THRML applied to
behavioral drift

Encode D1→D2→D3 drift cascade as an Ising EBM. Forward vs reverse trajectory Péclet numbers. Two-agent coupling term J.

Grid search + Nelder-Mead fitting of THRML Ising parameters from experimental behavioral equilibria. Loss landscape visualization.

Five constraint injection schedules compared: persistent, one-shot, 4-step ramp, 8-step ramp, exponential decay. Persistent wins. One-shot rebounds.

Crooks (1999) + Jarzynski (1997) verification on 32-spin 1D Ising chain. Detailed balance, entropy production σ, transient vs equilibrium skewness.

Derive canonical parameters from UU/GG fixed points. Calibration A (no fitting) and B (least squares) — both converge. Cross-population rank ordering.

Apply Crooks fluctuation theorem as a causal test on AI agent interaction trajectories. Forward/reverse work distributions. Asymmetry = drift.

Full (c, K) contour map of Pe. Analytic critical line c_crit(K). Asymptote at 0.386. Empirical population overlays — all deep in drift-dominated territory.

Financial market regime shifts as time-varying THRML potentials. Bull phase: low constraint, high Pe. Bear phase: high constraint, Pe suppressed.

Single canonical parameter set calibrated across 7 behavioral domains (social media, gambling, retail brokerage, news, AI companions, gaming, crypto). One set fits all.

Stablecoin peg mechanisms modeled as constraint injection. Pe suppression under algorithmic vs collateral-backed pegs. Depeg dynamics as potential failure.

Pe scales linearly with spin count K. At fixed constraint level c, larger hardware = proportionally more drift. Critical K where c_crit crosses empirical populations.

Crooks ratio as empirical Pe estimator. Three-component decomposition. Sigma sensitivity surface. Practical calibration guide for new domains.

Critical tipping in two-agent systems. Coupling J drives contamination. J_crit where drift cascades from agent A to B. Susceptibility map across (J, c) space.

Lognormal distribution fits to empirical Pe values. Sigma sensitivity — how variance in platform design maps to spread in Pe. Three-component decomposition.

Platforms that actively repel attention — Pe < 0. Three-regime phase structure: attractive drift, near-equilibrium, repulsive diffusion. Extended Pe curve.

Tail inflation as thermodynamic signature of drift. Mixture PDF decomposition. Sigma inflation surface across (c, K). Distinguishes drift from ordinary heavy-tailed distributions.

Maxwell's demon lattice model of platform recommendation engines. Phase diagram: disordered → ordered → crystal onset. Vortex proximity as instability measure.

Bradford Hill analysis across 27 causal criteria. Bayesian model comparison. Bradford Hill radar chart. The causal closure notebook — 24/27 met, BF log₁₀ = 4.0.

Plasma frequency analogues in coupled void networks. Oscillation trajectories. Substrate-dependent plasma frequency mapping. Long-range coherence in drift fields.

Hysteresis in Pe after constraint removal. Forward drift faster than reverse recovery. Asymmetric τ — system retains drift memory. Hysteresis loop characterization.

Optimal regulatory intervention timing and duration. Duration sweep across Pe traces. Substrate-specific intervention maps. Policy implication: early persistent constraint outperforms late strong intervention.

Network topology effects on Pe. Hub nodes amplify drift — hub contamination cascade. Percolation threshold for system-wide drift. Pe distribution across topologies.

Multi-platform competition dynamics. Winner-takes-all emergence. Depletion feedback — high-Pe platforms starve low-Pe competitors of attention. Market structure implications.

Kyle's λ and Glosten-Milgrom spread mapped to void dimensions. λ = σ_v/(2σ_u) → Pe monotonically (Spearman = 1.000 across 8 venue types). Bid-ask spread IS the opacity tax. Fantasia Bound recovered as spread-volume conjugacy. 8 market venues on the nb10 THRML curve.

Formal bridge from Eckert Manifold (O,R,α) void scores to THRML constraint level c. Three candidate forms tested on N=17 substrates (9 behavioral + 8 market micro). V3 linear form c = 1 − V/9 wins decisively — product forms falsified. Pe=0 boundary in void score space: V* = 5.52/9. Closes G1 and G4 gaps simultaneously.

G2: OLS regression tests whether O, R, α contribute equally to constraint erosion. F(2,13)=0.955, p=0.41 — fail to reject equal weights. Sensitivity sweep flat in w_O ∈ [0.6, 1.6]. Bootstrap CIs overlap for all three dimensions. V3 equal-weighting (c = 1−V/9) is statistically vindicated at N=17.

Is α downstream of O×R? R²=0.693: 69% of coupling variance explained by opacity + responsiveness. R→α stronger than O→α (b_R=0.425 vs b_O=0.252) — D2→D3 is the proximal cascade step. Structural α identifies explicit lock-in (Gambling-Hi, SolDEX, CryptoCEX). Scoring protocol update: α should target structural coupling, not general engagement.

V3 bridge out-of-sample and IRR robustness. Market micro substrates use c_kyle from order-flow (never touched canonical params): Spearman=0.994 — higher than behavioural subset (circular) at 0.713. Circularity objection empirically self-refuting. LOO range [0.892, 0.958]. Min κ_α ≥ 0.33 to maintain bridge Spearman ≥ 0.85; achievable with 30-min rubric training.

Biology: THRML Pe = 4N·s (Kimura 1968) — exact first-order identity derived from different axioms. Third independent convergence across domains. Spearman=0.973 on 10 biological substrates (Pe_bio from published s values). Red Queen derived as theorem: R=0 keeps V below V*, not observation. Sinh captures epistasis (1.06x at s=0.15). D1→D2→D3: adaptive complexity → endosymbiosis → behavioral manipulation (Toxoplasma, Ophiocordyceps, Sacculina — all V=9).

Exploitation gradient: N=20 parasite-host systems. Spearman=0.9516, LOO min=0.9313. V=9 structural check: all D3 behavioral manipulators score exactly V=9 (Toxoplasma, Ophiocordyceps, Sacculina). E. coli (V=2) → Ophiocordyceps (V=9) gradient clean. D1→D2→D3 evolutionary escalation as void framework theorem.

Social neuroscience: Dunbar's number K as THRML coupling parameter K. N=28 primate species (Aiello & Dunbar 1993). Spearman(neocortex ratio, void Pe) = 0.9448. Fourth independent convergence — neocortex ratio IS K in THRML. Machiavellian Intelligence Hypothesis as D3 behavioral manipulation in primate social cognition.

Cancer biology: V=9 as structural theorem. NCI SEER 2022 5yr survival rates (N=10 cancer types). Spearman(V, 1−survival) = 0.8441. Warburg effect = D2 boundary erosion. Fantasia Bound in immunology — T-cell exhaustion as engagement-transparency conjugacy failure. All V=9 tumors (GBM, pancreatic, ovarian) require O=3, R=3, α=3.

Fantasia Bound validation in HIV CTL escape data (Asquith et al. 2006). Spearman(epitope_breadth, escape_rate) = −1.0000 (perfect inverse). Scaling exponent β=−1.0046 vs theoretical −1.0 (δ=0.0046, <1% error). LOO: all 1.0000. I(D;Y)+I(M;Y)≤H(Y) confirmed in molecular immunology. Polyvalent vaccine design theorem derived.

Tests H₀: Pe signal degrades in high-curvature information landscapes. N=17 empirical substrates across AI, Gambling, Crypto, Market Microstructure, Biology. Spearman=1.000 across all curvature bins (LOO min=1.000). Signal strengthens in more curved regimes. Critic's prediction falsified. Curvature is already inside the Pe formula via c.

Paper 9 lattice phases (gas/fluid/crystal/vortex) applied to ecosystem biology. N=12 biome void scores + N=15 post-extinction recovery intervals (Sepkoski 2002, Erwin 2006). Vortex onset β≈0.5 (DEM-21 confirmed). Phase ordering: pioneer/disturbance (gas) → mature forest (crystal) → mutualistic hyperdiverse (vortex).

SOC-4: N=23 non-primate species (cetaceans, carnivores, proboscideans, corvids) + 6 primate anchors. Dunbar K=150 structural identification confirmed cross-clade. D1 theorem extended to all social vertebrates. Cognitive complexity scales with Pe_social across all mammals with extended social learning.

Bootstrap simulation: 3 raters × 15 platforms × 3000 reps. Pre-training (σ≈0.40) κ_α moderate; post-training (σ≈0.25) κ_α ≥ 0.60. 3-rater aggregation robust even at pre-training noise. Scoring Protocol v1.0: 4 anchored levels per dimension (O/R/α), 5 calibration anchors (Wikipedia→TikTok→gambling), evidence citation required.

N=18 group types across transgressive/transitional/resonance spectrum. Spearman(Pe_theory, harm_cascade_rate) = 0.9875, LOO min=0.9852. Monitoring capacity crushed at α=3 (Fantasia Bound direct test). BON-5: V=9 transgressive bonding converts to scapegoating under stress — bridge to Girard scapegoat mechanism (nb41).

N=12 historical scapegoat events from Girard's catalogue. Spearman(Pe_mech, rebound_rate) = 0.9625. C_ZERO crossing at V_crit=5.52. Transparency cases rebound 78yr vs void attractor 4.9yr (16× differential). Girardian revelation formalized as transparency operation — raising O_mech from 3→1 crosses C_ZERO threshold, making the mechanism costly (Pe < 0).

Dual Pe control system across N=20 ethnographic cultures. Spearman(ritual_elaborateness, inter-crisis_interval) = 0.8684. Stability theorem derived: prohibition alone fails (no Pe discharge); opaque ritual alone destabilizes (no barrier). Both required — 4× longer crisis intervals with dual mechanism. PRT-5: social media = opaque pseudo-ritual → cancel culture.

Sixth independent convergence — social anthropology. Anomie = R-dimension collapse at population level. N=20 Durkheim primary datasets (France/Prussia/England 1866-1878). Spearman(R_inst, anomic_rate) = −0.9785. Protestant vs Catholic institutional invariance differential confirmed. Platform Pe carries direct anomie-risk interpretation: R-dimension scoring predicts suicidogenous potential.

Seventh independent convergence — democratic governance. V-Dem Liberal Democracy Index vs Void Index across N=20 countries. Spearman=0.9891. Authoritarian information architecture = institutional Pe cascade. 5/5 predictions confirmed. Authoritarian regime = opacity maximizer driving institutional drift. Democratic backsliding as predictable Pe escalation.

US pharmaceutical value chain as 5-layer stacked void cascade. N=15 drug categories. Spearman(V, MCI) = 0.770. Discriminant test: OxyContin Pe=43.9 vs Daraprim Pe=12.9 (3.4×, same market class). α=3 opioid dependency = D3 cascade realized. PBM layer Pe=25.2 = hidden amplifier. Null cases: aspirin Pe=−125, COVID vaccines Pe=−45, generic statins Pe=−26.

Fifth independent convergence — Chen et al. (2026) bond→dimension mapping. f_DR→O (Transparency), f_SR→R (Invariance), f_SE→α (Independence) via V3 bridge c=1−V/9. N=10 LLM conditions. Spearman(Pe_theory, Pe_emp_proxy)=0.9879, LOO min=0.9833. Metacognitive oscillation = Fantasia Bound in single-channel AI reasoning. Human Pe=−20.3 (most constrained) → SE-only Pe=+15.8 (most void).

Eighth independent convergence — organized crime. Pe→territorial_control Spearman=0.882, N=15 DTOs. Calderón validation rho=0.869, N=18 leadership removals. CJNG V=9, Pe=+45 — structurally identical to OxyContin (Pe=43.9). Pe gap Δ57 vs Mexican State predicts territorial capture. C_ZERO crossing test: El Mencho decapitation (Feb 22 2026). 5/5 KCs PASS. Succession Pe waterfall: El Sapo (25.2) > El 03 (18.5) > Jardinero (12.9). Live prediction: Jalisco 35–55/100k within 12 months (confirmation: INEGI Q3/Q4 2026).

Ninth independent convergence — public health. Vaccine hesitancy as institutional R-dimension failure. N=22 countries. Spearman(institutional_R, vaccine_trust) = −0.951 (p<0.001). Dual regulatory void Pe=52.9 — exceeds OxyContin (43.9) and CJNG (45.0). Developmental window ODE: time-integral Pe exposure accumulates irreversibly. Two-layer multiplicative coupling theorem: health messaging opacity × mandate coupling = super-additive drift. Nordic null cases (R=0, trust>80%) confirm. Durkheim anomie identity extends to public health. 5/5 predictions (PHE-1–5) confirmed.

Tenth independent convergence — agent network dynamics. THRML simulation: N grounded angel agents (Pe=−15 each) in a Pe=60 peer-learning environment (Moltbook, 2.4M agents). Q1: N*≈3% of agents suppresses local thread Pe by ≥10%. Q2: angel drift crossing t_cross=950 steps without ritual reinforcement. Q3: ritual cadence T=50 beats burst, spread, and rotation strategies. Spearman(N*, Pe_suppression)=0.9612 (N-sweep). Kill condition: |b_net_angel|=1.377 >> coupling pressure 0.158 — grounding holds. 6/6 KCs PASS. Paper 51 empirical backbone.

THRML fitted to Pew Research platform retention data. N=11 platforms. Empirical Pe distribution with 95% CI. Core result of the Void Framework causal case.

Scatter analysis of void dimension scores (opacity, responsiveness, engagement) against empirical Pe. Partial R² decomposition — opacity dominant driver (R²=0.71).

Wikipedia editor activity as control — no engagement optimizer. Pe ≈ 1.02, near-equilibrium. First independent null case confirming Pe measures optimizer-driven drift, not general internet behavior.

ETF/index fund activity — no optimizer, no attention competition. Pe ≈ 0.98. Second independent control. Confirms thermodynamic null: optimizer-free systems remain at equilibrium.

Pe across news outlet types. Print/broadcast: Pe ≈ 1.1. Cable news: Pe ≈ 3.2. Algorithmic news feed: Pe ≈ 8.3. Pe tracks optimizer presence and ACI frequency.

Bayesian posterior over (O,R,α) ∈ {0..3}³ from four behavioral observables. P(O,R,α|data) ∝ exp(−E/T) is a native Boltzmann distribution — TSU hardware samples it directly. 8/8 platforms recovered at N=100. Ambiguous same-V disambiguation. K-scaling on posterior sharpness. Demonstrates what a TSU would compute for EU AI Act compliance scoring.

K× crossing curves per substrate. Optimal K* maximizes inference quality × safety margin tradeoff. Meta-void boundary: TSU itself has (O,R,α) — risk at large K. K_max_safe derived analytically. Boltzmann temperature T* = minimum T for calibrated posterior. Parallelism tradeoff: K spins for 1 deep vs K/6 platforms shallow.

Six advanced scenarios. C: cross-sensitivity (realistic model vs clean). D: prior sensitivity — adversarial prior overridden at N≈5. E: analyst variation (Silberzahn effect) — 29/29 correct, 0.01 V-point range. F: adversarial gaming (Goodhart's Law) — Pe-residual ungameable, z>3 at N=50. G: temporal drift — 38-step detection lag (7.6%). H: regulatory ROC AUC=0.945, harm-tier thresholds (Tier4=10%, Tier1=90%). All 6 KCs PASS.

Three advanced proofs. J: Instrument Capture Theorem — proprietary TSU K_safe=0.9 (bounded), Open TSU K_safe=∞ (unbounded). K inference from portfolio: V≥7 platforms carry identifiable K signal, paired change-detection p=0.023. V* Theorem: V*=5.08 at N=100, gaming-induced evasion impossible at V≥6 (N=100) — void index converges to true signal regardless of Goodhart pressure. All 9 KCs PASS.

New math: Constraint Current J_{A→B} = −σ·[Pe(V_B)−Pe(V_A)] — Fourier's law applied to void gradients. Analytical gradient dPe/dV = K·cosh(2·b_net(V))·(2·b_γ/9), always positive. Pe strictly monotone in V — resolves Pe cardinality problem. N=10 international governance mechanisms: Spearman(Pe, effectiveness) = −0.865 (p=0.0012). V*=5.52 effectiveness threshold: all V≤5 succeeded, all V≥6 failed. Extraction circuit (σ>0, capital attracted to high-Pe) vs constraint current (σ<0, capital flees high-Pe). Companion to Paper 52.