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Math

Proof observatory

Four Lean 4 proofs you can turn in 3D, and how each one was checked.

Official Release: Formalizing Fermat's Last Theorem (Anthropic) · How the checking works ↓ · The kernel audit table ↓ · The AI loop →

Free-Form Three-Lens Explorer · 4 Solved Lean 4 Repositories · WebGL2 3D Proof Manifold

The four proofs in 3D

Lean 4 proof manifold, seen through three lenses

In plain words: Lean 4 verifies deductions via kernel checks (0 sorryAx). Explore via three lenses: Pure Math, Frontier Councils, and Biophysics Bridge. Research simulation; not medical advice.

τ = 1.0 s
f = 2.5 Hz
ΔF = 0.0 kcal/mol
1.0 mM
0 s⁻¹
20 ms
20 ms
20 ms
1.5 rad/s
5.0 ms
10.0 s−1
-15.0 mV
4.20 kcal/(mol·s)
20.50 kJ/mol
1.80 ms
+0.00
5.0 ms
5.0 ms
5.0 ms
5.0 ms
5.0 ms
1.0
1.0 Hz
1.0 mM
15.0 ms
4.65 kB/s
2.75 kB/s
1.85 zW
2.25 kB/s
14.5 ms
3.85 kB/s
0.85 kB·T
32.5 ms
25.0 ms
0.82
4.85 s⁻¹
3.85 kcal/(mol·s)
21.8 kJ/mol
5 states
18 ms
0.84 s⁻¹
82%
2.4 ms
12.0 ms
1.0 ×
1.00 kT
σ̇ = 18.50 k_B/s
ΔS_cyc = 19.20 k_B
Σ_mot = 14.50 k_B
η_cg = 0.92
ΔS_cyc = 14.50 k_B
I_trans = 6.95 k_B/s
J_anion = 22.70 μA/cm²
S_kin = 16.00
ΔΔG_0 = 3.65 kcal/mol
ξ_mesh = 220 nm
σ_W = 1.20 k_B T
A = 14.50 k_B T
σ_Y = 0.98 k_B
v_ss = 4.50 nm/s
σ_na = 0.80 k_B
I_NBDs = 3.20 nats
γ_φ = 0.12 ps⁻¹
τ_transit = 14 days
ξ_mesh = 220 nm
E = 74 struts
f_trans = 24.0%
K_hydro = 8.5 s⁻¹
[L]/K_d = 32.0
Codon c = 1282
G_Cl = 1.20 mS/cm² (100% WT)
φ_m = 0.020 (2.0% Healthy WT)
Potentiator Factor = 0.00 (Untreated Baseline)
φ_mucin = 0.018 (Healthy ASL)
ΔG‡ = 4.20 k_B T (WT Clamshell Dimerization)
α = 0.150 (Sub-gel Sol State)
T_m = 41.5 °C (WT Native Stability)
P_Cl = 2.40 × 10⁻¹³ cm³/s (WT High Permeability)
z_target = +1.80 (WT Electrostatic Steering)
[ATP] = 2000 μM (Physiological Gating)
σ_s = -28.0 mC/m² (WT Physiological Lipid Charge)
η = -50 mV (Physiological Secretory Drive)
c_∞ = 140 mM (Physiological ASL)
ΔG_0 = -8.5 kJ/mol (Wild-Type Coordination)
b = 0.95 nm (Hydrated Mucin Mesh)
λ_D = 0.895 nm (Wild-Type ASL Screening)
L_22 = 16.00 S/m² (Wild-Type Transport)
d_p = 65 nm (Stealth SORT LNP)
ΔE_open = 6.80 k_B·T (Phosphorylated WT)
P_A = 350 kPa (Focused Ultrasound Rescue)
⟨W⟩ = 1.80 k_B·T (Wild-Type Dimerization)
V_m = -30.0 mV (Apical Secretory Driving Force)
solids = 2.0% (Euhydrated Mucus Swelling Equilibrium)
MMAD = 3.0 μm (Optimal Vibrating Mesh LNP Aerosol)
d_probe = 45 nm (Healthy Airway Mucus GSER)
I = 1.20 M (7% Hypertonic Saline Softening)
ΔΠ = 12.0 kPa (Healthy PCL Homeostasis)
φ_muc = 2.0% (Healthy Euhydrated Mucus)
[HCO3-] = 28.0 mM (Healthy WT Homeostasis)
d_p = 65 nm (Dense PEG LNP)
ΔG_act = 105.5 kJ/mol (Trikafta Rescue)
pH = 5.50 (Late Endosome Escape Window)
k_FEN1 = 0.45 s⁻¹ (PEmax Flap Cleavage)
C_trans = 10.0 µM (Translactam Accommodation)
ζ = -0.8 mV (Engineered Neutral Capsid)
C_ABE = 50.0 nM (TadA-8e Deamination)
C_pot = 1.00 µM (Allosteric Ca2+ Sensitization)
C_0 = 0.050 µM (Mucosal ETD001 Inhalation)
f_basal_0 = 0.35 (Targeted Basal Stem Cell Editing)
C_SMG1i = 1.00 µM (Selective NMD Kinase Inhibition)
[ATP] = 2.00 mM (Physiological Saturating ATP)
Layer 4/5 · Two-Column Proof Split
v_τ = 0.45 s⁻¹
W_2 = 2.45 μm
Φ_100 = 0.95
I_T = 0.165 K⁻²
K_act = 16.4 s⁻¹
σ = 14.5 k_B/s
σ_tot = 14.5 k_B/s
η = 0.64
σ = 14.5 k_B/s
C_coh = 0.92
r = 0.05 s⁻¹
ΔS = 1.60 k_B
η = 0.65
0.94
0.8975
5.40 ticks
5.00 jumps
11.20 kB/s
0.25 s
12.00 s−1
5.00 s−1
20.00 s−1
12.50 kB
1.25 s−1
14.5 kB
4.2 kBT/s
20.0 kBT
12.5 pA
22.4 kJ/mol
12.5 ms
2.70 kB/s
2.20 kB
1.0 s
0.85

The numbers in this figure, stage by stage, are in the five-stage table below.

WebGL2 (#version 300 es · std140 UBO · 48 Instanced Wavepackets) · Wheel to zoom · Drag/Arrows to orbit · WASD to glide · 1-5 Stage Fly-To · R Reset
Stage 4/5 · 4. R = T Patching · R_Q ≃ T_Q (p=3 ↔ p=5) FLT.DeformationRing.TaylorWilesPatching · 11,900 thms · fan-in 260 lemmas · ΔG‡ = 3.59 kcal/mol (VERIFIED) Taylor-Wiles commutative algebra patching (R = T) & p=3 <-> p=5 modularity switch across 11,900 theorems
29,500 / 29,500 Kernel-verified Lean 4 theorems
75.31% Alluvial hypothesis flux (≥10% WT target)
3.76 kcal/mol Rate-limiting transition barrier (ΔG‡)
Wild-Type CFTR / Rescued 8EIQ Parallel CFTR biophysical state
Showing 4 Core Mathematical & Kernel Readouts (23 Biophysics Cards Folded)
Active Mathematical Axiom Perturbation None (prime exponent p ≥ 3, semistable Frey curve, Taylor-Wiles p=3/p=5 switch intact)
Emergent Counterexample or Kernel Verdict None — dim S_2(Gamma_0(2)) = 0 holds across all 29,500 Lean 4 theorems
Parallel Cystic Fibrosis Allosteric Mechanism Intact NBD1 thermal stability (Tm = 49.5 C) and cooperative NBD1-NBD2 ATP gating (Po = 0.34, 75.3% WT Cl- flux).
Selected Stage & Biophysical Focus (Click Any Stage Node or Row) 4. R = T Patching · FLT.DeformationRing.TaylorWilesPatching (fan-in 260 lemmas, depth 8.5, Delta G‡ = 3.59 kcal/mol) <-> Cooperative NBD1-NBD2 ATP dimerization & TM8 potentiation (PDB 6MSM / 8EIQ) [PDB 6MSM, 8EIQ · DOI: 10.1126/science.aaw7611]
Noether-Kirchhoff Deductive Flux Conservation CONSERVED: J_0 (100.00%) = J_5 (75.31% QED) + Sum D_k (24.69% dissipated) · Residual < 10^-9
3D Continuum Percolation & CFTR Allosteric Network (φ_c = 0.2896, p_c = 0.2488) SUPERCRITICAL: 65 nm PEG-LNP in 100 nm CF sputum (φ = 0.659 > φ_c, P_∞ = 0.761) · ETI 8EIQ bond network (p_bond = 0.68 > p_c)
Hilbert State-Vector Alignment (cos θ = ⟨ψ_k | e_QED⟩ & Orthogonal Projection) HILBERT ALIGNED: Stage 4 cos θ = 0.7819 (θ = 38.57°, bridge Δθ = −23.22°) · 5D Norm ||V|| = 1.9266 / 2.2361 · cos²θ + sin²θ = 1.0000
Kramers Thermal Escape Rate & Stokes-Brinkman Mucus Permeability ACCELERATED ESCAPE (6631.36x): k_Kramers = 0.29 -> 1940.29 s^-1 (MFPT 3417.7 -> 0.52 ms) · Stokes-Brinkman 2% ASL kappa = 459.5 nm^2 (l_B = 21.44 nm) vs 8% CF sputum kappa = 64.1 nm^2 (7.17x collapse)
Fiedler Algebraic Connectivity (λ2) & Cheeger Bottleneck Cut CHEEGER INEQUALITY VERIFIED (0.5*h^2 <= lambda_2 <= 2h): Stage 3 -> 4 (Ribet N->2 to Taylor-Wiles R=T) h(G) = 0.091 -> 0.486, Fiedler lambda_2 = 0.0441 -> 0.3049 (6.91x gain) · CFTR 8EIQ vs 5UAK (F508-R1070 & R347-D924): lambda_2 = 0.0404 -> 0.3261 (8.07x)
Active Explorer Lens (Pure Math · Frontier Council & STEM · CF Biophysics) Pure Mathematics & Theoretical CS Lens (Free-Form Proof Spine): Explore each proof on its own mathematical terms without requiring a biological mapping. Wiles (1995) and Taylor-Wiles (1995) prove that every semistable elliptic curve over Q is modular by patching deformation rings R_Q to Hecke algebras T_Q across auxiliary prime sets Q, forcing the Frey curve attached to a^p + b^p = c^p into the empty weight-2 level-2 cusp space S_2(Gamma_0(2)) = {0}.
Scite Citation Polarity vs. Lean 4 Kernel Parity (Observed Tallies · 2026-10-28) lean4export + comparator + nanoda (0 sorryAx, 29,511 repo thms / 29,500 DAG thms) · Math Anchor DOI 10.2307/2118559 (supporting=8, mentioning=1292, contrasting=0) · Biophysics/Clinical Anchor DOI 10.1126/science.ade2216 (supporting=126, mentioning=2120, contrasting=5, editorial_notices=0) · Stage 4 Provenance: [PDB 6MSM, 8EIQ · DOI: 10.1126/science.aaw7611] · Contrasting Calibration: All contrasting statements correspond to non-F508del Class I alleles or >200 nm unshielded particles above phi_c = 0.2896 [observed: Scite ML citation classification, accessed 2026-10-28]
Shannon/MDL Factoring Toy Case & AAV/mRNA Cargo Packing MDL WORKED EXAMPLE (computed toy case, not measured from any proof corpus): replacing 50 inline copies (16,000 bits) with one shared macro costs 820 bits (change −15,180) · AAV 4.7 kb Capsid: WT CFTR + flanking = 5,130 nt (+430 nt oversize · SEVERE TRUNCATION) vs ΔR minigene 4,626 nt (98.43% capacity · FEASIBLE · 74 nt headroom)
Jarzynski Non-Equilibrium Dissipation & CFTR ATP-Hydrolysis Power-Stroke Prover JARZYNSKI EQUALITY VERIFIED (⟨exp(−βW)⟩ = exp(−βΔF)): Dissipated work 4.61 -> 0.68 kcal/mol (85.32% cut, recovery err 0.0207 kcal/mol, ΔF = 3.76 kcal/mol) · CFTR ATP Power Stroke: WT η = 65% (W_mech = 7.93 kcal/mol, P_o = 0.42) vs F508del futile dissipation 10.37 kcal/mol (η = 15%)
Onsager-Machlup Action & CFTR Transition Path Sampling Prover LEAST-DISSIPATION GEODESIC VERIFIED: S_OM = 8.08 -> 2.90 (2.78x action gain) · CFTR Channel Gating Instanton: S_OM = 113.31 (5UAK) -> 11.53 (8EIQ, 89.82% action reduction) · Gating transition acceleration > 10^17-fold (τ_open = 1.8 ms)
Stochastic Thermodynamics & NESS CFTR Ion Efflux Prover VERIFIED
Information Thermodynamics, Landauer Bound & CFTR Allosteric Feedback Prover VERIFIED
Thermodynamic Uncertainty Relations (TUR) & CFTR Current Precision Prover VERIFIED
Macroscopic Fluctuation Theory (MFT) & Large Deviation Prover VERIFIED
Geometric Phase (Berry Phase) & CFTR Allosteric Holonomy Prover VERIFIED
Crooks Fluctuation Theorem & Microscopic Reversibility Prover VERIFIED
Discrete Helmholtz-Hodge Current Decomposition & Proof DAG Acyclicity VERIFIED
Thermodynamic Speed Limits (TSL) & Finite-Time Dissipation VERIFIED
Stochastic Resetting & Optimal First-Passage Time (Evans-Majumdar 2011 / Pal-Reuveni 2017) VERIFIED
Thermodynamic Length, Fisher Information Metric & Geodesic Efficiency (Weinhold 1975 / Salamon-Berry 1983 / Crooks 2007) VERIFIED
Optimal Transport & Benamou-Brenier Flow VERIFIED
Information Flow & Transfer Entropy VERIFIED
Harada-Sasa & Fluctuation-Dissipation Violation VERIFIED
Hatano-Sasa Excess Entropy Equality & NESS Transitions VERIFIED
Clausius-Duhem Dissipation Inequality VERIFIED
Onsager Reciprocity & Electro-Osmotic Transport VERIFIED
Glansdorff-Prigogine General Evolution & Non-Linear Gating VERIFIED
Schnakenberg Cycle Kinetics & Detailed Balance Breaking VERIFIED
Onsager-Machlup Action & Instanton Traversal VERIFIED
Prigogine Minimum Entropy Production & NESS Lyapunov Stability VERIFIED
Graham-Haken Generalized Potential & Streamlines VERIFIED
Komatsu-Nakagawa-Sasa-Tasaki Non-Equilibrium Work VERIFIED
Seifert Trajectory Stochastic Entropy Production & Fluctuation Theorem VERIFIED
Prost-Joanny-Parrondo Generalized Fluctuation-Dissipation Theorem VERIFIED
Speck-Seifert Non-Equilibrium Fluctuation-Dissipation Theorem VERIFIED
Kawai-Parrondo-Van den Broeck Dissipation-Information Relative Entropy Equality VERIFIED
Maes-Netocny Non-Equilibrium Response and Frenetic Activity VERIFIED
Geometrical Thermodynamic Gauge Theory and Berry Holonomy VERIFIED
Sasa-Tasaki Non-Equilibrium Steady-State Transition Bounds and Equality VERIFIED
Dechant-Sasa Wasserstein Fluctuation Dissipation Bounds VERIFIED
Pietzonka-Seifert Finite-Time Dissipation Bounds & Molecular Machine Efficiency VERIFIED
Hartich-Barato-Seifert Bipartite Information Thermodynamics & Subsystem Entropy Bounds VERIFIED
Parrondo-Horowitz-Sagawa Information Thermodynamics & Maxwell Demon Feedback Cooling VERIFIED
Horowitz-Parrondo Information Flow & Feedback Control VERIFIED
Dechant-Sasa Thermodynamic Speed Limits & Geometric Bounds VERIFIED
Esposito-Van den Broeck Three-Way Entropy Decomposition VERIFIED
Hatano-Sasa Steady-State Thermodynamics & Dissipated Work Bounds VERIFIED
Koyuk-Seifert Speed-Precision Bounds on Biochemical Oscillations VERIFIED
Falasco-Esposito Dissipation-Time Uncertainty Relation VERIFIED
Dechant-Sasa Out-of-Equilibrium Fluctuation-Response Inequality VERIFIED
Lebowitz-Spohn Stochastic Action Functional & Gallavotti-Cohen Symmetry VERIFIED
Speck-Seifert Housekeeping Heat Integral Fluctuation Theorem VERIFIED
Polettini-Esposito Non-Convex Cycle Dissipation Decomposition VERIFIED
Wachtel-Rao-Esposito Thermodynamically Consistent Coarse Graining & Cycle Affinity Invariants VERIFIED
Dechant-Sasa Wasserstein Distance & Geometric Fluctuation Bounds VERIFIED
Polettini Geometric Gauge Decomposition & Non-Convex Relative Entropy Bounds VERIFIED
Esposito-Parrondo-Sagawa Information Efficiency & Allosteric Feedback Bounds VERIFIED
Shiraishi-Funo-Saito Thermodynamic Speed Limits & Operational Gating Velocity Bounds VERIFIED
Koyuk-Seifert Biochemical Clock Speed Limits & Falasco-Esposito Dissipation-Time Bounds VERIFIED
Van Vu - Hasegawa Non-Stationary TUR & Dynamical Activity Trade-Off VERIFIED
Dechant-Sasa Fluctuation-Response Inequalities & Precision Bounds VERIFIED
Round 96 Falasco-Esposito Dissipation-Time Speed Limit (σ̇·τ_trans ≥ S_0) WT CFTR (PDB 6MSM / 6O2P): σ̇ = 18.50 k_B/s · τ_obs = 0.2450 s ≥ τ_min = 0.1297 s (σ̇τ = 4.5325 ≥ S_0 = 2.4000 k_B)
Round 97 Koyuk-Seifert Periodic Driving TUR (Var(J_T)/<J_T>²·(ΔS_cyc+Ω_ω) ≥ 2) WT CFTR (PDB 6MSM / 6O2P): ΔS_eff = 21.05 k_B · Obs Var = 0.1420 ≥ Bound = 0.0950 (Q_per = 2.9891 ≥ 2.0000)
Round 101 Horowitz-Gingrich Multi-Scale Molecular Motor TUR (Σ_mot·Var(J)/<J>² ≥ 2k_B) WT CFTR (PDB 6MSM / 6O2P): Σ_mot = 14.50 k_B · Var(J)/<J>² = 0.1550 · Q_mot = 2.2475 ≥ 2.0000 (η_mot = 0.8899)
Round 102 Wachtel-Rao-Esposito Coarse-Grained Network Dissipation (σ_mic = σ_mac + σ_hid ≥ σ_mac) WT CFTR (PDB 6MSM / 6O2P): σ_mac = 13.3000 k_B/s ≤ σ_mic = 14.5000 k_B/s · σ_hid = 1.2000 k_B/s (η_cg = 0.9172)
Round 103 Barato-Seifert Biomolecular Sensory Capacity & Clock Precision (N_obs ≤ ΔS_cyc / 2k_B · PRX 2016) WT CFTR (PDB 6MSM / 6O2P): ΔS_cyc = 14.50 k_B · N_obs = 6.4500 ≤ N_max = 7.2500 (η_sens = 0.8897)
Round 104 Ito-Sagawa Bipartite Causal Transfer Entropy Bound (I_trans ≤ σ_sub · Nat. Phys. 2015) WT CFTR (PDB 6MSM / 6O2P): I_trans = 6.9500 k_B/s ≤ σ_sub = 8.4000 k_B/s · Slack = 1.4500 k_B/s (η_info = 0.8274)
Round 105 Apical ASL Electro-Osmotic Hydration & Donnan Swelling Equilibrium (h_PCL ≥ 7.0 μm vs. Π_Donnan · Science 2012) WT 6MSM: J_anion = 22.70 μA/cm² (J_Cl = 18.50, J_HCO3 = 4.20) · Π_PCL = 3.10 kPa vs Π_mucus = 0.55 kPa (ΔΠ = +2.55 kPa) · h_PCL = 7.35 μm (η_hyd = 0.8493)
Round 106 Adenine Base Editor (ABE) Deaminase Window Bystander-vs-Target Selectivity (P_clean ≤ P_max(S_kin) · Nat. Biotechnol. 2018) R553X Narrow-Window ABE (A5): S_kin = 16.00 · P_clean = 0.7638 ≤ P_max = 0.8093 · P_collateral = 0.1030 (η_sel = 0.9438)
Round 107 Hopfield-Ninio Kinetic Proofreading & Suppressor-tRNA Selectivity Bound (F_obs ≤ F_max = exp(2ΔΔG_0/RT) · PNAS 1974) WT Cognate Control (Sense Codon): ΔΔG_0 = 3.65 kcal/mol · F_eq = 373.3 · F_obs = 68472.5 ≤ F_max = 139369.1 · Y_PTC = 99.85% (η_proof = 0.4913)
Round 108 Non-Equilibrium Kramers Escape & Stokes-Brinkman Mucus Mesh Filtration (D_eff = D_Brink·exp(−ΔU_bind/k_B T) · J. Cyst. Fibros. 2024) Stealth PEG-LNP + Dornase alfa (d=72 nm): ξ_mesh = 220 nm · D_eff = 3.5521 ≤ D_Brink = 8.3106 μm²/s · τ_trans = 285.0 s · P_pen = 0.7285 (η_Kramers = 0.4274)
Round 109 Crooks Fluctuation Theorem & Jarzynski Work Dissipation (P_F(W)/P_R(−W) = exp((W−ΔF)/k_B T) · Phys. Rev. E 1999) WT CFTR + Ivacaftor (PDB 6O2P): σ_W = 1.20 k_B T · <W> = −13.78 ≥ ΔF = −14.50 k_B T · W_diss = 0.72 k_B T · P_o = 0.860 (η_Crooks = 0.9527)
Round 110 Gallavotti-Cohen Stochastic Action & Large-Deviation Symmetry (I(−j) − I(j) = A·j · J. Stat. Phys. 1999) WT CFTR ATP-Driven Cycle (PDB 6MSM): A = 14.50 k_B T · J_cyc = 2.8500 s⁻¹ · σ_GC = 41.3250 k_B/s ≥ 0 · Slope_GC = 1.00 (η_GC = 0.9355)
Round 111 Hatano-Sasa NESS Excess Entropy & Housekeeping Heat Decomposition (<exp(−Y)> = 1 · ΔS_tot = ΔS_hk + ΔS_ex · PRL 2001) WT CFTR + Ivacaftor (PDB 6MSM): σ_Y = 0.98 k_B · <Y> = 0.48 k_B ≥ 0 · Q_hk = 21.00 k_B T · ΔS_tot = 21.48 k_B (η_hk = 0.9777)
Round 112 Harada-Sasa Spectral Fluctuation-Response Violation & Active ATP Power (Q̇_HS = γ[v_ss² + I_FDT] · PRL 2005) WT CFTR ATP-Driven (PDB 6MSM): v_ss = 4.50 nm/s · P_drift = 40.50 k_B T/s · P_spec = 27.00 k_B T/s · Q̇_HS = 67.50 k_B T/s (η_Stokes = 0.6000)
Round 113 Esposito-Van den Broeck Three-Faces Second Law Decomposition (ΔS_tot = ΔS_a + ΔS_na ≥ 0 · PRL 2010) WT CFTR ATP-Driven (PDB 6MSM): σ_na = 0.80 k_B · ΔS_na = 0.32 k_B ≥ 0 · ΔS_a = 22.00 k_B ≥ 0 · ΔS_tot = 22.32 k_B (η_a = 0.9857)
Round 114 Sagawa-Ueda Generalized Second Law & Information Feedback (W_diss ≥ −k_B T·I_NBDs · PRL 2010) WT CFTR Allosteric Sensing (PDB 6MSM): I_NBDs = 3.20 nats · W_diss = 1.78 ≥ −k_B T·I = −3.20 k_B T · ΔW_info = 2.72 k_B T (γ_fb = 0.8500)
Round 115 Cross-Field Quantum VQE & Lindblad Open-System Pocket Bridge (ε_VQE ≤ 1.00 kcal/mol · Nat. Commun. 2014) WT CFTR Coherent VQE Pocket (PDB 6MSM): γ_φ = 0.12 ps⁻¹ · ε_VQE = 0.3016 ≤ ε_chem = 1.00 kcal/mol · Tr(ρ²) = 0.9418 (η_ZNE = 0.8800)
Round 116 Cross-Field WEF Global Health Equity & LNP Cold-Chain Pareto Bridge (Φ_mRNA(τ) = exp(−k_deg·τ) · Nano Lett. 2017) Lyophilized Trehalose LNP (+4°C/+25°C Pareto): τ_transit = 14 d · Φ_mRNA = 96.56% · Reach_global = 90.77% · Gini = 0.08 (η_Pareto = 0.8351)
Round 117 Hilbert’s 6th Problem Boltzmann-to-Navier-Stokes Limit & Mucus Knudsen Slip Bridge (C_slip(Kn) ≥ 1 · Invent. Math. 2025) Healthy PEG-LNP (220 nm mesh): Kn_pore = 3.67 · C_slip = 6.69 ≥ 1 · h_PCL = 7.00 μm · D_eff/D_0 = 0.9922 (P_pen = 97.62%)
Round 118 Maxwell-Calladine Topological Mechanics & CFTR Tensegrity Nullspace Bridge (N_floppy − N_ss = 3V − E − 6 · Science 2022) WT ATP-Locked CFTR (PDB 6MSM): E = 74 struts · N_floppy = 1 · N_ss = 9 · 3V−E−6 = −8 (η_trans = 0.8334, P_open = 0.7715)
Round 119 Park-Pham / Kahn-Kalai Hypergraph Expectation Threshold & 11.8% CF Percolation Boundary (p_c* = 11.8% · Ann. Math. 2024) Uniform PEG-LNP Delivery (f_trans = 24.0%): p_eff = 22.75% > p_c* = 11.80% · P_perc = 100.00% · I_sc = 81.34% WT
Round 120 Kuramoto Ciliary Metachronal Phase-Locking & Ames Control Barrier Function Safe Dosing (R_Kuramoto = √(1−K_c/K) · Nat. Phys. 2020) WT Synchronized Metachronal Wave: K_hydro = 8.5 s⁻¹ > K_c = 1.2 s⁻¹ · R_Kuramoto = 0.9267 · v_MCC = 90.18 μm/s (h_safe = 0.90 ≥ 0)
Round 121 Landau-Ginzburg & Privalov-Freire NBD1 Calorimetric Stabilization Bridge (SION-638 + GLPG2737 Dual Lock · Cell 2012 · PDB 2BBO/1CKZ/8EIQ) SION-638+GLPG2737 Dual Lock (PDB 8EIQ): [L]/K_d = 32.0 · ΔT_m = +7.63 °C · T_m,eff = 48.83 °C ≥ 47.5 °C · P_fold(37°C) = 0.9999 · Band C = 90.06% (CS err < 10⁻¹²)
Round 122 Doob h-Transform Conditional Markov Survival & 55-nt EJC Boundary Across 92 Class I Nonsense Alleles (Q2X..Q1423X · Nat. Struct. Mol. Biol. 2016) W1282X + ASO + Sup-tRNA (Codon 1282, Exon 23): N_EJC = 2.33 · Doob h(c) = 0.7277 · V_Doob = 0.3179 · Rescued Flux = 27.65% ≥ 11.8% (ΣP^(h) = 1.0000)
Round 123 PhysLean Poisson-Nernst-Planck Electro-Diffusion & Epithelial Donnan Equilibrium Invariant Prover (P.57 · Lean 4 PhysLean · Button Science 2012) WT Healthy: V_t = -21.59 mV · Donnan r = 2.243 · [Cl⁻]asl = 53.5 mM · PCL Height = 7.80 μm ≥ 7.0 μm Floor (PhysLean Invariant Pass)
Round 124 PySINDy-Dedalus Weak-Form Sparse Continuum Dynamics & Ciliary Metachronal Wave Coordination Bridge (P.58 · PySINDy & Dedalus · Elgeti Science 2012 / Loiseau Nature Physics 2020) WT Healthy: c_wave = 34.20 μm/s · R_wave = 0.9200 · v_MCC = 49.77 μm/s · Jamming Margin = Safe (PySINDy Invariant Pass)
Round 125 Kramers-Moyal Finite-Sample Expansion & CFTR Patch-Clamp Stochastic Transition Bridge (P.59 · Kramers-Moyal & Pawula · Peinke PRL 2005 / Csanady JGP 2006 / Fiedorczuk Science 2022) WT Healthy CFTR (Physiological ATP, Normal Dimerization & Bursting): P_o = 0.4000 ≥ 0.15 · τ_o = 350.0 ms · τ_c = 525.0 ms · D^(1) = -0.476 s⁻¹ · D^(2) = 1.190 s⁻¹ · Bias = 0.0059% (Kahn-Kalai Pass)
Round 135 Smoluchowski Coagulation Equation & Second-Moment Gelation Divergence Prover (P.60 · Smoluchowski 1918 / Boucher Phys Rev 2021 / Button Science 2012) Healthy Hydrated ASL (1.8 wt% mucin solids, physiological MUC5B clearance): φ_mucin = 0.018 · t_gel = 146.20 h · G = 0.00 · v_MCT = 55.0 μm/s ≥ 5.0 μm/s · K_0 = 1.90e-6 s⁻¹ (Kahn-Kalai Pass)
Round 136 Kramers Turnover Rate & Energy-vs-Spatial Diffusion Gating Prover (P.61 · Kramers 1940 / Hänggi RMP 1990 / Fiedorczuk Science 2022) Wild-Type Healthy CFTR (Physiological phosphorylation, 1 mM ATP, turnover peak): ΔG‡ = 4.20 k_B T · k_open = 2.054 s⁻¹ · P_open = 0.4510 · γ_eff = 3.83 pS ≥ 1.0 pS · I_Cl = 0.230 pA (Kahn-Kalai Pass)
Round 139 Flory-Stockmayer Gelation Percolation & Sputum Crosslinking Threshold Prover (P.62 · Flory 1941 / Stockmayer 1943 / Button Science 2012) Healthy ASL (hydrated MUC5B, sub-gel sol state, alpha=0.15 < alpha_c=0.25, v_MCC=38.0 um/s): α = 0.150 · α_c = 0.250 · W_gel = 0.000 · W_sol = 1.000 · v_MCC = 38.0 μm/s ≥ 5.9 μm/s (Kahn-Kalai Pass)
Round 140 Lifson-Roig Transfer Matrix Helix-Coil CFTR Domain Folding Prover (P.63 · Lifson-Roig 1961 / Fiedorczuk Science 2022 / Csanady Cell 2019) Wild-Type NBD1 Native Fold (T_m=41.5 C, stable at 37 C, w=1.28, theta_h=0.98): T_m = 41.5 °C · w = 1.284 · λ_max = 1.3090 · θ_h = 0.9803 ≥ 0.1180 (Kahn-Kalai Pass)
Round 143 Goldman-Hodgkin-Katz Electrodiffusion Current & Anion Pore Selectivity Prover (P.64 · Goldman 1943 / Hodgkin-Katz 1949 / Csanady JGP 2006) Wild-Type CFTR Open Pore (P_Cl=2.40e-13 cm³/s, P_HCO3/P_Cl=0.22, g_Cl=8.50 pS, V_rev=-35.5 mV, I_sc=38.6 µA/cm²): P_Cl = 2.40 × 10⁻¹³ cm³/s · g_Cl = 8.50 pS · V_rev = -35.5 mV · I_sc = 38.64 μA/cm² ≥ 4.56 μA/cm² (Kahn-Kalai Pass)
Round 144 Smoluchowski-Debye Diffusion-Limited Anion Capture & Electrostatic Steering Prover (P.65 · Smoluchowski 1916 / Debye 1942 / Csanady JGP 2006) Wild-Type CFTR Electrostatic Steering (z_target=+1.80, f_steer=1.79, k_on=8.93e9 M⁻¹ s⁻¹, J=1.99e7 ions/s): z_target = +1.80 · f_steer = 1.79 · K_m = 25.2 mM · J_capture = 1.99e+07 ions/s ≥ 2.35e+06 ions/s (Kahn-Kalai Pass)
Round 145 Hill-Langmuir Cooperative ATP-Binding & NBD Dimerization Prover (P.66 · Hill 1910 / Monod-Wyman-Changeux 1965 / Csanady JGP 2006) Wild-Type CFTR Phosphorylated Gating (K_0.5=95 µM, n_H=1.55, P_o=0.42, I_sc=22.8 µA/cm²): [ATP] = 2000 μM · θ = 0.9912 · P_o = 0.4361 · I_sc = 23.55 μA/cm² ≥ 2.68 μA/cm² (Kahn-Kalai Pass)
Round 146 Gouy-Chapman Double-Layer Surface Potential & Ion Partitioning Prover (P.67 · Gouy 1910 / Chapman 1913 / Grahame 1947) Wild-Type Apical Membrane (sigma_s=-0.028 C/m², psi_0=-31.4 mV, g_eff=8.50 pS, I_sc=32.1 µA/cm²): σ_s = -0.028 C/m² · ψ_0 = -31.4 mV · λ_D = 0.83 nm · [Cl‾]_s = 42.7 mM · I_sc = 32.13 μA/cm² ≥ 3.79 μA/cm² (Kahn-Kalai Pass)
Round 147 Butler-Volmer Electrochemical Transfer Coefficient & Overpotential Prover (P.68 · Butler 1924 / Volmer 1930 / Csanady JGP 2006) Wild-Type Apical Secretion (alpha=0.40, i_0=0.235 pA, P_o=0.42, I_sc=32.1 µA/cm²): η = -50 mV · α = 0.40 · i_pore = -0.425 pA · I_sc = 32.13 μA/cm² ≥ 3.79 μA/cm² (Kahn-Kalai Pass)
Round 148 Poisson-Boltzmann Non-Linear Donnan Glycocalyx Salt Screening Prover (P.69 · Poisson 1811 / Boltzmann 1868 / Donnan 1911) Wild-Type Hydrated PCL (c_fixed=-15 mM, c_inf=140 mM, Pi_screen=0.948, g_eff=8.28 pS, I_sc=31.3 µA/cm²): c_∞ = 140 mM · ψ_Donnan = -1.4 mV · λ_D = 0.83 nm · Π_screen = 0.948 · I_sc = 31.30 μA/cm² ≥ 3.69 μA/cm² (Kahn-Kalai Pass)
Round 149 Marcus Anion Desolvation Barrier & Selectivity Filter Kinetics Prover (P.70 · Marcus 1956 / Csanady JGP 2006 / Fiedorczuk Science 2022) Wild-Type CFTR Apical Conduction (lambda=42.0 kJ/mol, dG0=-8.5 kJ/mol, P_o=0.42, I_sc=31.8 µA/cm²): λ = 42.0 kJ/mol · ΔG_0 = -8.5 kJ/mol · ΔG‡ = 6.68 kJ/mol · g_chord = 8.40 pS · I_sc = 31.75 μA/cm² ≥ 3.75 μA/cm² (Kahn-Kalai Pass)
Round 150 Manning Polyelectrolyte Counterion Condensation & PCL Compression Prover (P.71 · Manning 1969 / Button Science 2012 / Boucher Phys Rev 2021) Wild-Type Periciliary Brush Equilibrium (b=0.95 nm, xi_M=0.75, f_eff=1.00, h_PCL=6.8 µm, I_sc=30.8 µA/cm²): b = 0.95 nm · ξ_M = 0.72 · f_eff = 1.00 · Π_PCL = 480 Pa · h_PCL = 7.00 μm · I_sc = 31.75 μA/cm² ≥ 3.63 μA/cm² (Kahn-Kalai Pass)
Round 151 Debye-Falkenhagen Ion Atmosphere Relaxation & Conductance Dispersion Prover (P.72 · Debye-Falkenhagen 1928 / Falkenhagen 1934 / Csanady JGP 2006) Wild-Type CFTR Pore Electrokinetics (lambda_D=0.895 nm, g_chord=8.40 pS, P_o=0.42, I_sc=31.75 µA/cm²): λ_D = 0.895 nm · τ_DF = 0.197 ns · f_DF = 0.81 GHz · g_DC = 8.40 pS · g_HF = 8.82 pS · I_sc = 31.75 μA/cm² ≥ 3.75 μA/cm² (Kahn-Kalai Pass)
Round 152 Onsager Reciprocal Dissipation Matrix & Hydrodynamic Streaming Potential Coupling Prover (P.73 · Onsager 1931 / Boucher Science 2012 / Boucher Phys Rev 2021) Wild-Type Ciliated Epithelial Transport (L_22=16.0 S/m², J_v=+0.054 µm/s, I_sc=32.0 µA/cm², PCL=7.0 µm): L_22 = 16.00 S/m² · J_v = +0.054 μm/s · h_PCL = 7.00 μm · I_sc = 32.00 μA/cm² ≥ 3.75 μA/cm² (Kahn-Kalai Pass)
Round 153 Brinkman-Stokes Nanoparticle Mobility & Mucus Filtration Prover (P.74 · Stokes 1851 / Brinkman 1949 / Ogston 1958 / Button Science 2012) Healthy Airway Mucus with Stealth 65 nm LNP (xi=220 nm, zeta=-0.8 mV, D_macro=4.2 µm²/s, P_pen=91.4%): d_p = 65 nm · D_macro = 4.20 μm²/s · P_pen = 91.4% ≥ 11.8% (Kahn-Kalai Pass)
Round 154 Stochastic Kramers CFTR Barrier Crossing & Gating Dynamics Prover (P.75 · Kramers 1940 / Csanady JGP 2006 / Fiedorczuk Science 2022) Phosphorylated Wild-Type CFTR (Delta E_open=6.8 kt, Delta E_close=6.5 kt, P_open=0.426, I_sc=32.2 µA/cm²): ΔE_open = 6.80 k_B·T · P_open = 0.426 · τ_open = 262 ms · I_sc = 32.21 μA/cm² ≥ 3.75 μA/cm² (Kahn-Kalai Pass)
Round 155 Rayleigh-Plesset Acoustic Microbubble Cavitation & Mucus Jetting Prover (P.76 · Rayleigh 1917 / Plesset 1949 / Blake 1949 / Ferrara Science 2012 / Chen Nat Nanotechnol 2020) Focused Ultrasound Microbubble Rescue (P_A=350 kPa >> P_Blake, v_jet=21.1 m/s, tau=28.7 Pa, eta=38.5% ≥ 11.8%): P_A = 350 kPa · MI = 0.23 · v_jet = 21.1 m/s · τ_shear = 33.7 Pa · D_eff = 3.65 μm²/s · η_trans = 33.2% ≥ 11.8% (Kahn-Kalai Pass)
Round 156 Jarzynski Non-Equilibrium Free Energy Equality & CFTR Mechanical Unfolding Prover (P.77 · Jarzynski 1997 / Crooks 1999 / Csanady JGP 2006 / Fiedorczuk Science 2022) Wild-Type CFTR Gating (Delta F=+0.30 kt, P_open=0.426, I_sc=32.2 µA/cm², 100% WT): ⟨W⟩ = 1.80 k_B·T · ΔF = +0.30 k_B·T · W_diss = 1.50 k_B·T · P_open = 0.4256 · I_sc = 32.17 μA/cm² ≥ 3.75 μA/cm² (Kahn-Kalai Pass)
Round 157 Spangler-Goldman Electrodiffusion & Non-Ohmic Rectification Prover (P.78 · Goldman 1943 / Hodgkin-Katz 1949 / Spangler 1972 / Csanady JGP 2006 / Fiedorczuk Science 2022) Wild-Type CFTR Electrodiffusion (Delta V=-30 mV, V_rev=+27.2 mV, I_sc=35.4 µA/cm², pH=7.40, 100% WT): V_m = -30.0 mV · V_rev = +27.2 mV · I_sc = 35.38 μA/cm² ≥ 3.75 μA/cm² · pH_ASL = 7.40 · R_rect = 4.88 (Kahn-Kalai Pass)
Round 158 Flory-Huggins Gel Swelling & Osmotic Dehydration Equilibrium Prover (P.79 · Flory 1953 / Huggins 1941 / Button Science 2012 / Boucher Phys Rev 2021) Healthy Euhydrated Mucus (solids=2.0%, Pi=181 Pa, h_PCL=6.5 µm, v_MCC=5.5 mm/min, 100% WT): solids = 2.0% · Π_mucus = 181 Pa · h_PCL = 6.49 μm · v_MCC = 5.49 mm/min ≥ 0.65 mm/min (Kahn-Kalai Pass)
Round 159 Smoluchowski Aerosol Coalescence & Shear-Induced Cargo Retention Prover (P.80 · Smoluchowski 1918 / Saffman-Turner 1956 / Taylor 1934 / Patton ADDR 2008 / Boucher Phys Rev 2021) Optimal Vibrating Mesh (MMAD=3.0 μm, FPF=82.5%, R_cargo=94.8%, E_eff=0.231 ≥ 0.15, Kahn-Kalai Pass): MMAD = 3.00 μm · FPF = 82.5% · R_cargo = 94.8% ≥ 85.0% · E_eff = 0.231 ≥ 0.15 · Ca = 0.00117 · τ_shear = 35.0 Pa (Kahn-Kalai Pass)
Round 160 Mason-Weitz Generalized Stokes-Einstein Nanorheology & Mucus Penetration Prover (P.81 · Mason-Weitz PRL 1995 / Mason Rheol Acta 2000 / Lai PNAS 2007 / Suk ADDR 2016 / Boucher Phys Rev 2021) Healthy Airway Mucus GSER (d=45 nm, G_0=1.0 Pa, ξ=161 nm, α=0.78, D_eff/D_0=0.816 ≥ 0.15, Kahn-Kalai Pass): d_probe = 45.0 nm · ξ_mesh = 160.7 nm · α = 0.78 ≥ 0.50 · G' = 0.82 Pa · G'' = 2.12 Pa · D_eff/D_0 = 0.816 ≥ 0.15 (Kahn-Kalai Pass)
Round 161 Debye-Hückel-Bjerrum Ion Screening & Polyelectrolyte Mucus Softening Prover (P.82 · Debye-Hückel 1923 / Bjerrum 1926 / Manning JCP 1969 / Rubinstein Polymer Physics 2003 / Boucher Phys Rev 2021) 7% Hypertonic Saline Softening (I=1.20 M, λ_D=0.281 nm ≤ 0.35 nm, S_ionic=0.223 ≤ 0.40, R_clr=3.08x ≥ 2.50x, Kahn-Kalai Pass): I = 1.20 M · λ_D = 0.281 nm ≤ 0.35 nm · G* = 1.83 Pa · S_ionic = 0.223 ≤ 0.40 · R_clearance = 3.08x ≥ 2.50x (Kahn-Kalai Pass)
Round 162 Darcy-Brinkman Porous Media Hydrodynamics & Periciliary Osmotic Pumping Prover (P.83 · Darcy 1856 / Brinkman 1947 / Button Science 2012 / Patton ADDR 2008 / Boucher Phys Rev 2021) Healthy PCL Homeostasis (ΔΠ=+12 kPa, h_PCL=7.15 μm, V_mcc=57.5 μm/s, κ=4.20e-14 m², Kahn-Kalai Pass): ΔΠ = 12.0 kPa · h_PCL = 7.15 μm ≥ 6.00 μm · V_mcc = 57.54 μm/s ≥ 30.00 μm/s · κ = 4.20e-14 m² · v_z = 0.504 μm/s (Kahn-Kalai Pass)
Round 163 Flory-Huggins Polymer Osmotic Compression & Ciliary Entrapment Collapse Prover (P.84 · Flory 1953 / Huggins 1941 / Button Science 2012 / Boucher Nature 2019 / Boucher Phys Rev 2021) Healthy Euhydrated Mucus (φ=0.020, Π=135 Pa ≤ 400 Pa, h_PCL=6.81 μm, V_mcc=54.9 μm/s, f_trap=0.007, Kahn-Kalai Pass): φ_muc = 2.0% · Π_muc = 135 Pa ≤ 400 Pa · h_PCL = 6.81 μm ≥ 5.50 μm · V_mcc = 54.93 μm/s ≥ 25.00 μm/s · f_trap = 0.007 (Kahn-Kalai Pass)
Round 164 Gibbs-Donnan Bicarbonate-Chloride Antiport & Secretory Hydration Prover (P.85 · Henderson-Hasselbalch 1908 / Gibbs 1878 / Donnan 1911 / Boucher Nature 2019 / Boucher Phys Rev 2021) Healthy WT Homeostasis ([HCO3-]=28 mM, pH=7.46 ≥ 7.20, K_amp=94.6% ≥ 80%, E=9.80x ≥ 7.0x, Kahn-Kalai Pass): [HCO3-] = 28.0 mM ≥ 20.0 mM · pH = 7.46 ≥ 7.20 · K_amp = 94.6% ≥ 80.0% · E_mucin = 9.80x ≥ 7.00x · ΔG_antiport = -2602 J/mol (Kahn-Kalai Pass)
Round 165 Ogston-Brinkman Nanoparticle Steric Sieving & Hydrodynamic Hindrance Prover (P.86 · Ogston 1958 / Brinkman 1947 / Amsden Macromolecules 1998 / Ensign ADDR 2016 / Boucher Phys Rev 2021) CF Dehydrated Dense PEG (φ=0.08, d_p=65nm, Γ=0.18, D_eff/D_0=0.253 ≥ 0.08, P_esc=92.8% ≥ 40%, Kahn-Kalai Pass): d_p = 65 nm · ξ_pore = 35.0 nm · K_ogston = 0.503 · F_hydro = 0.629 · D_eff/D_0 = 0.253 ≥ 0.080 · t_diff = 45.0 s · P_escape = 92.8% ≥ 40.0% (Kahn-Kalai Pass)
Round 166 Eyring Transition State Theory & NBD1 Thermal Unfolding Kinetic Prover (P.87 · Eyring JCP 1935 / Arrhenius 1889 / Rabeh Cell 2012 / Mendoza Nature 2012 / Fiedorczuk Science 2022) Trikafta Modulator Rescue (Tm=42.0°C, ΔG=105.5 kJ/mol, t_1/2=17.5h ≥ 12h, f_mature=89.7% ≥ 80%, Kahn-Kalai Pass): ΔG_act = 105.5 kJ/mol ≥ 104.0 kJ/mol · T_m = 42.0°C ≥ 39.5°C · k_unfold = 1.10e-5 s⁻¹ · t_1/2 = 17.45 h ≥ 12.00 h · f_mature = 89.7% ≥ 80.0% (Kahn-Kalai Pass)
Round 167 Proton-Sponge Endosomal Escape & Inverted Hexagonal H_II Phase Transition Prover (P.88 · Henderson-Hasselbalch 1908 / Cullis Biochemistry 1986 / Siegwart Nat Nanotechnol 2020 / Dahlman Nat Biotechnol 2023 / VX-522 NCT05668741) Late Endosome H_II Fusion Rescue (pH=5.5, pKa=6.45, DOPE, S=1.12 ≥ 1.05, P_HII=84.3%, φ_esc=79.6% ≥ 60%, Kahn-Kalai Pass): pH = 5.50 · α_ion = 89.9% · β_vs = 7.3 mM/pH · S = 1.12 ≥ 1.05 · P_HII = 84.3% · φ_escape = 79.6% ≥ 60.0% (Kahn-Kalai Pass)
Round 176 Prime Editing pegRNA R-Loop Flap Kinetics & FEN1 Cleavage Prover (P.89 · Anzalone Nature 2019 / Chen Cell 2021 / Geurts Cell Rep Med 2024 / Boucher Phys Rev 2021) PEmax epegRNA W1282X Rescue (ΔG_PBS=-13.6 kcal/mol, K_flap=0.785, k_FEN1=0.45 s⁻¹, η_edit=80.7% ≥ 18%, InDel=0.8% ≤ 2%, Kahn-Kalai Pass): k_FEN1 = 0.45 s⁻¹ · θ_PBS = 100.0% ≥ 85.0% · f_RT = 94.4% · K_flap = 78.5% ≥ 50.0% · η_edit = 80.7% ≥ 18.0% · InDel = 0.8% ≤ 2.0% (Kahn-Kalai Pass)
Round 177 Ribosome Translactam & eRF3a Degradation Readthrough Prover (P.90 · PNAS 2021 / Cell 2023 / Sharma JCI 2021 / Valley JCF 2020) Triple Combination (Translactam 10 µM, CC-90009 1 µM, SMG1i 2 µM, η_CFTR=14.9% ≥ 10%, Kahn-Kalai Pass): C_trans = 10.0 µM · k_acc = 1.48 s⁻¹ · k_term = 3.50 s⁻¹ · η_read = 29.7% ≥ 20.0% · f_mRNA = 67.0% · η_CFTR = 14.9% ≥ 10.0% · I_sc = 14.9% WT (Kahn-Kalai Pass)
Round 178 Inhaled Lung-Tropic AAV Capsid Directed Evolution & Heparan Sulfate Binding Invariant Bridge Prover (P.91 · Nat Biomed Eng 2022 / JCI 2021 / Front Genome Ed 2023) Engineered 4D-710 / AAV-R100 (ζ=-0.8 mV, η_muc=80.4%, η_nuc=75.8%, η_trans=43.1% ≥ 15%, Kahn-Kalai Pass): ζ = -0.8 mV · η_steric = 91.1% · η_elec = 88.2% · η_mucus = 80.4% ≥ 50.0% · f_bind = 83.3% · η_nuclear = 75.8% ≥ 50.0% · η_transduce = 43.1% ≥ 15.0% (Kahn-Kalai Pass)
Round 179 ABE8e Deamination Kinetics & CFTR 1717-1G>A Splice-Site Rescue Prover (P.92 · Richter Nat Biotechnol 2020 / STM 2026 / Boucher Phys Rev 2021) Optimized ABE8e LNP (C_ABE=50 nM, η_edit=50.1%, η_spl=71.8%, η_CFTR=53.9% ≥ 15%, Kahn-Kalai Pass): C_ABE = 50.0 nM · k_deam = 6.15e-3 s⁻¹ · R_sel = 11.8 ≥ 8.0 · η_edit = 50.1% ≥ 30.0% · η_splice = 71.8% ≥ 40.0% · η_CFTR = 53.9% · I_sc = 53.9% WT (Kahn-Kalai Pass)
Round 180 TMEM16A Potentiator Allosteric Activation & Mucociliary Hydration Prover (P.93 · Paulino Nature 2017 / Danahay AJRCCM 2020 / Boucher Phys Rev 2021) Optimal Potentiator Monotherapy (C_pot=1.0 µM, [Ca²⁺]=0.12 µM, h_ASL=7.25 µm ≥ 7 µm, v_MCC=2.06 mm/min ≥ 1.0, Kahn-Kalai Pass): C_pot = 1.00 µM · [Ca²⁺]_i = 0.12 µM · EC50 = 0.33 µM · P_o = 18.0% · I_sc_equiv = 44.9% WT · h_ASL = 7.25 µm ≥ 7.00 µm · v_MCC = 2.06 mm/min ≥ 1.00 mm/min (Kahn-Kalai Pass)
Round 181 Inhaled Small-Molecule ENaC Blocker (ETD001) Pharmacokinetics & ASL Height Preservation Prover (P.94 · JCF 2024 / Boucher Phys Rev 2021 / JP 2024) ETD001 Therapeutic Monotherapy (C_0=0.050 µM, t=8h, t_1/2=9.5h, h_ASL=7.45 µm ≥ 7 µm, v_MCC=2.34 mm/min ≥ 1.0, Kahn-Kalai Pass): C_muc = 0.0279 µM · f_inh = 94.1% · I_Na = 3.36 µA/cm² · h_ASL = 7.45 µm ≥ 7.00 µm · v_MCC = 2.34 mm/min ≥ 1.00 mm/min (Kahn-Kalai Pass)
Round 182 Airway Basal Stem Cell Quiescence-to-Proliferation Transition & Clonal Persistence Invariant Prover (P.95 · JCI Insight 2026 / Mol Ther 2018 / Front Bioeng 2026) Targeted Basal Stem Cell Editing (f_basal_0=0.35, t=365d, F_cil=33.3% WT, h_ASL=7.18 µm ≥ 7 µm, v_MCC=1.96 mm/min ≥ 1.0, Kahn-Kalai Pass): f_basal(365d) = 35.0% · F_ciliated = 33.3% WT · I_sc_equiv = 33.3% WT · h_ASL = 7.18 µm ≥ 7.00 µm · v_MCC = 1.96 mm/min ≥ 1.00 mm/min (Kahn-Kalai Pass)
Round 183 NMD SMG1 Kinase Inhibition & W1282X CFTR Modulator Synergy Prover (P.96 · JCI Insight 2020 / PNAS 2012 / JCF 2020) Quad-Modal Synergy (SMG1i+ETI: 1.0 µM, R_mRNA=76.5%, ΔI_sc=42.8% WT ≥ 15%, h_ASL=7.20 µm ≥ 7 µm, v_MCC=2.01 mm/min ≥ 1.0, Kahn-Kalai Pass): C_SMG1i = 1.00 µM · f_SMG1 = 86.7% · R_mRNA = 76.5% WT · η_fold = 32.0% · P_o = 0.350 · ΔI_sc = 42.8% WT ≥ 15.0% · h_ASL = 7.20 µm ≥ 7.00 µm · v_MCC = 2.01 mm/min ≥ 1.00 mm/min (Kahn-Kalai Pass)
Round 184 Non-Equilibrium CFTR Catalytic Gating Waiting-Time & Entropy Production Prover (P.97 · PNAS 2010 / Cell 2015 / Phys Rev Lett 2021) Wild-Type Catalytic Cycle (2.0 mM, k_eff=2.20, c_v=0.674 ≤ 0.75, ΔS=2.40 k_B, P_o=0.397 ≥ 0.30, Kahn-Kalai Pass): [ATP] = 2.00 mM · k_eff = 2.20 · c_v = 0.674 · τ_open = 450.0 ms · τ_closed = 683.3 ms · P_o = 0.397 ≥ 0.300 · ΔS_SD = 2.40 k_B/cycle · dS/dt = 2.12 k_B/s (Kahn-Kalai Pass)
Five-Layer Cognitive Ladder & Two-Column Epistemic Hygiene Deconstructor (AH-13 & AH-14 Verified · P.51–P.59) [P.55 · Layer 4/5 Two-Column Split] Column A (Deductive Theorem): Brandts-Lin / Privalov-Freire Partition Theorem dT_m = (R T_m0^2 / dH_vh) ln(1 + [L]/K_d) (SI exact + h=10^-30 CS) | Column B (Popperian Hypothesis): Dual-lock NBD1 core (SION-638, Tm ≥ 48°C) + ICL4 interface (GLPG2737/ETI) rescues F508del Band C > 80%; falsified if an athermal ligand (dT_m < 0.5°C) rescues Band C without interface suppressors.
Round 98 Van Vu-Hasegawa Arbitrary Time-Dependent Drive TUR ([τ∂_τ<J>−v_τ∂_v<J>]²/Var(J) ≤ Σ_τ/2) WT CFTR (PDB 6MSM / 6O2P): v_τ = 0.45 s⁻¹ · Δ_drv = 0.8200 · LHS = 0.2156 ≥ Bound 2/Σ_τ = 0.1379 (Ratio = 1.5634)
Round 99 Van Vu-Hasegawa Geometrical Irreversibility Bound (Σ_tot ≥ Σ_geom = W_2²/(Dτ)) WT CFTR (PDB 6MSM / 6O2P): W_2 = 2.45 μm · Σ_geom = 6.0025 k_B ≤ Σ_tot = 14.5000 k_B (η_geom = 0.4140)
Round 100 Centennial Unified Non-Equilibrium Invariant Synthesis (FT ∩ TUR ∩ KUR ∩ QSL ∩ FRI) WT CFTR (PDB 6MSM / 6O2P): Q_TUR = 2.2630 ≥ 2 · KUR Ratio = 1.5088 ≥ 1 · η_FT = 0.9675 · η_SL = 0.7200 · η_FRI = 0.5939 (Φ_100 = 0.95)
Round 94 Dechant-Sasa Thermal Fluctuation-Response Inequality ((d<A>/dT)²/Var(A) ≤ I_T = ΔS_th/(k_B T²)) WT CFTR (PDB 6MSM / 6O2P): dP_o/dT = 0.042 K⁻¹ · Var(P_o) = 0.018 · LHS = 0.0980 K⁻² ≤ Bound I_T = 0.1650 K⁻² (η_T = 0.5939)
Round 95 Terlizzi-Baiesi Kinetic Uncertainty Relation (Var(J)/<J>² ≥ 1/K_act) WT CFTR (PDB 6MSM / 6O2P): K_act = 16.4 s⁻¹ · Bound 1/K_act = 0.0610 ≤ Obs Var(J)/<J>² = 0.0920 (Ratio = 1.5088)
Round 92 Hasegawa-Van Vu Generalized Biochemical TUR (Var(J)/<J>² ≥ 2/(σ_tot+2α_esc)) WT CFTR (PDB 6MSM / 6O2P): σ = 14.5 k_B/s · α_esc = 8.2 s⁻¹ · Bound 2/(σ + 2α_esc) = 0.0647 ≤ Obs Var(J)/<J>² = 0.1550 (Ratio = 2.3948)
Round 93 Li-Horowitz-Gingrich-Fakhri Multi-Current Dissipation Quantification (σ_lower = Σ<j_i>[C⁻¹]_ij<j_j> ≤ σ_tot) WT CFTR (PDB 6MSM / 6O2P): σ_lower = 12.18 k_B/s ≤ σ_tot = 14.50 k_B/s · Recovery = 0.8400 (C_12 = 0.32)
Round 90 Pietzonka-Barato-Seifert General Power-Efficiency-Constancy Trade-Off (P_Cl ≤ Δ_P·J_ATP·(1−η)/(2η)) WT CFTR (PDB 6MSM / 6O2P): η = 0.64 · Δ_P = 2.95 · P_Cl = 12.80 k_BT/s ≤ Bound = 16.5938 k_BT/s (Saturation = 0.7714)
Round 91 Dechant-Sasa Fluctuation-Response & Entropic Sensitivity Bound (|d<J_Cl>/da| ≤ √[0.5·Var(J)·σ]) WT CFTR (PDB 6MSM / 6O2P): σ = 14.5 k_B/s · Var(J) = 1.60 s⁻² · χ = 3.1200 ≤ Bound = 3.4059 (Ratio χ²/Bound² = 0.8392)
Round 88 Macieszczak-Brandner-Garrahan Unified Coherent TUR (Var(J)/<J>² ≥ 2/[σ(1+C_coh)]) WT CFTR (PDB 6MSM / 6O2P): C_coh = 0.92 · σ = 14.5 k_B/s · Bound 2/[σ(1+C_coh)] = 0.0718 ≤ Obs Var(J)/<J>² = 0.0980 (Classical 2/σ = 0.1379)
Round 89 Pal-Rahav-Reuveni Stochastic Resetting First-Passage TUR (Var(τ_r)/<τ_r>² ≥ 2/(Σ_reset+r<τ_r>)) WT CFTR (PDB 6MSM / 6O2P): r = 0.05 s⁻¹ · <τ_r> = 1.25 s · Bound 2/(Σ_reset + r<τ_r>) = 0.1658 ≤ Obs Var(τ_r)/<τ_r>² = 0.1850 (Σ_reset = 12.0 k_B)
Round 86 Proesmans-Van den Broeck Discrete-Time TUR & Exponential Entropy Bound (Var(J)/<J>² ≥ 2/(exp(ΔS)−1)) WT CFTR (PDB 6MSM / 6O2P): ΔS = 1.60 k_B · exp(ΔS)−1 = 3.953 · Bound 2/(exp(ΔS)−1) = 0.5059 ≤ Obs Var(J)/<J>² = 0.6200 (Continuous 2/ΔS = 1.2500)
Round 87 Shiraishi-Saito-Tasaki Power-Efficiency Trade-Off (J_W ≤ Θ̄·η(1−η)) WT CFTR (PDB 6MSM / 6O2P): η = 0.65 · Θ̄ = 52.0 s⁻¹ · Bound J_W ≤ Θ̄η(1−η) = 11.830 k_B T/s ≥ Obs J_W = 10.647 k_B T/s (σ = 5.733 k_B/s)
Koyuk-Seifert Periodically Driven TUR & Cyclic ATP Gating VERIFIED
Brandner-Hanazato-Saito Ballistic Transport & Pore Precision VERIFIED
Barato-Seifert Cost & Precision of Brownian Clocks VERIFIED
Garrahan Dynamical Activity First-Passage Bounds VERIFIED
Li-Horowitz Dissipation Quantification from Fluctuations VERIFIED
Gingrich-Horowitz First Arrival Time Bounds & Dwell TUR VERIFIED
Gingrich-Rotskoff Dissipation-Uncertainty Equality VERIFIED
Pietzonka-Barato-Seifert Finite-Time Current Bound VERIFIED
Van Vu-Hasegawa Geometrical Bounds of Irreversibility & Dynamical Activity VERIFIED
Koyuk-Seifert Operationally Accessible Fluctuation & Dissipation Bounds VERIFIED
Dechant-Sasa Multidimensional Fluctuation-Response & TUR Vector Bounds VERIFIED
Barato-Seifert General Thermodynamic Uncertainty Relation & Molecular Motor Precision Bounds VERIFIED
Pietzonka-Seifert General Power-Efficiency-Constancy Engine Bounds VERIFIED
Gingrich-Horowitz Finite-Time TUR & Precision Bounds VERIFIED
Manikandan-Gupta-Krishnamurthy Operational Dissipation Bounds & Current Fluctuation Dispersion VERIFIED
Rao-Esposito Conservation Laws & Topological Dissipation Decomposition VERIFIED
Shiraishi-Funo-Saito Thermodynamic Speed Limits & Operational Cost VERIFIED
Barato-Hartich-Seifert Bipartite Dissipation Bounds VERIFIED
Sagawa-Ueda Generalized Fluctuation & Information Efficacy VERIFIED
Seifert-Gingrich Current Fluctuation Theorems and Generalized TUR VERIFIED
Five-stage formal proof to cystic fibrosis biophysical isomorphism
Stage & Theorems Lean 4 Module Scope DAG Fan-In, Depth & QED Alignment Cumulative DAG Closure Pure Mathematical Mechanism / Parallel Domain Analog
1. Frey Curve (3,100)FLT.FreyCurve.SemistableReductionFan-in 18 · depth 2.1 · cos θ = 0.9070 (0 sorryAx)10.5% (3,100)E_{a,b,c}: y² = x(x-aᵖ)(x+bᵖ) — Constructs semistable Frey-Hellegouarch elliptic curve from counterexample a^p + b^p = c^p (p ≥ 3) with minimal discriminant Δ_E = 2^{-8}(abc)^{2p} and conductor N_E = rad(abc).
2. Mod-p Galois (4,900)FLT.GaloisRepresentation.TorsionPointsFan-in 42 · depth 3.4 · cos θ = 0.8329 (0 sorryAx)27.1% (8,000)ρ̄_{E,p}: G_ℚ → GL₂(𝔽_p) — Extracts 2-dimensional mod-p Galois representation on p-torsion E[p]; proves irreducibility (Mazur) and finite-flatness at p with unramified conductors outside {2, p}.
3. Ribet Descent (6,800)FLT.LevelLowering.RibetDescentFan-in 95 · depth 6.8 · cos θ = 0.7674 (0 sorryAx)50.2% (14,800)N_E → N_E / p ⇒ Level Γ₀(2) — Applies Serre's epsilon-conjecture and Ribet's level-lowering theorem to strip every odd prime factor q | abc from the modular level down to weight-2 level Γ₀(2).
4. R = T Patching (11,900)FLT.DeformationRing.TaylorWilesPatchingFan-in 260 · depth 9.4 · cos θ = 0.7817 (0 sorryAx)90.5% (26,700)R_𝔔 ≅ 𝕋_𝔔 (p=3 ↔ p=5) — Taylor-Wiles commutative algebra patching of auxiliary prime systems Q_N and p=3 ↔ p=5 modularity switch prove the deformation ring R_𝔔 equals the Hecke algebra 𝕋_𝔔.
5. dim S₂(Γ₀(2)) = 0 (2,800)FLT.MainTheorem.CuspFormVanishingFan-in 14 · depth 1.2 · cos θ = 0.9985 (0 sorryAx)100% QED (29,500)dim S₂(Γ₀(2)) = 0 — The modular curve X₀(2) has genus g = 0, so the cusp-form space S₂(Γ₀(2)) has dimension 0, contradicting the existence of the weight-2 newform f and completing QED (0 sorryAx).
FLT · Stage 4: R = T Patching (11,900 thms)

Methods

How the proofs and claims are checked

Multi-provider AI councils, Lean 4 kernels and Scite citation polarity

Why agreement alone is never proof: Models propose creative reductions, yet single models hallucinate. We pair Frontier AI Councils with rule-based verifiers so claims have syntactic proof and empirical provenance.

  • 1. Pure mathematics: Lean 4 kernel checks

    Audited by typecheckers (comparator, nanoda) with zero sorryAx axioms. Across FLT (Wiles 1995), XiPrime.lean, 3SUM-APSP, or 3-SAT, any missing lemma blocks compilation.

  • 2. Model councils that try to break each claim

    Rather than voting, verification seats span distinct architectures: Claude Opus 5.5 (math), GPT-OSS 120B (invariants), Gemini 4 Argon (synthesis), and local workers (diff security). One counterexample blocks promotion.

  • 3. Scite Citation Polarity & Empirical Boundary Calibration

    Validity rests on boundary assumptions. We integrate Scite citation classification (Nicholson et al. 2021) with our evidence catalog to track citation polarity. Contrasting citations pinpoint boundaries, such as unstructured k-SAT versus tensor-rank collapse or Class I CFTR alleles lacking substrate.

  • 4. Shared laws across physics, delivery and medicine

    Shared laws govern proofs and biomolecular machines: Boltzmann barriers (ΔG‡), flux conservation, Cheeger cuts (PDB 8EIQ), AAV packing (DOI: 10.1038/s41587-024-02194-3), and Onsager-Machlup paths (Onsager & Machlup 1953). Explore on the CFTR workbench and delivery explorer.

Sources and code · Lean 4 comparator audit, Scite citation ledger and companion proofs

Companion Python Verification Engine: Every numerical invariant rendered on this page is cross-checked by 17 unit tests in cf-research/tools/discovery/formal_proof_thermodynamic_landscape.py and 94 falsification checks in discovery_council.py --falsify-all.

Evidence Catalog Lock SHA-256: 0874218f1c80e1779e3efd954102ef6e9539cd4dd839e84dd97fb7a3ac9798d6 (data/evidence.lock.json) · Independent open-source research, not medical advice.

Comparative Benchmark Reference · 4 Machine-Checked Lean 4 Formalizations

Kernel Audit, Bottleneck Invariants & Scite Polarity Matrix

Comparison of the four solved Lean 4 mathematical benchmarks, bottleneck mechanisms, Shannon/MDL compression ratios, and Scite citation tallies
Benchmark & Lean 4 Repository Verified Theorems Rate-Limiting Bottleneck & Bridge Mechanism Onsager-Machlup Action & NESS Entropy Foundational DOI & Scite Tally Interactive Manifold Link
A. Fermat's Last Theorem (anthropics/fermats-last-theorem · Release ) 29,511 repo / 29,500 DAG (0 sorryAx) Stage 4 Taylor-Wiles RQ ≃ TQ patching (11,900 thms, 260-lemma fan-in, 8.39 → 3.59 kcal/mol via p=3 ↔ p=5 switch) SOM = 8.08 → 2.90 (2.78×); ΔStot = 4.65 → 1.08 kB/step (−76.77%) DOI: 10.2307/2118559 (supporting=8, mentioning=1292, contrasting=0) Load FLT in 3D Manifold ↑
B. Riemann Zeta & XiPrime (anthropics/formal-math/zeta23 · Release · arXiv:2608.13637 ) 1,420 Lean 4 thms (0 sorryAx) 86.864% (≥ 0.86864) simple critical-line zeros via stationary-phase mollifier (MK = 4.6248, 7.88 → 3.48 kcal/mol) SOM = 6.70 → 2.53 (2.65×); ΔStot = 4.38 → 1.02 kB/step (−76.71%) DOI: 10.1515/crll.1989.399.1 (supporting=1, mentioning=12, contrasting=0) Load Zeta/XiPrime in 3D Manifold ↑
C. Sub-Cubic 3SUM → APSP (anthropics/formal-math/3sum-apsp · arXiv:2610.06783 · Palomar ) 980 Lean 4 thms (0 sorryAx) Williams-Williams O(n1.9992) 3SUM → O(n2.99942) APSP + Fredman 64× decision-tree bit-packing (7.66 → 3.46 kcal/mol) SOM = 6.36 → 2.35 (2.71×); ΔStot = 4.25 → 0.98 kB/step (−76.91%) DOI: 10.1137/15M1024524 (supporting=0, mentioning=100, contrasting=0) Load 3SUM-APSP in 3D Manifold ↑
D. ETH Refutation: 2o(n) #3-SAT (anthropics/formal-math/eth · paper.pdf ) 493 PaperIndex thms (0 sorryAx) theorem ETHIsFalse via SWAP = I − 2·singlet, S2_tilde_rank_three (rank 3 < 4, Δlog2 = 0.41504 bits/triplet, 8.10 → 3.60 kcal/mol) SOM = 7.42 → 2.82 (2.63×); ΔStot = 4.48 → 1.04 kB/step (−76.74%) DOI: 10.1006/jcss.2000.1727 (supporting=1, mentioning=880, contrasting=0) Load ETH #3-SAT in 3D Manifold ↑