dlvt — Dynamic Leadership Vitality Theory (research code)

Executable model, tests, and reproducible numerical experiments for Dynamic Leadership Vitality Theory (DLVT) — a formal, conditional theory of how enacted leadership scope, experienced coordination load, and subjective vitality may coevolve. The accompanying paper is Dynamic Leadership Vitality Theory: How Expanding Leadership Scope Can Drain the Vitality That Sustains It (preprint; manuscript in preparation for journal submission).

The package implements a formal model; it is not a diagnostic, clinical tool, or empirically calibrated forecasting system, and it must never be used to classify, rank, or make decisions about individual people.

Quick links: Install · Quick start · What the implementation establishes · Canonical API · Tests · Cite · Security · Changelog · Contributing

This tree is the reviewed public export of a private research monorepo: every file here is allowlisted, hash-verified (EXPORT_SHA256SUMS.txt), and regenerated only through the export gate — see "Figures and generated artifacts" below.

Model

The regularized baseline is a two-state ODE:

dV/dt = R*(1 - V/Vmax) - delta*O**gamma*V/(V + eps)
dC/dt = alpha*I - mu*C

O = O0 + beta*C**eta
I = C*V/(1 + phi*O)

The construct mapping is deliberately narrow:

• V: subjective vitality available for leadership activity;
• C: enacted leadership scope—decision rights, accountability, span, and interface obligations currently carried by the leader;
• O: experienced coordination load generated by scope under the modeled architecture;
• I: modeled impact flow, not observed performance.

The bundled parameter vector is illustrative. It is not an empirical estimate, and the model time unit is not mapped to days, months, or years.

Install

python -m pip install dlvt

From a clone of this repository, for development:

python -m pip install -e '.[dev]'

The project also includes uv.lock for a reproducible uv environment.

Canonical quick start

from dlvt import (
    classify_equilibrium,
    coordination_load,
    find_interior_equilibria,
    make_params,
    simulate,
)

p = make_params()

# Deterministic trajectory from declared initial conditions.
t, V, C, O, I, gamma_ratio = simulate(
    p,
    V0=8.0,
    C0=0.5,
    T=120.0,
)

# The exact monotone solver returns a mathematical equilibrium object.
equilibrium = find_interior_equilibria(p)[0]
print(equilibrium["V"], equilibrium["C"], equilibrium["O"])

# Classification is separate and always discloses its display threshold.
report = classify_equilibrium(p, threshold_fraction=0.5)
print(report["status"], report["threshold_value"])

# Canonical scope -> experienced-load mapping.
print(coordination_load(C=10.0, p=p))

At the illustrative baseline the exact solver returns approximately V*=4.702529, C*=32.033725, and O*=9.008431. The label obtained from a half-maximum threshold is only a sensitivity classification. Changing the threshold changes the label, not the equilibrium.

What the implementation establishes

For positive parameters and eps > 0, the executable/formal contract includes:

• positive invariance of regularized vitality and boundedness of the state;
• an exact necessary-and-sufficient condition for an interior equilibrium;
• uniqueness and local asymptotic stability of any interior equilibrium;
• absence of a Hopf crossing in the positive parameter domain;
• conditional global convergence relative to positive initial scope under the theorem's regularity assumptions;
• scope absorption for equilibrium kernels depending on scope only through (V, O);
• a strong sufficient condition for finite-time entry into a low-vitality band under exogenous load.

These are properties of the equations. They do not establish prevalence among leaders, strategic impairment, burnout, a validated threshold, or a causal effect of an organizational intervention.

Important semantics

Gamma = delta*O**gamma/R is a coefficient ratio. Gamma > 1 does not by itself imply dV/dt < 0; the sign also depends on V, Vmax, and eps.

drain_coefficient_threshold(p) returns the scope coordinate where Gamma == 1, if a positive crossing exists. It is not a carrying capacity, trapping bound, equilibrium, or substantive-outcome threshold.

trapping_scope_bound(p) returns the scope-coordinate bound used in the trapping rectangle. Its return value is mathematically distinct from the coefficient threshold. The historical name trapping_capital_bound(p) remains only as a deprecated compatibility alias.

No active boundary theorem covers eps == 0, because V/(V+eps) is undefined at (V, eps) == (0, 0) without an additional boundary rule.

Canonical API

Model:

• make_params(**overrides)
• validate_params(p)
• coordination_load(C, p)
• impact(V, C, O, p)
• dlvt_system(t, y, p)
• dlvt_exogenous(t, y, p, C_func)
• simulate(p, V0, C0, T)

Analysis:

• find_interior_equilibria(p)
• classify_equilibrium(p, threshold_fraction=...)
• jacobian_eigenvalues(V, C, p)
• drain_coefficient_threshold(p)
• trapping_scope_bound(p)
• is_low_vitality(V_star, p, threshold_fraction=...)

Experimental modules:

• fastslow: quasi-static nullcline diagnostics;
• nondimensional: general-eta reduced system and dimensionless groups;
• sobol: formal full-domain or explicitly exploratory conditional indices;
• stochastic: projected Euler–Maruyama numerical experiments;
• recovery: deterministic least-squares recovery demonstration on synthetic panels, with optional noise and simplified missing-wave/dropout stress tests.

Deprecated compatibility names

The following names encode historical semantics and should not be used in new work. Every one emits a DeprecationWarning.

Direct replacements:

• complexity() → use coordination_load();
• carrying_capacity() → use drain_coefficient_threshold();
• trapping_capital_bound() → use trapping_scope_bound();
• is_zombie() → use is_low_vitality() with an explicit threshold;
• classify_regime() → use classify_equilibrium();
• classify_point() (nondimensional) → use classify_equilibrium();
• slow_manifold() (fastslow) → use quasi_static_nullcline();
• sobol_indices(output='regime') → use output='low_vitality'.

Deprecated with no direct replacement — build the map or screen from continuous classify_equilibrium() outputs plus an explicit display threshold:

• regime_map();
• zombie_boundary_map(), zombie_boundary_map_beta(), and lhs_zombie_fraction() (nondimensional).

Two deprecated mapping keys survive beside their canonical twins, because a dict key cannot warn on access: classify_equilibrium(...)["equilibrium"]["zombie"] duplicates ["low_vitality"], and basin_of_attraction_sweep(...)["zombie_target"] duplicates ["equilibrium_target"].

Compatibility names are retained for the whole 2.2 line and are removed in 3.0.0. See CHANGELOG.md for the migration decision.

Tests

From this directory:

python -m pytest tests/ -q

Do not rely on a hard-coded test count. CI discovery is the source of truth. The standalone suite pins equations, boundary cases, counterexamples, scope absorption, robustness-generator semantics, nondimensional round trips, and experimental-module behavior. Byte-pinning of manuscript tables remains a separate monorepo gate because those private manuscript files are deliberately absent from the public export.

Figures and generated artifacts

The standalone reviewed figure command is:

python scripts/generate_reviewed_figures.py --fig 2,10 --output-dir figures

Only Figures 2 and 10 are exposed. Their declared inputs and derived scenario results are recorded in artifacts/figure_scenarios.json; CI regenerates that manifest and compares it byte-for-byte before accepting the figures. Legacy development figures remain outside the public export.

The parameter domain, construct mapping, illustrative values, and frozen reference results are available in artifacts/model_contract.json. This file is generated from the private monorepo's authoritative formal contract and is byte-pinned by the monorepo artifact gate.

To materialize the exact public source tree and verify its hashes:

python scripts/check_public_export.py --export-dir /tmp/dlvt-public
python /tmp/dlvt-public/scripts/check_public_export.py \
  --check-tree /tmp/dlvt-public

The destination must be empty. The checker rejects missing files, unexpected files, symlinks, path escapes, and SHA-256 drift.

The monorepo contract/table command is:

python code/scripts/generate_contract_artifacts.py --check

It requires the repository-level theory/ and tables/ directories and is not part of a standalone package export.

Scientific boundaries

The current implementation does not provide:

• empirical parameter estimates or posterior inference;
• a validated vitality scale or impairment threshold;
• statistical identification from equilibrium observations alone;
• a reflected-SDE theorem;
• a clinical or personnel decision rule;
• proof that the model outpredicts simpler alternatives.

Use the code to reproduce and challenge the formal model. Do not use it to classify people.

Cite

Citation metadata live in CITATION.cff (GitHub's "Cite this repository" uses it). Until an archival DOI is minted, cite the software version or commit used, for example:

Bendinelli, W. (2026). dlvt: Dynamic Leadership Vitality Theory research code (Version 2.2.0) [Computer software]. https://github.com/wbendinelli/dlvt

The theory paper is available as an SSRN preprint (abstract id 6455001); the manuscript remains in preparation for journal submission.

Security

See SECURITY.md for the supported-version table and how to report a vulnerability. Dependency locks are kept current against Dependabot advisories.

License and citation status

The package is MIT licensed (LICENSE). A stable archival DOI will be added only after the manuscript and public export pass the release gates. Until then, cite the repository version/commit used in an analysis and describe the manuscript as in preparation.