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@tasmaniacal1 Avatar @tasmaniacal1 Agamemnon López

Agamemnon López posts on X about flow, drift, ds, flows the most. They currently have XXXXX followers and 1012 posts still getting attention that total XXX engagements in the last XX hours.

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Social topic influence flow, drift #1952, ds, flows #558, theory, $383800ks #327, for all, elon musk, what you, drone

Top accounts mentioned or mentioned by @tacospamfilter @janecaro @elonmusk @pokerpolitics @grayconnolly @peterfitz @grok @gochagogsadze @filiposophy @ryanmcbeth @artcandee @jojofromjerz @ospehu @cbeckj @malcolmnance @keirstarmer @moderntrailhead @dawniebrowning @grogsgamut @timwalz

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"12.1 Fixed points and notation Recall the (local-in-time) PSGUPE evolution: = L(X) + (X) grad_g (X) with (X) = D(X J(X)) and g the chosen Riemannian metric (Fisher HS BKM etc.). A fixed point X* satisfies: X. Dynamics: L(X*) + (X*) = X X. Projection: X* = J(X*) X. Gradient: grad_g (X*) = X The last two are equivalent: relative entropy is minimized at J(X) so the gradient vanishes exactly when X = J(X) (subject to trace / normalization constraints). We study stability of X* under small perturbations X. XXXX Linearization of the PSGUPE flow Let X(t) = X(t) X*. Define: A = D(L + )(X*) (Jacobian"
X Link 2025-12-11T20:24Z 1525 followers, XX engagements

"Then: Theorem XXXX (Local exponential stability with PSGUPE damping) If X (entropy Hessian has a spectral gap on _) and (A) / then all eigenvalues of L_psg satisfy Re X and the fixed point X* is locally exponentially stable in the metric g. Sketch of argument: On ker H B = X so the spectrum is inherited from A; by assumption those modes are already stable. On _ consider the quadratic form V(u) = u u_g. Its derivative along = (A B)X is: V = u (A B)u_g = u A u_g u B u_g. Using the bound on the symmetric part of A and the Rayleigh quotient of B: u A u_g (A) u u_g u B u_g u u_g. Hence V ((A)"
X Link 2025-12-11T20:26Z 1525 followers, XX engagements

"@EricRWeinstein A challenger appears"
X Link 2025-12-10T06:35Z 1524 followers, XX engagements

"2 QUANTUM PSGUPE FRAMEWORK Here we use HS metric but we track the exact scaling so the diagonal (commuting) reduction is correct. Quantum state manifold Density matrices: = X = Tr = X . Tangent space: T_ = A = A Tr A = X . Metric: A B_HS = Tr(A B). X. Quantum relative entropy gradient Quantum KL: D( ) = Tr (log log ) . Derivative w.r.t. : D_(A) = Tr A (log log ) . Let = log log . The HS gradient must satisfy Tr(grad D) A = Tr A for all A with Tr A = X. Thus grad D = _tr() = (Tr / d) I_d. This is exact. X. Projection J() J() = argmin_C() D( ). Projection Axiom (quantum): Exact analogue of"
X Link 2025-12-11T12:04Z 1525 followers, XX engagements

"3 STOCHASTIC PSGUPE DYNAMICS AND ENTROPY BALANCE We introduce stochastic dynamics into the PSGUPE framework. The goal is to understand how noise interacts with entropy-contractive flows and how the balance between contraction and stochastic forcing determines long-time behavior. We work entirely on the classical simplex and conclude with a complete two-state worked example. X. State Space Divergence and Geometry XXX Probability simplex = p : p X p = X . Tangent space at p: T = v : v = X . XXX KullbackLeibler divergence For p q D(p q) = p log(p / q). Its Euclidean gradient w.r.t. p is: (D)"
X Link 2025-12-11T12:21Z 1525 followers, XX engagements

"8 : Information Geometry in PSGUPE This unifies the geometric structures underlying the PSGUPE flow. We present: Fisher geometry (classical probability) HilbertSchmidt geometry (quantum density matrices) Quantum Wasserstein geometry Classical Wasserstein geometry The unified contraction theorem Worked examples The PSGUPE flow is always: = L(X) + (X) _g (X) where (X) = D(X J(X)) and _g denotes the natural gradient with respect to the geometry of the state space. The goal is to prove: X in each geometry. XXX Classical Fisher Geometry State space: the open simplex . KL divergence: D(p q) = p"
X Link 2025-12-11T13:39Z 1525 followers, XX engagements

"8.5 Unified Contraction Theorem Let ( g) be any smooth Riemannian state manifold. Let (X) = D(X J(X)). PSGUPE flow: = L + _g . Lyapunov derivative = _g _g = _g L + _g _g _g. Assume compatibility: _g L + _g X. Then: _g _g X. Thus in any geometry under the same structural assumption relative-entropy projection D( J()) is a Lyapunov function. XXX Examples Example 1: Fisher 2-state system Let p (01) q fixed. D(p q) = p log(p/q) + (1p) log((1p)/(1q)). dD/dp = log( p(1q) / (q(1p)) ). Fisher natural gradient: _F D = p(1p) dD/dp. PSGUPE flow: = L(p) + (p) p(1p) dD/dp. Entropy dissipation: = "
X Link 2025-12-11T13:39Z 1525 followers, XX engagements

"16 THERMALISATION MODULAR FIXED POINTS AND KMS STRUCTURE IN PSGUPE DYNAMICS X. Introduction PSGUPE dynamics combine three components: transport (L) memory (integral kernels) contraction toward a constraint projection J. In many physically important systems the constraint set is an energy surface: Tr( H) = constant. When J minimises relative entropy on this surface the result is the thermal Gibbs state also known as the KMS equilibrium state. This lecture explains: why PSGUPE selects thermal states how the modular gradient determines the flow why the KMS state is stable how memory kernels"
X Link 2025-12-11T21:29Z 1525 followers, XX engagements

"Write the density matrix as: = diag(p X p). The PSGUPE contraction direction is: ( log log _ ) = ( log( p / p* ) log( p / p* ) ). The scalar ODE becomes: = a(1 p) b p _eff(p) log( p / p* ) where _eff(p) comes from the metric and projection. Solutions converge monotonically to p(t) p*. XX. Example: Harmonic Oscillator Truncated to N Levels Let energy levels be E_n = n. The Gibbs state is: _(n) = Z exp( n). The contraction term pushes log _nn toward n (plus constants removed by ). Thus the PSGUPE flow suppresses deviations from the Boltzmann distribution. XX. Free Energy Interpretation"
X Link 2025-12-11T21:30Z 1524 followers, XX engagements

"6. Free-Energy Monotonicity Theorem (Corrected and Rigorous) Theorem. Consider the PSGUPE evolution = K + M(ts)(s) ds X(). Assume: Reversible Isometry (Condition R): X() K = X for all . Generalized Passivity of Memory (Condition M): X(t) M(ts)(s) X for all s t. Dissipation Coefficient X. Then the free energy F((t)) = D((t) J((t))) satisfies dF/dt = X((t)) + X(t) M(ts)(s) ds X((t)) X. Furthermore: dF/dt = X iff X((t)) = X and X(t) M(ts)(s) = X. Thus the only stationary points satisfy = J(). These equilibrium states are fixed points of the dynamics. X. Classical Example: Two-State System"
X Link 2025-12-11T22:13Z 1523 followers, XX engagements

"@DataRepublican @elonmusk @ncri_io @honestpollster @Grok could it be that he is actually wildly popular"
X Link 2025-12-09T20:38Z 1524 followers, XX engagements

"@AlboMP take note. This is what you are sending young Australians into"
X Link 2025-12-09T22:10Z 1523 followers, XX engagements

"@EricRWeinstein dear Mr Weinstein Could you please take a few minutes out of your life to give me some feedback on my theory. I have added a section to show how GU and PSG UPE complement one another. Regards AL"
X Link 2025-12-10T20:00Z 1524 followers, XX engagements

"Australian Prime Minister totally cooked by Community Notes. God I hate lying politicians"
X Link 2025-12-11T10:03Z 1523 followers, XX engagements

"3 STOCHASTIC PSGUPE (SDEs) AND ENTROPY INEQUALITIES The plan: X. Set up an SDE on a state manifold with drift + diffusion. X. Insert the PSGUPE drift term into that SDE. X. Apply Its formula to (X) = D(X J(X)). X. Derive the entropy inequality for (X). X. Give a simple 1D worked example. X. Stochastic dynamics on a state manifold For concreteness start with X taking values in (or a chart of a manifold). XXX General It SDE Let (W) be an m-dimensional standard Brownian motion. Consider the It SDE: dX = b(X) dt + (X) dW where X b: is the drift field : dm is the diffusion matrix field."
X Link 2025-12-11T10:37Z 1525 followers, XX engagements

"3.4 Interpretation Write it in the standard form used in your canonical list: d/dt D(X J(X)) grad_g D(X J(X))_g Tr( (X) Hess_g D(X J(X)) (X) ) . Qualitative interpretation: The term is the deterministic entropy contraction we already had. The + trace term is the entropy injection from the noise: diffusion can increase the expected divergence because of curvature (Hessian) of . If the diffusion is modest compared to the contraction (or aligned with small curvature directions) the net effect is still decay of . At balance you can reach a non-trivial stationary law where contraction and"
X Link 2025-12-11T10:39Z 1525 followers, X engagements

"17 VARIATIONAL PRINCIPLE FOR PSGUPE DYNAMICS We construct a unified action functional whose EulerLagrange equation reproduces the PSGUPE evolution law (t) = K(t) + M(ts)(s) ds ( log (t) log J((t)) ) where: (t) lies in the density-operator manifold K is a reversible (Hamiltonian-like) generator M is a causal memory kernel J is the constraint projection projects onto the traceless tangent space. All equations are written with the HilbertSchmidt inner product A B = Tr(A B) and tangent space T_ = X Herm(d) : Tr X = X . X. Total Action We define the total action as S = S_rev + S_mem + S_ent"
X Link 2025-12-11T21:56Z 1523 followers, XX engagements

"5. EulerLagrange Equation and PSGUPE Law Collecting the variational derivatives: from S_rev: a reversible (Hamiltonian-like) term which we denote K in the EulerLagrange equation leading after a linear change of variables to K in the evolution law; from S_mem: S_mem / (t) = M(ts)(s) ds; from S_ent: S_ent / (t) = ( log (t) log J((t)) ). Setting S = X for all (t) and resolving the reversible part into the desired form we obtain an evolution equation of the standard PSGUPE type: (t) = K(t) + M(ts)(s) ds ( log (t) log J((t)) ). Thus PSGUPE dynamics admit a unified variational representation. 6."
X Link 2025-12-11T21:57Z 1523 followers, XX engagements

"18 FREE-ENERGY MONOTONICITY ONSAGER STRUCTURE AND DISSIPATION IN PSGUPE This lecture analyzes the irreversible structure of PSGUPE dynamics clarifies the role of the reversible and memory components and states precise conditions under which the free-energy functional F() = D( J()) is guaranteed to decrease along the dynamics. We work on the state manifold of density operators with HilbertSchmidt inner product AB = Tr(A B). All gradients are projected onto the traceless tangent space via . X. PSGUPE Evolution Law The general non-equilibrium evolution is (t) = K(t) + M(ts)(s) ds X((t)) where:"
X Link 2025-12-11T22:13Z 1523 followers, XX engagements

"19 ENTROPY PRODUCTION IN PSGUPE DYNAMICS DECOMPOSITION METRIC STRUCTURE AND SUFFICIENT CONDITIONS FOR THE SECOND LAW This chapter gives a rigorous entropy-production formulation for PSGUPE dynamics. We derive the entropy-production functional decompose it into reversible memory and dissipative components and state sufficient structural conditions guaranteeing monotonic decay of the free-energy functional. All notation applies to classical and quantum systems. X. Free energy and projected gradient Define the free energy F() = D( J()) = Tr( (log log J())). The traceless projected gradient is"
X Link 2025-12-11T22:37Z 1523 followers, XX engagements

"7 EXAMPLES OF PSGUPE MODULAR FLOWS In X we established the modular-theoretic foundation of the PSGUPE framework. Now we illustrate these structures through three explicit and technically correct examples: Finite-dimensional type-I factor (22 qubit example). Block-diagonal subalgebra constraint (a noncommutative coarse-graining). Modular flow in a quantum field theory wedge (Rindler example). All examples explicitly compute: the Araki relative entropy its gradient the projection the PSGUPE flow and the Lyapunov decay in real algebraic form. XXX Example X Qubit in a Type-I Factor with Thermal"
X Link 2025-12-11T13:18Z 1525 followers, XX engagements

"7.3.1 Modular Hamiltonians For any reduced state on a region: K_ = log K_KMS = X K. 7.3.2 Contraction term log log KMS = K (K_KMS) = K_KMS K_. So the PSGUPE evolution becomes: = L() + () ( K_KMS K_ ). If L() is simply the commutator with the true Hamiltonian the new term drives toward the KMS state. 7.3.3 Physical meaning relaxes to the wedge thermal state. This respects the Unruh effect and BisognanoWichmann modular structure. Relative entropy to _KMS decays: d/dt D(_t _KMS) = log _t log _KMS X. This gives a fully operator-theoretic second law for a QFT region. This is the first"
X Link 2025-12-11T13:18Z 1525 followers, XX engagements

"9 Dual Connections Projections and the Information Geometry Underlying PSGUPE A central strength of the PSGUPE framework is that it unifies its contraction dynamics with the intrinsic geometry of statistical manifolds. This lecture introduces the dual affine structures (exponential and mixture geometries) explains why KL projection is naturally defined and unique on these spaces and shows how the Riemannian gradient in PSGUPE arises as the canonical steepest-descent direction with respect to this geometry. XXX Dual affine structures on statistical manifolds Let be a smooth statistical"
X Link 2025-12-11T19:40Z 1525 followers, XX engagements

"15.5 Laplace transform of the modular kernel The Laplace transform () is defined by: ()X = e ()X d. Insert the definition of (): ()X = e(+) _X d e(+) X d. The second term evaluates to: e(+) X d = X / ( + ). Thus: ()X = _+X X / ( + ) where _+X = e(+) _X d. 15.5.1 Spectral representation Let have spectral decomposition: = eE P. Then: _tX = _ij ei (E E) t P X P. Define E_ij = E E and X_ij = P X P. Then: _+X = ij Xij / ( + i E_ij) (P X P) and therefore the transformed kernel becomes: ()X = ij Xij / ( + i E_ij) X_ij / ( + ) P P. This expression makes explicit the frequency-resolved"
X Link 2025-12-11T21:07Z 1525 followers, XX engagements

"21 GLOBAL CONVERGENCE AND INVARIANCE IN PSGUPE DYNAMICS X. Setting and Standing Assumptions Let (t) evolve according to the PSGUPE equation (t) = K(t) + M(ts)(s) ds X((t)) where: X K is skew-adjoint M is a memory kernel X() = ( log log J() ) projects onto the traceless Hermitian tangent space. Define the free energy F() = Tr (log log J()) . We impose the following global assumptions: (A1) Interior invariance There exists X such that for all t X (t) I. (This ensures log (t) and X((t)) are well-defined and locally Lipschitz.) (A2) Regularity The vector field K X() and the memory term"
X Link 2025-12-12T00:31Z 1525 followers, XX engagements

"22 Fluctuations Linear Response and Large Deviations in the PSGUPE Framework Standing Assumptions (Finite-Dimensional Explicit) A1. State space is a smooth finite-dimensional manifold embedded in a linear space : Classical: = (simplex interior) Quantum: = Herm(d) (full-rank density matrices) A2. Tangent space: Classical: Tp = v : v = X Quantum: T = A Herm(d) : Tr A = X A3. Inner product is fixed on each tangent space (smooth in state): Classical: FisherRao (or chosen information metric) Quantum: HilbertSchmidt AB = Tr(AB) A4. Orthogonal projection : T exists and is C in . A5. Constraint map"
X Link 2025-12-12T01:08Z 1525 followers, XX engagements

"31 Nonlinear Response and Second-Order Effects in PSGUPE XXXX Setup Let be a smooth state manifold with Riemannian metric g. Let the entropic potential be (Xt) = D(X J_t(X)). Define the PSGUPE drift f(Xt) = L(Xt) grad_g (Xt). Consider a small external forcing: (t) = f(X(t)t) + u(t) Y(X(t)) where is small u(t) is a scalar input and Y is a smooth vector field on . Fix a reference equilibrium X* (or frozen-time reference point) such that f(Xt) = X. Let X(t) = X(t) X. Derivative convention (to avoid coordinate nitpicks): all derivatives are covariant derivatives with respect to g (equivalently:"
X Link 2025-12-12T07:14Z 1525 followers, XX engagements

"Chapter XX Nonlinear Response and Soft-Mode Amplification in PSGUPE XXXX Setting and conventions Let be a smooth finite-dimensional manifold equipped with a Riemannian metric g. Let the entropic potential be (Xt) = D(X J_t(X)). Define the PSGUPE drift f(Xt) = L(Xt) grad_g (Xt). We study the forced dynamics (t) = f(X(t)t) + u(t) Y(X(t)) where X is a bookkeeping parameter u(t) is a scalar input Y is a smooth vector field on . Fix a reference trajectory X*(t) (equilibrium or frozen-time reference) satisfying f(X*(t)t) = X. Define perturbations X(t) = X(t) X*(t). Convention: all derivatives"
X Link 2025-12-12T07:21Z 1525 followers, XX engagements

"Chapter XX Contraction Response Scaling and Constraint Degeneracy XXXX Linearisation of PSGUPE Flow Let X*(t) be a reference trajectory satisfying the deterministic PSGUPE evolution = L(Xt) grad_g (Xt) Define the perturbation X(t) T_X*(t). The linearised dynamics is = A(t) X with Jacobian A(t) = L(X*(t)t) Hess_g (X*(t)t) Because the LeviCivita connection is metric-compatible Hess_g is g-self-adjoint. XXXX g-Symmetric Decomposition Define the g-adjoint A via v A w_g = A v w_g The symmetric part is Sym_g(A) = (A + A)/2 = Sym_g(L) H where H = Hess_g is positive semidefinite by convexity of ."
X Link 2025-12-12T07:37Z 1525 followers, X engagements

"34 STOCHASTIC ENTROPY BALANCE AND NOISE-LIMITED STABILITY IN PSGUPE XXXX Stochastic PSGUPE Dynamics Let X(t) evolve on a Riemannian state manifold ( g) according to the It SDE dX_t = b(X_t t) dt + (X_t t) dW_t with drift b(X t) = L(X t) grad_g (X t) and entropy functional (X t) = D(X J_t(X)). The noise covariance is Q(X t) = (X t) (X t) with tangent-projected so trajectories remain on . XXXX It Entropy Balance Applying Its formula to the time-dependent functional (X_t t) yields d = + grad_g b_g + (/2) Tr_g(H Q) dt + grad_g dW_g where H = Hess_g is the self-adjoint Hessian. Taking"
X Link 2025-12-12T07:48Z 1525 followers, X engagements

"35 Forced Response Noise and EntropyResponse Balance (Final) XXXX Framework and Scope Let ( g) be a finite-dimensional Riemannian state manifold and let : be a smooth nonnegative divergence-type Lyapunov functional. We study the combined effect of: deterministic forcing stochastic noise time-dependent constraints on the response of the system and on the expected entropy . XXXX Forced Stochastic PSGUPE Dynamics Consider the It SDE dX_t = L(X_tt) grad_g (X_tt) + Y(X_tt) u(t) dt + (2) (X_tt) dW_t Assumptions: Passivity: grad_g L_g X on the basin of interest. Strong convexity: Hess_g g with "
X Link 2025-12-12T08:02Z 1525 followers, XX engagements

"Chapter XX JumpDiffusion Forcing and Entropy Bounds in PSGUPE XXXX JumpDiffusion PSGUPE Dynamics Let X(t) evolve according to the stochastic dynamics dX_t = L(X_tt) grad_g (X_tt) + Y(X_tt) u(t) dt (2) (X_tt) dW_t (X_tzt) (dtdz) where: (Xt) = D(X J_t(X)) W_t is standard Brownian motion (dtdz) is a compensated Poisson random measure with Lvy measure (dz) Q(Xt) = u(t) is bounded with u The jump update is defined via a smooth retraction R_X on written X = R_X(). XXXX Standing Assumptions All statements below hold on a forward-invariant basin on which: (A1) Strong convexity Hess_g (Xt) g"
X Link 2025-12-12T08:32Z 1525 followers, X engagements