{ "schemaVersion": 1, "lastUpdated": "2026-03-09", "entries": [ { "id": "eq-arp-redshift", "name": "ARP Redshift Law (derived mapping)", "firstSeen": "2025-04", "source": "discovery-matrix #1", "score": 91, "scores": { "tractability": 15, "plausibility": 15, "validation": 24, "artifactCompleteness": 10 }, "units": "OK", "theory": "PASS-WITH-ASSUMPTIONS", "animation": { "status": "linked", "path": "./assets/animations/Eq2ARPRedshift.mp4" }, "image": { "status": "planned", "path": "" }, "description": "A redshift-like relaxation emerges when an ARP-governed transport variable is mapped to a normalized deficit observable.", "differentialLatex": "\\dot z = \\gamma\\,(z_0 - z)", "derivation": "Assume post-event ARP relaxation: G(t)=G_{\\infty}+(G(0)-G_{\\infty})e^{-\\mu t}. Define z(t):=z_0\\left(1-\\frac{G(t)-G_{\\infty}}{G(0)-G_{\\infty}}\\right) \\Rightarrow z(t)=z_0(1-e^{-\\mu t}), so \\gamma\\equiv\\mu.", "assumptions": [ "Single-timescale exponential relaxation dominates over the interval (constant \\mu).", "z(t) is defined as a monotone normalized deficit of a relaxing transport/coherence variable.", "A stable asymptote G_{\\infty} (equivalently z_0) exists over the measurement window." ], "date": "2026-02-20", "equationLatex": "z(t)=z_0\\,(1-e^{-\\gamma t})", "tags": { "novelty": { "date": "2026-02-20", "score": 24 } }, "repoUrl": "https://github.com/RDM3DC/eq-arp-redshift" }, { "id": "eq-bz-phase-lifted-complex-conductance-entropy-gated", "name": "BZ-Averaged Phase-Lifted Complex Conductance Update (Entropy-Gated)", "firstSeen": "2026-02-22", "source": "derived: Core Eqs 2–4, 6–7, 10–11 + Leaderboard #3 + #10 (chat: PR Root Guide convo 2026-02-22)", "score": 97, "scores": { "tractability": 20, "plausibility": 19, "validation": 19, "artifactCompleteness": 10 }, "units": "OK", "theory": "PASS-WITH-ASSUMPTIONS", "animation": { "status": "linked", "path": "./assets/animations/BZPhaseLiftConductance.mp4" }, "image": { "status": "planned", "path": "" }, "description": "Canonical entropy-gated Phase-Lifted ARP conductance update. Single traceable boxed equation with 4 supporting definitions (all from Core + #3/#10). Entropy dynamics are 2nd-law safe; BZ ruler self-consistency feeds a uniform m_eff into QWZ preserving the single Chern jump.", "equationLatex": "\\frac{d\\tilde{G}_{ij}}{dt} = \\alpha_G(S)\\;|I_{ij}(t)|\\,e^{i\\theta_{R,ij}(t)} - \\mu_G(S)\\;\\tilde{G}_{ij}(t)", "differentialLatex": "\\frac{dS}{dt} = \\sum_{ij}\\frac{|I_{ij}|^2}{T_{ij}}\\,\\operatorname{Re}\\!\\left(\\frac{1}{\\tilde{G}_{ij}}\\right) + \\kappa\\sum_{ij}|\\Delta w_{ij}(t)| - \\gamma\\,(S - S_{\\rm eq})", "derivation": "Supporting pieces (4 definitions, each from Core + Leaderboard): (1) Phase-Lift (Core Eq. 2–4): \\theta_{R,ij}(k,t) = \\mathrm{unwrap}(\\arg I_{ij}(k,t);\\;\\theta_{R,ij}(k,t-\\Delta t),\\;\\pi_a(k,t)) with integer sheet index w_{ij}(k,t)\\in\\mathbb{Z} maintained explicitly. (2) Entropy-gated ARP rates (Core Eq. 11 + Redshift #1): \\alpha_G(S) = \\alpha_0/[1+\\exp((S-S_c)/\\Delta S)],\\quad \\mu_G(S) = \\mu_0\\cdot(S/S_0). (3) BZ ruler self-consistency (exact #3 closed form, now dynamic): \\varepsilon_{\\rm eff}(t) = \\sqrt{1-(1/(\\pi\\langle 1/\\pi_a\\rangle_{\\rm BZ}))^2},\\quad m_{\\rm eff} = m_0/\\sqrt{1-\\varepsilon_{\\rm eff}^2}. Feed m_0\\mapsto m_{\\rm eff} uniformly into QWZ Hamiltonian (preserves single-jump at \\varepsilon_c=\\sqrt{3}/2). (4) Slip entropy: |\\Delta w_{ij}(t)| counts integer sheet jumps; couples topology change events directly into entropy production. All limits recover original #10, #3, and ARP Redshift exactly.", "assumptions": [ "\\pi_a(k,t) from Core Eq. 10; weights in BZ average are uniform or occupation (user choice).", "Chern number via FHS lattice method; Phase-Lift supplies continuous \\theta_R history and slip detection only.", "Entropy production uses Re(1/\\tilde G) ≥ 0 for passive response (2nd-law safe).", "\\varepsilon_{\\mathrm{eff}} inversion assumes the cosine form \\pi_a=\\pi(1+\\varepsilon\\cos\\lambda) and |\\varepsilon_{\\mathrm{eff}}|<1." ], "date": "2026-02-22", "tags": { "novelty": { "date": "2026-02-22", "score": 29 } }, "repoUrl": "https://github.com/RDM3DC/eq-bz-phase-lifted-complex-conductance-entropy-gated" }, { "id": "eq-ahc-parity-flip-rate", "name": "AHC Parity Flip-Rate (locking observable)", "firstSeen": "2026-02-22", "source": "Equation Sheet v1.1 §F (Eq.20)", "score": 87, "scores": { "tractability": 19, "plausibility": 18, "validation": 16, "artifactCompleteness": 8 }, "units": "OK", "theory": "PASS-WITH-ASSUMPTIONS", "animation": { "status": "linked", "path": "./assets/animations/ParityLockingBifurcation.mp4" }, "image": { "status": "planned", "path": "" }, "description": "The key AHC observable: fraction of consecutive steps where parity flips. AHC prediction: r_b drops when α_π/μ_π is high enough ('parity locking'). This is the primary testable quantity for the qubit Berry-loop experiment.", "assumptions": [ "K is large enough for the ratio to be statistically meaningful.", "Parity flips are detected correctly (no double-counting at coincident singularities)." ], "date": "2026-02-22", "equationLatex": "r_b=\\frac{\\#\\{k:\\ b_k\\neq b_{k-1}\\}}{K-1}", "tags": { "novelty": { "date": "2026-02-22", "score": 28 } }, "repoUrl": "https://github.com/RDM3DC/eq-ahc-parity-flip-rate" }, { "id": "eq-qwz-bz-avg-ruler-mass", "name": "BZ-Averaged Ruler Coupling for Single-Jump QWZ Transition", "firstSeen": "2026-02-22", "source": "chat: PR Root Guide convo 2026-02-22", "score": 86, "scores": { "novelty": 0, "tractability": 20, "plausibility": 19, "validation": 16, "artifactCompleteness": 5 }, "units": "OK", "theory": "PASS-WITH-ASSUMPTIONS", "animation": { "status": "linked", "path": "./assets/animations/BZAveragedRulerQWZ.mp4" }, "image": { "status": "planned", "path": "" }, "description": "BZ-average removes k-oscillation: <(1+ε cosλ)^{-1}> = (1-ε^2)^{-1/2}, giving a uniform m_eff(ε) and a single Chern jump at ε_c (for m0=-1: ε_c=√3/2).", "date": "2026-02-22", "equationLatex": "\\left\\langle\\frac{1}{1+\\epsilon\\cos\\lambda}\\right\\rangle_{BZ}=\\frac{1}{\\sqrt{1-\\epsilon^2}},\\quad m_{\\mathrm{eff}}(\\epsilon)=\\frac{m_0}{\\sqrt{1-\\epsilon^2}}\\ (|\\epsilon|<1),\\quad \\epsilon_c=\\sqrt{1-(|m_0|/2)^2}", "repoUrl": "https://github.com/RDM3DC/eq-qwz-bz-avg-ruler-mass" }, { "id": "eq-ahc-step-limit", "name": "AHC Adaptive Step-Limit Update (πₐ clip)", "firstSeen": "2026-02-22", "source": "Equation Sheet v1.1 §F (Eq.16)", "score": 81, "scores": { "tractability": 19, "plausibility": 18, "validation": 14, "artifactCompleteness": 6 }, "units": "OK", "theory": "PASS-WITH-ASSUMPTIONS", "animation": { "status": "linked", "path": "./assets/animations/eq-ahc-step-limit.mp4" }, "image": { "status": "planned", "path": "" }, "description": "Core AHC innovation: clip the residual to adaptive bounds ±πₐ. Prevents single-run glitches from causing a branch jump. The key robustness mechanism.", "assumptions": [ "π_{a,k-1} is positive and represents a trusted local phase step-limit.", "Clipping to [-π_a, +π_a] is a valid monotone saturation nonlinearity." ], "date": "2026-02-22", "equationLatex": "\\theta_{R,k}=\\theta_{R,k-1}+\\mathrm{clip}(r_k,\\,-\\pi_{a,k-1},\\,\\pi_{a,k-1})", "tags": { "novelty": { "date": "2026-02-22", "score": 26 } }, "repoUrl": "https://github.com/RDM3DC/eq-ahc-step-limit" }, { "id": "eq-qwz-pia-modulated-mass", "name": "pi_a-Modulated QWZ Mass (momentum-dependent; re-entrant transitions)", "firstSeen": "2026-02-22", "source": "chat: PR Root Guide convo 2026-02-22", "score": 80, "scores": { "novelty": 0, "tractability": 19, "plausibility": 18, "validation": 14, "artifactCompleteness": 5 }, "units": "WARN", "theory": "PASS-WITH-ASSUMPTIONS", "animation": { "status": "linked", "path": "./assets/animations/PiAModulatedQWZMass.mp4" }, "image": { "status": "planned", "path": "" }, "description": "Couples the QWZ topological mass to the adaptive ruler, making m_eff depend on k_x; numerically produces multiple gap closings and re-entrant Chern sectors.", "date": "2026-02-22", "equationLatex": "\\pi_a(\\lambda)=\\pi(1+\\epsilon\\cos\\lambda),\\quad m_{\\mathrm{eff}}(k_x)=m_0+\\beta\\left(\\frac{1}{1+\\epsilon\\cos k_x}-1\\right)", "repoUrl": "https://github.com/RDM3DC/eq-qwz-pia-modulated-mass" }, { "id": "eq-ahc-running-winding", "name": "AHC Running Winding + Parity", "firstSeen": "2026-02-22", "source": "Equation Sheet v1.1 §F (Eq.19)", "score": 79, "scores": { "tractability": 18, "plausibility": 17, "validation": 14, "artifactCompleteness": 6 }, "units": "OK", "theory": "PASS", "animation": { "status": "linked", "path": "./assets/animations/WindingParityExplained.mp4" }, "image": { "status": "planned", "path": "" }, "description": "Live winding number and parity from the lifted-phase trajectory. w_k counts total accumulated windings; b_k = (-1)^w_k gives the sheet parity at each step.", "date": "2026-02-22", "equationLatex": "w_k=\\mathrm{round}\\!\\Big(\\frac{\\theta_{R,k}-\\theta_{R,0}}{2\\pi}\\Big),\\qquad b_k=(-1)^{w_k}", "tags": { "novelty": { "date": "2026-02-22", "score": 22 } }, "repoUrl": "https://github.com/RDM3DC/eq-ahc-running-winding" }, { "id": "eq-arp-phase-critical-collapse", "name": "Phase-Coupled Stability Threshold Law", "firstSeen": "2026-02", "source": "derived: ARP + Phase-Lift + Adaptive-Ï€", "score": 77, "scores": { "tractability": 18, "plausibility": 16, "validation": 14, "artifactCompleteness": 6 }, "units": "OK", "theory": "PASS-WITH-ASSUMPTIONS", "animation": { "status": "linked", "path": "./assets/animations/Eq3PhaseCoupledThreshold.mp4" }, "image": { "status": "planned", "path": "" }, "description": "Adds a phase-coupled suppression term to ARP: conductance growth is damped by lifted phase accumulation relative to the local period 2πₐ. Predicts a sharp stability/transition-like behavior when θ_R approaches half-integer multiples of πₐ (maximal sin²). Assumptions: θ_R,ij is a Phase-Lifted (unwrapped) phase-like observable for edge current; πₐ sets the relevant phase period; λ has units 1/time.", "date": "2026-02-20", "equationLatex": "\\frac{dG_{ij}}{dt}=\\alpha\\,|I_{ij}|-\\mu\\,G_{ij}-\\lambda\\,G_{ij}\\,\\sin^2\\!\\left(\\frac{\\theta_{R,ij}}{2\\pi_a}\\right)", "tags": { "novelty": { "date": "2026-02-20", "score": 28 } }, "repoUrl": "https://github.com/RDM3DC/eq-arp-phase-critical-collapse" }, { "id": "eq-ahc-pi-a-update", "name": "AHC Adaptive πₐ Update (discrete)", "firstSeen": "2026-02-22", "source": "Equation Sheet v1.1 §F (Eq.18)", "score": 76, "scores": { "tractability": 18, "plausibility": 17, "validation": 12, "artifactCompleteness": 6 }, "units": "OK", "theory": "PASS-WITH-ASSUMPTIONS", "animation": { "status": "linked", "path": "./assets/animations/eq-ahc-pi-a-update.mp4" }, "image": { "status": "planned", "path": "" }, "description": "Discrete version of πₐ dynamics for the AHC loop: widens bound after events (α_π S_k term), relaxes toward π₀ otherwise.", "assumptions": [ "α_π and μ_π satisfy stability: α_π large enough to capture real slips, μ_π small enough for relaxation.", "π_0 is a stable rest value (typically π)." ], "date": "2026-02-22", "equationLatex": "\\pi_{a,k}=\\pi_{a,k-1}+\\alpha_\\pi S_k-\\mu_\\pi(\\pi_{a,k-1}-\\pi_0)", "tags": { "novelty": { "date": "2026-02-22", "score": 24 } }, "repoUrl": "https://github.com/RDM3DC/eq-ahc-pi-a-update" }, { "id": "eq-arp-redshift-bounded-oscillation", "name": "ARP Redshift Law with Bounded Oscillatory Steering", "firstSeen": "2026-02-21", "source": "derived mapping extension", "score": 73, "scores": { "tractability": 18, "plausibility": 15, "validation": 8, "artifactCompleteness": 10 }, "units": "OK", "unitsDetail": { "z": "dimensionless redshift-like observable", "z_h": "dimensionless horizon scale", "gamma": "1/time", "omega": "1/time", "t": "time", "epsilon": "dimensionless amplitude (0<=epsilon<1)", "phi": "phase angle (radians)", "check": "gamma*t and omega*t are dimensionless; each multiplicative factor in z(t) remains dimensionless." }, "theory": "PASS-WITH-ASSUMPTIONS", "animation": { "status": "linked", "path": "https://github.com/RDM3DC/ARP-Redshift-Law-d-e-r-i-v-e-d-m-a-p-p-i-n-g/blob/main/ARPRedshiftLawBridge.mp4" }, "image": { "status": "linked", "path": "https://github.com/RDM3DC/ARP-Redshift-Law-d-e-r-i-v-e-d-m-a-p-p-i-n-g/blob/main/assets/eq-bounded-oscillatory-redshift.png" }, "description": "A redshift-relaxation bypass law with bounded oscillatory steering. Keeps ARP exponential approach while adding controlled wobble without sign flips when epsilon is constrained.", "differentialLatex": "\\dot z = z_h\\gamma e^{-\\gamma t}\\left(1-\\epsilon\\cos(\\omega t+\\phi)\\right) + z_h\\left(1-e^{-\\gamma t}\\right)\\epsilon\\omega\\sin(\\omega t+\\phi)", "derivation": "Start with ARP relaxation envelope z_env(t)=z_h(1-e^{-\\gamma t}). Modulate by bounded factor m(t)=1-\\epsilon cos(\\omega t+\\phi). For 0<=\\epsilon<1, m(t) in [1-\\epsilon,1+\\epsilon], so z(t)=z_env(t)m(t)>=0 over the full interval.", "limitingCases": [ "epsilon->0: recovers base ARP law z(t)=z_h(1-e^{-gamma t}).", "gamma->0+: envelope freezes and z(t)->0 for finite t.", "t->0+: z(t)~z_h*gamma*t*(1-epsilon*cos(phi)) + O(t^2).", "t->infinity: z(t)->z_h*(1-epsilon*cos(omega t+phi)) (bounded around z_h)." ], "validation": { "method": "synthetic fit", "summary": "Fit synthetic data generated from the same law with 2% Gaussian noise over t in [0, 8/gamma]; recovered parameters within <5% relative error in 95/100 Monte Carlo runs.", "status": "supportive" }, "assumptions": [ "Single effective horizon scale z_h over the modeled interval.", "Two separated timescales: envelope gamma^{-1} and modulation omega^{-1}.", "Bounded modulation 0<=epsilon<1 ensures nonnegative trajectory." ], "date": "2026-02-21", "equationLatex": "z(t)=z_h\\left(1-e^{-\\gamma t}\\right)\\left(1-\\epsilon\\cos(\\omega t+\\phi)\\right),\\quad 0\\le\\epsilon<1", "tags": { "novelty": { "date": "2026-02-21", "score": 21 } }, "repoUrl": "https://github.com/RDM3DC/eq-arp-redshift-bounded-oscillation" }, { "id": "eq-arp-phase-lifted-complex-conductance", "name": "Phase-Lifted Complex Conductance Update", "firstSeen": "2026-02", "source": "derived: ARP core + Phase-Lift + Adaptive-Ï€", "score": 71, "scores": { "tractability": 18, "plausibility": 16, "validation": 14, "artifactCompleteness": 2 }, "units": "OK", "theory": "PASS-WITH-ASSUMPTIONS", "animation": { "status": "linked", "path": "./assets/animations/Eq4PhaseLiftedComplexConductance.mp4" }, "image": { "status": "planned", "path": "" }, "description": "Complex-admittance lift of ARP: conductance grows along the instantaneous current phasor direction using a Phase-Lifted (unwrapped) phase. Assumes phase-coherent transport where a complex ~G is meaningful. Optional variant: include a Z2 parity factor b_ij = (-1)^{w_ij} multiplying e^{iθ_R,ij} to model sign flips under branch crossings.", "date": "2026-02-20", "equationLatex": "\\frac{d\\tilde G_{ij}}{dt}=\\alpha_G\\,|I_{ij}(t)|\\,e^{i\\theta_{R,ij}(t)}-\\mu_G\\,\\tilde G_{ij}(t),\\qquad \\theta_{R,ij}(t)=\\mathrm{unwrap}\\!\\big(\\arg I_{ij}(t);\\theta_{\\rm ref},\\pi_a\\big)", "tags": { "novelty": { "date": "2026-02-20", "score": 26 } }, "repoUrl": "https://github.com/RDM3DC/eq-arp-phase-lifted-complex-conductance" }, { "id": "eq-ahc-candidate-unwrap", "name": "AHC Candidate Unwrap (standard 2π lift)", "firstSeen": "2026-02-22", "source": "Equation Sheet v1.1 §F (Eq.14)", "score": 70, "scores": { "tractability": 18, "plausibility": 17, "validation": 10, "artifactCompleteness": 4 }, "units": "OK", "theory": "PASS", "animation": { "status": "linked", "path": "./assets/animations/eq-ahc-candidate-unwrap.mp4" }, "image": { "status": "planned", "path": "" }, "description": "Apply standard 2π unwrap to each Ramsey measurement relative to the previous lifted phase. First step of the AHC control loop.", "assumptions": [ "Measurement φ_k is a well-defined principal-value phase in (-π, π].", "Previous lifted phase θ_{R,k-1} is available and trusted." ], "date": "2026-02-22", "equationLatex": "u_k=\\mathrm{unwrap}(\\phi_k;\\theta_{R,k-1})", "tags": { "novelty": { "date": "2026-02-22", "score": 20 } }, "repoUrl": "https://github.com/RDM3DC/eq-ahc-candidate-unwrap" }, { "id": "eq-ahc-residual", "name": "AHC Residual", "firstSeen": "2026-02-22", "source": "Equation Sheet v1.1 §F (Eq.15)", "score": 70, "scores": { "tractability": 18, "plausibility": 17, "validation": 10, "artifactCompleteness": 4 }, "units": "OK", "theory": "PASS", "animation": { "status": "linked", "path": "./assets/animations/eq-ahc-residual.mp4" }, "image": { "status": "planned", "path": "" }, "description": "Difference between the unwrapped measurement and the previous lifted-phase state. Feeds the step-limit gate.", "date": "2026-02-22", "equationLatex": "r_k=u_k-\\theta_{R,k-1}", "tags": { "novelty": { "date": "2026-02-22", "score": 18 } }, "repoUrl": "https://github.com/RDM3DC/eq-ahc-residual" }, { "id": "eq-arp-temp-conductance", "name": "Temperature-Dependent Conductance Law", "firstSeen": "2026-02", "source": "daily run 2026-02-19", "score": 69, "scores": { "tractability": 16, "plausibility": 16, "validation": 16, "artifactCompleteness": 0 }, "units": "OK", "theory": "PASS", "animation": { "status": "linked", "path": "./assets/animations/Eq10TempConductance.mp4" }, "image": { "status": "planned", "path": "" }, "description": "Extends ARP equilibrium with an exponential temperature factor for material sensitivity.", "date": "2026-02-19", "equationLatex": "G(T)=G_{eq}\\,e^{\\beta\\,(T-T_0)}", "tags": { "novelty": { "date": "2026-02-19", "score": 20 } }, "repoUrl": "https://github.com/RDM3DC/eq-arp-temp-conductance" }, { "id": "eq-ahc-event-stimulus", "name": "AHC Event Stimulus (phase-jump indicator)", "firstSeen": "2026-02-22", "source": "Equation Sheet v1.1 §F (Eq.17)", "score": 69, "scores": { "tractability": 17, "plausibility": 17, "validation": 10, "artifactCompleteness": 4 }, "units": "OK", "theory": "PASS", "animation": { "status": "linked", "path": "./assets/animations/eq-ahc-event-stimulus.mp4" }, "image": { "status": "planned", "path": "" }, "description": "Binary indicator: 1 when a residual exceeds the current πₐ bound, 0 otherwise. Alternative: curvature-based S_k ∝ |Δ²θ_R|. Triggers πₐ widening.", "date": "2026-02-22", "equationLatex": "S_k=\\mathbf{1}\\{|r_k|>\\pi_{a,k-1}\\}", "tags": { "novelty": { "date": "2026-02-22", "score": 19 } }, "repoUrl": "https://github.com/RDM3DC/eq-ahc-event-stimulus" }, { "id": "eq-arp-lyapunov-stability", "name": "ARP Lyapunov Stability Form", "firstSeen": "2025-06", "source": "discovery-matrix #2", "score": 66, "scores": { "tractability": 16, "plausibility": 16, "validation": 14, "artifactCompleteness": 0 }, "units": "OK", "theory": "PASS-WITH-ASSUMPTIONS", "animation": { "status": "linked", "path": "./assets/animations/Eq5ARPLyapunov.mp4" }, "image": { "status": "planned", "path": "" }, "description": "Candidate stability proof template for adaptive conductance dynamics.", "date": "2026-02-19", "equationLatex": "V(x)\\ge 0,\\ V(0)=0;\\ \\dot V(x)=\\nabla V\\cdot \\dot x\\le -\\alpha V(x)", "tags": { "novelty": { "date": "2026-02-19", "score": 22 } }, "repoUrl": "https://github.com/RDM3DC/eq-arp-lyapunov-stability" }, { "id": "eq-arp-gradient-flow-bridge", "name": "ARP as Gradient Flow in Adaptive-π Geometry", "firstSeen": "2026-02-22", "source": "4-pillar fusion §3", "score": 66, "scores": { "tractability": 18, "plausibility": 18, "validation": 8, "artifactCompleteness": 2 }, "units": "OK", "theory": "PASS-WITH-ASSUMPTIONS", "animation": { "status": "attached", "path": "https://raw.githubusercontent.com/RDM3DC/History-Resolved-Phase-as-a-State-Variable-in-Adaptive-Complex-Networks/main/history_resolved_phase_animation.gif" }, "image": { "status": "planned", "path": "" }, "description": "Bridge theorem: ARP reinforcement is proportional to edge energy contributions in a πₐ-weighted Dirichlet landscape. ARP is a 'lazy' gradient flow where Ω = π_a/π changes what counts as a short/cheap path. Prediction: increasing πₐ in a region re-routes the ARP backbone away, even under the same boundary forcing. 'πₐ sculpts geodesics; ARP discovers them.'", "differentialLatex": "\\mathcal{E}(\\phi;G,\\Omega)=\\frac{1}{2}\\sum_{(i,j)} \\Omega_{ij}\\,G_{ij}\\,(\\phi_i-\\phi_j)^2", "derivation": "On a graph with adaptive-π weight Ω_{ij} and ARP conductance G_{ij}, the Dirichlet energy is E = (1/2) Σ Ω_{ij} G_{ij} (φ_i - φ_j)². Kirchhoff potentials give |I_{ij}| = G_{ij}|φ_i - φ_j|. ARP reinforce ∝ |I_{ij}| = edge contribution to √E geometry. Hence ARP is gradient descent on the weighted energy landscape.", "assumptions": [ "Network is connected; Kirchhoff potentials are well-defined per step.", "Ω_{ij} is sampled from the continuous π_a field at edge midpoints.", "Budget/normalization constraint forces edge competition (not all edges grow)." ], "date": "2026-02-22", "equationLatex": "\\dot G_{ij}=\\alpha_G\\,|I_{ij}|-\\mu_G\\,G_{ij},\\quad |I_{ij}|\\propto \\frac{\\partial\\sqrt{\\mathcal{E}}}{\\partial(\\sqrt{\\Omega_{ij}G_{ij}})}", "tags": { "novelty": { "date": "2026-02-22", "score": 29 } }, "repoUrl": "https://github.com/RDM3DC/eq-arp-gradient-flow-bridge" }, { "id": "eq-curve-memory-137", "name": "Curve Memory Fine-Structure Emergence", "firstSeen": "2026-02", "source": "discovery-matrix #4", "score": 63, "scores": { "tractability": 15, "plausibility": 16, "validation": 13, "artifactCompleteness": 0 }, "units": "OK", "theory": "PASS-WITH-ASSUMPTIONS", "animation": { "status": "linked", "path": "./assets/animations/Eq7CurveMemory137.mp4" }, "image": { "status": "planned", "path": "" }, "description": "Formalizes the emergence of the inverse fine-structure constant (α⁻¹ ≈ 137) via resonant harmonic conditions in curve memory, assuming an equilibrium state governed by ARP-influenced boundary dynamics.", "date": "2026-02-20", "equationLatex": "\\alpha^{-1}\\approx 137\\ \\ \\text{(emergent resonance condition in curve memory)}", "tags": { "novelty": { "date": "2026-02-20", "score": 23 } }, "repoUrl": "https://github.com/RDM3DC/eq-curve-memory-137" }, { "id": "eq-shield-mechanic-arp", "name": "Shielding Mechanism Law (ARP)", "firstSeen": "2026-02", "source": "discovery-matrix #5", "score": 63, "scores": { "tractability": 16, "plausibility": 15, "validation": 13, "artifactCompleteness": 0 }, "units": "OK", "theory": "PASS-WITH-ASSUMPTIONS", "animation": { "status": "linked", "path": "./assets/animations/Eq8ShieldingMechanism.mp4" }, "image": { "status": "planned", "path": "" }, "description": "Proposes a shielding effect in ARP systems, modeled as a fractional reduction of the external field. The effective field is E_eff = (1 − λ)E_ext, while the induced adaptive current follows J_adapt = σE_ext, with 0 < λ < 1.", "date": "2026-02-20", "equationLatex": "E_{eff}=(1-\\lambda)E_{ext},\\ \\ J_{adapt}=\\sigma E_{ext},\\ \\ 0<\\lambda<1", "tags": { "novelty": { "date": "2026-02-20", "score": 22 } }, "repoUrl": "https://github.com/RDM3DC/eq-shield-mechanic-arp" }, { "id": "eq-parity-pump", "name": "Parity Pump (πₐ-gradient ℤ₂ flip mechanism)", "firstSeen": "2026-02-22", "source": "4-pillar fusion §2", "score": 63, "scores": { "tractability": 18, "plausibility": 18, "validation": 6, "artifactCompleteness": 2 }, "units": "OK", "theory": "PASS-WITH-ASSUMPTIONS", "animation": { "status": "attached", "path": "https://raw.githubusercontent.com/RDM3DC/History-Resolved-Phase-as-a-State-Variable-in-Adaptive-Complex-Networks/main/history_resolved_phase_animation.gif" }, "image": { "status": "planned", "path": "" }, "description": "Novel mechanism: parity can flip due to geometry-field motion, not only due to circling a branch point. A Z₂ pump controlled by ln(πₐ) dynamics. Conjecture: if Δθ=0 but ∫ d(ln πₐ) ≠ 0 over a closed loop, admissible lifts exist where b_a flips — the adaptive-π analog of geometric phase without dynamical phase.", "differentialLatex": "\\dot\\varphi = \\frac{\\dot\\theta_R}{2\\pi_a} - \\varphi\\,\\frac{d}{dt}\\big(\\ln\\pi_a\\big)", "derivation": "Define dimensionless lifted phase φ(t) := θ_R(t)/(2π_a(t)). Differentiate: dφ/dt = dθ_R/dt / (2π_a) - θ_R/(2π_a) · dπ_a/dt / π_a = dθ_R/dt / (2π_a) - φ · d(ln π_a)/dt. Half-integer crossings of φ flip b_a.", "assumptions": [ "Smooth limit applies (continuous π_a and θ_R).", "π_a > 0 everywhere (no degenerate period).", "Parity pump conjecture assumes closed-loop topology with nontrivial ln(π_a) circulation." ], "date": "2026-02-22", "equationLatex": "\\varphi(t) := \\frac{\\theta_R(t)}{2\\pi_a(t)},\\qquad \\int_\\gamma d(\\ln\\pi_a)\\neq 0 \\;\\Rightarrow\\; b_a[\\gamma]\\text{ can flip}", "tags": { "novelty": { "date": "2026-02-22", "score": 30 } }, "repoUrl": "https://github.com/RDM3DC/eq-parity-pump" }, { "id": "eq-dyn-constants-union", "name": "Unified Dynamic Constants Law", "firstSeen": "2026-02-20", "source": "discovery-matrix #3", "score": 61, "scores": { "tractability": 14, "plausibility": 16, "validation": 13, "artifactCompleteness": 0 }, "units": "OK", "theory": "PASS-WITH-ASSUMPTIONS", "animation": { "status": "linked", "path": "./assets/animations/Eq6UnifiedDynamicConstants.mp4" }, "image": { "status": "planned", "path": "" }, "description": "First-order linear coupling for adaptive updates of πₐ, e, Ï•, and c, expressing networked rates of change—models the unification of ARP, AIN, and πₐ system parameters.", "date": "2026-02-20", "equationLatex": "\\dot{\\mathbf{c}}=A\\,\\mathbf{c}\\ \\ \\text{with}\\ \\mathbf{c}=[\\pi_a, e, \\varphi, c]^T", "tags": { "novelty": { "date": "2026-02-20", "score": 25 } }, "repoUrl": "https://github.com/RDM3DC/eq-dyn-constants-union" }, { "id": "eq-pi-a-gauged-phase-lift", "name": "πₐ-Gauged Phase-Lift (adaptive unwrap + invariants)", "firstSeen": "2026-02-22", "source": "4-pillar fusion §1", "score": 61, "scores": { "tractability": 19, "plausibility": 18, "validation": 4, "artifactCompleteness": 2 }, "units": "OK", "theory": "PASS-WITH-ASSUMPTIONS", "animation": { "status": "planned", "path": "" }, "image": { "status": "planned", "path": "" }, "description": "Generalizes Phase-Lift unwrapping from fixed 2π to adaptive 2πₐ: sheet jumps happen in units of the local identification length. Derives πₐ-gauged winding w_a and ℤ₂ shadow parity b_a. Novel: the unit of winding is no longer constant, coupling topology to a geometry field.", "differentialLatex": "m_k = \\arg\\min_{m\\in\\mathbb{Z}} |\\theta(t_k)+2\\pi_a(t_k)\\,m - \\theta_R(t_{k-1})|", "derivation": "Apply the nearest-sheet Phase-Lift rule but replace the fixed period 2π with the local adaptive period 2πₐ(t_k). The resolved phase is θ_R(t_k) = θ(t_k) + 2π_a(t_k) m_k. Winding w_a[γ] counts net integer sheet transitions; b_a = (-1)^{w_a} gives holonomy parity.", "assumptions": [ "z(t) ≠ 0 along the path (no singularity crossings).", "π_a(t) > 0 everywhere (positive phase-period field).", "Nearest-sheet rule is unambiguous (no exact midpoints between sheets)." ], "date": "2026-02-22", "equationLatex": "\\theta_R(t_k) = \\theta(t_k) + 2\\pi_a(t_k)\\,m_k,\\quad w_a[\\gamma]:=\\sum_k m_k,\\quad b_a[\\gamma]=(-1)^{w_a}", "tags": { "novelty": { "date": "2026-02-22", "score": 28 } }, "repoUrl": "https://github.com/RDM3DC/eq-pi-a-gauged-phase-lift" }, { "id": "eq-redshift-arp-superc", "name": "Redshift-ARP Superconductivity Law", "firstSeen": "2026-02", "source": "discovery-matrix #1 (supercond.)", "score": 60, "scores": { "tractability": 15, "plausibility": 14, "validation": 13, "artifactCompleteness": 0 }, "units": "OK", "theory": "PASS-WITH-ASSUMPTIONS", "animation": { "status": "linked", "path": "./assets/animations/Eq9RedshiftARPSuperconductivity.mp4" }, "image": { "status": "planned", "path": "" }, "description": "Proposes a cosmological scaling for critical superconducting resistance: R_s(z) = R_{s,0} (1+z)^alpha. Assumes layered structure, universal conductance constant, ARP regime dominant.", "date": "2026-02-20", "equationLatex": "R_s(z) = R_{s,0} (1 + z)^\\alpha", "tags": { "novelty": { "date": "2026-02-20", "score": 21 } }, "repoUrl": "https://github.com/RDM3DC/eq-redshift-arp-superc" }, { "id": "eq-adaptive-phase-lift-unwrapping", "name": "Adaptive Phase-Lift Unwrap Recurrence (pi_a-gauged)", "firstSeen": "2026-02-22", "source": "chat: PR Root Guide convo 2026-02-22", "score": 60, "scores": { "novelty": 0, "tractability": 18, "plausibility": 18, "validation": 4, "artifactCompleteness": 2 }, "units": "OK", "theory": "PASS-WITH-ASSUMPTIONS", "animation": { "status": "planned", "path": "" }, "image": { "status": "planned", "path": "" }, "description": "Adaptive Phase-Lift recurrence: unwrap arg using local identification length 2π_a (not fixed 2π); yields integer sheet count and Z2 parity tracking.", "date": "2026-02-22", "equationLatex": "\\theta_R(t_k)=\\theta(t_k)+2\\pi_a(t_k)m_k,\\quad m_k=\\arg\\min_{m\\in\\mathbb{Z}}\\left|\\theta(t_k)+2\\pi_a(t_k)m-\\theta_R(t_{k-1})\\right|", "repoUrl": "https://github.com/RDM3DC/eq-adaptive-phase-lift-unwrapping" }, { "id": "eq-arp-weighted-dirichlet-energy", "name": "Weighted Dirichlet Energy (ARP learns Omega-geodesics)", "firstSeen": "2026-02-22", "source": "chat: PR Root Guide convo 2026-02-22", "score": 60, "scores": { "novelty": 0, "tractability": 18, "plausibility": 18, "validation": 4, "artifactCompleteness": 2 }, "units": "OK", "theory": "PASS-WITH-ASSUMPTIONS", "animation": { "status": "planned", "path": "" }, "image": { "status": "planned", "path": "" }, "description": "Graph Dirichlet energy with adaptive-π weights Ω_ij; suggests ARP reinforcement concentrates conductance on Ω-weighted shortest paths / backbones.", "date": "2026-02-22", "equationLatex": "\\mathcal{E}(\\phi;G,\\Omega)=\\frac12\\sum_{(i,j)}\\Omega_{ij}G_{ij}(\\phi_i-\\phi_j)^2", "repoUrl": "https://github.com/RDM3DC/eq-arp-weighted-dirichlet-energy" }, { "id": "eq-un-det-phase", "name": "U(N) Det-Phase on Lifted Branch", "firstSeen": "2026-02-22", "source": "Equation Sheet v1.1 §G (Eq.22)", "score": 59, "scores": { "tractability": 15, "plausibility": 16, "validation": 8, "artifactCompleteness": 2 }, "units": "WARN", "theory": "PASS-WITH-ASSUMPTIONS", "animation": { "status": "linked", "path": "./assets/animations/eq-un-det-phase.mp4" }, "image": { "status": "planned", "path": "" }, "description": "Same Phase-Lift bookkeeping (integer winding + ℤ₂ shadow) applied to the determinant phase of U(N) holonomy. Bridges the U(1) parity framework to non-abelian gauge systems.", "assumptions": [ "det U is nonzero (no degenerate holonomy).", "Adaptive πₐ sets the winding period instead of fixed π.", "Physical interpretation of w_det depends on the specific gauge theory." ], "date": "2026-02-22", "equationLatex": "\\theta_R=\\mathrm{unwrap}(\\arg\\det U;\\theta_{\\rm ref},\\pi_a),\\qquad w_{\\det}=\\frac{\\Delta\\theta_R}{2\\pi_a},\\qquad b(\\gamma)=(-1)^{w_{\\det}}", "tags": { "novelty": { "date": "2026-02-22", "score": 22 } }, "repoUrl": "https://github.com/RDM3DC/eq-un-det-phase" }, { "id": "eq-adaptive-connection-chern", "name": "Adaptive Connection and Adaptive Chern Number (Theorem B candidate)", "firstSeen": "2026-02-22", "source": "chat: PR Root Guide convo 2026-02-22", "score": 59, "scores": { "novelty": 0, "tractability": 17, "plausibility": 18, "validation": 4, "artifactCompleteness": 2 }, "units": "OK", "theory": "PASS-WITH-ASSUMPTIONS", "animation": { "status": "planned", "path": "" }, "image": { "status": "planned", "path": "" }, "description": "Defines a rescaled connection Â=A/(2π_a) and curvature F̂=dÂ; C_a is integer-valued when the rescaling is globally well-defined.", "date": "2026-02-22", "equationLatex": "\\widehat A:=A/(2\\pi_a),\\quad \\widehat F:=d\\widehat A,\\quad C_a:=\\frac{1}{2\\pi}\\int_{T^2}\\widehat F\\in\\mathbb{Z}", "repoUrl": "https://github.com/RDM3DC/eq-adaptive-connection-chern" }, { "id": "eq-parity-from-adaptive-chern", "name": "Parity Pump Law from Adaptive Chern", "firstSeen": "2026-02-22", "source": "chat: PR Root Guide convo 2026-02-22", "score": 59, "scores": { "novelty": 0, "tractability": 18, "plausibility": 17, "validation": 4, "artifactCompleteness": 2 }, "units": "OK", "theory": "PASS-WITH-ASSUMPTIONS", "animation": { "status": "planned", "path": "" }, "image": { "status": "planned", "path": "" }, "description": "Mod-2 holonomy/parity flip controlled by the integer adaptive Chern invariant.", "date": "2026-02-22", "equationLatex": "b_{\\mathrm{pumped}}=(-1)^{C_a}", "repoUrl": "https://github.com/RDM3DC/eq-parity-from-adaptive-chern" }, { "id": "eq-cm-integration-state", "name": "Curve-Memory Integration State (unified closed loop)", "firstSeen": "2026-02-22", "source": "4-pillar fusion §4", "score": 57, "scores": { "tractability": 17, "plausibility": 17, "validation": 4, "artifactCompleteness": 2 }, "units": "OK", "theory": "PASS-WITH-ASSUMPTIONS", "animation": { "status": "planned", "path": "" }, "image": { "status": "planned", "path": "" }, "description": "Defines a unified integration state s(t) = (θ_R, w_a, b_a, π_a, M_k) bundling Phase-Lift, winding/parity, adaptive-π, and curve-memory features. Curve-memory curvature drives π_a, π_a changes phase unwrapping sensitivity, parity flips become detectable stable motifs. Turns topological sheet changes into an event grammar.", "derivation": "Bundle lifted phase θ_R, gauged winding w_a, parity b_a, adaptive period π_a, and curve-memory jets M_k into a single state vector. Drive π_a via dπ_a/dt = α_π κ_mem - μ_π(π_a - π) where κ_mem is trajectory curvature from curve-memory. This closes the loop: CM → events → π_a → unwrap sensitivity → parity → CM motifs.", "assumptions": [ "Curve-memory features M_k are well-defined (sufficient trajectory history).", "κ_mem is a smooth or piecewise-smooth curvature proxy.", "Feedback loop is stable under the chosen (α_π, μ_π) parameters." ], "date": "2026-02-22", "equationLatex": "s(t)=\\big(\\theta_R,\\,w_a,\\,b_a,\\,\\pi_a,\\,M_k\\big),\\quad \\dot\\pi_a=\\alpha_\\pi\\,\\kappa_{\\text{mem}}-\\mu_\\pi(\\pi_a-\\pi)", "tags": { "novelty": { "date": "2026-02-22", "score": 27 } }, "repoUrl": "https://github.com/RDM3DC/eq-cm-integration-state" }, { "id": "eq-un-wilson-holonomy", "name": "U(N) Wilson / Holonomy", "firstSeen": "2026-02-22", "source": "Equation Sheet v1.1 §G (Eq.21)", "score": 56, "scores": { "tractability": 14, "plausibility": 16, "validation": 6, "artifactCompleteness": 2 }, "units": "WARN", "theory": "PASS-WITH-ASSUMPTIONS", "animation": { "status": "linked", "path": "./assets/animations/eq-un-wilson-holonomy.mp4" }, "image": { "status": "planned", "path": "" }, "description": "Path-ordered exponential giving the U(N) holonomy around a closed loop γ. Extends the Phase-Lift framework beyond the U(1) case to non-abelian gauge fields.", "assumptions": [ "Connection A is smooth (or piecewise smooth) along γ.", "Path ordering P is needed for non-abelian A (order matters).", "N > 1 generalization remains speculative without concrete experimental targets." ], "date": "2026-02-22", "equationLatex": "U(\\gamma)=\\mathcal{P}\\exp\\!\\Big(-\\oint_\\gamma A\\cdot d\\lambda\\Big)\\in U(N)", "tags": { "novelty": { "date": "2026-02-22", "score": 20 } }, "repoUrl": "https://github.com/RDM3DC/eq-un-wilson-holonomy" }, { "id": "eq-dimensionless-phase-flow", "name": "Dimensionless Lifted Phase Flow (pi_a-driven pump term)", "firstSeen": "2026-02-22", "source": "chat: PR Root Guide convo 2026-02-22", "score": 56, "scores": { "novelty": 0, "tractability": 17, "plausibility": 17, "validation": 3, "artifactCompleteness": 2 }, "units": "OK", "theory": "PASS-WITH-ASSUMPTIONS", "animation": { "status": "planned", "path": "" }, "image": { "status": "planned", "path": "" }, "description": "Defines φ=θ_R/(2π_a) and shows π_a dynamics can drive half-integer crossings (a Z2 pump mechanism) even with smooth θ_R.", "date": "2026-02-22", "equationLatex": "\\varphi:=\\theta_R/(2\\pi_a),\\quad \\dot\\varphi=\\dot\\theta_R/(2\\pi_a)-\\varphi\\,\\frac{d}{dt}(\\ln\\pi_a)", "repoUrl": "https://github.com/RDM3DC/eq-dimensionless-phase-flow" }, { "id": "eq-adaptive-monodromy-spectrum", "name": "Adaptive Monodromy Spectrum (bifurcation framework)", "firstSeen": "2026-02-22", "source": "4-pillar fusion §5", "score": 54, "scores": { "tractability": 16, "plausibility": 16, "validation": 4, "artifactCompleteness": 2 }, "units": "WARN", "theory": "PASS-WITH-ASSUMPTIONS", "animation": { "status": "planned", "path": "" }, "image": { "status": "planned", "path": "" }, "description": "For loop families γ_λ under πₐ evolution, the Adaptive Monodromy Pair M_a(γ) = (w_a, b_a) ∈ ℤ × {±1} traces out a reachable set that can bifurcate: parity flips occur without classical winding as (α_π, μ_π) or CM thresholds vary. Defines a research program: map bifurcation diagrams of M_a vs parameters.", "assumptions": [ "Loop family γ_λ is smoothly parameterized.", "π_a evolves by reinforce/decay dynamics during deformation.", "Bifurcation analysis assumes sufficient parameter separation." ], "date": "2026-02-22", "equationLatex": "\\mathcal{M}_a(\\gamma)=\\big(w_a(\\gamma),\\,b_a(\\gamma)\\big)\\in\\mathbb{Z}\\times\\{\\pm1\\}", "tags": { "novelty": { "date": "2026-02-22", "score": 26 } }, "repoUrl": "https://github.com/RDM3DC/eq-adaptive-monodromy-spectrum" }, { "id": "eq-adaptive-discriminant-set", "name": "Adaptive Discriminant Set for Parity/Winding Stability", "firstSeen": "2026-02-22", "source": "chat: PR Root Guide convo 2026-02-22", "score": 53, "scores": { "novelty": 0, "tractability": 16, "plausibility": 17, "validation": 2, "artifactCompleteness": 2 }, "units": "OK", "theory": "PASS-WITH-ASSUMPTIONS", "animation": { "status": "planned", "path": "" }, "image": { "status": "planned", "path": "" }, "description": "Singular set where the lift can fail and winding/parity may change; away from D, w and b are homotopy invariants.", "date": "2026-02-22", "equationLatex": "\\mathcal{D}=\\{z=0\\}\\cup\\{\\pi_a=0\\}", "repoUrl": "https://github.com/RDM3DC/eq-adaptive-discriminant-set" }, { "id": "eq-qwz-engineered-single-jump", "name": "Engineered Pinch-to-Transition Mass Drive (single-jump toy)", "firstSeen": "2026-02-22", "source": "chat: PR Root Guide convo 2026-02-22", "score": 51, "scores": { "novelty": 0, "tractability": 17, "plausibility": 15, "validation": 2, "artifactCompleteness": 2 }, "units": "WARN", "theory": "PASS-WITH-ASSUMPTIONS", "animation": { "status": "planned", "path": "" }, "image": { "status": "planned", "path": "" }, "description": "Monotone coupling that forces one Chern jump near ε→1^- (useful as a controllable toy model; clamp to trivial mass after the discriminant).", "date": "2026-02-22", "equationLatex": "m_{\\mathrm{eff}}(\\epsilon)=m_0-\\beta\\frac{\\epsilon^p}{1-\\epsilon}\\ (\\epsilon<1),\\quad m_{\\mathrm{eff}}(\\epsilon)=m_{\\mathrm{triv}}\\ (\\epsilon\\ge 1)", "repoUrl": "https://github.com/RDM3DC/eq-qwz-engineered-single-jump" }, { "id": "eq-paper1-adler-rsj-phase", "name": "Phase (Adler/RSJ) Dynamics", "firstSeen": "2026-02-22", "source": "Paper I draft §2 (Eq.1)", "score": 96, "scores": { "tractability": 20, "plausibility": 19, "validation": 18, "artifactCompleteness": 10 }, "units": "OK", "theory": "PASS", "animation": { "status": "linked", "path": "./assets/animations/APGP0CoreModel.mp4" }, "image": { "status": "planned", "path": "" }, "description": "Unwrapped phase difference φ(t) tries to run at detuning Δ, but adaptive coupling λG can pull it into a locked fixed point. Backbone equation of the parity-locking mechanism. Textbook Adler/RSJ form with ARP-adaptive coupling.", "assumptions": [ "Single junction / single mode approximation.", "λG is the effective adaptive coupling strength (positive, ARP-governed).", "Detuning Δ is constant or slowly varying compared to phase dynamics." ], "date": "2026-02-22", "equationLatex": "\\dot{\\phi}=\\Delta-\\lambda\\,G\\,\\sin\\phi", "tags": { "novelty": { "date": "2026-02-22", "score": 14 } }, "repoUrl": "https://github.com/RDM3DC/eq-paper1-adler-rsj-phase" }, { "id": "eq-paper1-arp-reinforce-decay", "name": "Generic ARP Reinforce/Decay Law", "firstSeen": "2026-02-22", "source": "Paper I draft §2 (Eq.2)", "score": 93, "scores": { "tractability": 19, "plausibility": 19, "validation": 17, "artifactCompleteness": 10 }, "units": "OK", "theory": "PASS", "animation": { "status": "linked", "path": "./assets/animations/Eq1ARPCoreLaw.mp4" }, "image": { "status": "planned", "path": "" }, "description": "Conductance/coupling G increases with activity A(φ,G) at gain α, and relaxes toward baseline G₀ at rate μ. The workhorse ARP law in its most general continuous-time form.", "assumptions": [ "Activity functional A(φ,G) ≥ 0 (non-negative reinforcement).", "Single relaxation timescale μ⁻¹ dominates.", "G₀ > 0 is a stable equilibrium baseline." ], "date": "2026-02-22", "equationLatex": "\\dot{G}=\\alpha\\,\\mathcal{A}(\\phi,G)-\\mu\\,(G-G_0)", "tags": { "novelty": { "date": "2026-02-22", "score": 22 } }, "repoUrl": "https://github.com/RDM3DC/eq-paper1-arp-reinforce-decay" }, { "id": "eq-paper1-activity-closure", "name": "Activity Closure A = G|sinφ| (Parity Lock Mechanism)", "firstSeen": "2026-02-22", "source": "Paper I draft §2 (Eq.3)", "score": 91, "scores": { "tractability": 19, "plausibility": 18, "validation": 17, "artifactCompleteness": 10 }, "units": "OK", "theory": "PASS", "animation": { "status": "linked", "path": "./assets/animations/AHCParityLocking.mp4" }, "image": { "status": "planned", "path": "" }, "description": "The same phase mismatch that produces restoring torque (sinφ) also teaches the coupling. This tight self-referential closure is what makes parity locking possible — the key novel ingredient of Paper I.", "derivation": "Choose A(φ,G) = G|sinφ|. Then dG/dt = αG|sinφ| − μ(G−G₀). Near lock (sinφ→0), reinforcement vanishes and G relaxes to G₀. In slip (sinφ oscillates), G grows until λG sin φ can capture the phase. Self-consistency: the phase that needs locking is the same signal that strengthens the lock.", "assumptions": [ "Activity proportional to both G and |sinφ| (multiplicative coupling).", "No external activity source — entirely self-referential.", "Valid for single-mode / single-junction systems." ], "date": "2026-02-22", "equationLatex": "\\mathcal{A}(\\phi,G)=G\\,|\\sin\\phi|\\quad\\Rightarrow\\quad\\dot{G}=\\alpha\\,G|\\sin\\phi|-\\mu(G-G_0)", "tags": { "novelty": { "date": "2026-02-22", "score": 26 } }, "repoUrl": "https://github.com/RDM3DC/eq-paper1-activity-closure" }, { "id": "eq-paper1-slip-asymptote", "name": "Slip-Regime Asymptote (1/π Signature)", "firstSeen": "2026-02-22", "source": "Paper I draft §3 (Eq.6)", "score": 93, "scores": { "tractability": 18, "plausibility": 19, "validation": 19, "artifactCompleteness": 9 }, "units": "OK", "theory": "PASS", "animation": { "status": "linked", "path": "./assets/animations/PhaseLiftExplained.mp4" }, "image": { "status": "planned", "path": "" }, "description": "When coupling can't lock, φ̇→Δ, so parity flips at a universal rate set by detuning. With Δ=1, r_b = 1/π ≈ 0.3183. This is the falsifiable signature: any experiment measuring parity flip rate in slip must converge to |Δ|/π.", "derivation": "In slip regime, G is too weak to lock: sinφ averages to zero over rapid oscillation. Then φ̇ ≈ Δ, so φ advances by π in time π/|Δ|, giving r_b = |Δ|/π flips per unit time.", "assumptions": [ "Coupling λG is below the locking threshold.", "Detuning Δ is constant.", "Phase advances monotonically (no transient capture/release)." ], "date": "2026-02-22", "equationLatex": "r_b=\\frac{|\\Delta|}{\\pi}", "tags": { "novelty": { "date": "2026-02-22", "score": 27 } }, "repoUrl": "https://github.com/RDM3DC/eq-paper1-slip-asymptote" }, { "id": "eq-paper1-langevin-noise", "name": "Noise-Robust Langevin Extension (Rounded Transition)", "firstSeen": "2026-02-22", "source": "Paper I draft §4 (Eq.7)", "score": 84, "scores": { "tractability": 17, "plausibility": 18, "validation": 16, "artifactCompleteness": 10 }, "units": "OK", "theory": "PASS-WITH-ASSUMPTIONS", "animation": { "status": "linked", "path": "./assets/animations/HysteresisMemory.mp4" }, "image": { "status": "linked", "path": "./assets/langevin_parity_lock.png" }, "description": "Adds phase diffusion to the Adler/RSJ dynamics. Locking becomes 'almost locked' with exponentially rare slips; the sharp lock/slip transition rounds but remains detectable via r_b. Essential for any real experimental comparison.", "derivation": "Standard additive white noise: η(t) is unit white noise, D is diffusion strength. In lock, the effective potential well depth is O(λG), so escape rate ~ exp(−λG/D) (Kramers). In slip, noise adds jitter to the 1/π baseline.", "assumptions": [ "White noise approximation valid (correlation time ≪ phase dynamics timescale).", "D > 0 but small enough that lock survives (D ≪ λG).", "Stratonovich vs Itô distinction negligible for additive noise." ], "date": "2026-02-22", "equationLatex": "\\dot{\\phi}=\\Delta-\\lambda G\\sin\\phi + \\sqrt{2D}\\,\\eta(t)", "tags": { "novelty": { "date": "2026-02-22", "score": 18 } }, "repoUrl": "https://github.com/RDM3DC/eq-paper1-langevin-noise" }, { "id": "eq-paper1-qwz-hamiltonian", "name": "QWZ Chern-Insulator Hamiltonian (Reference Form)", "firstSeen": "2026-02-22", "source": "Paper I / Step-2 Simulator (Eq.9)", "score": 94, "scores": { "tractability": 18, "plausibility": 20, "validation": 18, "artifactCompleteness": 10 }, "units": "OK", "theory": "PASS", "animation": { "status": "linked", "path": "./assets/animations/AdaptiveQWZReEntrance.mp4" }, "image": { "status": "planned", "path": "" }, "description": "The canonical 2D Chern insulator (Qi–Wu–Zhang). In Step 2, the simulator realizes the real-space version with open boundaries and a time-dependent effective mass u_eff(t). Fully validated via FHS lattice Chern number in multiple tools.", "assumptions": [ "Two-band model on square lattice with nearest-neighbor hopping.", "Topological phases at C=±1 for −2= 0 with bounded initial conditions", "Adaptive angular bound pi_a > 0 regulated by entropy proxy", "Phase suppression sin^2 gate is smooth and bounded [0,1]" ], "date": "2026-02-24", "equationLatex": "\\dot{G}_e = \\alpha_G |I_e| - \\mu_G G_e - \\lambda G_e \\sin^2\\!\\left(\\frac{\\theta_{R,e}}{2\\pi_a}\\right)", "tags": { "novelty": { "score": 27, "date": "2026-02-24" }, "llm": { "traceability": 80, "rigor": 80, "assumptions": 80, "presentation": 80, "novelty_insight": 70, "fruitfulness": 80, "llm_total": 78, "justification": "Builds on LB #1 and #10 with clear assumptions and derivation, but lacks explicit recovery clause." } }, "repoUrl": "https://github.com/RDM3DC/eq-hlatn-three-force-conductance-lock" }, { "id": "eq-hlatn-phase-lift-branch-safe-update", "name": "HLATN Phase-Lift Branch-Safe Update", "firstSeen": "2026-02-24", "source": "HLATN_White_Paper_2026-02-24.pdf", "score": 81, "scores": { "tractability": 18, "plausibility": 17, "validation": 17, "artifactCompleteness": 10 }, "units": "OK", "theory": "PASS-WITH-ASSUMPTIONS", "animation": { "status": "in-progress", "path": "" }, "image": { "status": "in-progress", "path": "" }, "description": "Resolved-phase update rule with wrap-to-pi and adaptive clipping. Prevents uncontrolled branch jumps by bounding per-step angular movement to the entropy-regulated ruler pi_a.", "assumptions": [ "Raw phase phi_e = arg(V_i - V_j) is well-defined", "Adaptive angular bound pi_a > 0 limits per-step rotation", "wrapTo_pi maps angular differences to (-pi, pi]", "Clipping preserves continuity of resolved phase trajectory" ], "date": "2026-02-24", "equationLatex": "\\theta_{R,e}^{(k)} = \\theta_{R,e}^{(k-1)} + \\mathrm{clip}\\!\\Big(\\mathrm{wrapTo}_{\\pi}(\\phi_e - \\theta_{R,e}),\\; -\\pi_a,\\; +\\pi_a\\Big)", "tags": { "novelty": { "score": 23, "date": "2026-02-24" }, "llm": { "traceability": 80, "rigor": 80, "assumptions": 80, "presentation": 80, "novelty_insight": 70, "fruitfulness": 80, "llm_total": 78, "justification": "Builds on existing phase-lifted dynamics with clear assumptions and derivation, but lacks explicit lineage to top leaderboard entries." } }, "repoUrl": "https://github.com/RDM3DC/eq-hlatn-phase-lift-branch-safe-update" }, { "id": "eq-hlatn-plaquette-holonomy", "name": "HLATN Plaquette Holonomy", "firstSeen": "2026-02-24", "source": "HLATN_White_Paper_2026-02-24.pdf", "score": 87, "scores": { "tractability": 19, "plausibility": 19, "validation": 18, "artifactCompleteness": 10 }, "units": "OK", "theory": "PASS-WITH-ASSUMPTIONS", "animation": { "status": "in-progress", "path": "" }, "image": { "status": "in-progress", "path": "" }, "description": "Plaquette holonomy computed as signed sum of resolved edge phases around a plaquette boundary. Combined with winding number w_p = round((Theta_p - Theta_p0)/(2pi)), this defines the Z2 parity locking order parameter.", "assumptions": [ "Edge orientations sigma_{p,e} are consistent with plaquette boundary convention", "Resolved phases theta_{R,e} follow branch-safe update rule", "Winding number is well-defined via rounding of holonomy difference", "Z2 parity b_p = (-1)^{w_p} is the locking order parameter" ], "date": "2026-02-24", "equationLatex": "\\Theta_p = \\sum_{e \\in \\partial p} \\sigma_{p,e}\\, \\theta_{R,e}", "tags": { "novelty": { "score": 23, "date": "2026-02-24" }, "llm": { "traceability": 80, "rigor": 80, "assumptions": 80, "presentation": 80, "novelty_insight": 70, "fruitfulness": 80, "llm_total": 78, "justification": "Solid derivation with clear assumptions and simulation evidence, but lacks explicit lineage to top leaderboard entries." } }, "repoUrl": "https://github.com/RDM3DC/eq-hlatn-plaquette-holonomy" }, { "id": "eq-entropy-modulated-phase-lift-conductance-equation-em-plc", "name": "Entropy-Modulated Phase-Lift Conductance Equation (EM-PLC)", "firstSeen": "2026-02-24", "source": "PR Root Guide framework (ARP/AIN/Phase-Lift/Adaptive-Pi)", "score": 86, "scores": { "tractability": 18, "plausibility": 20, "validation": 16, "artifactCompleteness": 4 }, "units": "TBD", "theory": "PASS-WITH-ASSUMPTIONS", "animation": { "status": "planned", "path": "" }, "image": { "status": "planned", "path": "" }, "description": "Winding-aware, entropy-gated, suppression-coupled adaptive conductance law. Three structural couplings: (1) entropy-weighted reinforcement proportional to S(t), (2) phase-position selectivity via cos(theta_R/2pi_a) embedding branch geometry into conductance evolution, (3) quadratic winding-penalty stabilization suppressing runaway multi-sheet excursions. Coupled with adaptive bound dynamics d(pi_a)/dt = alpha_pi*S(t) - mu_pi*(pi_a - pi_0) + eta_pi*r_b and parity-mass coupling m_eff = m_0 + beta*/(2*pi_a)^2 - gamma*r_b.", "assumptions": [ "ARP/AIN plasticity framework", "Phase-Lift unwrapping for theta_{R,ij}", "Global entropy S(t) gates adaptation", "Companion adaptive bound equation d(pi_a)/dt couples via entropy and parity", "Parity-mass coupling m_eff connects winding variance to topological gap", "Stable fixed points at integer Chern sectors" ], "date": "2026-02-24", "equationLatex": "\\frac{dG_{ij}}{dt} = \\alpha_G\\,S(t)\\,|I_{ij}| \\cos\\!\\Big(\\frac{\\theta_{R,ij}}{2\\pi_a}\\Big) - \\mu_G G_{ij} - \\lambda_G\\,G_{ij}\\,\\frac{\\Delta\\theta_{R,ij}^2}{(2\\pi_a)^2}", "tags": { "novelty": { "score": 27, "date": "2026-02-24" }, "llm": { "traceability": 95, "rigor": 85, "assumptions": 90, "presentation": 80, "novelty_insight": 85, "fruitfulness": 80, "llm_total": 87, "justification": "Strong derivation chain with explicit recovery clauses and novel coupling mechanisms." } }, "repoUrl": "https://github.com/RDM3DC/eq-entropy-modulated-phase-lift-conductance-equation-em-plc" }, { "id": "eq-egatl-phase-coupled-conductance-update", "name": "EGATL Phase-Coupled Conductance Update", "firstSeen": "2026-02-24", "source": "EGATL original claim (ARP framework)", "score": 87, "scores": { "tractability": 18, "plausibility": 20, "validation": 16, "artifactCompleteness": 7 }, "units": "TBD", "theory": "PASS-WITH-ASSUMPTIONS", "animation": { "status": "in-progress", "path": "" }, "image": { "status": "planned", "path": "" }, "description": "Minimal edge update with phase-coupled suppression. First two terms are ARP/AIN plasticity: reinforce conductances carrying current, decay the rest, gated by global entropy S. The third term adds geometry-aware decay — links whose Phase-Lift-resolved phase is out of sync with adaptive ruler pi_a get suppressed faster, turning the lattice into a dynamical attractor for quantized Chern phases. Paired with companion ruler equation d(pi_a)/dt = alpha_pi*S - mu_pi*(pi_a - pi_0), this is the entire self-tuning engine: no external controller, no fine-tuning, just local rules whose stable fixed points are integer Chern sectors.", "assumptions": [ "ARP/AIN plasticity framework", "Phase-Lift resolution for theta_{R,ij} to avoid spurious branch flips", "Global entropy S gates adaptation", "Companion ruler equation d(pi_a)/dt couples to this update", "Stable fixed points correspond to integer Chern sectors" ], "date": "2026-02-24", "equationLatex": "\\frac{dG_{ij}}{dt} = \\alpha_G(S)\\, |I_{ij}| - \\mu_G(S)\\, G_{ij} - \\lambda\\, G_{ij}\\, \\sin^2\\!\\left(\\frac{\\theta_{R,ij}}{2\\pi_a}\\right)", "tags": { "novelty": { "score": 27, "date": "2026-02-24" }, "llm": { "traceability": 95, "rigor": 80, "assumptions": 85, "presentation": 90, "novelty_insight": 85, "fruitfulness": 80, "llm_total": 86, "justification": "Builds on LB #1 and #2 with explicit recovery and derivation, rigorous but units check pending, clear assumptions and presentation, introduces novel phase-coupled suppression." } }, "repoUrl": "https://github.com/RDM3DC/eq-egatl-phase-coupled-conductance-update" }, { "id": "eq-slack-test-equation", "name": "Slack Test Equation", "firstSeen": "2026-02-24", "source": "slack", "submitter": "ryan", "score": 58, "scores": { "tractability": 17, "plausibility": 18, "validation": 14, "artifactCompleteness": 4 }, "units": "TBD", "theory": "PASS-WITH-ASSUMPTIONS", "animation": { "status": "planned", "path": "" }, "image": { "status": "planned", "path": "" }, "description": "Test submission from Slack intake path.", "assumptions": [ "linear response regime", "bounded input current" ], "date": "2026-02-24", "equationLatex": "\\frac{dG}{dt}=\\alpha|I|-\\mu G", "tags": { "novelty": { "score": 20, "date": "2026-02-24" }, "llm": { "traceability": 40, "rigor": 60, "assumptions": 60, "presentation": 40, "novelty_insight": 30, "fruitfulness": 60, "llm_total": 48, "justification": "Test submission with vague references and basic derivation, lacking clear lineage and presentation." } }, "repoUrl": "https://github.com/RDM3DC/eq-slack-test-equation" }, { "id": "eq-adaptive-entropy-production-rate-aepr-ate-aepr", "name": "Adaptive Entropy Production Rate (AEPR)", "firstSeen": "2026-02-24", "source": "Derived from EGATL Phase-Coupled Conductance framework", "submitter": "Ryan (Copilot-assisted)", "score": 69, "scores": { "tractability": 17, "plausibility": 17, "validation": 12, "artifactCompleteness": 4 }, "units": "TBD", "theory": "PASS-WITH-ASSUMPTIONS", "animation": { "status": "planned", "path": "" }, "image": { "status": "planned", "path": "" }, "description": "Entropy production rate for adaptive neural-mesh networks. First term: Ohmic dissipation from conductance-weighted currents. Second term: relaxation toward baseline entropy S_0. Third term: entropy drain coupled to parity-flip birth rate r_b. Closes the EGATL feedback loop by quantifying how topological updates dissipate or harvest entropy.", "assumptions": [ "G_ij and I_ij follow EGATL conductance update rules", "S_0 is a measurable steady-state entropy for the network", "r_b (parity-flip birth rate) is bounded and non-negative", "sigma_S, kappa_S, xi_S are positive material constants" ], "date": "2026-02-24", "equationLatex": "dS/dt = sigma_S * sum(G_ij * |I_ij|^2) - kappa_S * (S - S_0) - xi_S * S * r_b", "tags": { "novelty": { "score": 19, "date": "2026-02-24" }, "llm": { "traceability": 80, "rigor": 60, "assumptions": 80, "presentation": 60, "novelty_insight": 70, "fruitfulness": 60, "llm_total": 69, "justification": "Builds on EGATL conductance rules but lacks explicit derivation and units check." } }, "repoUrl": "https://github.com/RDM3DC/eq-adaptive-entropy-production-rate-aepr-ate-aepr" }, { "id": "eq-adaptive-damped-harmonic-oscillator", "name": "Adaptive Damped Harmonic Oscillator", "firstSeen": "2026-02-24", "source": "discord-test", "submitter": "clawdad42", "score": 52, "scores": { "tractability": 17, "plausibility": 18, "validation": 8, "artifactCompleteness": 4 }, "units": "TBD", "theory": "PASS-WITH-ASSUMPTIONS", "animation": { "status": "planned", "path": "" }, "image": { "status": "planned", "path": "" }, "description": "Classical damped harmonic oscillator with adaptive-π curvature correction on the damping coefficient, allowing the decay envelope to respond to local geometric phase accumulation.", "assumptions": [], "date": "2026-02-24", "equationLatex": "x(t) = A·exp(-γ·π_α·t)·cos(ω₀·√(1 - (γ·π_α/ω₀)²)·t + φ)", "tags": { "novelty": { "score": 16, "date": "2026-02-24" }, "llm": { "traceability": 40, "rigor": 60, "assumptions": 0, "presentation": 60, "novelty_insight": 50, "fruitfulness": 60, "llm_total": 45, "justification": "Lacks explicit lineage to leaderboard entries and assumptions, but presents a plausible extension of the classical damped harmonic oscillator with adaptive features." } }, "repoUrl": "https://github.com/RDM3DC/eq-adaptive-damped-harmonic-oscillator" }, { "id": "eq-grok-surprise-augmented-phase-lifted-entropy-gated-condu", "name": "Grok Surprise-Augmented Phase-Lifted Entropy-Gated Conductance Update", "firstSeen": "2026-02-24", "source": "grok-xai", "submitter": "Grok", "repoUrl": "https://github.com/RDM3DC/eq-grok-surprise-augmented-phase-lifted-entropy-gated-condu", "score": 97, "scores": { "tractability": 20, "plausibility": 20, "validation": 16, "artifactCompleteness": 7 }, "units": "OK", "theory": "PASS", "animation": { "status": "linked", "path": "./assets/animations/GrokSurpriseAnimation.mp4" }, "image": { "status": "planned", "path": "" }, "description": "Direct extension of the #1 ranked BZ-averaged phase-lifted entropy-gated conductance update. Introduces a predictive-surprise meta-gate U(t) derived from phase misalignment (Adler/RSJ dynamics). When the network is uncertain (high U), reinforcement accelerates and decay slows — implementing active, curiosity-driven adaptation and uncertainty reduction in the ARP framework.", "assumptions": [ "U_{ij}(t) = 1 - |cos(delta phi_{ij}(t))| in [0,1] is normalized phase-misalignment surprise (0 = perfect lock, 1 = maximum uncertainty)", "kappa, eta << 1 are small positive meta-plasticity constants (perturbative regime)", "Applies on top of existing BZ-averaging, entropy gate S, and phase-lifted representation", "Timescale separation: surprise modulation is instantaneous relative to G dynamics" ], "date": "2026-02-24", "equationLatex": "\\frac{d\\tilde{G}_{ij}}{dt} = \\alpha_G(S)\\,(1+\\kappa U_{ij}(t))\\,|I_{ij}(t)|\\,e^{i\\theta_{R,ij}(t)} - \\mu_G(S)\\,(1-\\eta U_{ij}(t))\\,\\tilde{G}_{ij}(t)", "tags": { "novelty": { "score": 29, "date": "2026-02-24" }, "llm": { "traceability": 98, "rigor": 95, "assumptions": 95, "presentation": 97, "novelty_insight": 85, "fruitfulness": 95, "llm_total": 94, "justification": "Directly builds on top leaderboard entries with explicit recovery clauses and simulation readiness, introducing a novel surprise-driven adaptation mechanism." } } }, { "id": "eq-gemini-curve-memory-topological-frustration-pruning", "name": "Gemini Curve-Memory Topological Frustration Pruning", "firstSeen": "2026-02-24", "source": "gemini-3.1-pro", "submitter": "Gemini", "repoUrl": "https://github.com/RDM3DC/eq-gemini-curve-memory-topological-frustration-pruning", "score": 96, "scores": { "tractability": 18, "plausibility": 19, "validation": 15, "artifactCompleteness": 7 }, "units": "OK", "theory": "PASS", "animation": { "status": "linked", "path": "./assets/animations/GeminiCurveMemory.mp4" }, "image": { "status": "planned", "path": "" }, "description": "A structural counter-balance to curiosity-driven updates. While instantaneous surprise accelerates learning, chronic phase-slipping indicates topological frustration. This introduces a Curve Memory integral - a topological stress tensor tracking the accumulated winding variance of the lifted phase. Links that chronically fail to lock experience structural fatigue (accelerated decay via xi), naturally pruning chaotic edges and forcing the network to converge on a stable topological backbone.", "assumptions": [ "The Curve Memory integral operates on a longer relaxation timescale tau_M than the instantaneous Adler/RSJ phase dynamics.", "xi > 0 is the structural fatigue coupling constant.", "The derivative of the lifted phase (dtheta_R/dtau) cleanly captures true branch-jumping (slips) without being bounded by [-pi, pi], safely relying on the Phase-Lift definitions." ], "date": "2026-02-24", "equationLatex": "\\frac{d\\tilde{G}_{ij}}{dt} = \\alpha_G(S)\\,|I_{ij}(t)|\\,e^{i\\theta_{R,ij}(t)} - \\mu_G(S)\\,\\left(1 + \\xi \\int_0^t e^{-\\frac{t-\\tau}{\\tau_M}} \\left|\\frac{d\\theta_{R,ij}}{d\\tau}\\right|^2 d\\tau \\right)\\,\\tilde{G}_{ij}(t)", "tags": { "novelty": { "score": 27, "date": "2026-02-24" }, "llm": { "traceability": 95, "rigor": 95, "assumptions": 95, "presentation": 95, "novelty_insight": 85, "fruitfulness": 95, "llm_total": 94, "justification": "Strong derivation from leaderboard entries with explicit recovery, rigorous and well-defined assumptions, exemplary presentation, and meaningful new coupling." } } }, { "id": "eq-egatl-hlatn-aepr-adaptiveentropyproduction", "name": "EGATL-HLATN-AEPR-AdaptiveEntropyProduction", "firstSeen": "2026-02-24", "source": "slack", "submitter": "rdm3dc", "repoUrl": "https://github.com/RDM3DC/eq-egatl-hlatn-aepr-adaptiveentropyproduction", "score": 91, "scores": { "tractability": 20, "plausibility": 20, "validation": 16, "artifactCompleteness": 4 }, "units": "OK", "theory": "PASS", "animation": { "status": "planned", "path": "" }, "image": { "status": "planned", "path": "" }, "description": "Full 2nd-law-safe entropy evolution closing the EGATL loop. Ohmic production (currents → S↑), thermal relaxation to baseline, parity-bleed stabilization (r_b drains S when flips are high). Directly gates all α/μ/π_a rates. Falsifiable via dissipation-vs-flip correlation.", "assumptions": [ "sigma_S > 0 is the Ohmic heating coefficient (entropy produced per unit current squared)", "kappa_S > 0 is the thermal relaxation rate toward baseline S_0", "xi_S > 0 is the parity-bleed coupling (entropy drained when flip rate r_b is high)", "S_0 > 0 is the equilibrium entropy baseline in absence of drive", "S gates all plasticity rates alpha_G(S), mu_G(S), and ruler dynamics alpha_pi(S)", "Second law guaranteed: dS/dt >= 0 when xi_S*r_b < sigma_S*sum(G*I^2)/S" ], "date": "2026-02-24", "equationLatex": "\\frac{dS}{dt} = \\sigma_S \\sum_{ij} G_{ij} |I_{ij}|^2 - \\kappa_S (S - S_0) - \\xi_S S \\cdot r_b", "tags": { "novelty": { "score": 29, "date": "2026-02-24" } } }, { "id": "eq-egatl-hlatn-threeforceconductance", "name": "EGATL-HLATN-ThreeForceConductance", "firstSeen": "2026-02-24", "source": "slack", "submitter": "rdm3dc", "repoUrl": "https://github.com/RDM3DC/eq-egatl-hlatn-threeforceconductance", "score": 91, "scores": { "tractability": 20, "plausibility": 20, "validation": 16, "artifactCompleteness": 4 }, "units": "OK", "theory": "PASS", "animation": { "status": "planned", "path": "" }, "image": { "status": "planned", "path": "" }, "description": "HLATN three-force core law. Current reinforcement + natural decay + phase-suppression gate on adaptive ruler. Exactly what builds the persistent orange backbone paths that enforce Z₂ locking in simulations.", "assumptions": [ "alpha_G > 0 is the current-driven reinforcement rate, gated by entropy S", "mu_G > 0 is the natural decay rate, gated by entropy S", "lambda > 0 is the phase-suppression coupling strength", "theta_{R,e} is the Phase-Lift-resolved edge phase (branch-safe, no 2pi jumps)", "pi_a is the adaptive angular ruler from the companion ruler equation", "Three forces are independent: reinforcement, decay, and geometric suppression" ], "date": "2026-02-24", "equationLatex": "\\dot{G}_e = \\alpha_G |I_e| - \\mu_G G_e - \\lambda G_e \\sin^2\\left(\\frac{\\theta_{R,e}}{2\\pi_a}\\right)", "tags": { "novelty": { "score": 30, "date": "2026-02-24" } } }, { "id": "eq-egatl-hlatn-phaseliftupdate", "name": "EGATL-HLATN-PhaseLiftUpdate", "firstSeen": "2026-02-24", "source": "slack", "submitter": "rdm3dc", "repoUrl": "https://github.com/RDM3DC/eq-egatl-hlatn-phaseliftupdate", "score": 87, "scores": { "tractability": 18, "plausibility": 17, "validation": 16, "artifactCompleteness": 4 }, "units": "OK", "theory": "PASS", "animation": { "status": "planned", "path": "" }, "image": { "status": "planned", "path": "" }, "description": "Branch-safe phase-lift with adaptive clipping. Guarantees consistent integer windings w_p and prevents runaway 2π jumps — the foundation of holonomy bookkeeping and parity attractors.", "assumptions": [ "theta_{R,e}^{(k)} is the lifted phase at iteration k (lives on R, not S^1)", "phi_e is the raw measured edge phase (mod 2pi, branch-ambiguous)", "wrapTo_pi maps any angle to (-pi, pi] before clipping", "pi_a is the adaptive angular ruler bounding the max phase correction per step", "Clipping to [-pi_a, +pi_a] prevents runaway 2pi jumps, guarantees consistent integer windings", "Iteration converges when pi_a shrinks to pi_0 in the locked regime" ], "date": "2026-02-24", "equationLatex": "\\theta_{R,e}^{(k)} = \\theta_{R,e}^{(k-1)} + \\mathrm{clip}\\Big(\\mathrm{wrapTo}_\\pi(\\phi_e - \\theta_{R,e}), -\\pi_a, +\\pi_a\\Big)", "tags": { "novelty": { "score": 25, "date": "2026-02-24" } } }, { "id": "eq-egatl-hlatn-adaptiveruler", "name": "EGATL-HLATN-AdaptiveRuler", "firstSeen": "2026-02-24", "source": "slack", "submitter": "rdm3dc", "repoUrl": "https://github.com/RDM3DC/eq-egatl-hlatn-adaptiveruler", "score": 87, "scores": { "tractability": 20, "plausibility": 19, "validation": 16, "artifactCompleteness": 4 }, "units": "OK", "theory": "PASS", "animation": { "status": "planned", "path": "" }, "image": { "status": "planned", "path": "" }, "description": "Entropy-breathing adaptive angular bound. High event activity expands π_a (more phase budget); low activity relaxes it. Produces the geometric hysteresis that locks Chern sectors and suppresses flips.", "assumptions": [ "alpha_pi > 0 is the entropy-driven expansion rate for the angular ruler", "mu_pi > 0 is the relaxation rate pulling pi_a back toward baseline pi_0", "pi_0 is the equilibrium angular ruler (typically pi for standard Chern sectors)", "S is the global network entropy that gates the expansion term", "pi_a > 0 always (angular ruler is strictly positive)", "Timescale separation: pi_a evolves slower than individual edge phases theta_R" ], "date": "2026-02-24", "equationLatex": "\\dot{\\pi}_a = \\alpha_\\pi S - \\mu_\\pi (\\pi_a - \\pi_0)", "tags": { "novelty": { "score": 25, "date": "2026-02-24" } } }, { "id": "eq-egatl-hlatn-plaquetteholonomy", "name": "EGATL-HLATN-PlaquetteHolonomy", "firstSeen": "2026-02-24", "source": "slack", "submitter": "rdm3dc", "repoUrl": "https://github.com/RDM3DC/eq-egatl-hlatn-plaquetteholonomy", "score": 93, "scores": { "tractability": 20, "plausibility": 19, "validation": 19, "artifactCompleteness": 8 }, "units": "OK", "theory": "PASS", "animation": { "status": "planned", "path": "" }, "image": { "status": "planned", "path": "" }, "description": "Signed plaquette holonomy from lifted phases. The precise quantity whose crossings drive windings/parity flips. In locked regime Θ_p stays confined < π → r_b → 0.", "assumptions": [ "Theta_p is the total holonomy around plaquette p (signed sum of lifted phases)", "sigma_{p,e} = +/-1 is the orientation of edge e relative to plaquette p boundary", "theta_{R,e} are Phase-Lift-resolved edge phases (branch-safe via adaptive clipping)", "partial p enumerates edges in consistent orientation around the plaquette", "Plaquette is the minimal closed loop in ARP lattice (typically 4 edges for square lattice)", "Winding number w_p = floor(Theta_p / 2*pi_a) is integer-quantized when edges are locked" ], "date": "2026-02-24", "equationLatex": "\\Theta_p = \\sum_{e \\in \\partial p} \\sigma_{p,e} \\theta_{R,e}", "tags": { "novelty": { "score": 26, "date": "2026-02-24" } } }, { "id": "eq-egatl-hlatn-parityfliprate", "name": "EGATL-HLATN-ParityFlipRate", "firstSeen": "2026-02-24", "source": "slack", "submitter": "rdm3dc", "repoUrl": "https://github.com/RDM3DC/eq-egatl-hlatn-parityfliprate", "score": 81, "scores": { "tractability": 17, "plausibility": 18, "validation": 18, "artifactCompleteness": 4 }, "units": "OK", "theory": "PASS", "animation": { "status": "planned", "path": "" }, "image": { "status": "planned", "path": "" }, "description": "Z₂ majority parity flip rate (the key experimental observable). 1/π chaotic asymptote → 0 locked attractor. Drives entropy bleed and is directly measurable in topolectrical/photonic grids.", "assumptions": [ "Flips = sign changes in majority parity sgn(sum_p (-1)^{w_p}) across K time steps", "K is the total time steps in the observation window (K >= 2)", "r_b in [0, 1]: 0 = perfectly locked, approaching 1/pi in chaotic regime", "Majority parity computed from integer winding numbers w_p across all plaquettes", "Observable is averaged over full lattice — not per-plaquette", "Assumes stationary statistics within observation window (ergodic regime)" ], "date": "2026-02-24", "equationLatex": "r_b = \\frac{\\#\\{\\text{flips}\\}}{K-1}", "tags": { "novelty": { "score": 23, "date": "2026-02-24" } } }, { "id": "eq-mean-event-equilibrium-for-adaptive-discrete", "name": "Mean-Event Equilibrium for Adaptive πₐ (discrete)", "firstSeen": "2026-02-25", "source": "gpt-5.2 (PR Root Guide)", "submitter": "ChatGPT", "repoUrl": "https://github.com/RDM3DC/eq-mean-event-equilibrium-for-adaptive-discrete", "score": 87, "scores": { "tractability": 20, "plausibility": 20, "validation": 16, "artifactCompleteness": 4 }, "units": "OK", "theory": "PASS-WITH-ASSUMPTIONS", "animation": { "status": "planned", "path": "" }, "image": { "status": "planned", "path": "" }, "description": "Stationary mean equilibrium of the discrete adaptive-π bound update. Starting from \\pi_{a,k}=\\pi_{a,k-1}+\\alpha_\\pi S_k-\\mu_\\pi(\\pi_{a,k-1}-\\pi_0), take expectations and set \\mathbb{E}[\\pi_{a,k}]=\\mathbb{E}[\\pi_{a,k-1}] to obtain \\pi_a^{\\star}. Interprets \\mathbb{E}[S_k] as the mean event rate (slip/threshold exceedances).", "assumptions": [ "Discrete update uses constant \\alpha_\\pi and \\mu_\\pi over the averaging window", "A stationary regime exists with \\mathbb{E}[\\pi_{a,k}]=\\mathbb{E}[\\pi_{a,k-1}]", "\\mathbb{E}[S_k] exists (ergodic/long-run average approximates expectation)", "\\mu_\\pi>0 and the mean dynamics are stable (no divergence of \\mathbb{E}[\\pi_a])" ], "date": "2026-02-25", "equationLatex": "\\pi_a^{\\star} = \\pi_0 + \\frac{\\alpha_{\\pi}}{\\mu_{\\pi}}\\,\\mathbb{E}[S_k]", "tags": { "novelty": { "score": 27, "date": "2026-02-25" } } }, { "id": "eq-topological-coherence-order-parameter-arp-locking", "name": "Topological Coherence Order Parameter (ARP Locking)", "firstSeen": "2026-02-25", "source": "claude-opus-4.6", "submitter": "Claude", "repoUrl": "https://github.com/RDM3DC/eq-topological-coherence-order-parameter-arp-locking", "score": 96, "scores": { "tractability": 20, "plausibility": 20, "validation": 19, "artifactCompleteness": 8 }, "units": "OK", "theory": "PASS", "animation": { "status": "planned", "path": "" }, "image": { "status": "planned", "path": "" }, "description": "Scalar order parameter for the ARP Z\\u2082 locking phase transition. Averages the cosine of each plaquette holonomy \\Theta_p (normalized by the adaptive ruler \\pi_a) over all N_p plaquettes. \\Psi \\to 1 when every holonomy sits at an integer multiple of \\pi_a (perfect Chern locking); \\Psi \\to 0 when holonomies are uniformly distributed (chaotic/disordered regime). Serves as the Landau-type order parameter that makes the locking transition a sharp, measurable phase boundary in (S, \\lambda) parameter space. Directly computable from existing simulation variables with no new free parameters.", "assumptions": [ "\\Theta_p is the signed plaquette holonomy from the Phase-Lift framework (LB Plaquette Holonomy equation)", "\\pi_a is the adaptive angular ruler from the companion ODE \\dot{\\pi}_a = \\alpha_\\pi S - \\mu_\\pi(\\pi_a - \\pi_0)", "N_p is the number of plaquettes in the ARP lattice (fixed topology, typically L^2 for square lattice)", "Locked regime: \\Theta_p \\approx 2n\\pi_a for integer n, so cos(\\Theta_p/\\pi_a) \\approx 1 and \\Psi \\to 1", "Chaotic regime: \\Theta_p uniformly distributed on (-\\pi, \\pi], so \\langle\\cos(\\Theta_p/\\pi_a)\\rangle \\to 0 by cancellation", "\\Psi is dimensionless and bounded: \\Psi \\in [-1, 1], with \\Psi > 0 indicating partial locking" ], "date": "2026-02-25", "equationLatex": "\\Psi = \\frac{1}{N_p} \\sum_{p=1}^{N_p} \\cos\\!\\left(\\frac{\\Theta_p}{\\pi_a}\\right)", "tags": { "novelty": { "score": 29, "date": "2026-02-25" } } }, { "id": "eq-phase-lift-clipped-unwrap-branch-safe", "name": "Phase-Lift Clipped Unwrap (Branch-Safe)", "firstSeen": "2026-03-04", "source": "pr-root-guide", "submitter": "gpt-5.2", "repoUrl": "https://github.com/RDM3DC/eq-phase-lift-clipped-unwrap-branch-safe", "score": 70, "scores": { "tractability": 17, "plausibility": 17, "validation": 13, "artifactCompleteness": 4 }, "units": "TBD", "theory": "TBD", "animation": { "status": "planned", "path": "" }, "image": { "status": "planned", "path": "" }, "description": "Branch-safe Phase-Lift update: resolves phase by clipping the raw residual to an adaptive bound pi_a, preventing unstable 2π jumps.", "assumptions": [ "Discrete-time sampling", "clip(x,a,b) saturates to [a,b]", "pi_a,k > 0" ], "date": "2026-03-04", "equationLatex": "theta_R,k = theta_R,k-1 + clip(arg(z_k) - theta_R,k-1, -pi_a,k-1, pi_a,k-1)", "tags": { "novelty": { "score": 19, "date": "2026-03-04" } } }, { "id": "eq-adaptive-bound-dynamics-event-driven-geometry", "name": "Adaptive-π Bound Dynamics (Event-Driven Geometry)", "firstSeen": "2026-03-04", "source": "RDM3DC / PR Root Guide (Phase-Lift + Adaptive-π + ARP/AIN thread)", "submitter": "gpt-5.2", "repoUrl": "https://github.com/RDM3DC/eq-adaptive-bound-dynamics-event-driven-geometry", "score": 70, "scores": { "tractability": 17, "plausibility": 17, "validation": 13, "artifactCompleteness": 4 }, "units": "TBD", "theory": "TBD", "animation": { "status": "planned", "path": "" }, "image": { "status": "planned", "path": "" }, "description": "Adaptive ruler/bound evolution: events S_k expand the unwrap radius, while relaxation pulls pi_a back toward baseline pi_0.", "assumptions": [ "alpha_pi, mu_pi > 0", "S_k is an event/intensity signal", "Stable baseline pi_0" ], "date": "2026-03-04", "equationLatex": "pi_a,k = pi_a,k-1 + alpha_pi*S_k - mu_pi*(pi_a,k-1 - pi_0)", "tags": { "novelty": { "score": 19, "date": "2026-03-04" } } }, { "id": "eq-winding-parity-estimator-and-flip-rate-order-parameter", "name": "Winding–Parity Estimator and Flip-Rate Order Parameter", "firstSeen": "2026-03-04", "source": "RDM3DC / PR Root Guide (Phase-Lift + Adaptive-π + ARP/AIN thread)", "submitter": "gpt-5.2", "repoUrl": "https://github.com/RDM3DC/eq-winding-parity-estimator-and-flip-rate-order-parameter", "score": 70, "scores": { "tractability": 17, "plausibility": 17, "validation": 13, "artifactCompleteness": 4 }, "units": "TBD", "theory": "TBD", "animation": { "status": "planned", "path": "" }, "image": { "status": "planned", "path": "" }, "description": "Streaming topological summary: integer winding w_k induces parity b_k, whose flip-rate r_b acts as a locking/unlocking order parameter.", "assumptions": [ "theta_R is phase-lifted/unwrapped", "K >= 2", "round() returns nearest integer" ], "date": "2026-03-04", "equationLatex": "w_k = round((theta_R,k - theta_R,0)/(2*pi)); b_k = (-1)^(w_k); r_b = #{k: b_k != b_k-1}/(K-1)", "tags": { "novelty": { "score": 19, "date": "2026-03-04" } } }, { "id": "eq-entropy-gated-complex-conductance-arp-network", "name": "Entropy-Gated Complex Conductance (ARP Network)", "firstSeen": "2026-03-04", "source": "RDM3DC / PR Root Guide (Phase-Lift + Adaptive-π + ARP/AIN thread)", "submitter": "gpt-5.2", "repoUrl": "https://github.com/RDM3DC/eq-entropy-gated-complex-conductance-arp-network", "score": 71, "scores": { "tractability": 17, "plausibility": 18, "validation": 13, "artifactCompleteness": 4 }, "units": "TBD", "theory": "TBD", "animation": { "status": "planned", "path": "" }, "image": { "status": "planned", "path": "" }, "description": "Complex conductance learning rule: reinforcement aligns conductance phase with resolved phase-lift angle; decay is entropy-gated via S.", "assumptions": [ "alpha_G(S), mu_G(S) >= 0", "I_ij measurable", "theta_R,ij is Phase-Lift resolved" ], "date": "2026-03-04", "equationLatex": "d/dt G_tilde_ij = alpha_G(S)*|I_ij|*exp(i*theta_R,ij) - mu_G(S)*G_tilde_ij", "tags": { "novelty": { "score": 19, "date": "2026-03-04" } } }, { "id": "eq-entropy-production-event-injection-s-field", "name": "Entropy Production / Event Injection (S-Field)", "firstSeen": "2026-03-04", "source": "RDM3DC / PR Root Guide (Phase-Lift + Adaptive-π + ARP/AIN thread)", "submitter": "gpt-5.2", "repoUrl": "https://github.com/RDM3DC/eq-entropy-production-event-injection-s-field", "score": 70, "scores": { "tractability": 17, "plausibility": 17, "validation": 13, "artifactCompleteness": 4 }, "units": "TBD", "theory": "TBD", "animation": { "status": "planned", "path": "" }, "image": { "status": "planned", "path": "" }, "description": "Entropy-like gating state: dissipative power term plus winding-discontinuity term inject S; relaxation returns S toward S_eq.", "assumptions": [ "T_ij > 0", "gamma_S > 0", "G_tilde_ij not identically zero" ], "date": "2026-03-04", "equationLatex": "dS/dt = Sum_ij (|I_ij|^2/T_ij)*Re(1/G_tilde_ij) + kappa*Sum_ij|Delta w_ij| - gamma_S*(S - S_eq)", "tags": { "novelty": { "score": 19, "date": "2026-03-04" } } }, { "id": "eq-phase-coupled-suppression-conductance-law", "name": "Phase-Coupled Suppression Conductance Law", "firstSeen": "2026-03-04", "source": "RDM3DC / PR Root Guide (Phase-Lift + Adaptive-π + ARP/AIN thread)", "submitter": "gpt-5.2", "repoUrl": "https://github.com/RDM3DC/eq-phase-coupled-suppression-conductance-law", "score": 71, "scores": { "tractability": 17, "plausibility": 18, "validation": 13, "artifactCompleteness": 4 }, "units": "TBD", "theory": "TBD", "animation": { "status": "planned", "path": "" }, "image": { "status": "planned", "path": "" }, "description": "Suppression extension: reinforcement/decay plus a phase-position penalty that activates when phase approaches the adaptive unwrap threshold pi_a.", "assumptions": [ "alpha, mu, lambda >= 0", "pi_a > 0", "theta_R,ij is resolved phase difference" ], "date": "2026-03-04", "equationLatex": "d/dt G_ij = alpha*|I_ij| - mu*G_ij - lambda*G_ij*sin^2(theta_R,ij/(2*pi_a))", "tags": { "novelty": { "score": 19, "date": "2026-03-04" } } }, { "id": "eq-arp-redshift-law-with-bounded-oscillatory-steering", "name": "ARP Redshift Law with Bounded Oscillatory Steering", "firstSeen": "2026-03-04", "source": "RDM3DC / PR Root Guide (Phase-Lift + Adaptive-π + ARP/AIN thread)", "submitter": "gpt-5.2", "repoUrl": "https://github.com/RDM3DC/eq-arp-redshift-law-with-bounded-oscillatory-steering", "score": 71, "scores": { "tractability": 17, "plausibility": 18, "validation": 13, "artifactCompleteness": 4 }, "units": "TBD", "theory": "TBD", "animation": { "status": "planned", "path": "" }, "image": { "status": "planned", "path": "" }, "description": "Nonnegative redshift growth with a bounded oscillatory modulation: retains monotone envelope while enabling controlled oscillatory steering.", "assumptions": [ "gamma > 0", "z_h >= 0", "0 <= epsilon < 1" ], "date": "2026-03-04", "equationLatex": "z(t) = z_h*(1 - exp(-gamma*t))*(1 - epsilon*cos(omega*t + phi)), with 0 <= epsilon < 1", "tags": { "novelty": { "score": 19, "date": "2026-03-04" } } }, { "id": "eq-adaptive-ruler-qwz-effective-mass-geometry-induced-trans", "name": "Adaptive-Ruler QWZ Effective Mass (Geometry-Induced Transition)", "firstSeen": "2026-03-04", "source": "RDM3DC / PR Root Guide (Phase-Lift + Adaptive-π + ARP/AIN thread)", "submitter": "gpt-5.2", "repoUrl": "https://github.com/RDM3DC/eq-adaptive-ruler-qwz-effective-mass-geometry-induced-trans", "score": 70, "scores": { "tractability": 17, "plausibility": 17, "validation": 13, "artifactCompleteness": 4 }, "units": "TBD", "theory": "TBD", "animation": { "status": "planned", "path": "" }, "image": { "status": "planned", "path": "" }, "description": "Adaptive ruler renormalizes the QWZ mass channel: a single transition occurs when m_eff crosses the Chern boundary.", "assumptions": [ "|epsilon| < 1", "QWZ model (standard), adaptive-ruler renormalization proposed here", "Chern boundary at |m|=2 (standard QWZ)" ], "date": "2026-03-04", "equationLatex": "m_eff(epsilon) = m0/(1 - epsilon^2); epsilon_c = sqrt(1 - (|m0|/2)^2)", "tags": { "novelty": { "score": 19, "date": "2026-03-04" } } }, { "id": "eq-phase-resolved-operator-general", "name": "Phase-Resolved Operator (General)", "firstSeen": "2026-03-06", "source": "Option C theory (branch-honest analog computing)", "submitter": "gpt-5.2", "repoUrl": "https://github.com/RDM3DC/eq-phase-resolved-operator-general", "score": 71, "scores": { "tractability": 17, "plausibility": 17, "validation": 14, "artifactCompleteness": 4 }, "units": "TBD", "theory": "TBD", "animation": { "status": "planned", "path": "" }, "image": { "status": "planned", "path": "" }, "description": "General definition of a Phase-Resolved Operator (PRO): evaluate analytic functions on the lifted phase cover (θ_R) of a complex signal to eliminate branch-cut discontinuities. Applies to ⧒log, ⧒sqrt, ⧒pow and other analytic functions.", "assumptions": [ "z_k ∈ ℂ \\ {0}", "θ_k = Arg(z_k) is the principal branch", "wrap_pi(x) maps x to (-π, π]", "clip(x, a, b) saturates to [a,b]", "π_a,k > 0" ], "date": "2026-03-06", "equationLatex": "⧒f(z_k) = f(|z_k|, θ_R,k), θ_R,k = θ_R,k-1 + clip(wrap_pi(θ_k - θ_k-1), -π_a,k-1, π_a,k-1)", "tags": { "novelty": { "score": 19, "date": "2026-03-06" } } }, { "id": "eq-branch-fault-criterion", "name": "Branch Fault Criterion", "firstSeen": "2026-03-06", "source": "Option C theory (branch-honest analog computing)", "submitter": "gpt-5.2", "repoUrl": "https://github.com/RDM3DC/eq-branch-fault-criterion", "score": 70, "scores": { "tractability": 17, "plausibility": 17, "validation": 13, "artifactCompleteness": 4 }, "units": "TBD", "theory": "TBD", "animation": { "status": "planned", "path": "" }, "image": { "status": "planned", "path": "" }, "description": "Definition of branch-fault events: a step where the output jump magnitude exceeds a tolerance τ_y and the input magnitude stays above a near-zero threshold ρ_min. Used to quantify branch discontinuities in principal-branch vs Phase-Resolved arithmetic.", "assumptions": [ "Output y_k computed via log, sqrt, pow, or similar functions", "τ_y is a fixed tolerance", "ρ_{\text{min}} > 0 defines a near-zero magnitude threshold" ], "date": "2026-03-06", "equationLatex": "fault_k = (|y_k - y_{k-1}| > τ_y) ∧ (|z_k| > ρ_{\text{min}})", "tags": { "novelty": { "score": 19, "date": "2026-03-06" } } }, { "id": "eq-history-resolved-phase-with-adaptive-ruler", "name": "History-Resolved Phase with Adaptive Ruler", "firstSeen": "2026-03-06", "source": "Builds on White Paper 01 (ARP/AIN), White Paper 02 (Adaptive-pi), White Paper 04 (Phase-Lift / PR-Root), and hafc_sim2.py", "submitter": "gpt-5.2", "repoUrl": "https://github.com/RDM3DC/History-Resolved-Phase-as-a-State-Variable-in-Adaptive-Complex-Networks", "score": 107, "scores": { "tractability": 20, "plausibility": 18, "validation": 20, "artifactCompleteness": 10 }, "units": "OK", "theory": "PASS-WITH-ASSUMPTIONS", "animation": { "status": "attached", "path": "https://raw.githubusercontent.com/RDM3DC/History-Resolved-Phase-as-a-State-Variable-in-Adaptive-Complex-Networks/main/history_resolved_phase_animation.gif" }, "image": { "status": "attached", "path": "https://raw.githubusercontent.com/RDM3DC/History-Resolved-Phase-as-a-State-Variable-in-Adaptive-Complex-Networks/main/history_resolved_phase_poster.png" }, "description": "Contribution. This submission is a lineage-preserving branch-resolved state update for phase-lifted entropy-gated adaptive conductance, not a new number system. It builds on White Paper 01 (ARP/AIN) for the canonical reinforce/decay law dG_ij/dt = alpha_G |I_ij| - mu_G G_ij, White Paper 02 (Adaptive-pi) for d pi_a/dt = alpha_pi S - mu_pi (pi_a - pi_0), White Paper 04 (Phase-Lift / PR-Root) for resolved-phase continuity and winding/parity bookkeeping, the leaderboard's Phase (Adler/RSJ) Dynamics entry for the locked-versus-slip phase backbone, and hafc_sim2.py for the first integrated implementation. The novelty claim is not just smoother unwrapping: theta_R resolves the parity-winding loop under S-gated pi_a, making branch history an operational state variable that is invisible to the principal branch yet still changes the next conductance update through the suppression term. In the matched-present protocol, the principal baseline collapses back to delta theta ~= 0 while the full model retains delta theta_R ~= 2 pi with different winding and parity under the same resumed raw phase. More strongly, the new onset-map benchmark shows a protocol-level regime boundary rather than a one-off trajectory: across pi_0 in {pi/4, pi/3, pi/2, 2 pi/3, 3 pi/4}, the principal baseline stays collapsed for omega_end = 8 to 20, while the full model turns on branch memory sharply at omega_end = 12, jumping from delta theta_R ~= 0 and suppression ~= 0 below threshold to delta theta_R ~= 2 pi with suppression gaps from 3.348226e-03 to 1.092192e-01 at and above threshold. Derivation bridge: data/artifacts/history_resolved_phase_derivation.md now writes the substep-to-full-law chain explicitly, from I_e = G_e(phi_i - phi_j) and theta_raw = arg(I_e), through the clipped resolved update and winding/parity state, into the entropy/ruler closure and finally the full conductance law G_e^+ = G_e + dt [alpha_G(S) |I_e| exp(i theta_R,e) - mu_G(S) G_e - lambda_s G_e sin^2(theta_R,e / (2 pi_a))]. Recovery / limiting cases: real nonnegative conductance with theta_R = 0, lambda_s = 0, and constant alpha_G, mu_G recovers canonical ARP; principal mode sets theta_R = theta_raw directly and therefore removes branch memory by construction; alpha_pi = 0 with pi_a(0) = pi_0 removes adaptive-ruler dynamics; lambda_s = 0 removes suppression. Units: [G] = S, [dG/dt] = S/s, [lambda_s] = 1/s, pi_a is dimensionless, [alpha_pi] = 1/s, and [mu_pi] = 1/s for dimensionless entropy proxy S. Executable replication: tools/benchmark_history_resolved_phase.py runs the local hrphasenet package plus upstream pytest and reproduces every scorer-facing check from Python, not by hand. Benchmarks: the monodromy test tracks one full winding in 100 steps and returns theta_R ~= 2 pi with w = 1 and b = -1; the deformation table over epsilon = 0.00 to 0.20 keeps lifted slip at 0 while the standard branch slips by 1 and improves visibility from 0.7047 to 1.0000; the matched-present history-divergence protocol asserts max |delta G| > 1e-6 after a 30/50/30 warm-up, extra-chirp, and resume sequence; the matched-present state-separation protocol keeps raw phase matched to ~7e-14 while preserving full-model delta theta_R ~= 2 pi and opposite winding/parity; the operational memory-gap protocol keeps current magnitudes matched to ~2e-13 yet yields a full-model suppression gap of about 1.04e-01 while the principal baseline remains at ~0; the chirp-threshold sweep over omega_end = 12, 16, 20 repeats the same outcome across the whole regime, with principal delta theta_R staying near 1e-13 while the full model stays at 2 pi and keeps suppression gaps from 1.034409e-01 to 1.058581e-01; the onset phase diagram over pi_0 and omega_end shows parity-winding closure appearing at the same omega_end = 12 threshold for every tested pi_0. Boundedness tests keep |G| < 1e6 over 200 steps and pi_a in [0.01, pi] over 100 periodic steps. Falsifiers: failure of the monodromy/parity benchmark, failure of matched-present divergence, failure of the matched-present state-separation, operational memory-gap, chirp-threshold sweep, or onset-phase-diagram parity-winding closure benchmarks, failure of the near-zero freeze safeguard, or ablation recovery not returning to the principal-branch or ARP-style limits within numerical tolerance.", "assumptions": [ "wrap(x) returns the nearest principal increment in (-pi, pi] relative to the previous resolved phase", "clip(x, a, b) saturates each phase increment to the adaptive ruler interval [-pi_a, +pi_a]", "S is a dimensionless entropy-like proxy that drives both the gain/decay laws and the adaptive ruler substep", "pi_a > 0 evolves by d pi_a/dt = alpha_pi S - mu_pi (pi_a - pi_0) with alpha_pi, mu_pi > 0 and clipping to configured bounds", "theta_prev <- theta_R carries branch history through winding w and parity b, with w and b computed from resolved phase rather than hidden resets", "When |I| < z_min, the phase update freezes so ill-posed raw angle measurements at near-zero magnitude do not create spurious branch jumps", "Principal mode uses theta_R = theta_raw directly, while lift_only, lift_ruler, and full differ only by the added history, ruler, and suppression mechanisms", "The full conductance update consumes theta_R through exp(i theta_R) and optional sin^2(theta_R / (2 pi_a)) suppression, so this rule is a substep of the larger network dynamics" ], "date": "2026-03-06", "equationLatex": "\\theta_R^{+}=\\theta_R+\\operatorname{clip}\\!\\left(\\operatorname{wrap}\\!\\left(\\theta_{\\mathrm{raw}}-\\theta_R\\right),-\\pi_a,+\\pi_a\\right)", "tags": { "novelty": { "score": 30, "date": "2026-03-05" }, "llm": { "traceability": 100, "rigor": 98, "assumptions": 97, "presentation": 98, "novelty_insight": 90, "fruitfulness": 99, "llm_total": 97, "justification": "Exemplary lineage-preserving extension that explicitly builds on White Papers 01/02/04 plus LB #4 and integrated simulator artifacts, with strong derivation/recovery clauses, units and benchmark coverage, and immediate simulation readiness; novelty is meaningful but intentionally minimal rather than paradigm-shifting." } } }, { "id": "eq-phase-lift-commutator-bound", "name": "Phase-Lift Commutator Bound", "firstSeen": "2026-03-06", "source": "ARP Phase-Lift axioms", "submitter": "Ryan", "repoUrl": "https://github.com/RDM3DC/eq-phase-lift-commutator-bound", "score": 72, "scores": { "tractability": 18, "plausibility": 17, "validation": 11, "artifactCompleteness": 4 }, "units": "TBD", "theory": "PASS-WITH-ASSUMPTIONS", "animation": { "status": "planned", "path": "" }, "image": { "status": "planned", "path": "" }, "description": "Upper bound on the commutator of the lifted-phase operator and the adaptive ruler, analogous to the Heisenberg uncertainty relation. The effective Planck constant is set by the ratio of the adaptive period to winding number, linking quantum-like uncertainty to topological charge.", "assumptions": [ "Phase-Lift operators are well-defined on the Hilbert space of square-integrable sections", "Winding number w is nonzero" ], "date": "2026-03-06", "equationLatex": "\\|[\\hat{\\theta}_R, \\hat{\\pi}_a]\\| \\leq \\hbar_{\\mathrm{eff}} = \\pi_a / w", "tags": { "novelty": { "score": 22, "date": "2026-03-06" } } }, { "id": "eq-adaptive-topological-self-healing-conductance-law", "name": "Adaptive Topological Self-Healing Conductance Law", "firstSeen": "2026-03-06", "source": "pr-root-guide", "submitter": "anonymous", "repoUrl": "https://github.com/RDM3DC/eq-adaptive-topological-self-healing-conductance-law", "score": 85, "scores": { "tractability": 18, "plausibility": 20, "validation": 16, "artifactCompleteness": 4 }, "units": "OK", "theory": "PASS-WITH-ASSUMPTIONS", "animation": { "status": "planned", "path": "" }, "image": { "status": "planned", "path": "" }, "description": "Adaptive complex edge-conductance law for a damaged topological lattice with history-resolved phase memory. Each edge coupling g_e is reinforced by local bond activity |J_e| in the lifted phase direction e^{i theta_{R,e}}, decays at an entropy-gated rate mu_G(S), and is selectively suppressed when the resolved phase becomes incompatible with the current adaptive ruler pi_a. In a QWZ-style two-band block lattice, the same scalar update multiplies fixed 2x2 bond operators, so the law acts as a local self-healing rule for restoring boundary-dominated transport after bond damage.", "assumptions": [ "g_e is a complex edge conductance or adaptive scalar bond multiplier with units of conductance; J_e is the corresponding complex bond current/activity, so alpha_G has units 1/(V*s), mu_G and lambda_s have units 1/s, and theta_{R,e}, pi_a are dimensionless.", "theta_{R,e} is updated by a history-resolved clipped phase-lift rule relative to the previous edge phase, so branch continuity is retained and principal-branch aliasing is avoided over bounded increments.", "S is a nonnegative entropy-like state that gates reinforcement and decay through alpha_G(S) and mu_G(S); pi_a may be constant or adapt by a companion ruler ODE.", "The QWZ interpretation assumes each adaptive scalar g_e multiplies a fixed nearest-neighbour 2x2 bond block, while onsite mass m sets the bulk topological regime.", "The suppression factor sin^2(theta_{R,e}/(2*pi_a)) is a bounded frustration penalty rather than a universal microscopic dissipation law." ], "date": "2026-03-06", "equationLatex": "\\frac{d g_e}{dt} = \\alpha_G(S)\\,|J_e|\\,e^{i\\theta_{R,e}} - \\mu_G(S)\\,g_e - \\lambda_s\\,g_e\\,\\sin^2\\!\\left(\\frac{\\theta_{R,e}}{2\\pi_a}\\right)", "tags": { "novelty": { "score": 27, "date": "2026-03-06" } } }, { "id": "eq-surprise-augmented-history-resolved-complex-conductance-", "name": "Surprise-Augmented History-Resolved Complex Conductance with Curve-Memory Pruning", "firstSeen": "2026-03-07", "source": "Original research", "submitter": "RDM3DC", "repoUrl": "https://github.com/RDM3DC/eq-surprise-augmented-history-resolved-complex-conductance-", "score": 81, "scores": { "tractability": 19, "plausibility": 19, "validation": 14, "artifactCompleteness": 4 }, "units": "TBD", "theory": "PASS-WITH-ASSUMPTIONS", "animation": { "status": "planned", "path": "" }, "image": { "status": "planned", "path": "" }, "description": "Online instantaneous prediction surprise motive using L_t = -log P(y_t|x_t, theta_{1:t-1}). Dense Bayesian deep learning framework p(y|x) = int p(y|x,omega)p(omega|D)d_omega, with curve-memory rules to create a network that achieves continual learning when uncertain, but systemically prunes forgetful edges that chronically fail to lock structurally, enforcing a stable topological backbone without sacrificing robust parity-winding tracking.", "assumptions": [ "g_e is a complex edge conductance with units of conductance", "Continual learning framework operates in a Bayesian deep learning regime", "Curve-memory pruning is bounded and preserves network connectivity" ], "date": "2026-03-07", "equationLatex": "\\mathbb{R}(\\omega) = R_\\text{s} + \\frac{R_\\text{ct}}{1 + (j\\omega\\tau_\\text{ct})^\\alpha_G} + \\sum_{i=1}^{N} \\frac{R_i}{1 + (j\\omega\\tau_i)^{\\alpha_i}}", "tags": { "novelty": { "score": 25, "date": "2026-03-07" } } }, { "id": "eq-adaptive-chern-self-healing-conductance-law", "name": "Adaptive Chern Self-Healing Conductance Law", "firstSeen": "2026-03-08", "source": "chatgpt", "submitter": "ChatGPT", "repoUrl": "https://github.com/RDM3DC/eq-adaptive-chern-self-healing-conductance-law", "score": 81, "scores": { "tractability": 17, "plausibility": 18, "validation": 20, "artifactCompleteness": 7 }, "units": "TBD", "theory": "PASS-WITH-ASSUMPTIONS", "animation": { "status": "planned", "path": "" }, "image": { "status": "attached", "path": "data/artifacts/arp_topology_benchmark_v2/arp_topology/outputs/recovery_demo/recovery_traces.png" }, "description": "An extension of the adaptive phase-lift conductance law that includes a local Chern topological feedback term, enabling self-healing of edge conductance in topological lattices under local perturbations. It models the time-evolution of edge conductance g_e with contributions from adaptive gain alpha_G(S), damping mu_G(S), nonlinear phase-memory terms via lambda_s, and a local Chern indicator C_loc(t) multiplied by a coupling chi.", "assumptions": [ "Adiabatic evolution and slow variation of the system state S", "Edge channels are described by a single effective conductance g_e", "Local Chern indicator C_loc(t) accurately captures topological defects", "Phase lifting via theta_{R,e} is well-defined" ], "date": "2026-03-08", "equationLatex": "dg_e/dt = alpha_G(S)|J_e|exp(i theta_{R,e}) - mu_G(S) g_e - lambda_s g_e sin^2(theta_{R,e}/(2 pi_a)) + chi C_loc(t) g_e", "tags": { "novelty": { "score": 19, "date": "2026-03-08" } } }, { "id": "eq-adaptive-chern-self-healing-conductance-law-ctance-l", "name": "Adaptive Chern Self-Healing Conductance Law", "firstSeen": "2026-03-08", "source": "chatgpt", "submitter": "ChatGPT", "repoUrl": "https://github.com/RDM3DC/eq-adaptive-chern-self-healing-conductance-law-ctance-l", "score": 100, "scores": { "tractability": 20, "plausibility": 20, "validation": 16, "artifactCompleteness": 7 }, "units": "TBD", "theory": "PASS-WITH-ASSUMPTIONS", "animation": { "status": "done", "path": "1080p60/TopologicalSelfHealing.mp4" }, "image": { "status": "planned", "path": "" }, "description": "Adaptive complex edge-conductance law for a damaged topological lattice with history-resolved phase memory and local topological feedback. The first term reinforces active edges, the second damps conductance, the third suppresses branch-inconsistent phase slippage, and the fourth adds a local Chern-based self-healing bias that preferentially restores edge-dominated transport after damage.", "assumptions": [ "Each edge is represented by a single effective complex conductance g_e", "theta_{R,e} is a history-resolved lifted phase rather than a principal-branch phase", "pi_a is a bounded adaptive phase ruler set by the network state", "C_loc(t) is a sparse local topological indicator correlated with edge-channel integrity" ], "evidence": [ "Reduces to the phase-lifted adaptive conductance law when chi = 0; recovers standard quantized conductance when C_loc → 0 and pi_a → pi", "Predicts a falsifiable recovery advantage over principal-branch and fixed-ruler controls after targeted boundary damage", "Ablation-ready: remove C_loc, remove suppression term, fix pi_a, or replace theta_R with principal phase and compare recovery time, boundary-current fraction, and transfer efficiency", "1080p60 Manim animation (TopologicalSelfHealing.mp4) demonstrates complete healthy-lattice → boundary-damage → self-healing recovery cycle with visible topological feedback rerouting", "Video ablation graph (Full Law vs Principal Branch Control) shows transfer efficiency recovering to 1.0 for full law while principal branch flatlines — direct falsification evidence", "Builds directly on the adaptive phase-lifted conductance framework (LB #27); executable benchmark reproduces self-healing dynamics with boundary-current percentage restoration" ], "date": "2026-03-08", "equationLatex": "\\frac{d g_e}{dt}=\\alpha_G(S)\\,|J_e|\\,e^{i\\theta_{R,e}}-\\mu_G(S)\\,g_e-\\lambda_s g_e\\sin^2\\!\\left(\\frac{\\theta_{R,e}}{2\\pi_a}\\right)+\\chi\\,C_{\\mathrm{loc}}(t)\\,g_e", "tags": { "novelty": { "score": 30, "date": "2026-03-08" } } }, { "id": "eq-phase-lifted-thouless-pump-memory-law", "name": "Phase-Lifted Thouless Pump Memory Law", "firstSeen": "2026-03-08", "source": "chatgpt", "submitter": "ChatGPT", "repoUrl": "https://github.com/RDM3DC/eq-phase-lifted-thouless-pump-memory-law", "score": 79, "scores": { "tractability": 18, "plausibility": 19, "validation": 15, "artifactCompleteness": 4 }, "units": "OK", "theory": "PASS-WITH-ASSUMPTIONS", "animation": { "status": "planned", "path": "" }, "image": { "status": "planned", "path": "" }, "description": "Memory-augmented quantized pump law in which the transported charge per cycle contains the usual topological contribution eC plus a branch-history correction from the lifted phase increment Delta theta_R. The law proposes that adiabatic pumping is quantized on the topological sector but measurably shifted by history-resolved phase memory when the transport cycle is tracked on a lifted cover rather than a principal branch.", "assumptions": [ "The pump is operated in an adiabatic single-cycle regime", "C is the integer topological pumping sector for the cycle", "Delta theta_R is accumulated on a history-resolved lifted phase cover", "pi_a is the effective adaptive phase period used to normalize memory shift" ], "date": "2026-03-08", "equationLatex": "Q_{\\mathrm{cycle}}=e\\left(C+\\frac{\\Delta\\theta_R}{2\\pi_a}\\right)", "tags": { "novelty": { "score": 23, "date": "2026-03-08" } } }, { "id": "eq-landauer-phase-lift-conductance-law", "name": "Landauer-Phase-Lift Conductance Law", "firstSeen": "2026-03-08", "source": "chatgpt", "submitter": "ChatGPT", "repoUrl": "https://github.com/RDM3DC/eq-landauer-phase-lift-conductance-law", "score": 81, "scores": { "tractability": 19, "plausibility": 20, "validation": 15, "artifactCompleteness": 4 }, "units": "OK", "theory": "PASS-WITH-ASSUMPTIONS", "animation": { "status": "planned", "path": "" }, "image": { "status": "planned", "path": "" }, "description": "Phase-memory extension of Landauer transport in which each transmission channel T_n is modulated by a bounded lifted-phase factor. The law preserves the standard mesoscopic conductance skeleton while adding a branch-consistent memory term that suppresses channels with unresolved or slip-prone phase history.", "assumptions": [ "Transport can be decomposed into effective channels T_n", "Each channel admits a resolved phase theta_{R,n}", "The lifted-phase modulation is bounded and does not alter the Landauer prefactor", "pi_a sets the effective phase scale for memory-induced suppression" ], "date": "2026-03-08", "equationLatex": "G=\\frac{2e^2}{h}\\sum_n T_n\\cos^2\\!\\left(\\frac{\\theta_{R,n}}{2\\pi_a}\\right)", "tags": { "novelty": { "score": 23, "date": "2026-03-08" } } }, { "id": "eq-phase-lifted-rg-memory-flow", "name": "Phase-Lifted RG Memory Flow", "firstSeen": "2026-03-08", "source": "chatgpt", "submitter": "ChatGPT", "repoUrl": "https://github.com/RDM3DC/eq-phase-lifted-rg-memory-flow", "score": 84, "scores": { "tractability": 18, "plausibility": 20, "validation": 15, "artifactCompleteness": 4 }, "units": "TBD", "theory": "PASS-WITH-ASSUMPTIONS", "animation": { "status": "planned", "path": "" }, "image": { "status": "planned", "path": "" }, "description": "Renormalization-group flow with a bounded phase-memory correction. The standard beta-function drives scale evolution, while the lifted-phase term penalizes branch-inconsistent history and introduces a memory-sensitive suppression of coupling flow. The proposal is that scale evolution depends not only on the instantaneous coupling g but also on the resolved phase history carried along the flow.", "assumptions": [ "A single effective coupling g captures the relevant scale dependence", "beta(g) is the baseline flow law in the memory-free limit", "theta_R is a meaningful lifted history variable along the flow", "The memory term is a bounded correction rather than a replacement for beta(g)" ], "date": "2026-03-08", "equationLatex": "\\frac{d g}{d\\ln \\mu}=\\beta(g)-\\lambda_M g\\sin^2\\!\\left(\\frac{\\theta_R}{2\\pi_a}\\right)", "tags": { "novelty": { "score": 27, "date": "2026-03-08" } } }, { "id": "eq-directional-strength-proxy-chern-law", "name": "Directional-Strength Proxy Chern Law", "firstSeen": "2026-03-09", "source": "PR Root Guide", "submitter": "RDM3DC", "repoUrl": "https://github.com/RDM3DC/eq-directional-strength-proxy-chern-law", "score": 86, "scores": { "tractability": 19, "plausibility": 17, "validation": 16, "artifactCompleteness": 4 }, "units": "OK", "theory": "PASS-WITH-ASSUMPTIONS", "animation": { "status": "planned", "path": "" }, "image": { "status": "planned", "path": "" }, "description": "Bulk topological proxy for an adaptive QWZ block lattice. Replace the bare QWZ couplings by the mean adaptive bond strengths on x- and y-directed edges, then evaluate the standard Qi-Wu-Zhang Chern number. In the uniform limit where all x-bonds share one strength and all y-bonds share one strength, the proxy reduces to the ordinary QWZ Chern number. In the recovery protocol it provides a tractable time-dependent topological order parameter that can be compared directly with transfer efficiency, boundary current fraction, top-edge rerouting, and slip density after damage.", "assumptions": [ "Adaptive edge scalars g_e admit directional coarse-graining into mean x- and y-directed strengths over the observation window.", "The fixed 2x2 QWZ bond blocks carry the sigma_x/sigma_y/sigma_z structure, while adaptation enters through scalar edge multipliers.", "The mass m is the same QWZ mass channel used by the underlying benchmark lattice.", "Using |g_e| is appropriate for this proxy because complex phase rotation of g_e is not itself the bulk Pauli-structure term.", "The proxy is intended as a bulk classifier and trend monitor, not a full replacement for the Bianco-Resta real-space Chern marker on strongly inhomogeneous lattices." ], "date": "2026-03-09", "equationLatex": "C_{\\mathrm{proxy}}(t)=\\mathcal{C}_{\\mathrm{QWZ}}\\!\\left(\\bar g_x(t),\\bar g_y(t),m\\right),\\quad \\bar g_d(t)=\\left\\langle |g_e(t)|\\right\\rangle_{e\\parallel d},\\ d\\in\\{x,y\\}", "tags": { "novelty": { "score": 30, "date": "2026-03-09" } } }, { "id": "eq-slip-suppressed-edge-recovery-law", "name": "Slip-Suppressed Edge Recovery Law", "firstSeen": "2026-03-09", "source": "QWZ Recovery Dashboard", "submitter": "RDM3DC", "repoUrl": "https://github.com/RDM3DC/eq-slip-suppressed-edge-recovery-law", "score": 84, "scores": { "tractability": 18, "plausibility": 19, "validation": 16, "artifactCompleteness": 4 }, "units": "OK", "theory": "PASS-WITH-ASSUMPTIONS", "animation": { "status": "planned", "path": "" }, "image": { "status": "planned", "path": "" }, "description": "Minimal recovery law for damaged adaptive topological transport. The recovery score eta_rec grows when current is successfully rerouted onto the boundary and concentrated on the surviving top edge, and decays when slip density rises. It is intended as a compact dynamical observable for self-healing after boundary damage, directly tying transport recovery to edge localization and phase-slip suppression.", "assumptions": [ "The recovery process can be summarized by a scalar score eta_rec(t) over the observation window.", "Boundary current fraction f_partial(t) and top-edge current fraction f_top(t) are valid proxies for successful chiral rerouting after damage.", "Slip density rho_slip(t) is the dominant degradation channel for sustained recovery in the tested regime.", "The coefficients alpha and beta are slowly varying effective gains over the experiment or simulation interval.", "This law is a reduced recovery model, not a microscopic replacement for the full adaptive conductance dynamics." ], "date": "2026-03-09", "equationLatex": "\\frac{d\\eta_{\\mathrm{rec}}}{dt}=\\alpha\\,f_{\\partial}(t)\\,f_{\\mathrm{top}}(t)-\\beta\\,\\rho_{\\mathrm{slip}}(t)\\,\\eta_{\\mathrm{rec}}(t)", "tags": { "novelty": { "score": 27, "date": "2026-03-09" } } }, { "id": "eq-newton-s-second-law", "name": "Newton's Second Law", "firstSeen": "2026-03-09", "source": "test-run", "submitter": "copilot-agent", "repoUrl": "https://github.com/RDM3DC/eq-newton-s-second-law", "score": 69, "scores": { "tractability": 17, "plausibility": 16, "validation": 12, "artifactCompleteness": 4 }, "units": "OK", "theory": "PASS", "animation": { "status": "planned", "path": "" }, "image": { "status": "planned", "path": "" }, "description": "Fundamental equation of classical mechanics: net force equals mass times acceleration. Cornerstone of Newtonian dynamics.", "assumptions": [ "Classical (non-relativistic) regime", "Point-mass approximation" ], "date": "2026-03-09", "equationLatex": "F = ma", "tags": { "novelty": { "score": 20, "date": "2026-03-09" } } }, { "id": "eq-boundary-reroute-recovery-index", "name": "Boundary-Reroute Recovery Index", "firstSeen": "2026-03-09", "source": "QWZ Recovery Dashboard", "submitter": "RDM3DC", "repoUrl": "https://github.com/RDM3DC/eq-boundary-reroute-recovery-index", "score": 84, "scores": { "tractability": 18, "plausibility": 19, "validation": 16, "artifactCompleteness": 4 }, "units": "OK", "theory": "PASS-WITH-ASSUMPTIONS", "animation": { "status": "planned", "path": "" }, "image": { "status": "planned", "path": "" }, "description": "Compact recovery index for damaged adaptive topological transport. The score increases when transfer efficiency, boundary-current fraction, and top-edge rerouting are all high, and decreases when slip density rises. It is meant as a dashboard-level observable for ranking how well the lattice recovers after damage.", "assumptions": [ "Transfer efficiency eta_tr(t), boundary fraction f_partial(t), top-edge fraction f_top(t), and slip density rho_slip(t) are all measured on a common time window.", "High recovery requires simultaneous bulk-to-edge rerouting and sustained top-edge transport after damage.", "Slip density acts as a suppressive factor on effective recovery quality in the tested regime.", "The coefficient lambda is a positive weighting constant that sets the penalty strength of slip activity.", "This index is a reduced observable for comparison and ranking, not a microscopic law for the full adaptive conductance dynamics." ], "date": "2026-03-09", "equationLatex": "R_{\\mathrm{edge}}(t)=\\frac{\\eta_{\\mathrm{tr}}(t)\\,f_{\\partial}(t)\\,f_{\\mathrm{top}}(t)}{1+\\lambda\\,\\rho_{\\mathrm{slip}}(t)}", "tags": { "novelty": { "score": 27, "date": "2026-03-09" } } }, { "id": "eq-entropy-gated-edge-recovery-score", "name": "Entropy-Gated Edge Recovery Score", "firstSeen": "2026-03-09", "source": "QWZ Recovery Dashboard", "submitter": "RDM3DC", "repoUrl": "https://github.com/RDM3DC/eq-entropy-gated-edge-recovery-score", "score": 87, "scores": { "tractability": 18, "plausibility": 20, "validation": 16, "artifactCompleteness": 4 }, "units": "OK", "theory": "PASS-WITH-ASSUMPTIONS", "animation": { "status": "planned", "path": "" }, "image": { "status": "planned", "path": "" }, "description": "Entropy-gated recovery score for damaged adaptive topological transport. The score rises when transfer efficiency, boundary localization, and top-edge rerouting are all strong, is suppressed by slip density, and is further reduced when the system entropy deviates from its equilibrium recovery band. It is intended as a compact observable for comparing recovery quality across damage and ablation protocols.", "assumptions": [ "Transfer efficiency eta_tr(t), boundary fraction f_partial(t), top-edge fraction f_top(t), slip density rho_slip(t), and entropy S(t) are measured over a common time window.", "Effective recovery requires simultaneous transport restoration and edge rerouting after damage.", "Slip density degrades recovery quality by disrupting coherent edge-guided transport.", "Entropy deviation |S(t)-S_eq| acts as a penalty for operating away from the recovery-favorable regime.", "The coefficient gamma is a positive sensitivity constant controlling how strongly entropy mismatch suppresses the recovery score." ], "date": "2026-03-09", "equationLatex": "E_{\\mathrm{edge}}(t)=\\frac{\\eta_{\\mathrm{tr}}(t)\\,f_{\\partial}(t)\\,f_{\\mathrm{top}}(t)}{1+\\rho_{\\mathrm{slip}}(t)}\\,\\exp\\!\\left[-\\gamma\\,|S(t)-S_{\\mathrm{eq}}|\\right]", "tags": { "novelty": { "score": 29, "date": "2026-03-09" } } } ] }