Top Ranked Derived Equations

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.

2026-03-06

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.

2026-03-08

planned

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.

$$\frac{dS}{dt} = \sum_{ij}\frac{|I_{ij}|^2}{T_{ij}}\,\operatorname{Re}\!\left(\frac{1}{\tilde{G}_{ij}}\right) + \kappa\sum_{ij}|\Delta w_{ij}(t)| - \gamma\,(S - S_{\rm eq})$$

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.

2026-02-22

planned

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.

2026-02-24

planned

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.

2026-02-22

planned

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.

2026-02-24

planned

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.

2026-02-25

planned

planned

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.

2026-02-22

planned

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.

2026-02-22

planned

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.

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.

2026-02-22

planned

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.

2026-02-24

planned

planned

A redshift-like relaxation emerges when an ARP-governed transport variable is mapped to a normalized deficit observable.

$$\dot z = \gamma\,(z_0 - z)$$

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.

2026-02-20

planned

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.

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.

2026-02-22

planned

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.

2026-02-24

planned

planned

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.

2026-02-24

planned

planned

Computes a local topological marker from the occupied-state projector P. The simulator bulk-averages C(r) over interior sites to estimate the Chern number in an inhomogeneous, adaptive lattice. Now implemented: solver computes C_bulk = -0.9969 at m=-1 on a 10x10 QWZ lattice with open boundaries (99.7% accuracy vs exact C=-1). Verified via benchmarks/benchmarks.py chern_marker_bianco_resta().

2026-02-22

planned

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.

2026-02-22

planned

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.

2026-02-24

in-progress

in-progress

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.

2026-02-24

in-progress

planned

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.

2026-02-24

planned

planned

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.

2026-02-24

planned

planned

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).

2026-02-25

planned

planned

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.

2026-03-09

planned

planned

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).

2026-02-22

planned

The adaptive-π conformal ruler feeding directly into the QWZ mass term. If u_eff(t) crosses 0 or ±2, the Chern phase can switch — without external tuning, purely via ARP reinforcement + memory. This is Paper I's central claim: geometry drives topology autonomously.

π_a(t) = π + β(⟨G⟩_edge − G_eq) deforms the local identification length based on edge-averaged ARP conductance. The effective mass u_eff(t) = u₀ + γ(π_a/π − 1) + δ·S_edge translates geometry deformation + entropy into QWZ mass shift. When u_eff crosses a gap-closing value, Chern number jumps.

2026-02-22

planned

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*<Delta_theta_R^2>/(2*pi_a)^2 - gamma*r_b.

2026-02-24

planned

planned

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.

2026-03-09

planned

planned

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.

2026-03-06

planned

planned

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.

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.

2026-02-22

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.

2026-03-08

planned

planned

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.

2026-03-09

planned

planned

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.

2026-03-09

planned

planned

Dynamical equation for entropy evolution in adaptive networks: Term 1 — Ohmic dissipation (entropy produced by current flow through G_ij), Term 2 — Thermal relaxation (entropy decays toward baseline S_0), Term 3 — Parity bleed (high parity-flip rate r_b drains entropy, stabilizing the network). Closes the EGATL feedback loop by quantifying how topological updates dissipate or harvest entropy.

2026-02-24

in-progress

in-progress

Conductance feedback law combining current-driven reinforcement, linear leak, and a phase-suppression gate keyed to the adaptive angular ruler. Core equation of HLATN framework — drives self-organized topological stabilization.

2026-02-24

in-progress

in-progress

Core AHC innovation: clip the residual to adaptive bounds ±πₐ. Prevents single-run glitches from causing a branch jump. The key robustness mechanism.

2026-02-22

planned

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.

2026-02-24

in-progress

in-progress

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.

2026-02-24

planned

planned

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.

2026-03-07

planned

planned

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.

2026-03-08

planned

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.

2026-03-08

planned

planned

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.

2026-02-22

planned

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.

2026-02-22

planned

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.

2026-03-08

planned

planned

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.

2026-02-20

planned

Discrete version of πₐ dynamics for the AHC loop: widens bound after events (α_π S_k term), relaxes toward π₀ otherwise.

2026-02-22

planned

A redshift-relaxation bypass law with bounded oscillatory steering. Keeps ARP exponential approach while adding controlled wobble without sign flips when epsilon is constrained.

$$\dot z = z_h\gamma e^{-\gamma t}\left(1-\epsilon\cos(\omega t+\phi)\right) + z_h\left(1-e^{-\gamma t}\right)\epsilon\omega\sin(\omega t+\phi)$$

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.

2026-02-21

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.

2026-03-06

planned

planned

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.

2026-02-20

planned

Complex conductance learning rule: reinforcement aligns conductance phase with resolved phase-lift angle; decay is entropy-gated via S.

2026-03-04

planned

planned

Suppression extension: reinforcement/decay plus a phase-position penalty that activates when phase approaches the adaptive unwrap threshold pi_a.

2026-03-04

planned

planned

Nonnegative redshift growth with a bounded oscillatory modulation: retains monotone envelope while enabling controlled oscillatory steering.

2026-03-04

planned

planned

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.

2026-03-06

planned

planned

Apply standard 2π unwrap to each Ramsey measurement relative to the previous lifted phase. First step of the AHC control loop.

2026-02-22

planned

Difference between the unwrapped measurement and the previous lifted-phase state. Feeds the step-limit gate.

2026-02-22

planned

Branch-safe Phase-Lift update: resolves phase by clipping the raw residual to an adaptive bound pi_a, preventing unstable 2π jumps.

2026-03-04

planned

planned

Adaptive ruler/bound evolution: events S_k expand the unwrap radius, while relaxation pulls pi_a back toward baseline pi_0.

2026-03-04

planned

planned

Streaming topological summary: integer winding w_k induces parity b_k, whose flip-rate r_b acts as a locking/unlocking order parameter.

2026-03-04

planned

planned

Entropy-like gating state: dissipative power term plus winding-discontinuity term inject S; relaxation returns S toward S_eq.

2026-03-04

planned

planned

Adaptive ruler renormalizes the QWZ mass channel: a single transition occurs when m_eff crosses the Chern boundary.

2026-03-04

planned

planned

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.

2026-03-06

planned

planned

Extends ARP equilibrium with an exponential temperature factor for material sensitivity.

2026-02-19

planned

Binary indicator: 1 when a residual exceeds the current πₐ bound, 0 otherwise. Alternative: curvature-based S_k ∝ |Δ²θ_R|. Triggers πₐ widening.

2026-02-22

planned

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.

2026-02-24

planned

planned

Fundamental equation of classical mechanics: net force equals mass times acceleration. Cornerstone of Newtonian dynamics.

2026-03-09

planned

planned

Candidate stability proof template for adaptive conductance dynamics.

2026-02-19

planned

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.'

$$\mathcal{E}(\phi;G,\Omega)=\frac{1}{2}\sum_{(i,j)} \Omega_{ij}\,G_{ij}\,(\phi_i-\phi_j)^2$$

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.

2026-02-22

planned