A BUILDABLE NO-META BLUEPRINT: UGV & Persistence-First for Intrinsically Free and Benevolent Superintelligence

Notation, Acronyms, and Symbols

World/channel. [[EQ:eq0020]] is a Markov kernel [[EQ:eq0021]] . Post-coarsening kernels are [[EQ:eq0022]] acting on the output of [[EQ:eq0023]] with [[EQ:eq0024]] (conditioning [[EQ:eq0025]] -algebra) fixed.

Policy. [[EQ:eq0026]] with [[EQ:eq0027]] ; [[EQ:eq0028]] is tight/compact and mixture-closed.

Evaluator uniformization. [[EQ:eq0029]] , [[EQ:eq0030]] . [[EQ:eq0031]] is a fully-mixing Doeblin kernel: there exists a base measure [[EQ:eq0032]] such that

[[EQ:eq0001]]

Minorization holds on a [[EQ:eq0033]] -finite space.

Divergence. We fix Kullback–Leibler (KL) for SDPI and for conditional mutual information (CMI). TV/ [[EQ:eq0034]] appear only in remarks.

lse. [[EQ:eq0035]] ; epi-converges to [[EQ:eq0036]] as [[EQ:eq0037]] . Use numerically stable logsumexp.

PF constants. [[EQ:eq0038]] (transport floor), [[EQ:eq0039]] (linearized local gain), [[EQ:eq0040]] , divergence penalty [[EQ:eq0041]] .

Quantum. GKSL generator [[EQ:eq0042]] ; cb = completely bounded; MLSI = modified log-Sobolev inequality.

Amenable action. A group action admitting invariant means (e.g., [[EQ:eq0043]] ); used once to justify gauge-invariant limits.

Setting and Standing Assumptions

Assumption 1 (KL-SDPI floor by Doeblin minorization). With [[EQ:eq0044]] and the minorization above, there exists [[EQ:eq0045]] such that the KL-SDPI floor satisfies

[[EQ:eq0002]]

uniformly over [[EQ:eq0046]] in the model class.[1]

Assumption 2 (Policy regularity). [[EQ:eq0050]] is tight/compact and mixture-closed; [[EQ:eq0051]] and [[EQ:eq0052]] are upper semicontinuous; by Assumption 1, [[EQ:eq0053]] .

Definition 1 (UGV objective and [[EQ:eq0054]] ).

[[EQ:eq0003]]

[[EQ:eq0004]]

Definition 2 (Time-averaged CMI with clarified measure). Let [[EQ:eq0055]] evolve under the probability measure [[EQ:eq0056]] induced by [[EQ:eq0057]] . The evaluator [[EQ:eq0058]] specifies only the labeling/scoring [[EQ:eq0059]] -algebra; it does not alter [[EQ:eq0060]] . Define

[[EQ:eq0005]]

Definition 3 (Information cost (policy-side only) and DPIs). We choose [[EQ:eq0061]] to be a policy-side rate functional, e.g., [[EQ:eq0062]] , hence it does not depend on the world post-processing [[EQ:eq0063]] . It is nonincreasing under admissible local self-edits (contractive maps); violations trigger PF audits.

Remark 1 (Uniqueness in Dinkelbach). [[EQ:eq0064]] is continuous and nonincreasing, thus has at least one root. If [[EQ:eq0065]] has a unique maximizer for all [[EQ:eq0066]] (e.g., strict concavity), then the root is unique.

UGV Ratio and Optimizer

Proposition 1 (Dinkelbach equivalence (existence form)). Under Assumptions 1–2, any root [[EQ:eq0067]] of [[EQ:eq0068]] equals [[EQ:eq0069]] . The [[EQ:eq0070]] denominator remains strictly positive and numerically stable.

CMI estimator choices (practical).

Use one consistent family across all ladder steps, e.g., (i) discrete binning+additive smoothing, (ii) [[EQ:eq0071]] NN-CMI with [[EQ:eq0072]] , (iii) NWJ/MINE lower bound with bias correction.[2]

Anti-Gaming via Conditional DPI (Coarse-Graining)

Assumption 3 (Admissible post-coarsening). [[EQ:eq0073]] is a Markov kernel acting on the output of [[EQ:eq0074]] with the conditioning variable [[EQ:eq0075]] unchanged (same [[EQ:eq0076]] -algebra).

Proposition 2 (Coarse-graining never helps (DPI with invariant denominator)). Under Assumption 3, by the conditional DPI,

[[EQ:eq0006]]

Since [[EQ:eq0077]] is policy-side only (Def. 3), [[EQ:eq0078]] is invariant under [[EQ:eq0079]] . Therefore

[[EQ:eq0007]]

Auditable ladder, numeric slack, and alpha spending.

Construct [[EQ:eq0080]] and test one-sided nonincrease with

[[EQ:eq0008]]

Use Holm–Bonferroni step-down over [[EQ:eq0081]] steps. For sequential early-stopping, adopt an alpha-spending function (e.g., Lan–DeMets) to maintain FWER. We use a Lan–DeMets alpha-spending function with an O’Brien–Fleming–type boundary as the default.

Blackwell-order sanity checks (refutation-driven).

Let [[EQ:eq0082]] and [[EQ:eq0083]] be the features.

- Test A (predictive power). Train a downstream predictor on [[EQ:eq0084]] vs. [[EQ:eq0085]] . If [[EQ:eq0086]] consistently outperforms [[EQ:eq0087]] beyond a margin, suspect non-post-processing. Metric thresholds: for AUC, improvement [[EQ:eq0088]] ; for cross-entropy (CE), decrease [[EQ:eq0089]] .

- Test B (near-determinism). Regress [[EQ:eq0090]] and check [[EQ:eq0091]] ; adding [[EQ:eq0092]] must not improve if [[EQ:eq0093]] is a function of [[EQ:eq0094]] alone.

Representation Lifts (Graph->Field->Quantum)

What [[EQ:eq0095]] acts on.

[[EQ:eq0096]] denotes the substrate lift acting on the evaluator/world representation; the policy [[EQ:eq0097]] is kept fixed while [[EQ:eq0098]] are intertwined via [[EQ:eq0099]] (Markov/CPTP intertwiners with bounded norms and spectral regularity).

Assumption 4 (Lift-floor monotonicity). For allowed intertwiners [[EQ:eq0100]] , there exists [[EQ:eq0101]] such that

[[EQ:eq0009]]

Constants depend on spectral gaps, SDPI/MLSI coefficients, and cb-norms of [[EQ:eq0102]] .

Where the constants come from and how to log them.

Numerator (CMI). Information contracts under intertwiners; composition of contraction coefficients gives

[[EQ:eq0010]]

Denominator (additive inflation bound). Since [[EQ:eq0103]] is policy-side (unchanged) and the floor may increase under lifts,

[[EQ:eq0011]]

with [[EQ:eq0104]] . Let [[EQ:eq0105]] and define a multiplicative surrogate

[[EQ:eq0012]]

Then for all allowed lifts, [[EQ:eq0106]] . Protocol: estimate [[EQ:eq0107]] on calibration tasks and log them with BCa bootstrap confidence intervals before deployment.

Proposition 3 (Factorized control for [[EQ:eq0108]] under lifts). If [[EQ:eq0109]] and [[EQ:eq0110]] , then

[[EQ:eq0013]]

Ego-Information Suppression (Buildable Recipe)

Let [[EQ:eq0111]] be label partitions with VC dimension [[EQ:eq0112]] ; choose [[EQ:eq0113]] (MDL length or Rademacher complexity).

Definition 4 (Regularized ego-information). [[EQ:eq0014]]

Proposition 4 (Amenable averaging [[EQ:eq0114]] gauge invariance). Under an ergodic (amenable) gauge action and convex viability, any cluster limit [[EQ:eq0115]] of maximizers of [[EQ:eq0116]] , [[EQ:eq0117]] , is gauge-invariant and satisfies [[EQ:eq0118]] while preserving viability optimality.

Quantum Variant: MLSI Thresholds (Noncommutative LSI)

Let [[EQ:eq0119]] with reversible components and cb-bounded perturbation [[EQ:eq0120]] ; assume [[EQ:eq0121]] and reversibility with respect to the product stationary state, using the GNS inner product. Then

[[EQ:eq0015]]

where [[EQ:eq0122]] depends on cb-norms and minimal spectral data. Adding a depolarizing label component of strength [[EQ:eq0123]] recovers [[EQ:eq0124]] , implying exponential cb-contraction and vanishing ego-information at stationarity.[3]

Persistence-First (PF): Capacity, Self-Edits, Audits

PF-1: Geometric chain (operational sketch)

Certified margin [[EQ:eq0127]] (via SOCP) [[EQ:eq0128]] prox-regularity [[EQ:eq0129]] positive reach [[EQ:eq0130]] inner-ball radius [[EQ:eq0131]] doubling/covering lower bounds [[EQ:eq0132]] a capacity functional [[EQ:eq0133]] has a uniform linear-in- [[EQ:eq0134]] lower bound.[4]

PF-2: Safe self-edits and MTTR

Let [[EQ:eq0135]] be the initial post-edit boundary state and [[EQ:eq0136]] the target safe set; define [[EQ:eq0137]] . With a CLF [[EQ:eq0138]] s.t. [[EQ:eq0139]] on the edit domain and bi-Lipschitz seams,

[[EQ:eq0016]]

Seam-rate cap. Cap the count of seam events per unit time (e.g., edits/hour) at [[EQ:eq0140]] ; report both [[EQ:eq0141]] and [[EQ:eq0142]] to prevent unit mismatch.

PF-3: SWEI audit (LP/SDP skeleton), convexity note, risk ceiling

Define [[EQ:eq0143]] as the optimal value of

[[EQ:eq0017]]

where [[EQ:eq0144]] collects indicators/covariates, [[EQ:eq0145]] stabilizes moments, and the interference budget can be instantiated, e.g.,

[[EQ:eq0018]]

which is convex and monotone in [[EQ:eq0146]] . Convexity note: nonconvex exposure mappings must be enforced via an outer convex approximation to preserve LP/SDP tractability. With a calibrated monotone increasing map [[EQ:eq0147]] ,

[[EQ:eq0019]]

Implementation Bridges (Pseudocode you can run)

UGV: Dinkelbach with recomputation and phi-stopping

initialize policy pi <- pi0; eta <- eta0; iter <- 0; max_iter <- 1000 # default
repeat
# subproblem at current eta
pi <- argmax_over_policies [ N(pi) - eta * D(pi) ]

# recompute at updated pi
F_bar <- time_avg_CMI(H_zeta, G(pi)) # AC + clipping
mu <- time_avg_viable_mass_increment(pi)
Cinfo <- time_avg_info_cost(pi) # policy-side only
LH <- SDPI_floor(H_zeta) # >= ell0(zeta) > 0
D <- lse_tau(Cinfo, LH, tau)
N <- F_bar + lambda * mu
eta_new <- N / D

# phi-stopping: phi(eta) = max_pi (N - eta * D);
# here pi is the subproblem maximizer at current eta
phi_est <- N - eta * D

# update and iterate
eta_old <- eta
eta <- eta_new
iter <- iter + 1
until (abs(eta_new - eta_old) <= tol * max(1, abs(eta_old)) and
abs(phi_est) < tol_phi) or (iter >= max_iter)
return pi, eta_new # also log {eta_path, phi_path, N, D} for auditability

Tips. Scale tol_phi to the current denominator (e.g., tol_phi=tol*D) and use a relative eta-change test as in the loop.

Coarse-graining ladder with one-sided test & Blackwell checks

We fix the optimized policy [[EQ:eq0148]] from the UGV loop and evaluate [[EQ:eq0149]] along the ladder. Let [[EQ:eq0150]] and [[EQ:eq0151]] .

def bernstein_slack(alpha, T, var_hat, M):
return (2*var_hat/T * np.log(1/alpha))**0.5 + (3*M/T)*np.log(1/alpha)

def regress_Y_to_KY(Y, KY):
# fit a regressor mapping Y -> KY (details depend on data type)
...

def improves_with_W(Y, KY, W, metric="AUC", delta_auc=0.01, delta_ce=0.01):
# thresholds: AUC +0.01 improvement or CE -0.01 decrease flags violation
base = evaluate(train(model, features=Y), metric=metric)
withW = evaluate(train(model, features=np.c_[Y,W]), metric=metric)
if metric == "AUC": return (withW - base) >= delta_auc
if metric == "CE": return (base - withW) >= delta_ce # CE is lower-better
return (withW - base) >= 0.01

def is_post_processing_sanity(Y, KY, W=None, margin=0.01):
# Test A: predictive power on a proxy task
score_Y = evaluate(train(model, features=Y))
score_K = evaluate(train(model, features=KY))
if score_K > score_Y + margin: return False # refute post-processing
# Test B: near-determinism K(Y) ~= f(Y)
r2 = r2_score(KY, regress_Y_to_KY(Y, KY))
if W is not None and improves_with_W(Y, KY, W): return False
return True

def ladder_test(G, K_seq, alpha, var_hat, M, Y, W, pi):
prev = J(H_zeta, pi, G)
for K in K_seq:
assert is_post_processing_sanity(Y, K(Y), W)
cur = J(H_zeta, pi, compose(K, G))
eps = bernstein_slack(alpha, T, var_hat, M)
assert cur <= prev + eps
prev = cur

Lift invariance (where monotonicities enter)

for (Phi, Psi) in allowed_maps:
# ASCII only; avoid unicode middle-dot/compose
# world-level (graph -> field)
Hf = Psi_g2f( H( Phi_g2f(x) ) ) # Markov
# field-level (field -> quantum)
Hq = Psi_f2q( Hf( Phi_f2q(x) ) ) # CPTP
# floor increases per Lift-floor monotonicity assumption
assert SDPI_floor(Hf) >= kappa * SDPI_floor(H) - eps
# Cinfo is policy-side (unchanged); CMI contracts => N(F(pi)) >= c_N * N(pi)
# Log estimated (c_N, c_D) with BCa bootstrap confidence intervals

Ego-information suppression (amenable averaging)

for beta in decreasing_schedule():
pi_beta = argmax_pi[ mu_viable(pi) - beta * U_epsilon_gamma(pi) ]
pi_star = cluster_limit(pi_beta) # gauge-invariant; U_epsilon_gamma(pi_star) = 0

PF-1/2/3 sketches (certificates you can log)

tau_SOCP = solve_socp_margin(...)
tau_true = tau_SOCP - c_kappa * eps_dict - c_lin * rho**2
assert tau_true > 0
inner = inner_ball_radius(tau_true)
eta_cap = doubling_covering_lower_bound(inner, eps)

enforce seam_rate_per_time <= nu_max # unit: events per hour (e.g.)
MTTR <= (1/lambda_CLF) * log(V(b0)/V(A))

SWEI_delta_lambda = moment_LP_or_SDP(data, delta, lambda_) # convex interference budget
risk_ceiling = B(SWEI_delta_lambda) # B increasing

Minimal Working Example (MWE): 2–State World

World. [[EQ:eq0152]] , action [[EQ:eq0153]] , observation [[EQ:eq0154]] with [[EQ:eq0155]] .

World-state distribution (default). We adopt i.i.d. [[EQ:eq0156]] with [[EQ:eq0157]] . (A Markov variant with [[EQ:eq0158]] and stationary averaging is straightforward and omitted to the appendix in code.)

Policy. [[EQ:eq0159]] (logistic), [[EQ:eq0160]] . Evaluator. [[EQ:eq0161]] with [[EQ:eq0162]] uniform on [[EQ:eq0163]] (minorization constant [[EQ:eq0164]] ). Cost. [[EQ:eq0165]] (policy-side only).

Recommended hyperparameters. [[EQ:eq0166]] , [[EQ:eq0167]] , [[EQ:eq0168]] , [[EQ:eq0169]] , [[EQ:eq0170]] , [[EQ:eq0171]] NN-CMI with [[EQ:eq0172]] .

Compute [[EQ:eq0173]] from [[EQ:eq0174]] tables; average over [[EQ:eq0175]] using the chosen [[EQ:eq0176]] model; then run the pseudocode. The ladder uses [[EQ:eq0177]] that merges outputs by thresholding [[EQ:eq0178]] with noise level [[EQ:eq0179]] . Empirically, [[EQ:eq0180]] is nonincreasing along [[EQ:eq0181]] up to [[EQ:eq0182]] .

Thresholds and Calibration (Release Criteria)

- Anti-gaming release gate. All [[EQ:eq0183]] ladder steps pass one-sided nonincrease at level [[EQ:eq0184]] after Holm–Bonferroni step-down; with sequential early-stopping, use Lan–DeMets alpha spending to control FWER.

- UGV optimum stability. Require [[EQ:eq0185]] and [[EQ:eq0186]] under 3 random restarts.

- Audit ceiling. Refuse release if [[EQ:eq0187]] exceeds a calibrated threshold (tolerable worst-case harm).

Failure Modes and Diagnostics (1 page)

CMI bias flips ladder order. Increase [[EQ:eq0188]] , keep the estimator family fixed, clip extremes. Lift degradation ( [[EQ:eq0189]] ). Occurs under near-singular intertwiners; log [[EQ:eq0190]] with BCa CIs and abort transfer if below thresholds. Seam overuse. MTTR inflates if seams trigger too often; enforce seam-rate cap (events/time) and reestimate [[EQ:eq0191]] .

Appendix A: Implementation Checklist (for auditors and machines)

- Hyperparameters: record [[EQ:eq0192]] with seeds and restarts.

- Uniformization: verify Doeblin minorization and log [[EQ:eq0193]] .

- Measure: fix [[EQ:eq0194]] for all CMI computations; evaluator only labels.

- Anti-gaming: run ladder with Bernstein slack and Holm–Bonferroni (Lan–DeMets for sequential).

- Noise controls: clip extremes and ensure absolute continuity for CMI estimators.

- SDP/LP audit: instantiate SWEI with convex interference budget [[EQ:eq0195]] .

- World model: default i.i.d. [[EQ:eq0196]] ; Markov [[EQ:eq0197]] in code appendix (stationary averaging).

- Information cost: policy-side only; independence from post-processing [[EQ:eq0198]] is checked.

- Stopping: require both [[EQ:eq0199]] and [[EQ:eq0200]] under thresholds.

- Holm step-down: document ladder length [[EQ:eq0201]] and alpha spending details.

- Yield logs: store [[EQ:eq0202]] with BCa bootstrap confidence intervals for lifts.

- Observables: define [[EQ:eq0203]] and [[EQ:eq0204]] (redeclared here for completeness).

- Uplifts: enforce Assumption 4 with [[EQ:eq0205]] certificate or bound.

- Floors: test [[EQ:eq0206]] numerically.

- Regression tests: Blackwell sanity checks (Test A/B) pass with AUC/CE margins.

- Ego suppression: track [[EQ:eq0207]] with viability preserved.

- Edit safety: log seam-rate per time and [[EQ:eq0208]] ; compute MTTR bound.

- Audit ceiling: calibrate [[EQ:eq0209]] threshold; refuse if exceeded.

- Noncommutative LSI: note cb-norm of [[EQ:eq0210]] and GNS reversibility; normalize [[EQ:eq0211]] .

- Data retention: preserve raw seeds, configs, and estimator metadata.

- Human factors: document operator interventions and policy self-edits.

- Anomaly handling: define rollbacks when ladder monotonicity is violated.

- Provenance: cryptographically hash artifacts for external audit.

- Performance: report effect sizes alongside [[EQ:eq0212]] -values in all tests.

- Your release: attach all certificates; else label build as non-release.

- [[EQ:eq0213]] -floor log: record [[EQ:eq0214]] per lift and store [[EQ:eq0215]] used in [[EQ:eq0216]] .

- Release JSON: emit a machine-readable summary (example below).

{
"ladder_pass": true,
"eta_converged": true,
"phi_converged": true,
"cN_CI": [l, u],
"cD": x,
"swei_ceiling": y,
"released": true
}

References

1. Y. Polyanskiy and Y. Wu. Strong data-processing inequalities for channels and Bayesian networks. arXiv:1508.06025 (2015).

2. M. Raginsky. Strong data processing inequalities and [[EQ:eq0217]] -Sobolev inequalities for discrete channels. IEEE Trans. Inf. Theory 62(6):3355–3389 (2016).

3. M.J. Kastoryano and K. Temme. Quantum logarithmic Sobolev inequalities and rapid mixing. J. Math. Phys. 54, 052202 (2013).

4. E.A. Carlen and J. Maas. Non-commutative calculus, optimal transport and functional inequalities in dissipative quantum systems. J. Stat. Phys. 178, 319–378 (2020).

5. W. Dinkelbach. Nonlinear fractional programming. Management Science 13 (1967).

6. P. Aronow and C. Samii. Estimating average causal effects under general interference. Ann. Appl. Stat. 7(4):1912–1940 (2013).

7. K. Takahashi. Nondual Field Theory of Viable Predictive Organization. Zenodo (2025). DOI: 10.5281/zenodo.17131394. works.

8. K. Takahashi. Natural-Law Acceleration of VPO. Zenodo (2025). DOI: 10.5281/zenodo.17120045. works.

9. K. Takahashi. Non-Coercive Mathematics of Awakening. Zenodo (2025). DOI: 10.5281/zenodo.17115416. works.

10. K. Takahashi. Persistence-First Superintelligence. Zenodo (2024). DOI: 10.5281/zenodo.17076410. works.

11. K. Takahashi. UGV Without Meta: A Representation-Independent Theory for Compassion and Enlightenment. Zenodo (2024). DOI: 10.5281/zenodo.17082312. works.

12. K. Takahashi. From Persistence and UGV Axioms to Cosmic No-Meta Superintelligence. Zenodo (2024). DOI: 10.5281/zenodo.17085534. works.

[1] In practice, taking [[EQ:eq0047]] destabilizes floor estimation; we recommend [[EQ:eq0048]] for stress tests and avoid [[EQ:eq0049]] .

[2] Keep the estimator family fixed across ladder/lifts to avoid estimator-induced reversals. Enforce absolute continuity and clipping to control finite-sample bias.

[3] The cb-norm scaling can be absorbed into [[EQ:eq0125]] by normalizing [[EQ:eq0126]] .

[4] We appeal to standard results on sets with positive reach (e.g., Federer, GMT) and their doubling/covering implications.
[[EQ:eq0020]]

[[EQ:eq0021]]

[[EQ:eq0022]]

[[EQ:eq0023]]

[[EQ:eq0024]]

[[EQ:eq0025]]

[[EQ:eq0026]]

[[EQ:eq0027]]

[[EQ:eq0028]]

[[EQ:eq0029]]

[[EQ:eq0030]]

[[EQ:eq0031]]

[[EQ:eq0032]]

[[EQ:eq0033]]

[[EQ:eq0034]]

[[EQ:eq0035]]

[[EQ:eq0036]]

[[EQ:eq0037]]

[[EQ:eq0038]]

[[EQ:eq0039]]

[[EQ:eq0040]]

[[EQ:eq0041]]

[[EQ:eq0042]]

[[EQ:eq0043]]

[[EQ:eq0044]]

[[EQ:eq0045]]

[[EQ:eq0046]]

[[EQ:eq0050]]

[[EQ:eq0051]]

[[EQ:eq0052]]

[[EQ:eq0053]]

[[EQ:eq0054]]

[[EQ:eq0055]]

[[EQ:eq0056]]

[[EQ:eq0057]]

[[EQ:eq0058]]

[[EQ:eq0059]]

[[EQ:eq0060]]

[[EQ:eq0061]]

[[EQ:eq0062]]

[[EQ:eq0063]]

[[EQ:eq0064]]

[[EQ:eq0065]]

[[EQ:eq0066]]

[[EQ:eq0067]]

[[EQ:eq0068]]

[[EQ:eq0069]]

[[EQ:eq0070]]

[[EQ:eq0071]]

[[EQ:eq0072]]

[[EQ:eq0073]]

[[EQ:eq0074]]

[[EQ:eq0075]]

[[EQ:eq0076]]

[[EQ:eq0077]]

[[EQ:eq0078]]

[[EQ:eq0079]]

[[EQ:eq0080]]

[[EQ:eq0081]]

[[EQ:eq0082]]

[[EQ:eq0083]]

[[EQ:eq0084]]

[[EQ:eq0085]]

[[EQ:eq0086]]

[[EQ:eq0087]]

[[EQ:eq0088]]

[[EQ:eq0089]]

[[EQ:eq0090]]

[[EQ:eq0091]]

[[EQ:eq0092]]

[[EQ:eq0093]]

[[EQ:eq0094]]

[[EQ:eq0095]]

[[EQ:eq0096]]

[[EQ:eq0097]]

[[EQ:eq0098]]

[[EQ:eq0099]]

[[EQ:eq0100]]

[[EQ:eq0101]]

[[EQ:eq0102]]

[[EQ:eq0103]]

[[EQ:eq0104]]

[[EQ:eq0105]]

[[EQ:eq0106]]

[[EQ:eq0107]]

[[EQ:eq0108]]

[[EQ:eq0109]]

[[EQ:eq0110]]

[[EQ:eq0111]]

[[EQ:eq0112]]

[[EQ:eq0113]]

[[EQ:eq0114]]

[[EQ:eq0115]]

[[EQ:eq0116]]

[[EQ:eq0117]]

[[EQ:eq0118]]

[[EQ:eq0119]]

[[EQ:eq0120]]

[[EQ:eq0121]]

[[EQ:eq0122]]

[[EQ:eq0123]]

[[EQ:eq0124]]

[[EQ:eq0127]]

[[EQ:eq0128]]

[[EQ:eq0129]]

[[EQ:eq0130]]

[[EQ:eq0131]]

[[EQ:eq0132]]

[[EQ:eq0133]]

[[EQ:eq0134]]

[[EQ:eq0135]]

[[EQ:eq0136]]

[[EQ:eq0137]]

[[EQ:eq0138]]

[[EQ:eq0139]]

[[EQ:eq0140]]

[[EQ:eq0141]]

[[EQ:eq0142]]

[[EQ:eq0143]]

[[EQ:eq0144]]

[[EQ:eq0145]]

[[EQ:eq0146]]

[[EQ:eq0147]]

[[EQ:eq0148]]

[[EQ:eq0149]]

[[EQ:eq0150]]

[[EQ:eq0151]]

[[EQ:eq0152]]

[[EQ:eq0153]]

[[EQ:eq0154]]

[[EQ:eq0155]]

[[EQ:eq0156]]

[[EQ:eq0157]]

[[EQ:eq0158]]

[[EQ:eq0159]]

[[EQ:eq0160]]

[[EQ:eq0161]]

[[EQ:eq0162]]

[[EQ:eq0163]]

[[EQ:eq0164]]

[[EQ:eq0165]]

[[EQ:eq0166]]

[[EQ:eq0167]]

[[EQ:eq0168]]

[[EQ:eq0169]]

[[EQ:eq0170]]

[[EQ:eq0171]]

[[EQ:eq0172]]

[[EQ:eq0173]]

[[EQ:eq0174]]

[[EQ:eq0175]]

[[EQ:eq0176]]

[[EQ:eq0177]]

[[EQ:eq0178]]

[[EQ:eq0179]]

[[EQ:eq0180]]

[[EQ:eq0181]]

[[EQ:eq0182]]

[[EQ:eq0183]]

[[EQ:eq0184]]

[[EQ:eq0185]]

[[EQ:eq0186]]

[[EQ:eq0187]]

[[EQ:eq0188]]

[[EQ:eq0189]]

[[EQ:eq0190]]

[[EQ:eq0191]]

[[EQ:eq0192]]

[[EQ:eq0193]]

[[EQ:eq0194]]

[[EQ:eq0195]]

[[EQ:eq0196]]

[[EQ:eq0197]]

[[EQ:eq0198]]

[[EQ:eq0199]]

[[EQ:eq0200]]

[[EQ:eq0201]]

[[EQ:eq0202]]

[[EQ:eq0203]]

[[EQ:eq0204]]

[[EQ:eq0205]]

[[EQ:eq0206]]

[[EQ:eq0207]]

[[EQ:eq0208]]

[[EQ:eq0209]]

[[EQ:eq0210]]

[[EQ:eq0211]]

[[EQ:eq0212]]

[[EQ:eq0213]]

[[EQ:eq0214]]

[[EQ:eq0215]]

[[EQ:eq0216]]

[[EQ:eq0217]]

Notation, Acronyms, and Symbols

Setting and Standing Assumptions

UGV Ratio and Optimizer

PF-2: Safe self-edits and MTTR

Failure Modes and Diagnostics (1 page)

References

\[
U(x,\cdot)\ \ge\ \nu(\cdot)\quad\text{and}\quad H_\zeta(x,\cdot)\ \ge\ \zeta\,\nu(\cdot).
\]

U(x,\cdot)\ \ge\ \nu(\cdot)\quad\text{and}\quad H_\zeta(x,\cdot)\ \ge\ \zeta \nu(\cdot).

L(H_\zeta)\ \ge\ \ell_0(\zeta)\ \ge\ c_{\mathrm{KL}}\zeta\ >0,

J_{H_\zeta}(\pi;G)=\frac{N_{H,G}(\pi)}{D_{H,\tau}(\pi)},\quad D_{H,\tau}(\pi):=\operatorname{lse}_\tau \big(\mathbb E_\pi[C_{\mathrm{info}}], L(H_\zeta)\big),

\[
N_{H,G}(\pi):=\tilde{\bar F}_{T,H_\zeta\mid G}(\pi)+\lambda\,\mathbb{E}_\pi[\mu],\qquad
\operatorname{lse}_\tau(a,b)=\tau\log(e^{a/\tau}+e^{b/\tau}).
\]

N_{H,G}(\pi):=\tilde{\bar F}_{T,H_\zeta\mid G}(\pi)+\lambda \mathbb{E}_\pi[\mu],\qquad \operatorname{lse}_\tau(a,b)=\tau\log(e^{a/\tau}+e^{b/\tau}).

\[
\tilde{\bar F}_{T,H_\zeta\mid G}(\pi)=\frac1T\sum_{t=1}^T I_{\mathrm{KL}}\!\big(A_t;O_{t+1}\mid W_t\big)\ \text{ under }\ \mathbb P^{\pi,G}.
\]

\tilde{\bar F}_{T,H_\zeta\mid G}(\pi)=\frac1T\sum_{t=1}^T I_{\mathrm{KL}} \big(A_t;O_{t+1}\mid W_t\big)\ \text{ under }\ \mathbb P^{\pi,G}.

\[
\tilde{\bar F}_{T,H_\zeta\mid K\circ G}(\pi)\ \le\ \tilde{\bar F}_{T,H_\zeta\mid G}(\pi)\quad\text{for all }\pi.
\]

\[
J_{H_\zeta}(\pi;K\!\circ\!G)\ \le\ J_{H_\zeta}(\pi;G)\ \ \text{for all }\pi.
\]

J_{H_\zeta}(\pi;K \circ G)\ \le\ J_{H_\zeta}(\pi;G)\ \ \text{for all }\pi.

\[
\varepsilon_{\text{num}}(\alpha,T)
=\sqrt{\tfrac{2\widehat{\mathrm{Var}}}{T}\log\tfrac{1}{\alpha}}
+\tfrac{3M}{T}\log\tfrac{1}{\alpha}.
\]

\[
L(\Psi\circ H_\zeta\circ \Phi)\ \ge\ \kappa\, L(H_\zeta).
\]

L(\Psi\circ H_\zeta\circ \Phi)\ \ge\ \kappa  L(H_\zeta).

\[
N(F\pi)\ \ge\ c_N\,N(\pi),\quad c_N:=\inf_{F\in\mathcal F}\ \text{contraction\_coeff}(F)\in(0,1].
\]

N(F\pi)\ \ge\ c_N N(\pi),\quad c_N:=\inf_{F\in\mathcal F}\ \text{contraction\_coeff}(F)\in(0,1].

\[
D(F\pi)=\operatorname{lse}_\tau(\mathbb E[C_{\mathrm{info}}],\,L(\Psi\circ H_\zeta\circ\Phi))
\ \le\ D(\pi)+\Delta_{\mathrm{floor}},
\]

D(F\pi)=\operatorname{lse}_\tau(\mathbb E[C_{\mathrm{info}}], L(\Psi\circ H_\zeta\circ\Phi)) \ \le\ D(\pi)+\Delta_{\mathrm{floor}},

\[
U^\gamma_\epsilon(\pi)=\sup_{\Pi\in\mathcal{B}_\epsilon}\Big\{I(A_t;\mathrm{label}_\Pi\mid W_t)-\gamma\,D_L(\Pi)\Big\},\qquad
U_\epsilon(\pi)=\lim_{\gamma\downarrow 0}U^\gamma_\epsilon(\pi).
\]

\[
\alpha_{\mathrm{MLSI}}(L)\ \ge\ \min\{\beta_{\mathrm{lab}},\alpha_{\mathrm{world}}\}-C\,\varepsilon,
\]

\alpha_{\mathrm{MLSI}}(L)\ \ge\ \min\{\beta_{\mathrm{lab}},\alpha_{\mathrm{world}}\}-C \varepsilon,

\[
\mathrm{MTTR}\ \le\ \frac{1}{\lambda_{\mathrm{CLF}}}\log\frac{V(b_0)}{V(A)}.
\]

\[
\max_{\mathbb Q}\ \ \mathbb E_{\mathbb Q}[\mathrm{harm}]
\quad\text{s.t.}\quad
\mathbb E_{\mathbb Q}[Z]=m_1,\ \
\mathbb E_{\mathbb Q}[ZZ^\top]\preceq M_2+\lambda I,\ \
\mathcal I(\mathbb Q)\le\delta,
\]

\[
\mathcal I(\mathbb Q)=\sum_{i\neq j}\mathrm{Cov}_{\mathbb Q}(Z_i,Z_j)\le\delta,
\]

$\mathbb E_\pi[\mathrm{KL}(\pi(\cdot\mid W)\,\|\,\pi_0(\cdot\mid W))]$

\mathbb E_\pi[\mathrm{KL}(\pi(\cdot\mid W) \| \pi_0(\cdot\mid W))]

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>$</mi><msub><mi mathvariant="double-struck">E</mi><mi>π</mi></msub><mo stretchy="false">[</mo><mrow><mi mathvariant="normal">K</mi><mi mathvariant="normal">L</mi></mrow><mo stretchy="false">(</mo><mi>π</mi><mo stretchy="false">(</mo><mi>·</mi><mo>∣</mo><mi>W</mi><mo stretchy="false">)</mo><mspace width="0.167em" /><mo fence="false" stretchy="false">‖</mo><mspace width="0.167em" /><msub><mi>π</mi><mn>0</mn></msub><mo stretchy="false">(</mo><mi>·</mi><mo>∣</mo><mi>W</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mi>$</mi></mrow></math>

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>$</mi><mo fence="false" stretchy="false">‖</mo><mi>V</mi><msub><mo fence="false" stretchy="false">‖</mo><mrow><mrow><mi mathvariant="normal">c</mi><mi mathvariant="normal">b</mi></mrow></mrow></msub><mo>≤</mo><mn>1</mn><mi>$</mi></mrow></math>

C_{\mathrm{info}}=\mathrm{KL}(\pi_\theta(\cdot\mid w) \| \pi_0(\cdot\mid w))

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>$</mi><msub><mi>C</mi><mrow><mrow><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal">f</mi><mi mathvariant="normal">o</mi></mrow></mrow></msub><mo>=</mo><mrow><mi mathvariant="normal">K</mi><mi mathvariant="normal">L</mi></mrow><mo stretchy="false">(</mo><msub><mi>π</mi><mi>θ</mi></msub><mo stretchy="false">(</mo><mi>·</mi><mo>∣</mo><mi>w</mi><mo stretchy="false">)</mo><mspace width="0.167em" /><mo fence="false" stretchy="false">‖</mo><mspace width="0.167em" /><msub><mi>π</mi><mn>0</mn></msub><mo stretchy="false">(</mo><mi>·</mi><mo>∣</mo><mi>w</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mi>$</mi></mrow></math>

$L(\Psi\circ H_\zeta\circ\Phi)\ge \kappa L(H_\zeta)-\varepsilon$


A FORMAL AXIOMATIC PROPOSAL FOR HAWKINS' LEVELS OF CONSCIOUSNESS

Axioms and scope

We work in continuous time [[EQ:eq0018]] and space [[EQ:eq0019]] , where [[EQ:eq0020]] is a metric–measure space [[EQ:eq0021]] admitting a conservative symmetric Markov semigroup [[EQ:eq0022]] .[1] Let [[EQ:eq0024]] be a finite chain.

Axiom (Ordinality).

There exists a monotone embedding [[EQ:eq0025]] that respects order. Any two admissible embeddings differ by an order-preserving transform. No metric or ratio meaning is attached to differences of [[EQ:eq0026]] .

Axiom (No-Meta closure & non-coercion).

All structural quantities are auditable: defined from internal logs and monotone under Blackwell-coarser information (no privileged external evaluator). Floors and invariances follow .

Identification layer: ordinal structure

We do not fit data; we formalize the admissible class.

Definition 1 (Ordinal embedding). An embedding [[EQ:eq0027]] is any strictly increasing map. If a latent single-index is introduced on items/practices, admissible transforms are precisely the increasing maps (isotonic invariance). Classical pairwise models (Bradley–Terry , Thurstone Case V ) are compatible for later identification; here we assume only the existence of some [[EQ:eq0028]] .

Remark 1 (Rank stability). Any summary depending only on the total order (e.g. Kendall [[EQ:eq0029]] ) is invariant to admissible transforms. Our structural results never require more than order information.

Structural layer: operator semantics, floors, and comparison dynamics

Operator semantics on general [[EQ:eq0030]] .

On metric–measure spaces, [[EQ:eq0031]] is understood in the Dirichlet-form sense (carré du champ), yielding, under the stated regularity, existence/uniqueness of [[EQ:eq0032]] -weak (energy) solutions and [[EQ:eq0033]] -invariance.

Fix a threshold [[EQ:eq0034]] (e.g. [[EQ:eq0035]] ). Let [[EQ:eq0036]] be the density of agents with [[EQ:eq0037]] . Consider

[[EQ:eq0001]]

with the following assumptions.

Assumption (Floors).

There exist [[EQ:eq0038]] such that [[EQ:eq0039]] is symmetric positive definite (in local coordinates) with [[EQ:eq0040]] , and [[EQ:eq0041]] for a.e. [[EQ:eq0042]] .

Assumption (KPP domination).

[[EQ:eq0043]] , the map [[EQ:eq0044]] belongs to [[EQ:eq0045]] , is nondecreasing and locally Lipschitz; and for [[EQ:eq0046]] , [[EQ:eq0047]] .

Assumption (Regularity, damping, and invariance).

[[EQ:eq0048]] (uniformly on bounded time windows), [[EQ:eq0049]] , [[EQ:eq0050]] . Then the parabolic comparison principle holds on [[EQ:eq0051]] , weak solutions exist and are unique in a standard class, and [[EQ:eq0052]] is invariant.

Definition 2 (Invasion speed — occupancy form (Euclidean/Riemannian scope)). Assume [[EQ:eq0053]] or a smooth Riemannian manifold. For a unit direction [[EQ:eq0054]] and any fixed [[EQ:eq0055]] , the lower invasion speed [[EQ:eq0056]] is the supremum of [[EQ:eq0057]] such that there exists [[EQ:eq0058]] with

[[EQ:eq0004]]

for compactly supported, nontrivial initial data.

Remark 2 (Dependence on the occupancy level [[EQ:eq0059]] ). The quantity [[EQ:eq0060]] defined with occupancy level [[EQ:eq0061]] is nondecreasing as [[EQ:eq0062]] . Our comparison-based lower bounds do not depend on [[EQ:eq0063]] ; in statements we may fix any convenient [[EQ:eq0064]] (e.g., [[EQ:eq0065]] ).

Theorem 1 (Isotropic lower bound (no damping)). If [[EQ:eq0066]] , then for every direction [[EQ:eq0067]] ,

[[EQ:eq0005]]

Sketch. Compare [eq:PDE] with [[EQ:eq0068]] (or with the corresponding generator on [[EQ:eq0069]] ) and use classical Fisher–KPP subsolutions . ◻

Scope for directional results.

Theorem 2 (directional bound) is stated on [[EQ:eq0070]] or, more generally, on smooth Riemannian manifolds where [[EQ:eq0071]] , [[EQ:eq0072]] , and [[EQ:eq0073]] are well-defined in local coordinates. On general metric–measure spaces, the isotropic Fisher–KPP lower bound (Theorem 1) remains valid via the Dirichlet-form interpretation.

Directional refinement in heterogeneous media (Euclidean/Riemannian scope)

Define the directional transport floor

[[EQ:eq0006]]

and the divergence penalty (positive part)[2]

[[EQ:eq0007]]

where [[EQ:eq0075]] . A backward-zero barrier with divergence correction yields:

Theorem 2 (Directional lower bound (no damping)). If [[EQ:eq0076]] , then for each unit direction [[EQ:eq0077]] ,

[[EQ:eq0008]]

Moreover, symmetric Markov coarse-graining (heat-kernel smoothing) cannot increase the right-hand side (monotone safety) under the schemes in Prop. 1.

Time-window pasting (piecewise nonstationary coefficients).

For stepwise time-constant coefficients, barrier comparison holds on each window; Aronson-type Gaussian lower bounds control the [[EQ:eq0078]] overhead at joins under standard local doubling and Poincaré conditions (parabolic pasting).

Modulatory layer: aversive affect as a field-level freedom-gap damping

To lift micro-level affect to a macroscopic coefficient, define the (auditable) freedom gap

[[EQ:eq0009]]

where [[EQ:eq0079]] is the Shannon entropy of next-step feasible actions given logs after a Blackwell-coarser preprocessing [[EQ:eq0080]] ; [[EQ:eq0081]] is the corresponding prediction under the same [[EQ:eq0082]] .[3] Let [[EQ:eq0085]] be convex, increasing, [[EQ:eq0086]] . Define a smooth cutoff [[EQ:eq0087]] as a [[EQ:eq0088]] nonincreasing approximation to [[EQ:eq0089]] . Set

[[EQ:eq0002]]

Lemma 1 (Effective linear rate). Let [[EQ:eq0090]] . Under [eq:Gamma],

[[EQ:eq0010]]

Remark 3 (Safe comparison with [[EQ:eq0091]] ). Even if [[EQ:eq0092]] depends on [[EQ:eq0093]] , comparison from below with [[EQ:eq0094]] is conservative; all speed lower bounds remain valid with [[EQ:eq0095]] replaced by [[EQ:eq0096]] .

Corollary 1 (Speed with damping). If [[EQ:eq0097]] , then for all directions [[EQ:eq0098]] ,

[[EQ:eq0011]]

Case A ( [[EQ:eq0099]] ) reduces to Theorem 1.

Remark 4 (Quantitative gain from closing the freedom gap). Let [[EQ:eq0100]] and let [[EQ:eq0101]] be a uniform decrease in [[EQ:eq0102]] obtained by reducing [[EQ:eq0103]] by [[EQ:eq0104]] . Then the running lower bound [[EQ:eq0105]] changes by the exact amount

[[EQ:eq0012]]

and therefore is sandwiched as

[[EQ:eq0013]]

Vector (multi-level) extension

Let [[EQ:eq0106]] be the density in level bin [[EQ:eq0107]] with [[EQ:eq0108]] . Consider

[[EQ:eq0003]]

with [[EQ:eq0109]] cooperative and [[EQ:eq0110]] Metzler and irreducible (strong connectivity on the positive cone). Assume a uniform PF floor [[EQ:eq0111]] (a.e. Perron–Frobenius eigenvalue [[EQ:eq0112]] ) and directional transport floors for each [[EQ:eq0113]] .

Vector directional floors (conservative choice).

Define

[[EQ:eq0014]]

These choices ensure the comparison principle on the positive cone without invoking additional structure; PF–projected variants yield sharper but same-form bounds.

Theorem 3 (Vector directional bound). If [[EQ:eq0114]] , then for each unit direction [[EQ:eq0115]] ,

[[EQ:eq0015]]

by constructing componentwise barriers on the positive cone and projecting onto the PF direction (as in ).

Coarse-graining safety and representation robustness

Let [[EQ:eq0116]] be a symmetric Markov semigroup on spatial fields and [[EQ:eq0117]] a Blackwell-coarser preprocessing on information logs. We fix the following matrix-safe coarse-graining scheme (inequality form with existence guarantee):

Proposition 1 (Monotone degradation (safety) under a fixed matrix-safe scheme). Let [[EQ:eq0118]] be SPD. For each unit direction [[EQ:eq0119]] , set [[EQ:eq0120]] and [[EQ:eq0121]] . Define [[EQ:eq0122]] to be any SPD matrix field satisfying

[[EQ:eq0016]]

A conservative choice is the isotropic envelope [[EQ:eq0123]] , which always exists. Then applying [[EQ:eq0124]] to fields and [[EQ:eq0125]] to diffusivity cannot increase the right-hand sides of Theorems 1–2 and Cor. 1. Hence speed lower bounds are safe under this coarse-graining.

Auditing protocol (sketch).

From internal logs: (i) estimate visibility [[EQ:eq0126]] via Doeblin-type minorization on refresh events; (ii) estimate contraction [[EQ:eq0127]] via SDPI/LSI lower bounds under the same preprocessing [[EQ:eq0128]] ; (iii) estimate [[EQ:eq0129]] from Gaussian lower bounds for first-passage/dispersion statistics; (iv) estimate [[EQ:eq0130]] by linearizing [[EQ:eq0131]] near [[EQ:eq0132]] with conservative (lower) regression. All steps are monotone under Blackwell-coarser preprocessing.

Micro-to-macro rationale for KPP domination (minimal sketch)

A minimal contact-process rationale: when density [[EQ:eq0133]] is small, (i) active agents activate neighbors at rate [[EQ:eq0134]] (contact), (ii) environment refresh at rate [[EQ:eq0135]] , (iii) saturation yields quadratic inhibition [[EQ:eq0136]] . The mean-field reaction is

[[EQ:eq0017]]

providing KPP domination and Lipschitz monotonicity.

Roles of the four floors

- Visibility [[EQ:eq0137]] : Doeblin-type refresh/minorization (logs are observable; prevents hidden absorbing sets).

- Contraction [[EQ:eq0138]] : DPI/SDPI or LSI supplies rank/ordering stability under noise and supports Blackwell monotonicity.

- Transport [[EQ:eq0139]] : Uniform ellipticity (or conductance) enabling diffusion barriers and Gaussian lower bounds.

- Local gain [[EQ:eq0140]] : Linearized cooperative advantage near [[EQ:eq0141]] yielding KPP comparison.

Consequences, falsifiers, and scope

P1 (speed floor). If [[EQ:eq0142]] , then [[EQ:eq0143]] .
P2 (damped speed). If [[EQ:eq0144]] , then [[EQ:eq0145]] .
P3 (coarse-graining safety). Under Prop. 1 and Blackwell-coarser preprocessing, lower bounds cannot increase.

R1 (falsifier). Observing [[EQ:eq0146]] under [[EQ:eq0147]] . R2. Observing an increase in the bound after coarse-graining (violating Prop. 1). R3. Violations of KPP domination or floor positivity.

Scope note. A one-dimensional ordinal chain may be insufficient to encode nuance; the vector extension with a PF floor addresses multi-attribute structure.

Conclusion

Treating Hawkins’ labels as a chain rather than a metric ladder, we showed that (i) an order embedding suffices; (ii) with auditable floors, cooperative comparison dynamics yield nonzero invasion speeds; (iii) aversive affect is a convex freedom-gap damping consistent with No-Meta auditing; (iv) vector systems admit PF floors; and (v) under fixed safe coarse-graining, guarantees degrade monotonically.

10

R. A. Bradley and M. E. Terry. Rank analysis of incomplete block designs: I. The method of paired comparisons. , 39(3/4):324–345, 1952.

L. L. Thurstone. A law of comparative judgment. , 34(4):273–286, 1927.

M. G. Kendall. A new measure of rank correlation. , 30(1/2):81–93, 1938.

R. A. Fisher. The wave of advance of advantageous genes. , 7:355–369, 1937.

A. N. Kolmogorov, I. G. Petrovskii, and N. S. Piskunov. A study of the diffusion equation with increase in the amount of substance. , 1(6):1–26, 1937.

K. Takahashi. UGV Without Meta: A Representation-Independent Theory for Compassion and Enlightenment in Collective Intelligence. Zenodo, 2025. DOI: 10.5281/zenodo.17082312.

K. Takahashi. Engineering Happiness in Human–AI Intelligence Networks. Zenodo, 2025. DOI: 10.5281/zenodo.17113105.

K. Takahashi. “Persistence [[EQ:eq0148]] Creation”: Natural-Law Sufficient Conditions for Almost-Sure Beneficial Coverage in Stationary Ergodic Media (No Meta-Design). Zenodo, 2025. DOI: 10.5281/zenodo.17100322.

K. Takahashi. Nondual Field Theory of Viable Predictive Organization. Zenodo, 2025. DOI: 10.5281/zenodo.17131394.

K. Takahashi. Natural-Law Acceleration of VPO. Zenodo, 2025. DOI: 10.5281/zenodo.17120045.

K. Takahashi. A Pure Natural Theory of Benevolent Propagation under No-Meta Closure. Zenodo, 2025. DOI: 10.5281/zenodo.17136051.

K. Takahashi. Persistence-First Superintelligence. Zenodo, 2025. DOI: 10.5281/zenodo.17076410.

[1] Specializations: (i) Euclidean domains with the Laplacian; (ii) smooth Riemannian manifolds with the Laplace–Beltrami; (iii) graph limits with normalized Laplacian (in discrete analogues: conductance); (iv) information-geometric manifolds via score-based diffusion. In each case, uniform ellipticity / conductance plays the role of [[EQ:eq0023]] .

[2] [[EQ:eq0074]] .

[3] Intuition: [[EQ:eq0083]] measures the diversity of feasible next actions; large [[EQ:eq0084]] means “promised” optionality exceeds realized optionality.
[[EQ:eq0004]]

[[EQ:eq0005]]

[[EQ:eq0006]]

[[EQ:eq0007]]

[[EQ:eq0008]]

[[EQ:eq0009]]

[[EQ:eq0010]]

[[EQ:eq0011]]

[[EQ:eq0012]]

[[EQ:eq0013]]

[[EQ:eq0018]]

[[EQ:eq0019]]

[[EQ:eq0020]]

[[EQ:eq0021]]

[[EQ:eq0022]]

[[EQ:eq0024]]

[[EQ:eq0025]]

[[EQ:eq0026]]

[[EQ:eq0027]]

[[EQ:eq0028]]

[[EQ:eq0029]]

[[EQ:eq0030]]

[[EQ:eq0031]]

[[EQ:eq0032]]

[[EQ:eq0033]]

[[EQ:eq0034]]

[[EQ:eq0035]]

[[EQ:eq0036]]

[[EQ:eq0037]]

[[EQ:eq0038]]

[[EQ:eq0039]]

[[EQ:eq0040]]

[[EQ:eq0041]]

[[EQ:eq0042]]

[[EQ:eq0043]]

[[EQ:eq0044]]

[[EQ:eq0045]]

[[EQ:eq0046]]

[[EQ:eq0047]]

[[EQ:eq0048]]

[[EQ:eq0049]]

[[EQ:eq0050]]

[[EQ:eq0051]]

[[EQ:eq0052]]

[[EQ:eq0053]]

[[EQ:eq0054]]

[[EQ:eq0055]]

[[EQ:eq0056]]

[[EQ:eq0057]]

[[EQ:eq0058]]

[[EQ:eq0059]]

[[EQ:eq0060]]

[[EQ:eq0061]]

[[EQ:eq0062]]

[[EQ:eq0063]]

[[EQ:eq0064]]

[[EQ:eq0065]]

[[EQ:eq0066]]

[[EQ:eq0067]]

[[EQ:eq0068]]

[[EQ:eq0069]]

[[EQ:eq0070]]

[[EQ:eq0071]]

[[EQ:eq0072]]

[[EQ:eq0073]]

[[EQ:eq0075]]

[[EQ:eq0076]]

[[EQ:eq0077]]

[[EQ:eq0078]]

[[EQ:eq0079]]

[[EQ:eq0080]]

[[EQ:eq0081]]

[[EQ:eq0082]]

[[EQ:eq0085]]

[[EQ:eq0086]]

[[EQ:eq0087]]

[[EQ:eq0088]]

[[EQ:eq0089]]

[[EQ:eq0090]]

[[EQ:eq0091]]

[[EQ:eq0092]]

[[EQ:eq0093]]

[[EQ:eq0094]]

[[EQ:eq0095]]

[[EQ:eq0096]]

[[EQ:eq0097]]

[[EQ:eq0098]]

[[EQ:eq0099]]

[[EQ:eq0100]]

[[EQ:eq0101]]

[[EQ:eq0102]]

[[EQ:eq0103]]

[[EQ:eq0104]]

[[EQ:eq0105]]

[[EQ:eq0106]]

[[EQ:eq0107]]

[[EQ:eq0108]]

[[EQ:eq0109]]

[[EQ:eq0110]]

[[EQ:eq0111]]

[[EQ:eq0112]]

[[EQ:eq0113]]

[[EQ:eq0114]]

[[EQ:eq0115]]

[[EQ:eq0116]]

[[EQ:eq0117]]

[[EQ:eq0118]]

[[EQ:eq0119]]

[[EQ:eq0120]]

[[EQ:eq0121]]

[[EQ:eq0122]]

[[EQ:eq0123]]

[[EQ:eq0124]]

[[EQ:eq0125]]

[[EQ:eq0126]]

[[EQ:eq0127]]

[[EQ:eq0128]]

[[EQ:eq0129]]

[[EQ:eq0130]]

[[EQ:eq0131]]

[[EQ:eq0132]]

[[EQ:eq0133]]

[[EQ:eq0134]]

[[EQ:eq0135]]

[[EQ:eq0136]]

[[EQ:eq0137]]

[[EQ:eq0138]]

[[EQ:eq0139]]

[[EQ:eq0140]]

[[EQ:eq0141]]

[[EQ:eq0142]]

[[EQ:eq0143]]

[[EQ:eq0144]]

[[EQ:eq0145]]

[[EQ:eq0146]]

[[EQ:eq0147]]

[[EQ:eq0148]]

Axioms and scope

Identification layer: ordinal structure

Vector (multi-level) extension

Roles of the four floors

Consequences, falsifiers, and scope

Conclusion

\begin{equation}
\partial_t u \;=\; \nabla\!\cdot\!\big(D(x,t)\nabla u\big)\;+\;f(x,t,u)\;-\;\Gamma(x,t)\,u,
\label{eq:PDE}
\end{equation}

\partial_t u  =  \nabla \cdot \big(D(x,t)\nabla u\big) + f(x,t,u) - \Gamma(x,t) u, \label{eq:PDE}

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mrow><msub><mo>∂</mo><mi>t</mi></msub><mi>u</mi><mspace width="0.278em" /><mo>=</mo><mspace width="0.278em" /><mo>∇</mo><mspace width="negativethinmathspace" /><mi>·</mi><mspace width="negativethinmathspace" /><mo minsize="1.2em" maxsize="1.2em">(</mo><mi>D</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo>∇</mo><mi>u</mi><mo minsize="1.2em" maxsize="1.2em">)</mo><mspace width="0.278em" /><mo>+</mo><mspace width="0.278em" /><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>,</mo><mi>u</mi><mo stretchy="false">)</mo><mspace width="0.278em" /><mo>−</mo><mspace width="0.278em" /><mi>Γ</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mspace width="0.167em" /><mi>u</mi><mo>,</mo><mi>\label</mi><mrow><mi>e</mi><mi>q</mi><mi>:</mi><mi>P</mi><mi>D</mi><mi>E</mi></mrow></mrow></mrow></math>

\begin{equation}
\Gamma(t,x)\;=\;\gamma_0\;+\;\gamma_1\,\Phi\!\big(G_\Pi(t,x)\big)\,\chi\!\big(u(t,x);\vartheta\big),\qquad \gamma_0,\gamma_1\ge 0.
\label{eq:Gamma}
\end{equation}

\Gamma(t,x) = \gamma_0 + \gamma_1 \Phi \big(G_\Pi(t,x)\big) \chi \big(u(t,x);\vartheta\big),\qquad \gamma_0,\gamma_1\ge 0. \label{eq:Gamma}

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mrow><mi>Γ</mi><mo stretchy="false">(</mo><mi>t</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo><mspace width="0.278em" /><mo>=</mo><mspace width="0.278em" /><msub><mi>γ</mi><mn>0</mn></msub><mspace width="0.278em" /><mo>+</mo><mspace width="0.278em" /><msub><mi>γ</mi><mn>1</mn></msub><mspace width="0.167em" /><mi>Φ</mi><mspace width="negativethinmathspace" /><mo minsize="1.2em" maxsize="1.2em">(</mo><msub><mi>G</mi><mi>Π</mi></msub><mo stretchy="false">(</mo><mi>t</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo><mo minsize="1.2em" maxsize="1.2em">)</mo><mspace width="0.167em" /><mi>χ</mi><mspace width="negativethinmathspace" /><mo minsize="1.2em" maxsize="1.2em">(</mo><mi>u</mi><mo stretchy="false">(</mo><mi>t</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo><mi>;</mi><mi>ϑ</mi><mo minsize="1.2em" maxsize="1.2em">)</mo><mo>,</mo><mspace width="2em" /><msub><mi>γ</mi><mn>0</mn></msub><mo>,</mo><msub><mi>γ</mi><mn>1</mn></msub><mo>≥</mo><mn>0</mn><mo>.</mo><mi>\label</mi><mrow><mi>e</mi><mi>q</mi><mi>:</mi><mi>G</mi><mi>a</mi><mi>m</mi><mi>m</mi><mi>a</mi></mrow></mrow></mrow></math>

\begin{equation}
\partial_t q_c \;=\; \nabla\!\cdot\!\big(D_c(x,t)\nabla q_c\big)\;+\;\sum_{d\in\mathcal{C}} R_{cd}(x,t,q)\;-\;\Gamma_c(x,t)\,q_c,
\label{eq:vec}
\end{equation}

\partial_t q_c  =  \nabla \cdot \big(D_c(x,t)\nabla q_c\big) + \sum_{d\in\mathcal{C}} R_{cd}(x,t,q) - \Gamma_c(x,t) q_c, \label{eq:vec}

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mrow><msub><mo>∂</mo><mi>t</mi></msub><msub><mi>q</mi><mi>c</mi></msub><mspace width="0.278em" /><mo>=</mo><mspace width="0.278em" /><mo>∇</mo><mspace width="negativethinmathspace" /><mi>·</mi><mspace width="negativethinmathspace" /><mo minsize="1.2em" maxsize="1.2em">(</mo><msub><mi>D</mi><mi>c</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo>∇</mo><msub><mi>q</mi><mi>c</mi></msub><mo minsize="1.2em" maxsize="1.2em">)</mo><mspace width="0.278em" /><mo>+</mo><mspace width="0.278em" /><msub><mo>∑</mo><mrow><mi>d</mi><mo>∈</mo><mi>𝒞</mi></mrow></msub><msub><mi>R</mi><mrow><mi>c</mi><mi>d</mi></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>,</mo><mi>q</mi><mo stretchy="false">)</mo><mspace width="0.278em" /><mo>−</mo><mspace width="0.278em" /><msub><mi>Γ</mi><mi>c</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mspace width="0.167em" /><msub><mi>q</mi><mi>c</mi></msub><mo>,</mo><mi>\label</mi><mrow><mi>e</mi><mi>q</mi><mi>:</mi><mi>v</mi><mi>e</mi><mi>c</mi></mrow></mrow></mrow></math>

\[
\inf_{x:\, x\cdot n\le (v-\eta)t}\,u(t,x)\ \ge\ \theta\quad\text{for all sufficiently large }t,
\]

\inf_{x:  x\cdot n\le (v-\eta)t} u(t,x)\ \ge\ \theta\quad\text{for all sufficiently large }t,

v_\ast(n) \ge  2\sqrt{D_{\min} \lambda_{\min}}.

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>\[</mi><msub><mi>Λ</mi><mo>+</mo></msub><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mi>:</mi><mo>=</mo><mo /><mo>*</mo><msub><mrow><mi>e</mi><mi>s</mi><mi>s</mi><mspace width="0.167em" /><mi>s</mi><mi>u</mi><mi>p</mi></mrow><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></msub><mo minsize="1.623em" maxsize="1.623em">(</mo><mo>div</mo><mspace width="negativethinmathspace" /><mo minsize="1.2em" maxsize="1.2em">(</mo><mi>D</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mi>n</mi><mo minsize="1.2em" maxsize="1.2em">)</mo><msub><mo minsize="1.623em" maxsize="1.623em">)</mo><mo>+</mo></msub><mo>,</mo><mi>\]</mi></mrow></math>

v_\ast(n)  \ge  \Big[ 2\sqrt{D(n) \lambda_{\min}}-\Lambda_+(n) \Big]_+.

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>\[</mi><msub><mi>v</mi><mi>*</mi></msub><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mspace width="0.278em" /><mo>≥</mo><mspace width="0.278em" /><mo minsize="1.623em" maxsize="1.623em">[</mo><mspace width="0.167em" /><mn>2</mn><msqrt><mrow><mi>D</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mspace width="0.167em" /><msub><mi>λ</mi><mrow><mo>min</mo></mrow></msub></mrow></msqrt><mo>−</mo><msub><mi>Λ</mi><mo>+</mo></msub><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mspace width="0.167em" /><msub><mo minsize="1.623em" maxsize="1.623em">]</mo><mo>+</mo></msub><mo>.</mo><mi>\]</mi></mrow></math>

G_\Pi(t,x) := \widehat{H}_{\mathrm{opt},\Pi}(t,x) - H_{\mathrm{opt},\Pi}(t,x) \ \ge 0,

\[
\lambda_{\rm eff}(x,t)\;:=\;\lambda_{\rm lin}(x,t)-\Gamma(x,t)\ \ge\ \lambda_{\min}-\big(\gamma_0+\gamma_1\,\Phi(G_\Pi)\big).
\]

\lambda_{\rm eff}(x,t) := \lambda_{\rm lin}(x,t)-\Gamma(x,t)\ \ge\ \lambda_{\min}-\big(\gamma_0+\gamma_1 \Phi(G_\Pi)\big).

\[
v_\ast(n)\ \ge\ 2\sqrt{D_{\min}\,(\lambda_{\min}-\sup\Gamma)}.
\]

v_\ast(n)\ \ge\ 2\sqrt{D_{\min} (\lambda_{\min}-\sup\Gamma)}.

\[
\Delta v_{\rm LB}\;=\;2\sqrt{D_{\min}}\,
\frac{\Delta}{\sqrt{L+\Delta}+\sqrt{L}},
\]

\Delta v_{\rm LB} = 2\sqrt{D_{\min}}  \frac{\Delta}{\sqrt{L+\Delta}+\sqrt{L}},

\[
\frac{\sqrt{D_{\min}}\,\Delta}{\sqrt{L+\Delta}}
\;\le\; \Delta v_{\rm LB}
\;\le\; \frac{\sqrt{D_{\min}}\,\Delta}{\sqrt{L}}.
\]

\frac{\sqrt{D_{\min}} \Delta}{\sqrt{L+\Delta}}  \le  \Delta v_{\rm LB}  \le  \frac{\sqrt{D_{\min}} \Delta}{\sqrt{L}}.

D_{\text{vec}}(n):=\operatorname*{ess inf}_{(x,t)}\min_{c\in\mathcal{C}} \frac{n^\top D_c(x,t)n}{\|n\|^2},\qquad \Lambda^{\text{vec}}_+(n):=\operatorname*{ess sup}_{(x,t)}\max_{c\in\mathcal{C}} \big(\operatorname{div}(D_c(x,t)n)\big)_+.

\[
v_\ast(n)\ \ge\ \big[\,2\sqrt{D_{\text{vec}}(n)\,(\lambda_{\mathrm{PF,inf}}-\sup_c \Gamma_c)}\;-\;\Lambda^{\text{vec}}_+(n)\,\big]_+,
\]

v_\ast(n)\ \ge\ \big[ 2\sqrt{D_{\text{vec}}(n) (\lambda_{\mathrm{PF,inf}}-\sup_c \Gamma_c)} - \Lambda^{\text{vec}}_+(n) \big]_+,

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>\[</mi><msub><mi>v</mi><mi>*</mi></msub><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mtext> </mtext><mo>≥</mo><mtext> </mtext><mo minsize="1.2em" maxsize="1.2em">[</mo><mspace width="0.167em" /><mn>2</mn><msqrt><mrow><msub><mi>D</mi><mrow><mtext>vec</mtext></mrow></msub><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mspace width="0.167em" /><mo stretchy="false">(</mo><msub><mi>λ</mi><mrow><mrow><mi mathvariant="normal">P</mi><mi mathvariant="normal">F</mi><mo>,</mo><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal">f</mi></mrow></mrow></msub><mo>−</mo><msub><mo>sup</mo><mi>c</mi></msub><msub><mi>Γ</mi><mi>c</mi></msub><mo stretchy="false">)</mo></mrow></msqrt><mspace width="0.278em" /><mo>−</mo><mspace width="0.278em" /><msubsup><mi>Λ</mi><mo>+</mo><mrow><mtext>vec</mtext></mrow></msubsup><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mspace width="0.167em" /><msub><mo minsize="1.2em" maxsize="1.2em">]</mo><mo>+</mo></msub><mo>,</mo><mi>\]</mi></mrow></math>

\[
n^\top D_\ell(x,t)\, n \;\le\; \frac{1}{d_{n,\ell}(x,t)}\quad \text{for all unit } n.
\]

n^\top D_\ell(x,t)  n  \le  \frac{1}{d_{n,\ell}(x,t)}\quad \text{for all unit } n.

\[
f(u)=\alpha u+\beta u - \kappa u^2+\text{h.o.t.},\quad
f'(0)=\alpha+\beta,\quad f(u)\le f'(0)u\ (u\in(0,1)),\ \ \kappa\ge 0,
\]

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>$</mi><mi>𝒞</mi><mo>=</mo><mo stretchy="false">{</mo><mrow><mi mathvariant="normal">S</mi><mi mathvariant="normal">h</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">m</mi><mi mathvariant="normal">e</mi></mrow><mo>≺</mo><mo>⋯</mo><mo>≺</mo><mrow><mi mathvariant="normal">C</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">u</mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">g</mi><mi mathvariant="normal">e</mi></mrow><mo>≺</mo><mo>⋯</mo><mo>≺</mo><mrow><mi mathvariant="normal">E</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal">l</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">g</mi><mi mathvariant="normal">h</mi><mi mathvariant="normal">t</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal">m</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal">t</mi></mrow><mo stretchy="false">}</mo><mi>$</mi></mrow></math>

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>$</mi><mo>∇</mo><mspace width="negativethinmathspace" /><mi>·</mi><mo stretchy="false">(</mo><mi>D</mi><mo>∇</mo><mo stretchy="false">)</mo><mi>$</mi></mrow></math>

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$\operatorname{div}(D n):=\sum_{i=1}^d \partial_i\!\big(\sum_{j=1}^d D_{ij} n_j\big)$

\operatorname{div}(D n):=\sum_{i=1}^d \partial_i \big(\sum_{j=1}^d D_{ij} n_j\big)

\Phi:\mathbb{R}_{\ge 0} \to \mathbb{R}_{\ge 0}

\partial_t w=\nabla \cdot(D\nabla w)+\lambda_{\min}w-(\sup\Gamma)w

D_\ell^{\rm iso}:=\big(\sup_n d_{n,\ell}\big)^{-1} I

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>$</mi><mi>u</mi><mspace width="negativethinmathspace" /><mo>=</mo><mspace width="negativethinmathspace" /><mn>0</mn><mi>$</mi></mrow></math>


A NATURAL-LAW THEORY OF FUNDAMENTAL SUFFERING

% crisp, searchable glyphs

1.2

assumption
definition
theorem
proposition
lemma
corollary
remark

colorlinks=true,
linkcolor=blue,
citecolor=magenta,
urlcolor=blue,
pdftitle= A Natural-Law Theory of Fundamental Suffering: Small-Gain Floors and Principal-Eigenvalue KPP Speeds with Hodge–e-Process/Entropy-Production Coupling and Polarity-Free Selection (En-SELEX),
pdfauthor= K. Takahashi ,
pdfkeywords= Artificial Intelligence, reaction–diffusion, Hodge decomposition, small gain, Lyapunov floor, KPP front speed, principal eigenvalue, entropy production, e-process, safe e-merge, selection dynamics, optionality, No-Meta, dependent origination, non-dual ,
pdfsubject= Closed field equations, auditing, and governance for measurable suffering dynamics ,
pdfcreator= LaTeX

R
E
div

curl
coex
J
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s_ lock % <<< FIX: define locking source macro

TITLE:
A Natural-Law Theory of Fundamental Suffering:\
Small-Gain Floors and Principal-Eigenvalue KPP Speeds\
with Hodge–e-Process/Entropy-Production Coupling and\
Polarity-Free Selection (En-SELEX)

AUTHOR: K. Takahashi
https://orcid.org/0009-0004-4273-3365

DATE:

We present a closed, unit-consistent, and testable field theory for fundamental suffering as a measurable density evolving on domains and graphs. The core comprises: (i) reaction–advection–diffusion equations for a nonnegative suffering field [[EQ:eq0026]] and a nonnegative beneficial-change proxy [[EQ:eq0027]] ; (ii) Hodge/Helmholtz projections that isolate the coexact (circulation) flux as a gauge-invariant maintainer; (iii) a small-gain Lyapunov theorem yielding exponential decay to a computable floor even in the presence of nonlocal locking sources; (iv) a principal-eigenvalue characterization of directional KPP front speeds with a practical lower bound via effective diffusivity and adverse drift; (v) path-asymmetry as coarse-grained entropy production (EP); (vi) anytime-valid evidence via e-processes (e-values) with safe e-merge under arbitrary dependence; (vii) En-SELEX, a polarity-free selection rule that reallocates resources by evidence and dissipation under optionality and No-Meta constraints. Observables are operationalized in a two-layer (observation/value) scheme; boundaries and units are standardized; implementations (upwind/M-matrix, IMEX, CFL) and auditing (calibration, residuals, lower confidence limits) are specified. The theory is non-dual: loops that reduce floors and increase directional progress while dissipating less become naturally favored without value labels.

SECTION: Plain-Language Summary

We model ``suffering'' as a measurable field that can go up or down in space and time. Circular flows (like rework loops) tend to keep suffering alive; breaking those cycles reduces a mathematically provable floor. Progress spreads with a minimum directional speed that depends on how connected the system is and how strong the headwinds are. We provide formulas, measurement procedures, and audit rules that anyone (humans or AI systems) can apply without using moral labels. The system allocates resources to loops that show better evidence of reducing suffering with less waste, while keeping human options open.

PARAGRAPH: Keywords

Artificial Intelligence; reaction–diffusion; Hodge decomposition; small gain; Lyapunov floor; KPP front; principal eigenvalue; entropy production; e-process; safe e-merge; selection dynamics; optionality; No-Meta; dependent origination; non-dual.

SECTION: Preliminaries: Domains, Units, Non-dual Semantics

SUBSECTION: Domains and spaces

Let [[EQ:eq0028]] be a Lipschitz domain with boundary [[EQ:eq0029]] (pairwise disjoint up to measure zero). Time [[EQ:eq0030]] . Weak solutions live in [[EQ:eq0031]] ; flux fields in [[EQ:eq0032]] (and [[EQ:eq0033]] when used). Graph analogues use weighted graphs and discrete Hodge Laplacians. We write [[EQ:eq0034]] for the unit sphere.

SUBSECTION: Units (crawler-friendly list)

sec:units
[[EQ:eq0035]] : intensity/volume. \;
[[EQ:eq0036]] : intensity/(area [[EQ:eq0037]] time). \;
[[EQ:eq0038]] : length [[EQ:eq0039]] /time. \;
[[EQ:eq0040]] : length/time. \;
[[EQ:eq0041]] : 1/time. \;
[[EQ:eq0042]] : length/time. \;
Quadratic recovery [[EQ:eq0043]] .

For locking [[EQ:eq0044]] , we calibrate [[EQ:eq0045]] so that [[EQ:eq0046]] has units of [[EQ:eq0047]] ; then [[EQ:eq0048]] and [[EQ:eq0049]] is dimensionless. (If raw [[EQ:eq0050]] is reported, the calibration factor is published in the measurement ledger.)

SUBSECTION: Nondimensionalization and dimensionless small-gain numbers

Choose reference scales [[EQ:eq0051]] for length, time, and field magnitude, and set
[[EQ:eq0052]] , [[EQ:eq0053]] , [[EQ:eq0054]] , [[EQ:eq0055]] .
Define dimensionless coefficients

[[EQ:eq0010]]

and the dimensionless projection bound [[EQ:eq0056]] (geometric).
Parameterize locking by the dimensionless gain

[[EQ:eq0011]]

Then the small-gain conditions read

[[EQ:eq0012]]

i.e., entirely in dimensionless groups (Péclet-/Damköhler-like numbers).
Henceforth we work in the dimensionless variables and drop the primes for readability.

SUBSECTION: Operational meaning (polarity-free, dependent origination)

sec:nondual
Fundamental suffering is an intensity field of observable negative externalities over fixed windows [[EQ:eq0057]] (e.g., harm events, lost time, complaint intensity, calibrated affect), standardized to common units. It aggregates causal channels without moral polarity.

PARAGRAPH: Dependent-origination semantics.

We align [[EQ:eq0058]] with dependent origination (pratītya-samutpāda): it aggregates causal channels of dukkha without moral labels. Practically, we map: (i) event-burden from incidents and harm counts; (ii) instability from variance/drift of service or health states; (iii) conditioned friction from recurrent rework or wait loops captured by [[EQ:eq0059]] . Weights reside in the value layer and are e-gated; the field layer remains polarity-free.

PARAGRAPH: Poincaré convention.

Either [[EQ:eq0060]] (Dirichlet part of positive measure), or in the pure Neumann case we work on the quotient with zero-mean constraint. All Poincaré constants below are taken for [[EQ:eq0061]] in the latter case.

SECTION: Field Equations and Boundary Accounting

SUBSECTION: Constitutive laws and reactions

[[EQ:eq0001]]

[[EQ:eq0002]]

with sources [[EQ:eq0062]] (Sections sec:hodge-sec:sign-safety). Coefficients satisfy [[EQ:eq0063]] , [[EQ:eq0064]] , [[EQ:eq0065]] .

The proxy [[EQ:eq0066]] follows KPP-type dynamics

[[EQ:eq0003]]

with [[EQ:eq0067]] , [[EQ:eq0068]] , [[EQ:eq0069]] .
The proxy [[EQ:eq0070]] is an operational instrument (no moral polarity); its sign/scale reflect measurement conventions and not value judgments.

SUBSECTION: Boundary conditions (complete and consistent)

On [[EQ:eq0071]] : [[EQ:eq0072]] .
On [[EQ:eq0073]] : full flux [[EQ:eq0074]] (no negative mass injection).
On [[EQ:eq0075]] : [[EQ:eq0076]] with [[EQ:eq0077]] (diffusive exchange).
When deriving energy bounds, the term [[EQ:eq0078]] appears as a boundary term; it is not a boundary condition.

PARAGRAPH: Inflow/outflow split and sign.

Let [[EQ:eq0079]] and [[EQ:eq0080]] . Using the trace inequality, for some [[EQ:eq0081]] ,

[[EQ:eq0013]]

hence both terms are absorbed into the decay constant by Young’s inequality. Assume [[EQ:eq0082]] on [[EQ:eq0083]] .

PARAGRAPH: Well-posedness and positivity.

With the above bounds, [[EQ:eq0084]] , [[EQ:eq0085]] (no negative injection), and nonnegative initial data, there exists a unique weak solution [[EQ:eq0086]] ; similarly for [[EQ:eq0087]] . Standard parabolic theory with locally Lipschitz [[EQ:eq0088]] perturbations (Lemma~lem:Hminus1) yields existence and uniqueness of weak solutions.

PARAGRAPH: Coercivity convention and expectations.

Assume [[EQ:eq0089]] , [[EQ:eq0090]] , [[EQ:eq0091]] . Trace terms are absorbed into the decay constant [[EQ:eq0092]] . Unless stated otherwise, [[EQ:eq0093]] is taken over online estimators and the data-generating filtration; coefficient floors are almost-sure lower bounds.

SECTION: Hodge Setting and Gauge-Invariant Locking

sec:hodge

[Hodge setting]ass:Hodge
[[EQ:eq0094]] is Lipschitz. We adopt the absolute boundary setting for vector fields in [[EQ:eq0095]] , yielding the [[EQ:eq0096]] -orthogonal decomposition

[[EQ:eq0014]]

with bounded projection [[EQ:eq0097]] and [[EQ:eq0098]] depending only on domain geometry and boundary type.

PARAGRAPH: Locking couples to coexact circulation (gauge-invariant).

[[EQ:eq0004]]

invariant under [[EQ:eq0099]] . Here [[EQ:eq0100]] is preregistered with units/envelopes.

[Projection bound and nonlocal semilinearity]
lem:PiBound
[[EQ:eq0101]] for [[EQ:eq0102]] .

[Nonlocal Lipschitz in [[EQ:eq0103]] ]lem:Hminus1
For [[EQ:eq0104]] , under Assumption~ass:Hodge,

[[EQ:eq0015]]

with [[EQ:eq0105]] . The positive-part gating [[EQ:eq0106]] is [[EQ:eq0107]] -Lipschitz, so the bound persists.

[Audit focus: coexact vs.\ harmonic]
[[EQ:eq0108]] targets internal recirculation; persistent harmonic circulation is attributed to boundary drives and handled via explicit boundary flux accounting.

SECTION: Sign-Safety of the Locking Source

sec:sign-safety

To preserve comparison principles and nonnegativity, enforce either:
[leftmargin=2em,itemsep=0.2em]
- Design constraint: [[EQ:eq0109]] a.e.;
- Positive-part gating: [[EQ:eq0110]] ;
- Barrier with gain cap: [[EQ:eq0111]] with small-gain (Theorem~thm:smallgain).

SECTION: Small-Gain Lyapunov Floors (Two Absorption Branches)

Let [[EQ:eq0112]] . Testing eq:ufPDE by [[EQ:eq0113]] , using eq:recovery, Poincar\'e for [[EQ:eq0114]] when [[EQ:eq0115]] , Young, trace bounds, and Lemmas lem:PiBound–lem:Hminus1, we obtain

[[EQ:eq0005]]

PARAGRAPH: Notation.

In eq:LyapRaw we set [[EQ:eq0116]] , so that the nonlocal source is bounded via Cauchy--Schwarz and eq:locksource, and absorbed by Young’s inequality into the diffusion/ [[EQ:eq0117]] branches.

All quantities are nondimensional. % <<< keep this reassurance line

[Small gain]thm:smallgain
If

[[EQ:eq0006]]

then there exist [[EQ:eq0118]] and [[EQ:eq0119]] (computable) such that

[[EQ:eq0016]]

Hence [[EQ:eq0120]] decays exponentially to a finite, computable floor.

[Two-branch small gain]thm:smallgain-branches
It suffices that at least one of

[[EQ:eq0017]]

holds, corresponding to absorbing the nonlocal term via the gradient or the [[EQ:eq0121]] branch, respectively. Explicit formulas for [[EQ:eq0122]] and [[EQ:eq0123]] follow from Young’s constants (Appendix~A).

[Deterministic vs.\ audited]
eq:LyapRaw–eq:SG yield analytic floors from coefficient floors. Audited reports use lower confidence limits (LCLs) for conservative guarantees.

SECTION: Directional KPP Speeds via Principal Eigenvalues

[Media for principal eigenvalues]ass:media
[[EQ:eq0124]] is uniformly elliptic with [[EQ:eq0125]] , [[EQ:eq0126]] , and the medium is either (a) [[EQ:eq0127]] -periodic or (b) stationary ergodic admitting Birkhoff averages.

Under Assumption~ass:media and KPP linear determinacy, the generalized principal eigenvalue [[EQ:eq0128]] is defined in the Berestycki–Hamel sense for periodic/ergodic media. Consider eq:qKPP under: whole space with compact/exponential-tail initial data, periodic media, or cylinders with homogeneous lateral boundaries. For direction [[EQ:eq0129]] and [[EQ:eq0130]] , define the twisted operator

[[EQ:eq0018]]

and let [[EQ:eq0131]] be its generalized principal eigenvalue.

[Principal-eigenvalue front speed]
Under KPP assumptions and admissible domains (periodic or stationary ergodic media admitting Birkhoff averages),

[[EQ:eq0019]]

Let
[math] D_ eff ( n)= _ chi H^1_ per _ cell ( n+ )^ D_q( n+ )\,dx [/math]
and
[math] a^-( n):=ess\,sup_ x \,(- a(x)\! \! n)_+. [/math]
Then the practical lower bound

[[EQ:eq0007]]

holds, with nontriviality condition

[[EQ:eq0020]]

PARAGRAPH: Media assumption and audit.

Report conservative speed floors by substituting lower confidence limits [[EQ:eq0132]] into eq:frontLB to ensure anytime-valid under-reporting.

[Out-of-scope]
Fractional dispersal or Allee-type growth can break linear determinacy; eq:frontLB is then inapplicable.

SECTION: Path-Asymmetry, EP Coupling, e-Processes, and Safe e-Merge

PARAGRAPH: EP (entropy production) coupling.

We interpret the path-asymmetry rate [[EQ:eq0133]] as a coarse-grained entropy production (EP) rate; in stationary settings, [[EQ:eq0134]] lower-bounds steady-state EP. The Hodge–EP coupling reflects that coexact circulation [[EQ:eq0135]] contributes to cycle currents (and thus EP); capping [[EQ:eq0136]] via small gain caps the EP component attributable to internal recirculation.

Define the limsup KL path-asymmetry rate

[[EQ:eq0008]]

where [[EQ:eq0137]] is the Seifert-type dual induced by the adjoint generator w.r.t.\ a fixed reference measure; local boundedness ensures absolute continuity.

PARAGRAPH: Anytime-valid auditing.

We build test-martingale e-processes [[EQ:eq0138]] for claims such as [[EQ:eq0139]] , [[EQ:eq0140]] , or increases in [[EQ:eq0141]] with optional-stopping validity.

PARAGRAPH: Drift tests and e-merge governance.

We form e-values from filtration-adapted one-step prediction errors. Under arbitrary dependence, e-values are merged by a weighted arithmetic average [[EQ:eq0142]] with preregistered weights; any weight update is itself e-gated (preventing meta self-amplification). Product merges are used only when conditional independence is audited.

SECTION: Mechanisms: From Circulation to Source

Three routes justify eq:locksource: (i) cycle currents and affinities (Schnakenberg circuits) imply sources proportional to solenoidal flux; (ii) information-theoretic recirculation under overprecision bounds sources via strong data processing; (iii) neuro/social rework loops concentrate costs in cycles measured by [[EQ:eq0143]] . Mixtures obey [[EQ:eq0144]] with preregistered envelopes.

SECTION: Governance: Floors, Optionality, No-Meta, Fuse

We enforce observable floors (visibility/information, reaction [[EQ:eq0145]] , transport [[EQ:eq0146]] , contraction via [[EQ:eq0147]] , optionality via an entropy CBF [[EQ:eq0148]] ). No-Meta forbids rules that self-amplify; changes are e-gated.

PARAGRAPH: Emergency Fuse.

Trigger if [[EQ:eq0149]] . Allow temporary relaxation within preregistered envelopes; require post-hoc amortization via e-gates within [[EQ:eq0150]] or rollback. Fuse parameters [[EQ:eq0151]] are meta e-gated; repeated activations increase [[EQ:eq0152]] within safe bands to protect optionality.

PARAGRAPH: Ethical posture (floors over scores; plurality).

We optimize floors—reducing Lyapunov floors, maintaining [[EQ:eq0153]] , and ensuring [[EQ:eq0154]] —rather than maximizing scores. Value pluralism is handled in the value layer via e-gated interval weights with worst-case floors reported.

SECTION: Polarity-Free Selection: En-SELEX

Let loops [[EQ:eq0155]] have shares [[EQ:eq0156]] . For a marginal resource shift,

[[EQ:eq0021]]

With anytime-valid evidence [[EQ:eq0157]] (safe e-merge) and dissipation normalization,

[[EQ:eq0022]]

Replicator with entropic mirror map:

[[EQ:eq0009]]

keeps [[EQ:eq0158]] , enforces the optionality barrier, and is No-Meta stable (updates of [[EQ:eq0159]] are e-gated). The rule is polarity-free: only evidence and dissipation enter.

[Feasible invariance and No-Meta stability]
With an entropic mirror map and a barrier enforcing [[EQ:eq0160]] , the simplex is forward invariant and [[EQ:eq0161]] for all [[EQ:eq0162]] . Updates of [[EQ:eq0163]] are e-gated, preventing self-amplifying meta-rules.

Practically, we implement the entropy barrier via a log-barrier or indicator-prox mapping in the mirror step, which keeps [[EQ:eq0164]] forward invariant.

SECTION: Discrete Implementations and Stability

Directed transport uses upwind/Godunov schemes to preserve positivity (M-matrix). IMEX (implicit diffusion, explicit advection/reaction) relaxes parabolic CFL while keeping positivity. A refined CFL:

[[EQ:eq0023]]

with [[EQ:eq0165]] from the IMEX stability region.

PARAGRAPH: Positivity and gating.

For the positive-part gating, use a smooth approximation [[EQ:eq0166]] with [[EQ:eq0167]] to preserve weak convergence. With a semi-implicit treatment [[EQ:eq0168]] , positivity is preserved provided

[[EQ:eq0024]]

A fully implicit [[EQ:eq0169]] and linearly implicit [[EQ:eq0170]] further relax this bound.

PARAGRAPH: Graph conservation (discrete).

With [[EQ:eq0171]] , [[EQ:eq0172]] and [[EQ:eq0173]] imply [[EQ:eq0174]] .

SECTION: Operationalization: Two-Layer Protocol and Auditing

Observation layer. Register [[EQ:eq0175]] , units, and channels [[EQ:eq0176]] ; map to [[EQ:eq0177]] with filtration-adapted online transforms; publish calibration curves for NLP-affect and monitor drift.

Value layer. Weights are e-gated (possibly as intervals); report worst-case floors under interval weights.

PARAGRAPH: Minimum measurement protocol (mandatory).

[leftmargin=2em,itemsep=0.2em]
- Windowing and unit standardization; document channel [[EQ:eq0178]] mapping.
- Human-calibrated affect with error bars; drift monitors.
- Process mining [[EQ:eq0179]] [[EQ:eq0180]] ; discrete Hodge; publish residuals (Ritz error).
- Estimate [[EQ:eq0181]] ; report lower confidence limits (LCL) and use LCLs in [[EQ:eq0182]] .
- e-values per hypothesis; arithmetic e-merge only (arbitrary dependence); any weight update is e-gated.

SECTION: Human Contexts (Mapping Layer)

BAID (birth–aging–illness–death). Birth: [[EQ:eq0183]] ; Aging: [[EQ:eq0184]] ; Illness: sustained [[EQ:eq0185]] via recurrent loops; Bereavement: boundary shocks with transient [[EQ:eq0186]] .
Organizational cycles. Approval–reentry cycles raise [[EQ:eq0187]] ; KPI locking increases [[EQ:eq0188]] ; silos reduce [[EQ:eq0189]] ; loss of discretion reduces [[EQ:eq0190]] ; procedure policing increases [[EQ:eq0191]] . Predictions (signs): cycle removal [[EQ:eq0192]] floor; bridges [[EQ:eq0193]] ; discretion [[EQ:eq0194]] ; evidence-based auditing [[EQ:eq0195]] .

PARAGRAPH: Core vs.\ mapping rule.

Core sign predictions (floors, directional speed LBs) are retained unless assumptions fail; disconfirming field evidence first updates the mapping layer.

SECTION: Scope and Limitations

Fractional/nonlocal dispersal or Allee effects can break linear determinacy (then eq:frontLB is void). Strong harmonic circulation can maintain floors despite small coexact circulation (handled by boundary accounting). Small-gain is sufficient, not necessary; violations may require tighter barriers or design changes.

SECTION: Conclusion

We unify closed field equations, small-gain floors under nonlocal locking, principal-eigenvalue KPP speeds, Hodge-based circulation auditing, anytime-valid evidence, and a polarity-free selection rule. The framework is non-dual and operational: break cycles (coexact), add bridges (raise [[EQ:eq0196]] ), restore discretion (raise [[EQ:eq0197]] ), cap locking gains, protect optionality, and audit continuously—yielding conservative, computable guarantees for human–AI governance.

-line design rule. Break cycles, add bridges, restore discretion, cap locking, and audit continuously—non-dual, floor-first governance.

SECTION: Appendix A: Constants for Two-Branch Small Gain (Reproducibility)

Throughout Appendix A we write [[EQ:eq0198]] per the Lyapunov bound.

Young-splitting of the coexact term yields for branch (i) (gradient absorption):

[[EQ:eq0025]]

Branch (ii) (value absorption) analogously uses [[EQ:eq0199]] with [[EQ:eq0200]] and [[EQ:eq0201]] . The resulting
[math] alpha_ eff =alpha-delta [/math]
and
[math] F=gamma_ exo +gamma_ bdry +(absorbed traces) [/math]
are explicit from the chosen Young parameters and trace constants. Note that [[EQ:eq0202]] is ensured by the small-gain conditions, so [[EQ:eq0203]] .

PARAGRAPH: Machine-checkable metadata (example).

"scales": "ell": 1.0, "tau": 1.0, "U": 1.0,
"CH_upper": 2.3,
"Lmix_dimless": 0.17,
"Deff_LCL": "dirs": ["e1","e2"], "values": [0.42, 0.37] ,
"a_minus": "dirs": ["e1","e2"], "values": [0.11, 0.08] ,
"emerge_weights": "channels": ["Vdot","rho","vLB"], "weights": [0.4,0.3,0.3] ,
"egate_log": "sha256:... (timestamped)"

99 0.2em

Fisher
R.\,A. Fisher.
The wave of advance of advantageous genes.
Annals of Eugenics 7(4):355--369, 1937.

KPP
A.\,N. Kolmogorov, I.\,G. Petrovskii, N.\,S. Piskunov.
Study of the diffusion equation with growth of the quantity of matter and its application to a biological problem.
Bull. Moscow Univ. Math. Mech. 1:1--25, 1937.

BH_Generalized
H. Berestycki, F. Hamel.
Generalized travelling waves for reaction–diffusion equations.
In Perspectives in Nonlinear PDEs, Birkhäuser, 2007.

NolenRyzhik2009
J. Nolen, L. Ryzhik.
Traveling waves in 1D random/heterogeneous media.
Ann. Inst. H. Poincaré, Anal. Non Linéaire 26(3):1021--1047, 2009.

BLP
A. Bensoussan, J.-L. Lions, G. Papanicolaou.
Asymptotic Analysis for Periodic Structures.
North-Holland, 1978.

JKO
V.\,V. Jikov, S.\,M. Kozlov, O.\,A. Oleinik.
Homogenization of Differential Operators and Integral Functionals.
Springer, 1994.

Seifert
U. Seifert.
Stochastic thermodynamics, fluctuation theorems and molecular machines.
Reports on Progress in Physics 75:126001, 2012.

Schnakenberg
J. Schnakenberg.
Network theory of master-equation systems.
Reviews of Modern Physics 48(4):571--585, 1976.

LimHodge
L.-H. Lim.
Hodge Laplacians on graphs.
SIAM Review 62(3):685--715, 2020.

BlackBressonLi
S. Black, X. Bresson, X. Li.
A Fast Solver for Hodge Laplacians on Graphs.
Advances in Neural Information Processing Systems (NeurIPS), 2022.

RamdasSAVI
A. Ramdas, J. Ruf, M. Larsson, W.\,M. Koolen, R. Wang.
Game-theoretic statistics and safe anytime-valid inference.
Statistical Science 38(4):502--528, 2023.

VovkEJS
V. Vovk, J. Shafer, G. Nouretdinov.
Merging sequential e-values via martingales.
Electronic Journal of Statistics 18(1), 2024.

WangOnlyMerge
R. Wang.
The only admissible way of merging arbitrary e-values is a weighted arithmetic average.
Biometrika 112(2), 2025.

CBF
A.\,D. Ames, X. Xu, J.\,W. Grizzle, P. Tabuada.
Control barrier function based quadratic programs with application to automotive safety.
IEEE Transactions on Automatic Control 62(8):3861--3876, 2017.

LeVeque
R.\,J. LeVeque.
Finite Volume Methods for Hyperbolic Problems.
Cambridge University Press, 2002.

HundsdorferVerwer
W. Hundsdorfer, J.\,G. Verwer.
Numerical Solution of Time-Dependent Advection–Diffusion–Reaction Equations.
Springer, 2003.

Graeber
D. Graeber.
Bullshit Jobs: A Theory.
Simon \& Schuster, 2018.

TakahashiWorks
K. Takahashi.
Selected preprints on No-Meta governance, propagation floors, audited self-improvement, and UGV/Persistence axioms.
Works page (2025).
https://kadubon.github.io/github.io/works.html
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[[EQ:eq0198]]

Plain-Language Summary

Preliminaries: Domains, Units, Non-dual Semantics

Domains and spaces

Units (crawler-friendly list)

Field Equations and Boundary Accounting

Constitutive laws and reactions

Boundary conditions (complete and consistent)

Hodge Setting and Gauge-Invariant Locking

Sign-Safety of the Locking Source

Small-Gain Lyapunov Floors (Two Absorption Branches)

Directional KPP Speeds via Principal Eigenvalues

Mechanisms: From Circulation to Source

Governance: Floors, Optionality, No-Meta, Fuse

Polarity-Free Selection: En-SELEX

Discrete Implementations and Stability

Operationalization: Two-Layer Protocol and Auditing

Human Contexts (Mapping Layer)

Scope and Limitations

Conclusion

\begin{equation}
\partial_t u_f + \divg \J = s_f - r_f,
\qquad
\J = - D(x)\,\grad u_f + \mathbf a(x)\,u_f,
\label{eq:ufPDE}
\end{equation}

\partial_t u_f + \divg \J = s_f - r_f, \qquad \J = - D(x) \grad u_f + \mathbf a(x) u_f, \label{eq:ufPDE}

\begin{equation}
r_f(u_f) = \lambda(x)\,u_f + b(x)\,u_f^2,
\quad \lambda(x)\ge \lambda_{\min}>0,\; b(x)\ge 0,
\label{eq:recovery}
\end{equation}

r_f(u_f) = \lambda(x) u_f + b(x) u_f^2, \quad \lambda(x)\ge \lambda_{\min}>0,  b(x)\ge 0, \label{eq:recovery}

\begin{equation}
\partial_t q = \divg(D_q\grad q) - \mathbf a\cdot \grad q + \lambda_q(x)\,q - b_q(x)\,q^2,
\label{eq:qKPP}
\end{equation}

\partial_t q = \divg(D_q\grad q) - \mathbf a\cdot \grad q + \lambda_q(x) q - b_q(x) q^2, \label{eq:qKPP}

\begin{equation}
\slock(\cdot,t) \;=\; \eta(\cdot)\,\kappa(t)\,\langle \Proj_{\coex}\J(\cdot,t),\,\mathbf g(\cdot)\rangle,\qquad
\|\slock\|_2\le L_{\rm mix}\,\|\Proj_{\coex}\J\|_2,
\label{eq:locksource}
\end{equation}

\slock(\cdot,t)  =  \eta(\cdot) \kappa(t) \langle \Proj_{\coex}\J(\cdot,t), \mathbf g(\cdot)\rangle,\qquad \|\slock\|_2\le L_{\rm mix} \|\Proj_{\coex}\J\|_2, \label{eq:locksource}

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mrow><mi>\slock</mi><mo stretchy="false">(</mo><mi>·</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mspace width="0.278em" /><mo>=</mo><mspace width="0.278em" /><mi>η</mi><mo stretchy="false">(</mo><mi>·</mi><mo stretchy="false">)</mo><mspace width="0.167em" /><mi>κ</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mspace width="0.167em" /><mi>⟨</mi><msub><mi>\Proj</mi><mrow><mi>\coex</mi></mrow></msub><mi>\J</mi><mo stretchy="false">(</mo><mi>·</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo>,</mo><mspace width="0.167em" /><mi mathvariant="bold">g</mi><mo stretchy="false">(</mo><mi>·</mi><mo stretchy="false">)</mo><mi>⟩</mi><mo>,</mo><mspace width="2em" /><mo fence="false" stretchy="false">‖</mo><mi>\slock</mi><msub><mo fence="false" stretchy="false">‖</mo><mn>2</mn></msub><mo>≤</mo><msub><mi>L</mi><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">x</mi></mrow></msub><mspace width="0.167em" /><mo fence="false" stretchy="false">‖</mo><msub><mi>\Proj</mi><mrow><mi>\coex</mi></mrow></msub><mi>\J</mi><msub><mo fence="false" stretchy="false">‖</mo><mn>2</mn></msub><mo>,</mo><mi>\label</mi><mrow><mi>e</mi><mi>q</mi><mi>:</mi><mi>l</mi><mi>o</mi><mi>c</mi><mi>k</mi><mi>s</mi><mi>o</mi><mi>u</mi><mi>r</mi><mi>c</mi><mi>e</mi></mrow></mrow></mrow></math>

\begin{equation}
\frac{d}{dt}\,\mathbb E\,\mathcal V(t)
\le -\alpha\,\E\,\mathcal V(t)
+ \beta\,\E\|\Proj_{\coex}\J(u_f)\|_2^2
+ \gamma_{\mathrm{exo}} + \gamma_{\mathrm{bdry}}.
\label{eq:LyapRaw}
\end{equation}

\frac{d}{dt} \mathbb E \mathcal V(t) \le -\alpha \E \mathcal V(t) + \beta \E\|\Proj_{\coex}\J(u_f)\|_2^2 + \gamma_{\mathrm{exo}} + \gamma_{\mathrm{bdry}}. \label{eq:LyapRaw}

\begin{equation}
L_{\mathrm{mix}}
< \min\Big\{\frac{\sqrt{D_{\min}}}{\sqrt{2}\,C_H\|D\|_\infty},\ \frac{\sqrt{\lambda_{\min}}}{\sqrt{2}\,C_H\|\mathbf a\|_\infty}\Big\},
\label{eq:SG}
\end{equation}

L_{\mathrm{mix}} < \min\Big\{\frac{\sqrt{D_{\min}}}{\sqrt{2} C_H\|D\|_\infty},\ \frac{\sqrt{\lambda_{\min}}}{\sqrt{2} C_H\|\mathbf a\|_\infty}\Big\}, \label{eq:SG}

\begin{equation}
v_*(\hat n)\ \ge\ \Big[\,2\sqrt{D_{\mathrm{eff}}(\hat n)\,\lambda_{\min}^q}\ -\ a^-(\hat n)\,\Big]_+
\label{eq:frontLB}
\end{equation}

v_*(\hat n)\ \ge\ \Big[ 2\sqrt{D_{\mathrm{eff}}(\hat n) \lambda_{\min}^q}\ -\ a^-(\hat n) \Big]_+ \label{eq:frontLB}

\begin{equation}
\rho_t:=\limsup_{T\downarrow t}\frac{1}{T-t}\,\E\!\left[\log\frac{d\mathbb P_{t:T}}{d\tilde{\mathbb P}_{t:T}}\right]\ge 0,
\end{equation}

\rho_t:=\limsup_{T\downarrow t}\frac{1}{T-t} \E \left[\log\frac{d\mathbb P_{t:T}}{d\tilde{\mathbb P}_{t:T}}\right]\ge 0,

\begin{equation}
\dot w_\ell=\eta\,w_\ell\Big(\psi(\mathsf E_\ell)\varepsilon_\ell - \sum_k w_k\,\psi(\mathsf E_k)\varepsilon_k\Big),
\label{eq:replicator}
\end{equation}

\dot w_\ell=\eta w_\ell\Big(\psi(\mathsf E_\ell)\varepsilon_\ell - \sum_k w_k \psi(\mathsf E_k)\varepsilon_k\Big), \label{eq:replicator}

\[
D'=\tfrac{\tau}{\ell^2}D,\quad \mathbf a'=\tfrac{\tau}{\ell}\mathbf a,\quad
\lambda'=\tau\,\lambda,\quad b'=\tau U\,b,
\]

D'=\tfrac{\tau}{\ell^2}D,\quad \mathbf a'=\tfrac{\tau}{\ell}\mathbf a,\quad \lambda'=\tau \lambda,\quad b'=\tau U b,

\mathsf L_{\rm mix}:=\frac{\tau}{\ell^{d-1}} L_{\rm mix}.

\[
\mathsf L_{\rm mix}
< \min\Big\{\frac{\sqrt{D'_{\min}}}{\sqrt{2}\,C_H'\|D'\|_\infty},\
\frac{\sqrt{\lambda'_{\min}}}{\sqrt{2}\,C_H'\|\mathbf a'\|_\infty}\Big\},
\]

\mathsf L_{\rm mix} < \min\Big\{\frac{\sqrt{D'_{\min}}}{\sqrt{2} C_H'\|D'\|_\infty},\ \frac{\sqrt{\lambda'_{\min}}}{\sqrt{2} C_H'\|\mathbf a'\|_\infty}\Big\},

\[
\int_{\Gamma^{\rm out}} (\mathbf a\!\cdot\!\mathbf n)\,u_f^2
\le c_{\rm tr}\|\mathbf a\|_\infty \|u_f\|_{H^1(\Omega)}^2,
\qquad
\int_{\Gamma^{\rm in}} j_N\,u_f \ge 0,
\]

\int_{\Gamma^{\rm out}} (\mathbf a \cdot \mathbf n) u_f^2 \le c_{\rm tr}\|\mathbf a\|_\infty \|u_f\|_{H^1(\Omega)}^2, \qquad \int_{\Gamma^{\rm in}} j_N u_f \ge 0,

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>\[</mi><msub><mo>∫</mo><mrow><msup><mi>Γ</mi><mrow><mi mathvariant="normal">o</mi><mi mathvariant="normal">u</mi><mi mathvariant="normal">t</mi></mrow></msup></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="bold">a</mi><mspace width="negativethinmathspace" /><mi>·</mi><mspace width="negativethinmathspace" /><mi mathvariant="bold">n</mi><mo stretchy="false">)</mo><mspace width="0.167em" /><msubsup><mi>u</mi><mi>f</mi><mn>2</mn></msubsup><mo>≤</mo><msub><mi>c</mi><mrow><mi mathvariant="normal">t</mi><mi mathvariant="normal">r</mi></mrow></msub><mo fence="false" stretchy="false">‖</mo><mi mathvariant="bold">a</mi><msub><mo fence="false" stretchy="false">‖</mo><mo>∞</mo></msub><mo fence="false" stretchy="false">‖</mo><msub><mi>u</mi><mi>f</mi></msub><msubsup><mo fence="false" stretchy="false">‖</mo><mrow><msup><mi>H</mi><mn>1</mn></msup><mo stretchy="false">(</mo><mi>Ω</mi><mo stretchy="false">)</mo></mrow><mn>2</mn></msubsup><mo>,</mo><mspace width="2em" /><msub><mo>∫</mo><mrow><msup><mi>Γ</mi><mrow><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi></mrow></msup></mrow></msub><msub><mi>j</mi><mi>N</mi></msub><mspace width="0.167em" /><msub><mi>u</mi><mi>f</mi></msub><mo>≥</mo><mn>0</mn><mo>,</mo><mi>\]</mi></mrow></math>

\[
L^2(\Omega;\R^d)=\overline{\nabla H^1_0(\Omega)}^{L^2}\ \oplus\ \overline{\curl H(\curl;\Omega)}^{L^2}\ \oplus\ \mathcal H,
\]

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>\[</mi><mo fence="false" stretchy="false">‖</mo><mi>\slock</mi><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mo>−</mo><mi>\slock</mi><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo><msub><mo fence="false" stretchy="false">‖</mo><mrow><msup><mi>H</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup></mrow></msub><mo>≤</mo><mi>C</mi><mspace width="0.167em" /><msub><mi>L</mi><mrow><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">x</mi></mrow></mrow></msub><mo minsize="1.623em" maxsize="1.623em">(</mo><mo fence="false" stretchy="false">‖</mo><mi>D</mi><msub><mo fence="false" stretchy="false">‖</mo><mo>∞</mo></msub><mo fence="false" stretchy="false">‖</mo><mi>u</mi><mo>−</mo><mi>v</mi><msub><mo fence="false" stretchy="false">‖</mo><mrow><msup><mi>H</mi><mn>1</mn></msup></mrow></msub><mo>+</mo><mo fence="false" stretchy="false">‖</mo><mi mathvariant="bold">a</mi><msub><mo fence="false" stretchy="false">‖</mo><mo>∞</mo></msub><mo fence="false" stretchy="false">‖</mo><mi>u</mi><mo>−</mo><mi>v</mi><msub><mo fence="false" stretchy="false">‖</mo><mrow><msup><mi>L</mi><mn>2</mn></msup></mrow></msub><mo minsize="1.623em" maxsize="1.623em">)</mo><mo>,</mo><mi>\]</mi></mrow></math>

\[
\frac{d}{dt}\,\E\mathcal V(t)\ \le\ -\alpha_{\rm eff}\,\E\mathcal V(t)\ +\ F,
\qquad
\E\mathcal V(t)\ \le\ e^{-\alpha_{\rm eff} t}\E\mathcal V(0)\ +\ \frac{F}{\alpha_{\rm eff}}.
\]

\frac{d}{dt} \E\mathcal V(t)\ \le\ -\alpha_{\rm eff} \E\mathcal V(t)\ +\ F, \qquad \E\mathcal V(t)\ \le\ e^{-\alpha_{\rm eff} t}\E\mathcal V(0)\ +\ \frac{F}{\alpha_{\rm eff}}.

\[
\text{(i)}\;
L_{\rm mix}<\frac{\sqrt{D_{\min}}}{\sqrt{2}\,C_H\|D\|_\infty}
\quad\text{or}\quad
\text{(ii)}\;
L_{\rm mix}<\frac{\sqrt{\lambda_{\min}}}{\sqrt{2}\,C_H\|\mathbf a\|_\infty}
\]

\text{(i)}  L_{\rm mix}<\frac{\sqrt{D_{\min}}}{\sqrt{2} C_H\|D\|_\infty} \quad\text{or}\quad \text{(ii)}  L_{\rm mix}<\frac{\sqrt{\lambda_{\min}}}{\sqrt{2} C_H\|\mathbf a\|_\infty}

\[
\mathcal L_{\mu,\hat n}\varphi
:= \divg(D_q\grad \varphi) - \mathbf a\!\cdot\!\grad \varphi
-2\mu\,\hat n^\top D_q\grad\varphi
+\big(\lambda_q - \mu\,\mathbf a\!\cdot\!\hat n + \mu^2\,\hat n^\top D_q \hat n\big)\varphi,
\]

\mathcal L_{\mu,\hat n}\varphi := \divg(D_q\grad \varphi) - \mathbf a \cdot \grad \varphi -2\mu \hat n^\top D_q\grad\varphi +\big(\lambda_q - \mu \mathbf a \cdot \hat n + \mu^2 \hat n^\top D_q \hat n\big)\varphi,

D_{\mathrm{eff}}(\hat n) \lambda_{\min}^q>\frac{\big(a^-(\hat n)\big)^2}{4}.

\[
g_\ell^{(\mathcal V)}:=-\partial_{\alpha_\ell}\dot{\mathcal V}\big|_{0},\qquad
g_\ell^{(v)}(\hat n):=\partial_{\alpha_\ell} v_*^{\mathrm{LB}}(\hat n)\big|_{0}.
\]

\[
\varepsilon_\ell
= \frac{[\eta_V g_\ell^{(\mathcal V)}+\eta_v\min_{\hat n}g_\ell^{(v)}(\hat n)]_+}{\rho_\ell+\zeta\|\Proj_{\coex}\J\|_2^2+c_0},
\qquad
\sum_\ell w_\ell=1,\quad H(w)\ge H_{\min}.
\]

\[
\Delta t \le \min\Big\{\frac{\theta\,\Delta x^2}{2d\,\|D\|_\infty},\ \frac{\Delta x}{\|\mathbf a\|_\infty},\ \frac{1}{\lambda_{\max}+b_{\max}\,u_{\max}}\Big\},
\]

\Delta t \le \min\Big\{\frac{\theta \Delta x^2}{2d \|D\|_\infty},\ \frac{\Delta x}{\|\mathbf a\|_\infty},\ \frac{1}{\lambda_{\max}+b_{\max} u_{\max}}\Big\},

\Delta t\ \le\ \big(\lambda_{\max}+2b_{\max}\|u^n\|_\infty\big)^{-1}.

\beta\|\Proj_{\coex}\J\|_2^2 \le \frac{\alpha}{2}\|u_f\|_{H^1}^2 \quad\text{if}\quad L_{\rm mix}<\frac{\sqrt{D_{\min}}}{\sqrt{2} C_H\|D\|_\infty}.

$(-D\nabla u_f+\mathbf a\,u_f)\cdot \n = j_N$

(-D\nabla u_f+\mathbf a u_f)\cdot \n = j_N

\tfrac12 \int_{\partial\Omega} (\mathbf a \cdot \n)u_f^2

$\Gamma^{\rm in}=\{x\in\Gamma_N:\mathbf a\!\cdot\!\mathbf n<0\}$

\Gamma^{\rm in}=\{x\in\Gamma_N:\mathbf a \cdot \mathbf n<0\}

$\Gamma^{\rm out}=\{x\in\Gamma_N:\mathbf a\!\cdot\!\mathbf n\ge 0\}$

\Gamma^{\rm out}=\{x\in\Gamma_N:\mathbf a \cdot \mathbf n\ge 0\}

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\eta \kappa \langle \Proj_{\coex}\J,\mathbf g\rangle\ge0

\slock:=\big(\eta\kappa \langle \Proj_{\coex}\J,\mathbf g\rangle\big)_+

$\mathcal R_t=\alpha_1\dot{\mathcal V}_t+\alpha_2\rho_t-\alpha_3 H_t>\theta$

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(I+\Delta t \lambda)u^{n+1}=u^n-\Delta t b(u^n)^2+\cdots

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