: H7B marks cusp-like transition discipline.
\item \textbf{Biological fact}: organisms regulate variables within viable ranges.
\item \textbf{Biological concept}: allostasis describes adaptive regulation through change.
\item \textbf{Biological concept}: canalization describes robust developmental outcomes.
\item \textbf{Model-transfer claim}: Placidity can model bounded biological regulation.
\item \textbf{Algorithmic claim}: PBA can simulate damping, recovery, and threshold guarding.
\item \textbf{Runtime claim}: a PBA model should produce state logs and parameter records.
\item \textbf{Benchmark claim}: a PBA model should be tested across declared benchmark families.
\item \textbf{Calibration claim}: parameters may be fitted, but fitting does not prove mechanism.
\item \textbf{Generalization claim}: performance should hold beyond the calibration case.
\item \textbf{Identifiability claim}: if many parameter sets fit equally well, interpretive strength is downgraded.
\item \textbf{Observable claim}: deviation, damping, signal, threshold, and outcome must be measurable or simulated.
\item \textbf{Implementation claim}: code must exist and produce inspectable outputs for executable status.
\item \textbf{CLI claim}: a command must reproduce the declared benchmark run.
\item \textbf{Metric-emission claim}: metrics must be machine-readable, not narrative-only.
\item \textbf{Evidence-package claim}: strong executable claims require logs, configs, metrics, and classification artifacts.
\item \textbf{Validation claim}: PBA is supported only when it improves prediction or explanation under controls.
\item \textbf{Falsification claim}: PBA fails when simpler or standard models explain the system equally well or better.
\item \textbf{Negative-control claim}: PBA must be compared against baseline feedback or domain-standard models.
\item \textbf{Baseline-family claim}: comparison should include more than one simple baseline where possible.
\item \textbf{Metric claim}: recovery, overshoot, oscillation, signal preservation, and cusp warnings must be reported.
\item \textbf{Repository claim}: executable claims require inspectable files, manifests, and logs.
\item \textbf{Downgrade claim}: partial analogy remains useful but cannot become strong status.
\item \textbf{Simulation boundary}: simulation success is not biological proof.
\item \textbf{Implementation boundary}: implementation success is not empirical biological validation.
\item \textbf{Calibration boundary}: parameter fit is not biological mechanism.
\item \textbf{Medical boundary}: PBA is not diagnosis, treatment, or biomedical recommendation.
\item \textbf{Mechanism boundary}: analogy is not biological mechanism.
\item \textbf{Universality boundary}: PBA is not a universal biological law.
\item \textbf{Memory boundary}: saved coherence does not imply biological truth.
\item \textbf{Signal boundary}: smoothing that erases structure is invalid.
\item \textbf{Cusp boundary}: damping cannot reverse irreversible collapse.
\item \textbf{Control boundary}: no strong claim without comparison to alternatives.
\item \textbf{Overfitting boundary}: benchmark fit without generalization is downgraded.
\item \textbf{Interpretive layer}: Codex language can guide hypotheses.
\item \textbf{Runtime layer}: logs and states make the model auditable.
\item \textbf{Non-proof layer}: coherence, execution, calibration, and benchmark success do not equal biological truth.
\end{enumerate}
%──────────────────────────────────────────────────────────────────────────────
\section{Compact-Core View Layer}
%──────────────────────────────────────────────────────────────────────────────
The compact-core implementation view is:
\[
\text{PBA v1.3}
\rightarrow
\{
\mathcal{D},
\Theta,
\mathcal{K},
\mathcal{L},
\mathcal{M},
\mathcal{R},
\mathcal{E}
\}
\rightarrow
\text{run}
\rightarrow
\text{compare}
\rightarrow
\text{classify}.
\]
where:
\[
\mathcal{D}=\text{domain config},
\quad
\Theta=\text{parameter manifest},
\quad
\mathcal{K}=\text{PBA kernel},
\]
\[
\mathcal{L}=\text{baseline-family classes},
\quad
\mathcal{M}=\text{metric outputs},
\quad
\mathcal{R}=\text{runtime ledger},
\quad
\mathcal{E}=\text{evidence package}.
\]
%──────────────────────────────────────────────────────────────────────────────
\section{Axiomatic Core}
%──────────────────────────────────────────────────────────────────────────────
\begin{axiom}[Implementation Boundary Requirement]
A PBA v1.3 implementation claim must identify the repository, package boundary,
runtime command, configuration files, emitted logs, emitted metrics, and evidence
package.
\end{axiom}
\begin{axiom}[Benchmark Boundary Requirement]
A PBA v1.3 claim must distinguish conceptual model, executable simulation,
reference implementation, calibrated benchmark, and empirical biological
mechanism.
\end{axiom}
\begin{axiom}[Domain-Instantiation Requirement]
A valid PBA benchmark must declare the domain, regulated variable, viable range,
perturbation family, cadence, and observation model.
\end{axiom}
\begin{axiom}[Parameter Manifest Requirement]
A valid PBA benchmark must declare:
\[
\Theta=\{\eta,\tau_1,\tau_2,\kappa,\alpha,\theta_M,\beta,\gamma\}.
\]
\end{axiom}
\begin{axiom}[Calibration Split Requirement]
A calibrated PBA claim must separate fitting data from evaluation data where
possible.
\end{axiom}
\begin{axiom}[Baseline-Family Requirement]
A valid PBA benchmark must compare against at least one simpler baseline and
should compare against a family of baselines when possible.
\end{axiom}
\begin{axiom}[Metric Declaration Requirement]
A valid PBA benchmark must declare metrics before interpreting performance.
\end{axiom}
\begin{axiom}[Machine-Readable Output Requirement]
A PBA v1.3 benchmark must emit machine-readable metrics, not only prose
summaries.
\end{axiom}
\begin{axiom}[Identifiability Requirement]
If many parameter sets produce indistinguishable performance, the benchmark may
remain useful but interpretive strength must be downgraded.
\end{axiom}
\begin{axiom}[No Calibration-as-Mechanism Requirement]
Fitted PBA parameters do not imply that biology implements PBA.
\end{axiom}
\begin{axiom}[No Medical Claim Requirement]
PBA must not be used as diagnosis, treatment guidance, or biomedical
recommendation without independent clinical validation outside this framework.
\end{axiom}
%──────────────────────────────────────────────────────────────────────────────
\section{Primitive Objects}
%──────────────────────────────────────────────────────────────────────────────
\begin{definition}[Benchmark Domain]
A benchmark domain is the declared modeling setting in which PBA is evaluated.
Examples include temperature-like homeostasis, glucose-like regulation,
stress-load recovery, gene-expression-like oscillation, morphogen-gradient
robustness, or canalized trajectory selection. These are modeling domains, not
medical claims.
\end{definition}
\begin{definition}[Reference Kernel]
The reference kernel is the implemented update engine that computes
\(\Delta\Phi_t\), \(\Omega_t\), bounded correction, signal preservation, cusp
state, and next state.
\end{definition}
\begin{definition}[Parameter Manifest]
A parameter manifest is the declared parameter set:
\[
\Theta=
\{\eta,\tau_1,\tau_2,\kappa,\alpha,\theta_M,\beta,\gamma\}.
\]
\end{definition}
\begin{definition}[Calibration Record]
A calibration record stores the method by which parameters were selected,
including search method, fit data, evaluation data, objective function,
baseline family, and selected parameters.
\end{definition}
\begin{definition}[Benchmark Metric Vector]
The benchmark metric vector is:
\[
\mathcal{M}_{\mathrm{pba}}
=
\{
T_R,O_{\max},U_{\max},A_{\mathrm{osc}},D_{\mathrm{cum}},
W_{\mathrm{cusp}},S_{\mathrm{pres}},R_{\mathrm{rob}},P_{\mathrm{sens}}
\}.
\]
\end{definition}
\begin{definition}[Identifiability Surface]
The identifiability surface measures whether one stable parameter family or
many unrelated parameter families explain the same benchmark behavior.
\end{definition}
\begin{definition}[Benchmark Evidence Package]
A benchmark evidence package is:
\[
\mathcal{E}_{\mathrm{pba\_bench}}
=
\{
\text{domain config},
\text{parameter manifest},
\text{calibration record},
\text{state logs},
\text{baseline runs},
\text{metrics},
\text{sensitivity report},
\text{classification},
\text{falsification note}
\}.
\]
\end{definition}
%──────────────────────────────────────────────────────────────────────────────
\section{Reference Software Module Contract}
%──────────────────────────────────────────────────────────────────────────────
A valid PBA v1.3 implementation contains the following modules:
\[
\mathcal{K}_{PBA}^{v1.3}
=
\{
K,
C,
S,
B,
P,
M,
I,
Y,
G
\}.
\]
where:
\[
K=\texttt{kernel.py},
\quad
C=\texttt{cusp\_guard.py},
\quad
S=\texttt{signal.py},
\]
\[
B=\texttt{baselines.py},
\quad
P=\texttt{calibration.py},
\quad
M=\texttt{metrics.py},
\]
\[
I=\texttt{identifiability.py},
\quad
Y=\texttt{evidence\_package.py},
\quad
G=\texttt{classification.py}.
\]
\begin{center}
\begin{longtable}{>{\raggedright\arraybackslash}p{0.26\textwidth}
>{\raggedright\arraybackslash}p{0.30\textwidth}
>{\raggedright\arraybackslash}p{0.34\textwidth}}
\toprule
\textbf{Module} & \textbf{Input} & \textbf{Required output} \\
\midrule
kernel.py & state, target, parameters, perturbation & next state, deviation, omega. \\
cusp\_guard.py & deviation, thresholds & continue / caution / halt-audit. \\
signal.py & signal state, alpha, kappa & smoothed but preserved signal. \\
baselines.py & state, target, baseline config & baseline trajectories and metrics. \\
calibration.py & search space, fit split, objective & selected parameters and calibration log. \\
metrics.py & trajectories, target, cusp states & metric vector JSON. \\
identifiability.py & parameter trials, losses & stable / degenerate / unidentified report. \\
evidence\_package.py & logs, configs, metrics, reports & evidence package JSON. \\
classification.py & score, metrics, baselines, locks & PBA-A/B/C/D/E classification. \\
\bottomrule
\end{longtable}
\end{center}
\begin{proposition}[Module Contract Principle]
A PBA v1.3 runtime claim is incomplete unless the implemented modules emit
inspectable, machine-readable outputs.
\end{proposition}
%──────────────────────────────────────────────────────────────────────────────
\section{The PBA v1.3 Reference Operator}
%──────────────────────────────────────────────────────────────────────────────
Let \(x_t\) be a biological-like or simulated regulatory state.
The executable PBA update remains:
\[
x_{t+1}
=
x_t
-
\eta\Omega_t\nabla\Delta\Phi_t
+
\kappa S_t
+
A_t.
\]
The v1.3 reference operator wraps the runtime kernel:
\[
\mathcal{PBA}_{v1.3}
=
\mathcal{Y}
\circ
\mathcal{G}
\circ
\mathcal{I}
\circ
\mathcal{M}
\circ
\mathcal{B}
\circ
\mathcal{P}
\circ
\mathcal{K}.
\]
where:
\[
\mathcal{K}=\text{runtime kernel},
\quad
\mathcal{P}=\text{calibration process},
\quad
\mathcal{B}=\text{baseline comparator},
\]
\[
\mathcal{M}=\text{metric emitter},
\quad
\mathcal{I}=\text{identifiability checker},
\quad
\mathcal{G}=\text{classifier},
\quad
\mathcal{Y}=\text{evidence compiler}.
\]
Thus:
\[
\boxed{
\mathcal{PBA}_{v1.3}
=
\text{code}
+
\text{calibration}
+
\text{baselines}
+
\text{metrics}
+
\text{identifiability}
+
\text{evidence package}.
}
\]
%──────────────────────────────────────────────────────────────────────────────
\section{Toy Benchmark Suite}
%──────────────────────────────────────────────────────────────────────────────
PBA v1.3 defines the minimum toy benchmark suite:
\[
\mathcal{T}_{PBA}^{v1.3}
=
\{
T_{temp},
T_{pulse},
T_{osc},
T_{noise}
\}.
\]
where:
\[
T_{temp}=\text{temperature-like homeostasis},
\quad
T_{pulse}=\text{pulse recovery},
\quad
T_{osc}=\text{oscillatory signal preservation},
\quad
T_{noise}=\text{noisy perturbation robustness}.
\]
Each benchmark task must declare:
\[
\{
\text{domain},
x_t,
x^\ast,
V,
p_t,
\Theta,
\mathcal{L},
\mathcal{M},
\text{fit split},
\text{eval split}
\}.
\]
\begin{remark}
Toy benchmark success is useful for software evidence. It is not biological
proof.
\end{remark}
%──────────────────────────────────────────────────────────────────────────────
\section{Baseline Class Contract}
%──────────────────────────────────────────────────────────────────────────────
Each baseline must implement:
\begin{lstlisting}
class Baseline:
name: str
def run(self, initial_state, target, perturbations, config):
...
def metrics(self):
...
\end{lstlisting}
The minimum baseline family is:
\[
\mathcal{L}_{v1.3}
=
\{
L_{\mathrm{prop}},
L_{\mathrm{PI}},
L_{\mathrm{threshold}},
L_{\mathrm{return}}
\}.
\]
where:
\[
L_{\mathrm{prop}}=\text{proportional feedback},
\quad
L_{\mathrm{PI}}=\text{proportional-integral feedback},
\]
\[
L_{\mathrm{threshold}}=\text{threshold-triggered switching},
\quad
L_{\mathrm{return}}=\text{simple return-to-setpoint}.
\]
Required baseline output:
\begin{lstlisting}
{
"baseline_name": "proportional_feedback",
"domain": "temperature_like",
"parameters": {},
"metrics": {
"recovery_time": null,
"overshoot": null,
"cumulative_deviation": null,
"signal_preservation": null
},
"failure_notes": []
}
\end{lstlisting}
\begin{proposition}[Baseline Fairness Principle]
PBA v1.3 cannot support a strong benchmark claim unless PBA and all baselines
run under the same domain config, perturbation family, evaluation split, and
metric rules.
\end{proposition}
%──────────────────────────────────────────────────────────────────────────────
\section{Parameter Calibration Layer}
%──────────────────────────────────────────────────────────────────────────────}
The calibration target is:
\[
\Theta^\ast
=
\arg\min_{\Theta}
\mathcal{J}_{\mathrm{pba}}(\Theta),
\]
where:
\[
\mathcal{J}_{\mathrm{pba}}
=
w_1D_{\mathrm{cum}}
+
w_2O_{\max}
+
w_3A_{\mathrm{osc}}
+
w_4W_{\mathrm{cusp}}
-
w_5S_{\mathrm{pres}}
+
w_6P_{\mathrm{sens}}.
\]
A valid calibration record must include:
\begin{enumerate}
\item parameter search space,
\item objective function,
\item fitting data or fitting simulation seeds,
\item evaluation data or held-out seeds,
\item baseline model family,
\item selected parameter set,
\item sensitivity summary,
\item and failure or downgrade notes.
\end{enumerate}
Minimal calibration JSON:
\begin{lstlisting}
{
"calibration_id": "PBA-CAL-0001",
"method": "grid_search",
"fit_domain": "temperature_like_fit",
"eval_domain": "temperature_like_eval",
"objective": "J_pba_v1_3",
"search_space": {
"eta": [0.05, 0.10, 0.15],
"tau_1": [0.25, 0.35],
"tau_2": [0.70, 0.90],
"kappa": [0.05, 0.10],
"alpha": [0.70, 0.85],
"theta_M": [0.70]
},
"selected_params": {},
"fit_loss": null,
"eval_loss": null,
"downgrade_notes": []
}
\end{lstlisting}
\begin{remark}
Parameter calibration improves model usability. It does not prove biological
mechanism.
\end{remark}
%──────────────────────────────────────────────────────────────────────────────
\section{Benchmark Metric Layer}
%──────────────────────────────────────────────────────────────────────────────
PBA v1.3 standardizes machine-readable benchmark metrics:
\[
T_R=\text{recovery time},
\]
\[
O_{\max}=\max_t \max(0,x_t-x^\ast_{\mathrm{upper}}),
\]
\[
U_{\max}=\max_t \max(0,x^\ast_{\mathrm{lower}}-x_t),
\]
\[
A_{\mathrm{osc}}=\text{oscillation amplitude after perturbation},
\]
\[
D_{\mathrm{cum}}=\sum_t |\Delta\Phi_t|,
\]
\[
W_{\mathrm{cusp}}=\sum_t \mathbf{1}[c_t\in\{\text{caution},\text{halt/audit}\}],
\]
\[
S_{\mathrm{pres}}=\text{preserved signal score},
\]
\[
R_{\mathrm{rob}}=\text{performance under perturbation variation},
\]
\[
P_{\mathrm{sens}}=\text{parameter sensitivity penalty}.
\]
Metric output JSON:
\begin{lstlisting}
{
"metrics_id": "PBA-METRICS-0001",
"domain": "temperature_like",
"model": "PBA",
"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
}
}
\end{lstlisting}
%──────────────────────────────────────────────────────────────────────────────
\section{Identifiability and Overfitting Layer}
%──────────────────────────────────────────────────────────────────────────────
Let:
\[
\mathcal{A}_{\epsilon}
=
\{\Theta_i:
|\mathcal{J}(\Theta_i)-\mathcal{J}(\Theta^\ast)|<\epsilon\}.
\]
If:
\[
|\mathcal{A}_{\epsilon}| \gg 1
\]
and the parameter sets are structurally diverse, then interpretive strength is
downgraded:
\[
\text{many equivalent fits}
\Rightarrow
\text{reduced interpretive strength}.
\]
Identifiability report JSON:
\begin{lstlisting}
{
"identifiability_report_id": "PBA-ID-0001",
"epsilon": null,
"near_equivalent_parameter_sets": 0,
"status": "stable/degenerate/unidentified",
"interpretive_downgrade": false,
"downgrade_note": ""
}
\end{lstlisting}
\begin{proposition}[Calibration Humility Principle]
A calibrated PBA model earns stronger status only when performance improves
against baselines and the relevant parameter structure is sufficiently stable
to support interpretation.
\end{proposition}
%──────────────────────────────────────────────────────────────────────────────
\section{Evidence Package v1.3}
%──────────────────────────────────────────────────────────────────────────────
A valid v1.3 evidence package is:
\[
\mathcal{E}_{v1.3}
=
\{
C_D,
C_P,
R_S,
R_B,
M_P,
M_B,
C_R,
I_R,
G_C,
F_N
\}.
\]
where:
\[
C_D=\text{domain config},
\quad
C_P=\text{parameter config},
\quad
R_S=\text{state run log},
\quad
R_B=\text{baseline run log},
\]
\[
M_P=\text{PBA metrics},
\quad
M_B=\text{baseline metrics},
\quad
C_R=\text{calibration record},
\]
\[
I_R=\text{identifiability report},
\quad
G_C=\text{classification},
\quad
F_N=\text{falsification note}.
\]
Minimal evidence package JSON:
\begin{lstlisting}
{
"evidence_package_id": "PBA-EVIDENCE-0001",
"version": "PBA-v1.3",
"domain": "temperature_like",
"files": {
"domain_config": "configs/domains/temperature_like.json",
"parameter_manifest": "configs/pba_params.json",
"baseline_params": "configs/baseline_params.json",
"state_log": "runs/run_timestamp/state_log.jsonl",
"pba_metrics": "runs/run_timestamp/pba_metrics.json",
"baseline_metrics": "runs/run_timestamp/baseline_metrics.json",
"calibration_record": "runs/run_timestamp/calibration_record.json",
"identifiability_report": "runs/run_timestamp/identifiability_report.json",
"classification": "runs/run_timestamp/classification.json"
},
"claim_boundary": {
"supports": "reference implementation and toy benchmark evidence",
"does_not_support": [
"medical guidance",
"biological law",
"mechanism proof",
"clinical validation",
"universal biological theory"
]
}
}
\end{lstlisting}
\begin{remark}
The evidence package is the boundary between a runnable artifact and a strong
claim. The package can support implementation evidence, not biological proof.
\end{remark}
%──────────────────────────────────────────────────────────────────────────────
\section{Domain-Instantiation Grammar}
%──────────────────────────────────────────────────────────────────────────────
Each benchmark run should declare:
\begin{enumerate}
\item benchmark domain,
\item regulated variable,
\item viable interval or target,
\item perturbation family,
\item observation cadence,
\item noise model,
\item parameter manifest,
\item calibration method,
\item fit / evaluation split,
\item baseline family,
\item metric vector,
\item runtime evidence package,
\item classification,
\item falsification note.
\end{enumerate}
Minimal domain JSON:
\begin{lstlisting}
{
"domain_id": "temperature_like",
"regulated_variable": "x_t",
"target": 0.0,
"viable_interval": [-0.25, 0.25],
"time_steps": 100,
"perturbation_family": "pulse_plus_noise",
"noise_model": "bounded_uniform",
"observation_cadence": 1,
"non_claim_locks": [
"not_medical",
"not_biological_law",
"not_mechanism_proof"
]
}
\end{lstlisting}
%──────────────────────────────────────────────────────────────────────────────
\section{Placidic Runtime Sub-Algorithms}
%──────────────────────────────────────────────────────────────────────────────
\subsection{PDD — Placidic Drift Dampener}
\[
\mathrm{PDD}(x_t)
=
x_t-\eta\Omega_t\nabla\Delta\Phi_t.
\]
\[
|\Delta\Phi_t|\uparrow
\Rightarrow
\Omega_t\downarrow
\Rightarrow
\text{correction remains bounded}.
\]
\subsection{PCG — Placidic Cusp Guard}
\[
\mathrm{PCG}(\Delta\Phi_t,\dot{\Delta\Phi}_t,\tau_1,\tau_2)
\rightarrow
\{\text{continue},\text{caution},\text{halt/audit}\}.
\]
\subsection{SPS — Signal-Preserving Smoother}
\[
S_{t+1}=\alpha S_t+(1-\alpha)S_{\mathrm{stable}},
\]
subject to:
\[
S_{t+1}\geq \kappa.
\]
\[
\boxed{
\text{smooth noise, not structure.}
}
\]
\subsection{CBS — Canalization-Basin Selector}
For candidate trajectories \(T_i\):
\[
B(T_i)
=
\frac{V(T_i)\cdot C(T_i)}
{1+E(T_i)+\sigma(T_i)}.
\]
\[
T^\ast=\arg\max_i B(T_i).
\]
\subsection{AAL — Allostatic Anticipation Layer}
\[
\widehat{\Delta\Phi}_{t+1}
=
\Delta\Phi_t+\beta(\Delta\Phi_t-\Delta\Phi_{t-1})+\gamma p_{t+1}.
\]
\subsection{BCS — Benchmark Calibration Search}
\[
\mathrm{BCS}:
\Theta_0
\rightarrow
\Theta^\ast
\rightarrow
\text{evaluation metrics}
\rightarrow
\text{sensitivity report}.
\]
\subsection{BIG — Benchmark Identifiability Gate}
\[
\mathrm{BIG}:
\mathcal{A}_{\epsilon}
\rightarrow
\{\text{stable},\text{degenerate},\text{unidentified}\}.
\]
\subsection{MPG — Memory Promotion Gate}
A runtime or benchmark pattern enters memory only if:
\[
U\cdot R\cdot S\cdot G>\theta_M,
\]
where \(U\) is downstream utility, \(R\) is recurrence, \(S\) is stability under
reuse, \(G\) is cross-benchmark generalization, and \(\theta_M\) is the
promotion threshold.
%──────────────────────────────────────────────────────────────────────────────
\section{Biological Crossing Layer}
%──────────────────────────────────────────────────────────────────────────────
The biological crossing remains:
\[
\boxed{
\text{Placidity}
\approx
\text{bounded adaptive regulation near viability boundaries}.
}
\]
The v1.3 refinement is:
\[
\boxed{
\text{a biological-facing PBA model becomes implementation-evaluable only}
\atop
\text{when benchmark behavior is produced by inspectable code and compared}
\atop
\text{against declared baselines under logged metrics.}
}
\]
\begin{center}
\begin{longtable}{>{\raggedright\arraybackslash}p{0.30\textwidth}
>{\raggedright\arraybackslash}p{0.60\textwidth}}
\toprule
\textbf{PBA v1.3 term} & \textbf{Biological modeling analogue} \\
\midrule
Benchmark domain & Biological-like regulatory scenario. \\
\(x_t\) & Regulated variable or modeled phenotype. \\
\(x^\ast\) & Target state, viable range, or expected reference state. \\
\(\Delta\Phi_t\) & Deviation from viable or expected range. \\
\(\Omega_t\) & Bounded response under increasing instability. \\
\(\eta\) & Correction / damping rate. \\
\(\tau_1,\tau_2\) & Recoverability and high-risk thresholds. \\
\(\kappa\) & Minimum preserved functional or structural signal. \\
Reference kernel & Executable modeling update, not biological mechanism. \\
Calibration record & Parameter fitting history. \\
Baseline family & Standard biological or control-model comparator. \\
Identifiability surface & Whether the model parameters are interpretable. \\
Benchmark evidence package & Audit trail for the comparative result. \\
\bottomrule
\end{longtable}
\end{center}
\begin{remark}
This table is a model-transfer map. It is not proof of biological mechanism.
\end{remark}
%──────────────────────────────────────────────────────────────────────────────
\section{Outcome Classification Layer}
%──────────────────────────────────────────────────────────────────────────────
\begin{definition}[PBA-A: Strong Implementation-Backed Benchmark Support]
A PBA-A case defines measurable variables, deviation, damping, signal
preservation, cusp thresholds, benchmark domain, parameter manifest,
calibration record, validation metrics, falsification surfaces, baseline-family
comparison, runtime logs, benchmark evidence package, and performs better than
baseline models on declared metrics without identifiability collapse. It remains
a modeling result, not biological proof.
\end{definition}
\begin{definition}[PBA-B: Partial Implementation-Backed Benchmark Support]
A PBA-B case has runnable implementation, useful simulation, partial
calibration, or promising benchmark performance but lacks complete controls,
held-out evaluation, identifiability, repeat-run evidence, or domain support.
\end{definition}
\begin{definition}[PBA-C: Better Standard Model Explanation]
A PBA-C case is better explained by ordinary feedback, PI/PID-like, stochastic,
developmental, mechanistic, or domain-standard models without requiring PBA.
\end{definition}
\begin{definition}[PBA-D: Null / Ambiguous Benchmark]
A PBA-D case lacks enough evidence, clarity, implementation completeness, or
comparative advantage to select PBA or an alternative model.
\end{definition}
\begin{definition}[PBA-E: Rejected Bioregulatory Claim]
A PBA-E case overclaims, lacks measurable variables, lacks controls, treats
metaphor as mechanism, makes medical claims, treats calibration as proof, or
uses simulation coherence as biological truth.
\end{definition}
%──────────────────────────────────────────────────────────────────────────────
\section{PBA v1.3 Scoring Surface}
%──────────────────────────────────────────────────────────────────────────────
PBA v1.3 expands the v1.2 scoring surface:
\[
\mathcal{O}^{\mathrm{pba}}_{v1.3}
=
\{
S_t,D_t,O_t,C_t,P_t,V_t,X_t,N_t,E_t,R_t,B_t,K_t,G_t,I_t,M_t,L_t,
P_{pkg},C_{cli},Y_{emit},Z_{test}
\}.
\]
where:
\[
P_{pkg}=\text{reference package boundary},
\quad
C_{cli}=\text{CLI run contract},
\quad
Y_{emit}=\text{machine-readable metric emission},
\quad
Z_{test}=\text{test / reproducibility scaffold}.
\]
\begin{center}
\begin{longtable}{>{\raggedright\arraybackslash}p{0.34\textwidth}
>{\centering\arraybackslash}p{0.14\textwidth}
>{\raggedright\arraybackslash}p{0.42\textwidth}}
\toprule
\textbf{Observable} & \textbf{Status (0 / 0.5 / 1)} & \textbf{Evidence} \\
\midrule
\(S_t\) System / Domain Boundary & & \\
\(D_t\) Deviation Measure \(\Delta\Phi\) & & \\
\(O_t\) Omega Weight \(\Omega\) & & \\
\(C_t\) Cusp Thresholds & & \\
\(P_t\) Signal Preservation & & \\
\(V_t\) Validation Metrics & & \\
\(X_t\) Falsification Surface & & \\
\(N_t\) Negative Controls & & \\
\(E_t\) Evidence / Assay Package & & \\
\(R_t\) Runtime / Repository Anchor & & \\
\(B_t\) Baseline-Family Comparison & & \\
\(K_t\) Calibration Record & & \\
\(G_t\) Generalization / Held-Out Evaluation & & \\
\(I_t\) Identifiability / Sensitivity Check & & \\
\(M_t\) Memory Promotion Rule & & \\
\(L_t\) Biological Non-Claim Locks & & \\
\(P_{pkg}\) Reference Package Boundary & & \\
\(C_{cli}\) CLI Run Contract & & \\
\(Y_{emit}\) Machine-Readable Metric Emission & & \\
\(Z_{test}\) Test / Reproducibility Scaffold & & \\
\bottomrule
\end{longtable}
\end{center}
\[
\mathrm{PBAScore}_{v1.3}
=
\frac{
S_t+D_t+O_t+C_t+P_t+V_t+X_t+N_t+E_t+R_t+B_t+K_t+G_t+I_t+M_t+L_t
+P_{pkg}+C_{cli}+Y_{emit}+Z_{test}
}{20}.
\]
\begin{remark}
PBAScore measures model-governance completeness, executable auditability,
benchmark discipline, calibration clarity, and reproducibility. It does not
measure biological truth.
\end{remark}
%──────────────────────────────────────────────────────────────────────────────
\section{Validation Layer}
%──────────────────────────────────────────────────────────────────────────────
A valid PBA v1.3 analysis must identify:
\begin{enumerate}
\item repository and package boundary,
\item CLI run command,
\item biological or simulated benchmark domain,
\item regulated variable \(x_t\),
\item target or viable reference \(x^\ast\),
\item perturbation family \(p_t\),
\item deviation \(\Delta\Phi_t\),
\item stability weight \(\Omega_t\),
\item damping parameter \(\eta\),
\item cusp thresholds \(\tau_1,\tau_2\),
\item signal-preservation floor \(\kappa\),
\item runtime update rule,
\item calibration method,
\item fit / evaluation split,
\item baseline-family comparison model,
\item validation metrics,
\item machine-readable metric outputs,
\item falsification criteria,
\item parameter sensitivity report,
\item identifiability note,
\item state logs,
\item benchmark evidence package,
\item downgrade path,
\item memory-promotion candidates.
\end{enumerate}
%──────────────────────────────────────────────────────────────────────────────
\section{Falsification Surface}
%──────────────────────────────────────────────────────────────────────────────
PBA v1.3 is weakened or rejected if:
\begin{itemize}
\item no repository or package boundary exists for implementation claims,
\item no CLI or reproducible command is declared,
\item no measurable or simulated variable is defined,
\item \(\Delta\Phi\) is metaphorical only,
\item \(\Omega\) has no computational role,
\item no cusp or recoverability boundary is specified,
\item signal smoothing erases meaningful structure,
\item baseline feedback explains the behavior equally well or better,
\item baseline-family comparison is absent,
\item calibration is reported without parameter manifest,
\item calibration uses the same data as evaluation without disclosure,
\item many unrelated parameter sets perform equally well and this is ignored,
\item validation is narrative-only,
\item machine-readable metric outputs are absent,
\item runtime logs are absent for executable claims,
\item evidence package is incomplete,
\item negative controls are absent,
\item medical claims are made,
\item universal biological-law claims are made,
\item calibration success is treated as mechanism proof,
\item implementation success is treated as empirical biological validation,
\item simulation success is treated as biological proof,
\item or Codex coherence is treated as biological truth.
\end{itemize}
Compact falsification condition:
\[
\text{PBA-A claim}
\wedge
\left(
V_t=0
\vee
X_t=0
\vee
N_t=0
\vee
R_t=0
\vee
B_t=0
\vee
K_t=0
\vee
I_t=0
\vee
P_{pkg}=0
\vee
Y_{emit}=0
\vee
L_t=0
\right)
\Rightarrow
\text{invalid strong classification}.
\]
%──────────────────────────────────────────────────────────────────────────────
\section{Upgrade and Downgrade Thresholds}
%──────────────────────────────────────────────────────────────────────────────
A candidate may be considered for PBA-A only if:
\[
\mathrm{PBAScore}_{v1.3}=1,
\]
and PBA performance exceeds declared baseline-family performance on meaningful
metrics without identifiability collapse.
A candidate should be classified as PBA-B if:
\[
0.80 \leq \mathrm{PBAScore}_{v1.3}<1
\]
and PBA benchmark behavior remains useful but incomplete.
A candidate should be classified as PBA-C if a simpler or standard biological
or control model explains the system equally well or better.
A candidate should be classified as PBA-D if the evidence is insufficient.
A candidate should be classified as PBA-E if the claim is post-hoc, unmeasured,
uncontrolled, overfit, medically framed, or dependent on Codex interpretation
rather than benchmark evidence.
%──────────────────────────────────────────────────────────────────────────────
\section{Rejection Surface}
%──────────────────────────────────────────────────────────────────────────────
A proposed PBA refinement should be rejected if it:
\begin{itemize}
\item converts model-transfer into biological proof,
\item removes biological-source caution,
\item removes non-medical boundaries,
\item removes falsification,
\item removes negative controls,
\item removes runtime logging for executable claims,
\item removes machine-readable metric outputs,
\item removes baseline comparison,
\item removes calibration disclosure,
\item removes identifiability caution,
\item treats metaphor as mechanism,
\item treats calibrated simulation as biological truth,
\item treats implementation success as empirical biological validation,
\item claims universal biological governance,
\item or promotes partial benchmark success to PBA-A.
\end{itemize}
%──────────────────────────────────────────────────────────────────────────────
\section{Repository Record Grammar}
%──────────────────────────────────────────────────────────────────────────────
A repository-ready PBA v1.3 project should preserve code, parameters,
calibration records, state logs, baseline runs, controls, metrics, and benchmark
evidence packages.
\begin{lstlisting}
pba_benchmarks/
README.md
LICENSE
pyproject.toml
.gitignore
docs/
theory/
pba_v1_3.tex
pba_v1_2_to_v1_3_mapping.md
biological_non_claim_locks.md
architecture/
reference_implementation_boundary.md
module_contracts.md
cli_contract.md
evidence_package_schema.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
kernel.py
cusp_guard.py
signal.py
baselines.py
calibration.py
metrics.py
identifiability.py
evidence_package.py
classification.py
configs/
domains/
temperature_like.json
pulse_recovery.json
oscillatory_signal.json
noisy_perturbation.json
pba_params.json
baseline_params.json
calibration_grid.json
metric_manifest.json
runs/
.gitkeep
evidence/
raw_inputs/
processed_outputs/
negative_controls/
benchmark_packages/
reports/
benchmark_summary.md
falsification_report.md
sensitivity_report.md
ledgers/
pba_evolution_ledger.jsonl
pba_runtime_ledger.jsonl
pba_decision_ledger.jsonl
scripts/
run_benchmark.py
run_suite.py
compile_evidence_package.py
repo_dump_light.ps1
tests/
test_kernel.py
test_cusp_guard.py
test_signal.py
test_baselines.py
test_calibration.py
test_metrics.py
test_identifiability.py
test_evidence_package.py
\end{lstlisting}
\begin{remark}
This repository grammar is optional for conceptual notes but required in spirit
for benchmark, executable, or strong PBA claims.
\end{remark}
%──────────────────────────────────────────────────────────────────────────────
\section{Minimal CLI Contract}
%──────────────────────────────────────────────────────────────────────────────
Single benchmark command:
\begin{lstlisting}
python -m pba.cli run-benchmark --domain ".\configs\domains\temperature_like.json"
\end{lstlisting}
Suite command:
\begin{lstlisting}
python -m pba.cli run-suite --config ".\configs\suite_v1_3.json"
\end{lstlisting}
Expected outputs:
\begin{lstlisting}
runs/run_<timestamp>/
domain_config.json
parameter_manifest.json
state_log.jsonl
pba_metrics.json
baseline_metrics.json
calibration_record.json
identifiability_report.json
classification.json
evidence_package.json
result_ledger.jsonl
\end{lstlisting}
\begin{remark}
The CLI is part of the evidence surface. If the run cannot be reproduced by a
declared command, it cannot support strong implementation claims.
\end{remark}
%──────────────────────────────────────────────────────────────────────────────
\section{Minimal Benchmark JSON State}
%──────────────────────────────────────────────────────────────────────────────
\begin{lstlisting}
{
"run_id": "PBA-BENCH-0001",
"version": "PBA-v1.3",
"benchmark_domain": "",
"regulated_variable": "x_t",
"target_reference": null,
"viable_interval": [null, null],
"perturbation_family": "",
"fit_eval_split": {
"fit_seed_or_data": "",
"eval_seed_or_data": "",
"split_note": ""
},
"parameters": {
"eta": null,
"tau_1": null,
"tau_2": null,
"kappa": null,
"alpha": null,
"theta_M": null,
"beta": null,
"gamma": null
},
"calibration": {
"method": "",
"objective": "",
"search_space": {},
"selected_params": {},
"calibration_loss": null,
"evaluation_loss": null
},
"baselines": [
{
"model": "proportional_feedback",
"parameters": {},
"metrics": {}
},
{
"model": "threshold_switch",
"parameters": {},
"metrics": {}
}
],
"pba_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
},
"identifiability": {
"near_equivalent_parameter_sets": 0,
"status": "",
"downgrade_note": ""
},
"classification": "",
"falsification_note": "",
"memory_promotion": {
"promote": false,
"reason": ""
},
"non_claim_locks": [
"not_medical",
"not_biological_law",
"not_mechanism_proof",
"simulation_success_not_biological_proof",
"implementation_success_not_empirical_validation",
"calibration_success_not_mechanism_proof"
]
}
\end{lstlisting}
%──────────────────────────────────────────────────────────────────────────────
\section{Minimal Runtime Pseudocode}
%──────────────────────────────────────────────────────────────────────────────
\begin{lstlisting}
Input:
domain_config
pba_params
baseline_params
calibration_grid
metric_manifest
Initialize:
load domain configuration
load parameter manifest
initialize PBA kernel
initialize cusp guard
initialize signal preservation module
initialize baseline classes
initialize calibration search
initialize metrics module
initialize identifiability checker
initialize evidence package compiler
Calibration phase:
for each parameter set in calibration_grid:
run PBA on fit split
compute objective J_pba
log calibration trial
select Theta_star
write calibration_record.json
Evaluation phase:
run PBA with Theta_star on evaluation split
write state_log.jsonl
compute pba_metrics.json
Baseline phase:
for each baseline in baseline family:
run baseline on same evaluation split
compute baseline metrics
write baseline_metrics.json
Identifiability phase:
collect near-equivalent parameter sets
classify identifiability:
stable / degenerate / unidentified
write identifiability_report.json
Classification phase:
compare PBA metrics against baseline family
score observables
compute PBAScore_v1_3
classify:
PBA-A / PBA-B / PBA-C / PBA-D / PBA-E
Evidence phase:
compile evidence package:
configs
logs
metrics
calibration record
identifiability report
classification
falsification note
Memory promotion:
promote only:
reproducible benchmark wins
stable thresholds
implementation constraints
failure lessons
cross-benchmark recurrent invariants
Reject:
medical claims
universal-law claims
mechanism-proof claims
simulation-as-biological-proof
implementation-as-empirical-validation
calibration-as-mechanism
coherence-as-biological-truth
\end{lstlisting}
%──────────────────────────────────────────────────────────────────────────────
\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 for implementation claim. \\
\(F_{cli}\) & Reproducible CLI command absent. \\
\(F_{domain}\) & Domain config missing or underspecified. \\
\(F_{param}\) & Parameter manifest missing. \\
\(F_{\Delta\Phi}\) & Deviation measure absent or metaphorical only. \\
\(F_{\Omega}\) & Omega weighting absent or computationally unused. \\
\(F_{cusp}\) & Cusp thresholds missing. \\
\(F_{signal}\) & Smoothing erases meaningful signal. \\
\(F_{baseline}\) & 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. \\
\(F_{medical}\) & Medical or treatment claim made. \\
\(F_{mechanism}\) & Calibration or simulation 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{Appendix A — Minimal PBA v1.3 Candidate Checklist}
%──────────────────────────────────────────────────────────────────────────────
\begin{enumerate}
\item What repository or package contains the implementation?
\item What CLI command reproduces the run?
\item What biological-like or simulated benchmark domain is being modeled?
\item What variable is regulated?
\item What target or viable range is used?
\item What perturbation family is introduced?
\item How is \(\Delta\Phi\) computed?
\item How is \(\Omega\) computed?
\item What damping rate is used?
\item What signal floor is used?
\item What cusp thresholds are used?
\item What calibration method is used?
\item What fit / evaluation split is used?
\item What baseline model family is compared?
\item What metrics determine improvement or failure?
\item What machine-readable outputs are emitted?
\item What parameter sensitivity tests are reported?
\item Is identifiability stable or degenerate?
\item What logs are preserved?
\item What benchmark evidence package is produced?
\item What falsifies the model?
\item What downgrade class applies?
\item What, if anything, is memory-promotable?
\item Are all non-medical and non-universal-law locks preserved?
\end{enumerate}
%──────────────────────────────────────────────────────────────────────────────
\section{Appendix B — Canonical Formula Summary}
%──────────────────────────────────────────────────────────────────────────────
\[
\Delta\Phi_t=|x_t-x^\ast|
\]
\[
\Omega_t=\frac{1}{1+|\Delta\Phi_t|}
\]
\[
x_{t+1}
=
x_t
-
\eta\Omega_t\nabla\Delta\Phi_t
+
\kappa S_t
+
A_t
\]
\[
c_t=
\begin{cases}
\text{continue}, & \Delta\Phi_t < \tau_1,\\
\text{caution}, & \tau_1 \leq \Delta\Phi_t < \tau_2,\\
\text{halt/audit}, & \Delta\Phi_t \geq \tau_2.
\end{cases}
\]
\[
\Theta^\ast
=
\arg\min_{\Theta}
\mathcal{J}_{\mathrm{pba}}(\Theta)
\]
\[
\mathcal{J}_{\mathrm{pba}}
=
w_1D_{\mathrm{cum}}
+
w_2O_{\max}
+
w_3A_{\mathrm{osc}}
+
w_4W_{\mathrm{cusp}}
-
w_5S_{\mathrm{pres}}
+
w_6P_{\mathrm{sens}}
\]
\[
\mathcal{A}_{\epsilon}
=
\{\Theta_i:
|\mathcal{J}(\Theta_i)-\mathcal{J}(\Theta^\ast)|<\epsilon\}
\]
\[
|\mathcal{A}_{\epsilon}| \gg 1
\Rightarrow
\text{identifiability downgrade}
\]
\[
U\cdot R\cdot S\cdot G>\theta_M
\Rightarrow
\text{memory promotion candidate}
\]
\[
\mathrm{PBAScore}_{v1.3}
=
\frac{
S_t+D_t+O_t+C_t+P_t+V_t+X_t+N_t+E_t+R_t+B_t+K_t+G_t+I_t+M_t+L_t
+P_{pkg}+C_{cli}+Y_{emit}+Z_{test}
}{20}
\]
\[
\mathcal{E}_{spec}
<
\mathcal{E}_{impl}
<
\mathcal{E}_{bench}
<
\mathcal{E}_{robust}
\]
%──────────────────────────────────────────────────────────────────────────────
\section{Concluding Compression}
%──────────────────────────────────────────────────────────────────────────────
PBA v1.3 names the implementation-evaluable form of biological-facing
Placidity:
\[
\boxed{
\text{Placidity becomes implementation-evaluable only when its reference}
\atop
\text{kernel, baselines, calibration logs, metrics, identifiability reports,}
\atop
\text{and evidence packages are repository-anchored and reproducible.}
}
\]
The operational statement is:
\[
\boxed{
\text{PBA models bounded adaptive regulation by measuring deviation,}
\atop
\text{weighting correction, preserving signal, guarding cusp thresholds,}
\atop
\text{and comparing against declared baselines under benchmark metrics.}
}
\]
The implementation statement is:
\[
\boxed{
\text{a PBA runtime claim requires code, configs, logs, metrics, baseline}
\atop
\text{outputs, calibration records, identifiability reports, and an evidence}
\atop
\text{package.}
}
\]
The calibration statement is:
\[
\boxed{
\text{parameter fit improves usability, but calibration is not mechanism proof.}
}
\]
The biological caution statement is:
\[
\boxed{
\text{PBA may model homeostasis, allostasis, canalization, robustness,}
\atop
\text{and pre-critical stability, but it does not prove a universal}
\atop
\text{biological law or medical mechanism.}
}
\]
The CITA statement is:
\[
\boxed{
\text{v1.3 completes the reference-implementation evidence surface by adding}
\atop
\text{package contracts, CLI grammar, baseline classes, metric outputs,}
\atop
\text{identifiability reports, and evidence packages.}
}
\]
Thus, PBA v1.3 upgrades the Placidic biological bridge from benchmark-
calibrated specification to reference implementation and benchmark evidence
layer while preserving biological caution, CITA governance, non-medical
boundaries, falsification, negative controls, baseline discipline,
identifiability caution, and downgrade-preserving classification.
\end{document}
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