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Kime-Representation Formulations of Three Open Problems in the Foundations of Classical Mechanics: Uncertainty, Invariant Entropy, and Directional Degrees of Freedom
Ivo D. Dinov
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Summary
The paper formulates three open problems in classical mechanics—entropic uncertainty, invariant entropy, and directional degrees of freedom—using a complex-time (kime) representation. It establishes a symplectic identification between the kime cone and the action-angle chart of a one-degree-of-freedom phase space, mapping the kime measure to the Liouville measure. The work proves sharp entropic uncertainty relations, characterizes coordinate-invariant entropies, and constructs a classical analogue of spin-1/2 systems, leveraging circular statistics, Fisher information, and symplectic geometry to bridge statistical variability with classical mechanical principles.
Entities (10)
Relation Signals (8)
Kime measure → correspondsto → Liouville measure
confidence 95% · under which the kime measure is the Liouville measure
Kime cone → identifiedwith → Action-angle chart
confidence 95% · exact symplectic identification of the kime cone with the action–angle chart of a one-degree-of-freedom phase space
Kime representation → characterizes → Invariant entropy
confidence 90% · the characterization of coordinate-invariant measures and entropies
Kime representation → constructs → Directional degrees of freedom
confidence 90% · the construction of a classical relativistic directional degree of freedom
Kime representation → extends → Entropic uncertainty principle
confidence 90% · the extension of the classical entropic uncertainty principle to non-canonical variables and to multiple degrees of freedom
Phase law → models → Intrinsic trial-to-trial variability
confidence 90% · models the intrinsic trial-to-trial variability of repeated, identically controlled experiments
Circular Fisher information → bounds → Entropic uncertainty
confidence 85% · sharp circular Fisher-information inequality saturated exactly by von Mises laws
Williamson normal form → proves →
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Abstract
Abstract:We give mathematically self-contained formulations, in the complex-time (kime) representation, of three open problems from the foundations of classical mechanics: (I) the extension of the classical entropic uncertainty principle to non-canonical variables and to multiple degrees of freedom; (II) the characterization of coordinate-invariant measures and entropies, i.e., the question of why continuous physical quantities must be paired for an invariant entropy to exist; and (III) the construction of a classical relativistic directional degree of freedom (a classical analogue of a spin-1/2 system). Throughout, the kime phase is interpreted {statistically as a latent circular random variable whose law \Phi models the intrinsic trial-to-trial variability of repeated, identically controlled experiments indexed by the kime magnitude. The mathematical bridge is an exact symplectic identification of the kime cone with the action-angle chart of a one-degree-of-freedom phase space, under which the kime measure is the Liouville measure and the phase law becomes the angular conditional of a Liouville density. Specifically, we (i) prove a sharp entropic uncertainty relation on the kime cylinder whose extremal family is von Mises x Gaussian, together with a sharp circular Fisher-information inequality saturated exactly by von Mises laws; (ii) prove an exact non-canonical uncertainty relation in which the correction term is the geometric mean of the Poisson bracket, clarifying the conjectured role of the expected bracket; (iii) prove aggregate multi-degree-of-freedom bounds via the Williamson normal form and Fischer's inequality, and isolate the per-degree-of-freedom refinement as a precise open problem of symplectic Schur-Horn type; (iv) prove that diffusion of the kime phase produces monotone entropy growth with the equipartitioned (Haar-uniform) phase law.
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- Source: https://arxiv.org/abs/2607.07851v1
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Kime-Representation Formulations of Three Open Problems in the Foundations of Classical Mechanics: Uncertainty, Invariant Entropy, and Directional Degrees of Freedom Ivo D. Dinov Statistics Online Computational Resource (SOCR) University of Michigan, Ann Arbor, MI 48109, US SOCR Spacekime Group, University of Michigan. The formulations below are stated relative to the open-problem chapter of the Assumptions of Physics project [1, 2] (Problems 1.16, 1.20, and 1.21 in the numbering of that draft) and to the kime manuscripts [4, 5, 6, 3]. Abstract We give mathematically self-contained formulations, in the complex-time (kime) representation, of three open problems from the foundations of classical mechanics: (I) the extension of the classical entropic uncertainty principle to non-canonical variables and to multiple degrees of freedom; (I) the characterization of coordinate-invariant measures and entropies, i.e., the question of why continuous physical quantities must be paired for an invariant entropy to exist; and (I) the construction of a classical relativistic directional degree of freedom (a classical analogue of a spin-12 12 system). Throughout, the kime phase θ∈1θ ^1 is interpreted statistically as a latent circular random variable whose law Φ(θ∣t) (θ t) models the intrinsic trial-to-trial variability of repeated, identically controlled experiments indexed by the kime magnitude t=|κ|t=|κ|, κ=teiθκ=te^iθ. The mathematical bridge is an exact symplectic identification of the kime cone with the action–angle chart of a one-degree-of-freedom phase space, under which the kime measure tdtdθt\,dt\,dθ is the Liouville measure and the phase law becomes the angular conditional of a Liouville density. Within this dictionary we (i) prove a sharp entropic uncertainty relation on the cylinder 1×ℝS^1×R whose extremal family is von Mises ⊗ Gaussian, together with a sharp circular Fisher-information inequality saturated exactly by von Mises laws; (i) prove an exact non-canonical uncertainty relation in which the correction term is the geometric mean of the Poisson bracket, clarifying the conjectured role of the expected bracket; (i) prove aggregate multi-degree-of-freedom bounds via the Williamson normal form and Fischer’s inequality, and isolate the per-degree-of-freedom refinement as a precise open problem of symplectic Schur–Horn type; (iv) prove that diffusion of the kime phase produces monotone entropy growth with the equipartitioned (Haar-uniform) phase law as the unique attractor, giving rigorous content to the “equipartition of entropy” conjecture; (v) prove that a diffeomorphism-covariant theory of continuous quantities admits an invariant entropy if and only if quantities are canonically paired, with the Liouville entropy unique up to an additive constant, and exhibit the kime chart as the Kähler normal form of the resulting pairing; and (vi) formulate the relativistic directional problem on Poincaré coadjoint orbits, prove the fibered circular uncertainty relation in the nonrelativistic sector, and translate the “four-vector versus two-form” dichotomy into a precise moment-map criterion informed by the absence of chirality in the Cl(3,2)Cl(3,2) kime compactification. Open problems and conjectures are stated in a form directly addressable by the kime-phase tomography inference framework. 1 Introduction The Carcassi and Aidala Assumptions of Physics project [1] derives classical Hamiltonian mechanics from informational premises: states are identified with distributions over a continuum of possibilities, the count of states must be independent of the coordinates used to label them, and deterministic-and-reversible evolution must preserve that count. Within this program the classical uncertainty principle appears as an entropic statement [2]. As Hamiltonian evolution preserves the Liouville measure, the differential entropy of a state is an invariant, and the product of marginal uncertainties of a canonical pair is bounded below by a function of that invariant. The specific three open problems explored in ti study are described below. (I) (Problem 1.16, uncertainty.) Extending the classical uncertainty principle from canonical pairs of a single degree of freedom (DOF) to (a) non-canonical variable pairs, with the minimum uncertainty conjecturally governed by the (expectation of the) Poisson bracket, and (b) multiple DOF, where one asks how uncertainty and correlation migrate between DOF under the symplectic group, and whether “equipartition of entropy over uncorrelated DOF” is a lower bound. (I) (Problem 1.21, invariant entropy.) Explaining, more generally than the classical derivation, why continuous quantities must come in pairs for a coordinate-invariant entropy to exist, with suggested connections to measures on the complex plane and to generalized complex structures. (I) (Problem 1.20, directional DOF.) Constructing a classical relativistic directional degree of freedom, a classical analogue of a spin-12 12 system, including the identification of the correct phase space, the correct conjugate variables generalizing θxy,Sz=1\θ^xy,S_z\=1, and the resolution of whether spin generalizes to a four-vector or to a two-form. The kime (complex-time) representation [3, 4, 5, 6] replaces the ordering variable t of repeated experiments by a complex coordinate κ=teiθκ=te^iθ on the time cone ℳ=[0,T]×1M=[0,T]×S^1, where the magnitude t orders observations and the phase θ is a latent circular variable. The present paper adopts throughout the statistical interpretation of the phase, which we fix as assumption 1.1. Assumption 1.1 (Statistical interpretation of the kime phase). The kime phase θ is not a directly controllable or directly observable coordinate. It is a latent random variable on 1S^1 whose conditional law Φ(θ∣t) (θ t) models the intrinsic domain variability exhibited by repeated measurements of the same controlled experiment at clock reading t: independent repetitions j=1,…,Nj=1,…,N of the experiment correspond to independent draws Θj(t)∼Φ(⋅∣t) _j(t) (· t), and observables are functions (possibly noisy) of (t,Θj(t))(t, _j(t)), as in the kime-phase-tomography (KPT) observation model Yj,k=(tk,Θj(tk))+εj,kY_j,k=S(t_k, _j(t_k))+ _j,k of [4]. All theorems below are statements about this representation; no claim is made that θ is an ontic mechanical coordinate. Where a mechanical reading is used (the action–angle dictionary of Section 2), it is introduced as an explicit, falsifiable modeling identification. The contribution of this white paper is to show that, under Assumption 1.1 plus one exact symplectic identification (Lemma 2.3, the kime cone with its canonical measure is the action–angle chart of a one-DOF phase space with its Liouville measure), the three open problems (I)–(I) acquire kime-native formulations where (a) a nontrivial portion of each problem becomes a theorem provable with the circular-statistics and information-geometric tools already developed in [4, 5, 6]; and (b) the genuinely open remainder becomes a sharply stated problem or conjecture, expressed in terms of objects (phase laws, trigonometric moments, circular Fisher information, symplectic spectra) that are estimable from repeated-measurement data by kime-phase tomography, so that partial numerical evidence is obtainable in principle. Section 2 fixes the kime-representation foundations. Section 3 treats Problem (I), Section 4 treats Problem (I), and Section 5 treats Problem (I). Section 6 collects interpretations and conclusions. All proofs are given in full except where a result is classical and explicitly cited. Notational conventions. 1=ℝ/2πℤS^1=R/2 is parametrized by θ∈[−π,π)θ∈[-π,π); dθdθ denotes Lebesgue measure on 1S^1 (total mass 2π2π) and dθ/2πdθ/2π the Haar probability measure, the normalization used in [4]. Densities on 1S^1 are taken with respect to dθdθ unless stated otherwise; the conversion to the Haar convention multiplies densities by 2π2π and shifts entropies by log2π 2π, and we indicate this wherever both conventions appear. For a probability density ρ with respect to a reference measure λ on a measurable space X, the (differential) entropy relative to λ is λ[ρ]=−∫Xρlogρdλ, S_λ[ρ]\;=\;- _Xρ\, ρ\;dλ, whenever the integral is well defined in [−∞,+∞)[-∞,+∞); the subscript is dropped when the reference measure is clear [15]. KL(⋅∥⋅)KL(·\|·) and χ2(⋅∥⋅)χ^2(·\|·) denote the Kullback–Leibler and chi-squared divergences. Also, all logarithms are natural, Sp(2n,ℝ)Sp(2n,R) is the real symplectic group, Ω=(0In−In0) = pmatrix0&I_n\\ -I_n&0 pmatrix the standard symplectic form in coordinates z=(q1,…,qn,p1,…,pn)z=(q^1,…,q^n,p_1,…,p_n), and for smooth f,gf,g, f,g=∑i(∂qif∂pig−∂pif∂qig)\f,g\= _i( _q^if\, _p_ig- _p_if\, _q^ig). 2 Kime-representation foundations 2.1 The kime cone and the phase law Definition 2.1 (Kime coordinate and time cone). The kime coordinate is κ=teiθ∈ℂκ=te^iθ with kime magnitude t=|κ|≥0t=|κ|≥ 0 and kime phase θ∈1θ ^1. The time cone is the manifold-with-apex ℳ=[0,T]×1M=[0,T]×S^1 (apex t=0t=0), equipped with the cone metric and canonical measure g0=dt2+t2dθ2,dμg0=tdt⊗dθ2π,g_0=dt^2+t^2\,dθ^2, _g_0=t\,dt dθ2π, (1) as in [4]. Definition 2.2 (Phase law). A phase law is a measurable family Φ(⋅∣t)t∈[0,T]\ (· t)\_t∈[0,T] of probability densities on 1S^1 with respect to dθdθ: Φ(⋅∣t)≥0 (· t)≥ 0 and ∫−πΦ(θ∣t)dθ=1 _-π^π (θ t)\,dθ=1 for each t. Its trigonometric moments are αn(t)=[einΘt]=∫−πeinθΦ(θ∣t)dθ _n(t)=E[e^in _t]= _-π^πe^inθ (θ t)\,dθ, n∈ℤn . The mean resultant length is r(t)=|α1(t)|∈[0,1]r(t)=| _1(t)|∈[0,1], and when r(t)>0r(t)>0 the mean direction θ¯(t) θ(t) is defined by α1(t)=r(t)eiθ¯(t) _1(t)=r(t)e^i θ(t). The circular variance is V(t)=1−r(t)V(t)=1-r(t) [11]. Under Assumption 1.1, Φ(⋅∣t) (· t) is the object estimated by kime-phase tomography from the repeated-measurement records Yj,k\Y_j,k\; the identifiability, deconvolution, and Cramér–Rao theory for this estimation problem is developed in [4] and is taken as given here. 2.2 The action–angle dictionary The following elementary lemma, which does not appear explicitly in [4, 5, 6], although all of its ingredients do and it’s important in this study. Lemma 2.3 (The kime cone is an action–angle chart). Let J=12t2=12|κ|2J= 12t^2= 12|κ|^2 and define Ψ:1×(0,∞)⟶ℝ2∖0,Ψ(θ,J)=(q,p)=(2Jsinθ,2Jcosθ). :S^1×(0,∞) ^2 \0\, (θ,J)=(q,p)= ( 2J\, θ,\; 2J\, θ ). Then Ψ is a diffeomorphism and Ψ∗(dq∧dp)=dθ∧dJ,henceθ,J=1 ^*(dq )=dθ , \θ,J\=1 (2) with respect to the standard symplectic structure ω0=dq∧dp _0=dq on the punctured plane. Moreover the Liouville measure corresponds to the kime measure: dqdp=dJdθ=tdtdθ=2πdμg0dq\,dp=dJ\,dθ=t\,dt\,dθ=2π\,d _g_0. Proof. With t=2Jt= 2J, we compute dq=2Jcosθdθ+(2J)−1/2sinθdJdq= 2J θ\,dθ+(2J)^-1/2 θ\,dJ and dp=−2Jsinθdθ+(2J)−1/2cosθdJdp=- 2J θ\,dθ+(2J)^-1/2 θ\,dJ. Wedging, dq∧dp=cos2θdθ∧dJ−sin2θdJ∧dθ=dθ∧dJ.dq = ^2θ\,dθ - ^2θ\,dJ θ=dθ . Smooth invertibility on the stated domains is clear (polar coordinates). The bracket statement follows because in any chart in which ω0 _0 takes the Darboux form dx∧dydx one has x,y=1\x,y\=1; here (x,y)=(θ,J)(x,y)=(θ,J). Finally dJ=tdtdJ=t\,dt, and (1) carries the Haar factor 1/2π1/2π, giving the last display. ∎ Orientation convention. Two natural identifications of the punctured kime plane with a one-DOF phase space differ by orientation. The map Ψ above is chosen so that θ,J=+1\θ,J\=+1, matching the sign convention of the conjugate pair θxy,Sz=1\θ^xy,S_z\=1 in Problem (I) [1]; the alternative identification (q,p)=(Reκ,Imκ)=(tcosθ,tsinθ)(q,p)=(Reκ,Imκ)=(t θ,\,t θ) is holomorphic in κ and gives J,θ=+1\J,θ\=+1, i.e., the opposite orientation. The two choices are exchanged by the reflection κ↦κ¯κ κ and carry the same unsigned area measure |dθ∧dJ|=tdtdθ|dθ |=t\,dt\,dθ, so every measure-theoretic and entropic statement below is independent of the choice. The distinction matters only for the Kähler normal form of Section 4.3, where it is made explicit (Proposition 4.5). Scope of the dictionary. Lemma 2.3 states that the kime chart (θ,t)(θ,t) with the cone measure is exactly the action–angle chart of one classical DOF (for the harmonic oscillator, J is the action and θ the angle; for a general one-DOF Hamiltonian with compact regular energy levels, the Liouville–Arnold theorem [14, Ch. 10] supplies an action–angle chart with the same symplectic normal form). Under Assumption 1.1 the identification of the latent statistical phase with the mechanical angle is a modeling step: it asserts that trial-to-trial variability of a repeated experiment is variability of the angle variable at (approximately) fixed action. This is the precise, falsifiable sense in which the kime representation “lives on” the phase spaces of Problems (I)–(I), and every theorem below separates cleanly into a representation-level statement (unconditional) and this identification (a postulate, in the same spirit as the ground-state matching postulate of [5]). Definition 2.4 (Kime representation of a state). A state of one DOF is a probability density ρ on (ℝ2,dqdp)(R^2,dq\,dp). Its kime representation is the density ρ~=ρ∘Ψ ρ=ρ on 1×(0,∞)S^1×(0,∞) with respect to dθdJdθ\,dJ (no Jacobian appears, by Lemma 2.3). The induced phase law at action J is the conditional density Φ(θ∣J)=ρ~(θ,J)ρJ(J),ρJ(J)=∫−πρ~(θ,J)dθ, (θ J)= ρ(θ,J) _J(J), _J(J)= _-π^π ρ(θ,J)\,dθ, defined for ρJ(J)>0 _J(J)>0. A state is phase-equipartitioned if Φ(⋅∣J) (· J) is the uniform (Haar) law for ρJ _J-a.e. J, i.e., ρ~=ρ~(J) ρ= ρ(J). 2.3 Circular Fisher information and phase diffusion Definition 2.5 (Circular Fisher information). For a strictly positive (kime-phase) density Φ∈C1(1) ∈ C^1(S^1), ℐ[Φ]=∫−π(Φ′(θ))2Φ(θ)dθ=∫−πΦ(θ)(∂θlogΦ(θ))2dθ.I[ ]= _-π^π ( (θ) )^2 (θ)\,dθ= _-π^π (θ) ( _θ (θ) )^2\,dθ. Lemma 2.6 (Amplitude/kinetic identity; [5]). For strictly positive Φ∈C1(1) ∈ C^1(S^1), ∫−π|∂θΦ(θ)|2dθ=14ℐ[Φ],hence⟨p^θ 22μ⟩Φ=ℏ28μℐ[Φ], _-π^π | _θ (θ) |^2dθ= 14\,I[ ], p_θ^\,22μ _\! = ^28μ\,I[ ], (3) where p^θ=−iℏ∂θ p_θ=-i _θ acts on the periodic Sobolev space H1(1)H^1(S^1) and μ>0μ>0 is the phase inertia of [5], in which (3) appears as the Fisher–kinetic identity underlying the potential-reconstruction theorem. Proof. ∂θΦ=Φ′/(2Φ) _θ = /(2 ), so |∂θΦ|2=(Φ′)2/(4Φ)| _θ |^2=( )^2/(4 ); integrate. The second identity is the expectation of p^θ2/(2μ) p_θ^2/(2μ) in the real state Φ , using integration by parts on 1S^1 (boundary terms vanish by periodicity). ∎ Lemma 2.7 (Phase diffusion: moment decay, entropy production, equipartition attractor). Let Φt _t solve the Fokker–Planck (heat) equation on 1S^1 [18], ∂tΦt=D∂θ2Φt _t _t=D\, _θ^2 _t, D>0D>0, with strictly positive C2C^2 initial datum Φ0 _0. Then, (i) (Moment decay; the stochastic-phase suppression law of [6]) αn(t)=e−Dn2tαn(0) _n(t)=e^-Dn^2t _n(0) for all n∈ℤn . (i) (de Bruijn identity on the circle; cf. [17, 16]) dt[Φt]=Dℐ[Φt]≥0 ddt\, S[ _t]=D\,I[ _t]\ ≥ 0, with equality at time t iff Φt _t is the uniform density Φ∞≡1/2π _∞≡ 1/2π. (i) (Exponential equipartition) χ2(Φt∥Φ∞):=2π∫−π(Φt−12π)2dθ≤e−2Dtχ2(Φ0∥Φ∞)χ^2( _t\,\|\, _∞):=2π\! _-π^π\! ( _t- 12π )^2dθ≤ e^-2Dt\,χ^2( _0\,\|\, _∞), and [Φt]↑log2π=max S[ _t] 2π= S. Proof. (i) Multiply the equation by einθe^inθ and integrate by parts twice on 1S^1 (all boundary terms vanish by periodicity): α˙n=−Dn2αn α_n=-Dn^2 _n. (i) Positivity of Φt _t for t>0t>0 is standard (strong maximum principle for the heat semigroup on 1S^1). Differentiating, dt[Φt]=−∫(∂tΦt)(1+logΦt)dθ=−D∫Φt′logΦtdθ=D∫(Φt′)2Φtdθ=Dℐ[Φt], ddt S[ _t]=- ( _t _t)(1+ _t)\,dθ=-D _t \, _t\,dθ=D ( _t )^2 _t\,dθ=D\,I[ _t], where we used ∫Φt′dθ=0 _t \,dθ=0 and one integration by parts; all boundary terms vanish by periodicity. ℐ[Φ]=0I[ ]=0 iff Φ′≡0 ≡ 0 iff Φ is uniform. (i) With trigonometric moments normalized as in Definition 2.2, Parseval’s identity gives χ2(Φt∥Φ∞)=∑n≠0|αn(t)|2χ^2( _t\| _∞)= _n≠ 0| _n(t)|^2, so by (i) χ2(Φt∥Φ∞)=∑n≠0e−2Dn2t|αn(0)|2≤e−2Dtχ2(Φ0∥Φ∞).χ^2( _t\| _∞)= _n≠ 0e^-2Dn^2t| _n(0)|^2≤ e^-2Dt\,χ^2( _0\| _∞). For the entropy limit, note first that [Φ]=log2π−KL(Φ∥Φ∞) S[ ]= 2π-KL( \,\|\, _∞) by direct computation, and second that KL≤log(1+χ2)≤χ2KL≤ (1+χ^2)≤χ^2 by Jensen’s inequality applied to the concave logarithm [15]. Hence log2π≥[Φt]≥log2π−χ2(Φt∥Φ∞)⟶log2π 2π\ ≥\ S[ _t]\ ≥\ 2π-χ^2( _t\| _∞)\ \ 2π, and monotonicity of the convergence is (i). Uniqueness of the maximizer ([Φ]≤log2π S[ ]≤ 2π with equality iff Φ uniform) is Jensen’s inequality applied to −log- . ∎ Lemma 2.7 interpretation. Under Assumption 1.1, Lemma 2.7 is a statement about experimental reproducibility: if the latent trial phase diffuses between repetitions (the stochastic-projection mechanism of [6]), then the empirical phase law loses structure at the universal rate e−Dn2te^-Dn^2t per harmonic, its entropy grows monotonically, and the maximally reproducibility-agnostic (Haar) law is the unique attractor. Lemma 2.3 will convert this into the dynamical/thermodynamic half of the uncertainty problem in Section 3.5. 3 Problem I (uncertainty): the classical uncertainty principle in the kime representation We first recall the flat-space entropic uncertainty principle in the form used by [1, 2], then prove its exact analogue on the kime cylinder 1×ℝS^1×R (Theorem 3.7), a sharp circular Fisher inequality (Theorem 3.8), and an exact non-canonical uncertainty relation (Theorem 3.11). We then treat multiple DOF (Theorems 3.13–3.16) and the dynamical/thermodynamic conjecture (Theorem 3.21), and state the remaining open problems (Problems 3.12–3.19). 3.1 Maximum-entropy lemmas and the flat benchmark Lemma 3.1 (Gaussian maximum entropy; [15]). Let ρ be a probability density on ℝR (w.r.t. Lebesgue) with finite variance σ2>0σ^2>0 and [ρ]>−∞ S[ρ]>-∞. Then [ρ]≤12log(2πeσ2) S[ρ]≤ 12 (2π eσ^2), with equality iff ρ is Gaussian with variance σ2σ^2. Proof. Let g be the Gaussian density with the same mean m and variance σ2σ^2. Then 0≤KL(ρ∥g)=−[ρ]−∫ρlogg0 (ρ\|g)=- S[ρ]- ρ g, and −∫ρlogg=12log(2πσ2)+ρ[(x−m)2]/(2σ2)=12log(2πσ2)+12- ρ g= 12 (2πσ^2)+E_ρ\! [(x-m)^2 ]/(2σ^2)= 12 (2πσ^2)+ 12. Rearranging gives the bound; equality in KL≥0KL≥ 0 holds iff ρ=gρ=g a.e. ∎ Lemma 3.2 (Entropy subadditivity; [15]). Let ρ be a probability density on a product measure space (X×Y,λX⊗λY)(X× Y, _X _Y) with marginals ρX,ρY _X, _Y and [ρ]>−∞ S[ρ]>-∞. Then [ρ]≤[ρX]+[ρY] S[ρ]≤ S[ _X]+ S[ _Y], with equality iff ρ=ρX⊗ρYρ= _X _Y a.e. Proof. KL(ρ∥ρX⊗ρY)=∫ρlogρ−∫ρlog(ρXρY)=−[ρ]+[ρX]+[ρY]≥0KL(ρ\,\|\, _X _Y)= ρ ρ- ρ ( _X _Y)=- S[ρ]+ S[ _X]+ S[ _Y]\ ≥ 0, using that the X- and Y-marginal integrals of ρlogρXρ _X and ρlogρYρ _Y reduce to marginal entropies by Fubini. Equality iff ρ=ρX⊗ρYρ= _X _Y a.e. ∎ Theorem 3.3 (Flat entropic uncertainty principle; cf. [1, 2]). Let ρ be a probability density on (ℝ2,dqdp)(R^2,dq\,dp) with finite marginal variances σq2,σp2 _q^2, _p^2 and [ρ]>−∞ S[ρ]>-∞. Then σqσp≥12πee[ρ], _q\, _p\;≥\; 12π e\,e S[ρ], (4) with equality iff ρ is a Gaussian product density. Since Hamiltonian flows preserve dqdpdq\,dp, the right-hand side is a constant of motion, while the factors on the left individually evolve. Proof. Lemma 3.2 and then Lemma 3.1 on each marginal give [ρ]≤12log(2πeσq2)+12log(2πeσp2)=log(2πeσqσp) S[ρ]≤ 12 (2π e _q^2)+ 12 (2π e _p^2)= (2π e\, _q _p). Then, exponentiate and notice that equality in (4) requires equality in both lemmas 3.1 and 3.2. Invariance of S under measure-preserving flows is the change-of-variables identity [ρ∘ϕ−1]=[ρ] S[ρ φ^-1]= S[ρ] for ϕφ preserving dqdpdq\,dp (Liouville’s theorem). ∎ Theorem 3.3 interpretation. Inequality (4) is a purely classical, Liouville-based statement; it should be distinguished from the Hirschman–Beckner entropic uncertainty principle for Fourier-conjugate quantum observables, for which see [19]. The two share the maximum-entropy mechanism but not the underlying invariance. 3.2 The circular sector: sharp inequalities on 1S^1 and on the kime cylinder The kime phase is compact, and the pair (θ,pθ)(θ,p_θ), an angle and its conjugate momentum, is the simplest pair for which the flat formulation of Theorem 3.3 is not directly meaningful (θ is multivalued; its “variance” is convention-dependent). This is exactly the non-canonical obstruction isolated in Problem (I)(a). The circular-statistics objects of Definition 2.2 resolves it. Lemma 3.4 (von Mises maximum entropy; cf. [11]). Let Φ be a probability density on (1,dθ)(S^1,dθ) with mean resultant length r∈[0,1)r∈[0,1) and (if r>0r>0) mean direction θ¯ θ, and [Φ]>−∞ S[ ]>-∞. Let A(κ)=I1(κ)/I0(κ)A(κ)=I_1(κ)/I_0(κ) denote the Bessel ratio, a strictly increasing bijection A:[0,∞)→[0,1)A:[0,∞)→[0,1) [11, Sec. 3.5], and set κ(r)=A−1(r)κ(r)=A^-1(r). Then, [Φ]≤hc(r):=log(2πI0(κ(r)))−κ(r)r, S[ ]\;≤\;h_c(r)\;:=\; \! (2π I_0(κ(r)) )-κ(r)\,r, (5) with equality iff Φ=vM(⋅;θ¯,κ(r))=expκ(r)cos(θ−θ¯)/(2πI0(κ(r))) =vM(·; θ,κ(r))= \κ(r) (θ- θ)\/ (2π I_0(κ(r)) ). The function hch_c is strictly decreasing on (0,1)(0,1) with hc′(r)=−κ(r)h_c (r)=-κ(r), hc(0)=log2πh_c(0)= 2π, and hc(r)→−∞h_c(r)→-∞ as r↑1r 1. Proof. Let g=vM(⋅;θ¯,κ(r))g=vM(·; θ,κ(r)) (for r=0r=0, g is uniform and the bound is Jensen’s inequality). Then 0≤KL(Φ∥g)=−[Φ]−∫Φlogg0 ( \|g)=- S[ ]- g and −∫Φlogg=log(2πI0(κ(r)))−κ(r)Φ[cos(Θ−θ¯)]=log(2πI0(κ(r)))−κ(r)r,- g= \! (2π I_0(κ(r)) )-κ(r)\,E_ [ ( - θ) ]= \! (2π I_0(κ(r)) )-κ(r)\,r, because Φcos(Θ−θ¯)=rE_ ( - θ)=r by definition of (r,θ¯)(r, θ). Equality iff Φ=g =g. For the derivative, write hc(r)=log(2πI0(κ))−κrh_c(r)= (2π I_0(κ))-κ r with A(κ)=rA(κ)=r; then, using I0′=I1I_0 =I_1, hc′(r)=A(κ)κ′(r)−rκ′(r)−κ(r)=−κ(r)<0h_c (r)=A(κ)κ (r)-rκ (r)-κ(r)=-κ(r)<0 for r>0r>0. The boundary values are immediate from κ(0)=0κ(0)=0 and κ(r)→∞κ(r)→∞. ∎ Definition 3.5 (Circular entropy width). For r∈[0,1)r∈[0,1) define Λ(r):=ehc(r)∈(0,2π] (r):=e^h_c(r)∈(0,2π]. By Lemma 3.4, Λ is strictly decreasing, Λ(0)=2π (0)=2π, and Λ(r)↓0 (r) 0 as r↑1r 1. That is, Λ(r) (r) is the effective support length of the most disordered circular law compatible with concentration r. Proposition 3.6 (Flat limit of the width). As r↑1r 1 (equivalently κ=κ(r)→∞κ=κ(r)→∞), Λ(r)=2πeκ(r)(1+O(κ−1)), (r)= 2π eκ(r)\, (1+O(κ^-1) ), and for the extremal law vM(⋅;θ¯,κ)vM(·; θ,κ) one has Var(Θ−θ¯)=κ−1(1+O(κ−1))Var( - θ)=κ^-1 (1+O(κ^-1) ), so that Λ(r)∼2πeσΘ (r) 2π e\; _ and the cylinder theorem below degenerates to the flat bound (4). Proof. Inserting the standard asymptotics I0(κ)=eκ(2πκ)−1/2(1+O(κ−1))I_0(κ)=e^κ(2πκ)^-1/2(1+O(κ^-1)) and A(κ)=1−12κ+O(κ−2)A(κ)=1- 12κ+O(κ^-2) [12, 9.7.1] into (5) yields hc=log2π+κ−12log(2πκ)−κA(κ)+O(κ−1)=12log2πeκ+O(κ−1).h_c= 2π+κ- 12 (2πκ)-κ A(κ)+O(κ^-1)= 12 \! 2π eκ+O(κ^-1). Exponentiating gives the first claim. Observe that the variance asymptotics follow from the Laplace approximation of the von Mises law around θ¯ θ (Gaussian with variance 1/κ1/κ), justified by the same Bessel asymptotics. Matching with Theorem 3.3 is then the computation Λ(r)σp∼2πeσΘσp (r) _p 2π e\, _ _p. ∎ Theorem 3.7 (Kime-cylinder entropic uncertainty principle). Let ρ be a probability density on (1×ℝ,dθdp) (S^1×R,\;dθ\,dp ) with [ρ]>−∞ S[ρ]>-∞, angular marginal Φ with mean resultant length r∈[0,1)r∈[0,1), and momentum marginal with finite variance σp2>0 _p^2>0. Then, Λ(r)⋅σp≥e[ρ]2πe, (r)· _p\;≥\; e S[ρ] 2π e, (6) with equality iff ρ=vM(⋅;θ¯,κ(r))⊗(m,σp2)ρ=vM(·; θ,κ(r)) (m, _p^2) for some m∈ℝm . Moreover, if ρ evolves by any Hamiltonian flow on the cylinder T∗1T^*S^1 (symplectic form dθ∧dpdθ ), the right-hand side of (6) is a constant of motion. Proof. By Lemma 3.2, [ρ]≤[Φ]+[ρp] S[ρ]≤ S[ ]+ S[ _p]. Bounding the two summands by Lemmas 3.4 and 3.1, [ρ]≤hc(r)+12log(2πeσp2)=log(Λ(r)σp2πe) S[ρ]≤ h_c(r)+ 12 (2π e _p^2)= ( (r)\, _p 2π e ). Again, we exponentiate the terms. Equality forces equality in all three inequalities, i.e., independence with extremal marginals. Any Hamiltonian flow preserves the Liouville measure dθdpdθ\,dp, hence preserves [ρ] S[ρ] by the change-of-variables identity. ∎ Compactness is a feature, not a artifact. Since Λ(r)≤2π (r)≤ 2π, inequality (6) contains an absolute momentum floor σp≥e[ρ]/(2π2πe) _p≥ e S[ρ]/(2π 2π e). On a compact angle, total delocalization of the phase cannot absorb an unbounded share of the invariant entropy. This is the precise structural difference from the flat case that Problem (I)(a) anticipated for non-canonical variables. Compact topology converts the uncertainty trade-off into a trade-off with a hard floor. Under Assumption 1.1, r is the concentration of the empirical phase law of the repeated experiment and is directly estimable [4]. Therefore, the kime entropic uncertainty principle (6) is a testable inequality. Theorem 3.8 (Sharp circular Fisher (Cramér–Rao–type) uncertainty relation). Let Φ∈C1(1) ∈ C^1(S^1) be strictly positive with mean resultant length r>0r>0 and mean direction θ¯ θ. Then ℐ[Φ]⋅Φ[sin2(Θ−θ¯)]≥r2,I[ ]·E_ \! [ ^2( - θ) ]\;≥\;r^2, (7) with equality iff Φ=vM(⋅;θ¯,κ) =vM(·; θ,κ) for some κ>0κ>0. Equivalently, in terms of trigonometric moments, ℐ[Φ]≥2r2/(1−Ree−2iθ¯α2)I[ ]≥ 2r^2/ (1-Re\,e^-2i θ _2 ). Proof. Integrating by parts on 1S^1 (periodic boundary terms vanish), ∫−πΦ′(θ)sin(θ−θ¯)dθ=−∫−πΦ(θ)cos(θ−θ¯)dθ=−r. _-π^π (θ)\, (θ- θ)\,dθ=- _-π^π (θ) (θ- θ)\,dθ=-r. By Cauchy–Schwarz, r2=(∫Φ′Φ⋅Φsin(θ−θ¯)dθ)2≤∫(Φ′)2Φdθ∫Φsin2(θ−θ¯)dθ,r^2= ( · \, (θ- θ)\,dθ )^\!2≤ ( )^2 \,dθ\; ^2(θ- θ)\,dθ, which is (7). Equality holds iff Φ′/Φ=csin(θ−θ¯) / =c\, (θ- θ) a.e. for some constant c, i.e., logΦ=−ccos(θ−θ¯)+const =-c (θ- θ)+const, a von Mises law. Consistency with cos(Θ−θ¯)=r>0E ( - θ)=r>0 forces c<0c<0, i.e., concentration at θ¯ θ with κ=−c>0κ=-c>0. Note that for Φ=vM(⋅;θ¯,κ) =vM(·; θ,κ) we can directly check ℐ=κ2sin2(Θ−θ¯)I=κ^2\,E ^2( - θ) and sin2(Θ−θ¯)=A(κ)/κE ^2( - θ)=A(κ)/κ. Hence, ℐ⋅sin2=A(κ)2=r2I·E ^2=A(κ)^2=r^2, see the von Mises Fisher computation in [5]. The corresponding trigonometric moment form follows from sin2x=12(1−cos2x) ^2x= 12(1- 2x). ∎ Corollary 3.9 (Kime-native kinetic uncertainty bound). Under the ground-state matching postulate of [5] (φ0=Φ _0= ), the kinetic energy of the reconstructed phase state obeys ⟨p^θ 22μ⟩Φ=ℏ28μℐ[Φ]≥ℏ28μr2Φ[sin2(Θ−θ¯)]≥ℏ2r28μ, p_θ^\,22μ _\! \;=\; ^28μ\,I[ ]\;≥\; ^28μ\, r^2E_ [ ^2( - θ)]\;≥\; ^2r^28μ, with the first inequality saturated exactly by von Mises phase laws. Concentration of the empirical phase law of a repeated experiment therefore imposes a quantitative lower bound on the kinetic part of the reconstructed phase Hamiltonian of [5]. Proof. Combine Lemma 2.6 with Theorem 3.8 and sin2≤1E ^2≤ 1. ∎ 3.3 Non-canonical variables: the geometric-mean bracket theorem Problem (I)(a) conjectures a minimum-uncertainty relation for a non-canonical pair (u,v)(u,v) “based on the Poisson bracket (more precisely, the expectation of the Poisson bracket).” Theorem 3.11 resolves the diffeomorphic case exactly and shows that the correct universal correction is the geometric mean expρlog|u,v| _ρ |\u,v\| of the bracket, not its arithmetic expectation. The two coincide precisely in the linear (in particular canonical) case. Lemma 3.10 (Entropy under phase-space reparametrization). Let U⊆ℝ2U ^2 be open, ρ a probability density supported in U (w.r.t. dqdpdq\,dp) with [ρ] S[ρ] finite, and (u,v):U→ℝ2(u,v):U ^2 a C1C^1 diffeomorphism onto its image with Jacobian determinant det∂(u,v)∂(q,p)=u,v≠0 ∂(u,v)∂(q,p)=\u,v\≠ 0 on U. Let ρ(u,v)ρ^(u,v) denote the pushforward density w.r.t. dudvdu\,dv. If ρ|log|u,v||<∞E_ρ | |\u,v\| |<∞, then [ρ(u,v)]=[ρ]+ρlog|u,v|. S [ρ^(u,v) ]= S[ρ]+E_ρ |\u,v\ |. (8) Proof. Write Ψ=(u,v) =(u,v) and JΨ=u,vJ_ =\u,v\. The pushforward density is ρ(u,v)=(ρ/|JΨ|)∘Ψ−1ρ^(u,v)= (ρ/|J_ | ) ^-1. By the change of variables formula, [ρ(u,v)]=−∫Ψ(U)ρ|JΨ|logρ|JΨ||Ψ−1dudv=−∫Uρ(logρ−log|JΨ|)dqdp, S[ρ^(u,v)]=- _ (U) ρ|J_ | ρ|J_ | |_ ^-1du\,dv=- _Uρ\, ( ρ- |J_ | )\,dq\,dp, which is (8); the integrability hypothesis justifies splitting the integral. ∎ Theorem 3.11 (Non-canonical uncertainty relation with geometric-mean bracket). In the setting of Lemma 3.10, suppose additionally that the pushforward marginals of u and v have finite variances σu2,σv2 _u^2, _v^2. Then σuσv≥12πee[ρ]exp(ρlog|u,v|), _u\, _v\;≥\; 12π e\;e S[ρ]\; (E_ρ |\u,v\ | ), (9) with equality iff the pushforward of ρ under (u,v)(u,v) is a Gaussian product density. In particular: (a) if (u,v)(u,v) is canonical (u,v≡1\u,v\≡ 1), (9) reduces to (4); (b) if (u,v)(u,v) is linear, u,v\u,v\ is constant and the correction equals |u,v|=ρ|u,v||\u,v\|=E_ρ|\u,v\|; (c) in general, by Jensen’s inequality expρlog|u,v|≤ρ|u,v| _ρ |\u,v\| _ρ|\u,v\|, so the conjectured bound with the arithmetic expectation of the bracket is a strictly stronger statement than (9) and does not follow from entropy methods alone. Proof. Apply Theorem 3.3 to the pushforward density (a legitimate density on ℝ2R^2) and substitute (8). Items (a)–(b) are immediate; (c) is Jensen applied to the concave logarithm. ∎ Problem 3.12 (Status of the expected-bracket form). Determine the largest class of pairs (u,v)(u,v) and states ρ for which the strengthened inequality σuσv≥(2πe)−1e[ρ]ρ|u,v| _u _v≥(2π e)^-1e S[ρ]\,E_ρ|\u,v\| holds, and exhibit either a proof for a natural class beyond the linear one or an explicit counterexample. In the kime representation the natural test family is (u,v)=(a circular function of θ,J)(u,v)=(a circular function of θ,\;J), for which the compactness corrections are controlled by Theorem 3.7. In particular, formulate and prove the correct statement when (u,v)(u,v) is not injective (winding angle), the conjectured mechanism is a holonomy correction quantized in units of the circulation ∮dθ=2π θ=2π, i.e., an additive term log(2πw) (2π w) for winding number w, whose precise form should follow by applying Lemma 3.10 on a fundamental domain and Lemma 3.4 on the quotient. 3.4 Multiple degrees of freedom: Williamson form, Fischer inequality, and the symplectic Schur–Horn problem Fix n≥1n≥ 1 and coordinates z=(q1,p1,…,qn,pn)∈ℝ2nz=(q^1,p_1,…,q^n,p_n) ^2n ordered by DOF, so that a covariance matrix Σ∈ℝ2n×2n ^2n× 2n, Σ≻0 0, decomposes into 2×22× 2 blocks Σjk _jk, j,k=1,…,nj,k=1,…,n, with Σjj _j the within-DOF block of DOF j and Σjk _jk (j≠kj≠ k) the cross-DOF correlation blocks, the decomposition singled out in Problem (I)(b), Eq. (1.229) of [1]. In this ordering the symplectic form is Ωn=⨁j=1n(01−10) _n= _j=1^n pmatrix0&1\\ -1&0 pmatrix and Sp(2n,ℝ)=S:SΩnS⊤=ΩnSp(2n,R)=\S:\,S _nS^\! = _n\. Theorem 3.13 (Williamson normal form; [7]). For every Σ≻0 0 there exists S∈Sp(2n,ℝ)S (2n,R) and unique ν1≥⋯≥νn>0 _1≥·s≥ _n>0 (the symplectic eigenvalues, the positive spectrum of iΩn−1Σi _n^-1 up to sign) with SΣS⊤=diag(ν1,ν1,…,νn,νn)S S^\! =diag( _1, _1,…, _n, _n). Lemma 3.14 (Gaussian entropy and symplectic invariants). If ρ is the Gaussian density on ℝ2nR^2n with covariance Σ , then [ρ]=12log((2πe)2ndetΣ) S[ρ]= 12 ((2π e)^2n ) and detΣ=∏j=1nνj2 = _j=1^n _j^2. Both [ρ] S[ρ] and the multiset νj\ _j\ are invariant under every linear Hamiltonian evolution Σ↦SΣS⊤ S S^\! , S∈Sp(2n,ℝ)S (2n,R). Proof. The entropy formula is the standard Gaussian computation (diagonalize Σ orthogonally and apply Lemma 3.1 coordinatewise, plus subadditivity with equality for independent coordinates). Since detS=1 S=1 for S∈Sp(2n,ℝ)S (2n,R), detΣ=det(SΣS⊤)=∏jνj2 = (S S^\! )= _j _j^2 by Theorem 3.13. Invariance of the symplectic spectrum: if S0∈SpS_0 , then Ωn−1(S0ΣS0⊤)=S0−⊤(Ωn−1Σ)S0⊤ _n^-1(S_0 S_0^\! )=S_0^-\! ( _n^-1 )S_0^\! using S0⊤ΩnS0=ΩnS_0^\! _nS_0= _n, a similarity transformation, so the spectrum of iΩn−1Σi _n^-1 is unchanged. ∎ Lemma 3.15 (Fischer inequality; see [13]). For Σ≻0 0 partitioned into diagonal blocks Σ11,…,Σnn _11,…, _n (any sizes), detΣ≤∏j=1ndetΣjj ≤ _j=1^n _j, with equality iff Σ is block diagonal. Proof. It suffices to treat n=2n=2 and induct. Writing the Schur complement Σ/Σ11=Σ22−Σ21Σ11−1Σ12 / _11= _22- _21 _11^-1 _12, one has detΣ=detΣ11det(Σ/Σ11) = _11 ( / _11) and 0≺Σ/Σ11⪯Σ220 / _11 _22. Monotonicity of the determinant on the positive-semidefinite order [13, Cor. 7.7.4] gives det(Σ/Σ11)≤detΣ22 ( / _11)≤ _22, with equality iff Σ21Σ11−1Σ12=0 _21 _11^-1 _12=0, i.e., Σ12=0 _12=0. ∎ Theorem 3.16 (Aggregate multi-DOF uncertainty floor). Let ρ be a probability density on ℝ2nR^2n (with respect to d2nzd^2nz) with finite covariance Σ≻0 0 and [ρ]>−∞ S[ρ]>-∞, and let ν1≥⋯≥νn>0 _1≥…≥ _n>0 be the symplectic eigenvalues of Σ . Define the within-DOF uncertainty of DOF j as uj:=detΣjj=σqj2σpj2−Cov(qj,pj)2,u_j:= _j= σ^2_q^jσ^2_p_j-Cov(q^j,p_j)^2, the area scale of the jjth marginal covariance ellipse. Then, ∏j=1nuj≥detΣ=∏j=1nνj≥e[ρ](2πe)n, _j=1^nu_j\;≥\; \;=\; _j=1^n _j\;≥\; e S[ρ](2π e)^n, (10) where the first inequality is an equality iff there are no cross-DOF correlations (Σjk=0 _jk=0 for j≠kj≠ k); the middle quantity is invariant under all linear Hamiltonian flows; and the last inequality is an equality iff ρ is Gaussian. Consequently, along any linear Hamiltonian evolution of a state that is initially Gaussian, uncorrelated across DOF, and equipartitioned (νj=ν _j=ν for all j, uj(0)=νu_j(0)=ν), ∏j=1nuj(t)≥νn=∏juj(0)for all t, _j=1^nu_j(t)\;≥\;ν^\,n= _ju_j(0) all t, with equality at time t iff the state is again uncorrelated across DOF at time t. The product of within-DOF uncertainties can only be raised above its initial equipartition value, and only by the creation of cross-DOF correlations. For n=1n=1 the product is a single factor and u1(t)=ν1u_1(t)= _1. Identically, within one DOF, linear Hamiltonian flow conserves the uncertainty area exactly. Proof. The first inequality is Lemma 3.15 applied to the 2×22× 2 block partition, together with detΣ=∏jνj2 = _j _j^2 (Lemma 3.14). The last inequality is the maximum-entropy bound [ρ]≤12log((2πe)2ndetΣ) S[ρ]≤ 12 ((2π e)^2n ), proved exactly as in Lemma 3.1 with the matching Gaussian of covariance Σ (equality iff ρ Gaussian). Invariance of ∏νjΠ _j under Σ↦SΣS⊤ S S^\! is Lemma 3.14. For the dynamical statement, Σ(t)=StΣ(0)St⊤ (t)=S_t (0)S_t^\! with St∈Sp(2n,ℝ)S_t (2n,R) preserves the symplectic spectrum ν\ν\, so ∏juj(t)≥∏jνj=νn _ju_j(t)≥ _j _j=ν^n by the first inequality, with the stated equality case. For n=1n=1, Sp(2,ℝ)=SL(2,ℝ)Sp(2,R)=SL(2,R), so u1(t)2=detΣ(t)=detΣ(0)=ν12u_1(t)^2= (t)= (0)= _1^2. ∎ Relevance to Problem (I)(b). Theorem 3.16 proves the aggregate (product) version of both conjectures in Problem (I)(b). The equi-partitioned uncorrelated state minimizes the total within-DOF uncertainty over its entire symplectic orbit, and any excess is exactly accounted for by cross-DOF correlation (Fischer defect). What it does not decide is the per-DOF refinement, whether an individual uj(t)u_j(t) can dip below ν while others rise. That is the genuinely open-problem content discussed in Problems 3.17 – 3.19. Problem 3.17 (Symplectic Schur–Horn problem for within-DOF uncertainties). Characterize, for fixed symplectic spectrum ν1≥⋯≥νn>0 _1≥·s≥ _n>0, the attainable set (ν)=(u1(Σ),…,un(Σ)):Σ∈the Sp(2n,ℝ)-orbit with spectrum ν⊂ℝ>0n,U(ν)= \ (u_1( ),…,u_n( ) ): Sp(2n,R)-orbit with spectrum ν \ _>0^n, where uj(Σ)=detΣjju_j( )= _j. In particular, we need to explore (a) is minjuj≥νn _ju_j≥ _n on the whole orbit (so that no DOF can be squeezed below the smallest symplectic eigenvalue)? (b) in the equipartitioned case νj≡ν _j≡ν, is uj≥νu_j≥ν for every j (the per-DOF form of the conjecture in [1])? And (c) describe (ν)U(ν) by majorization-type inequalities, in analogy with the Schur–Horn theorem, using the symplectic eigenvalue technology of [8]. Some partial cues include (i) ∏juj≥∏jνj _ju_j≥ _j _j (Theorem 3.16); (i) uj≥νnu_j≥ _n would follow from the interlacing-type bound “every 2×22× 2 symplectic compression of Σ has symplectic eigenvalue ≥νn≥ _n,” a statement of exactly the kind studied in [8]; and (i) for n=1n=1, (ν)=νU(ν)=\ν\ (Theorem 3.16). Problem 3.18 (Kime/torus reformulation and estimability). Via Lemma 2.3 applied per DOF, a state on ℝ2nR^2n (off the coordinate axes) is a density on n×(0,∞)nT^n×(0,∞)^n in variables (θ1,…,θn,J1,…,Jn)( _1,…, _n,J_1,…,J_n), and cross-DOF correlations at fixed actions are encoded in the joint phase law on nT^n, e.g., in the relative-phase moment matrix Rjk=[ei(Θj−Θk)]R_jk=E [e^i( _j- _k) ], which is Hermitian, positive semidefinite, unit diagonal. The solution may require progress in three directions. (a) Express the quantities uj,νju_j, _j of Problem 3.17, for Gaussian states, in terms of (R,J1,…,Jn)(R,EJ_1,…,EJ_n) in the small-dispersion regime, and determine which functions of (ν)U(ν) are identifiable from repeated-measurement data under the multivariate extension of the KPT observation model and its convolution identity M=FΦM=F of [4] (circular convolution now acting on nT^n); (b) prove the multivariate anchored-identifiability theorem (gauge group: independent rigid rotations of each θj _j, i.e., the torus nT^n acting diagonally on R by conjugation with unimodular diagonals, note R itself is gauge-covariant while specRspecR and |Rjk||R_jk| are gauge-invariant); (c) derive the Cramér–Rao bound for estimating (|Rjk|)j<k(|R_jk|)_j<k, extending the parametric Cramér–Rao theorem of [4], so that Problem 3.17(b) acquires a statistically testable surrogate. Relation to symplectic capacities. For the covariance ellipsoid EΣ=z:z⊤Σ−1z≤1E_ =\z:z^\! ^-1z≤ 1\, the linear symplectic capacity equals πνnπ _n (the smallest symplectic eigenvalue sets the Gromov width of the ellipsoid) [9, 10]. Thus, Problem 3.17(a) asks whether the within-DOF uncertainty of every DOF dominates the capacity scale of the total state. Under nonlinear Hamiltonian flows Σ is no longer transported by Sp(2n,ℝ)Sp(2n,R), but Gromov non-squeezing still bounds the projection of the evolved support onto each conjugate plane. Problem 3.19 (Nonlinear invariant interpolating entropy and capacity). Construct a functional [ρ]C[ρ] of states on ℝ2nR^2n such that: (i) C is invariant under all (possibly nonlinear) Hamiltonian flows; (i) C reduces to πνnπ _n on Gaussian states; (i) C lower-bounds 2πminjuj2π _ju_j up to a universal constant; and (iv) C is expressible through the kime-torus data of Problem 3.18 (hence estimable). Consider the following candidate functional, a sublevel-set capacity of ρ at the entropy-calibrated level e−[ρ]e^- S[ρ], i.e., [ρ]=c(ρ≥e−[ρ])C[ρ]=c (\ρ≥ e^- S[ρ]\ ) for a normalized symplectic capacity c. Properties (i) and (i) then hold by symplectomorphism-invariance of c and a direct Gaussian computation, while (i)–(iv) are open. 3.5 Dynamics and thermodynamics: equipartition as the kime-diffusive attractor The final component of Problem (I) is the conjecture that “Hamiltonian dynamics preserves, at least locally, the entropic relationships that one would find at equilibrium,” with “equipartition of entropy over uncorrelated DOF” as the conjectured floor. The kime representation supplies both an exactly solvable relaxation model (Lemma 2.7) and a clean statement of what Hamiltonian flow does and does not preserve. Definition 3.20 (Kime-deformed evolution). Let H be a one-DOF Hamiltonian admitting a global action–angle chart (θ,J)∈1×(θ,J) ^1×J, ⊆(0,∞)J (0,∞) open, with frequency ω(J)=H′(J)ω(J)=H (J) (Liouville–Arnold, [14, Ch. 10]). For ε≥0 ≥ 0, the kime-deformed evolution of a density ρ~ ρ on 1×S^1×J is ∂tρ~=−ω(J)∂θρ~+ε∂θ2ρ~. _t ρ=-\,ω(J)\, _θ ρ+ \,∂^2_θ ρ. (11) For ε=0 =0 this is the Liouville equation of H in action–angle variables; for ω≡0ω≡ 0 it is the kime-phase Fokker–Planck equation of [6] fiberwise in J. At the level of classical densities, the one-parameter family (11) interpolates between an entropy-conserving (transport) and an entropy-producing (diffusive) sector, in direct analogy with the unitary/contractive interpolation of the kime-ray factorization of [4] recalled in Proposition 4.6 below. Theorem 3.21 (Entropy production and the equipartition attractor). Let ρ~t ρ_t solve (11) with strictly positive C2C^2 initial datum of finite entropy, rapid decay in J, and ε>0 >0. Then, (i) The action marginal ρJ _J is conserved: ∂tρJ(J)=0 _t _J(J)=0 for all J. (i) Entropy production is purely diffusive and nonnegative dt[ρ~t]=ε∫−π(∂θρ~t)2ρ~tdθdJ≥ 0, ddt\, S[ ρ_t]= _J\! _-π^π ( _θ ρ_t)^2 ρ_t\,dθ\,dJ\;≥\;0, with instantaneous equality iff ρ~t ρ_t is phase-equipartitioned (Definition 2.4). In particular, for ε=0 =0 (pure Hamiltonian flow) [ρ~t] S[ ρ_t] is exactly conserved. (i) The phase-equipartitioned state ρ~∞(θ,J)=ρJ(J)/2π ρ_∞(θ,J)= _J(J)/2π with the initial action marginal is the unique stationary solution with that marginal, is the entropy maximizer among all densities with action marginal ρJ _J, namely [ρ~]≤[ρJ]+log2π(equality iff phase-equipartitioned), S[ ρ]\;≤\; S[ _J]+ 2π (equality iff phase-equipartitioned), and ρ~t→ρ~∞ ρ_t→ ρ_∞ in L2L^2 with fiberwise moment decay |αn(t∣J)|=e−εn2t|αn(0∣J)|| _n(t J)|=e^- n^2t\,| _n(0 J)|. (iv) Pure Hamiltonian flow (ε=0 =0) preserves the class of phase-equipartitioned states and every functional of the action marginal; i.e., equilibrium entropic relationships are exactly invariant under the Hamiltonian sector of the kime-deformed family. Proof. (i) By integrating (11) over θ∈1θ ^1, both terms are exact θ-derivatives and vanish by periodicity. (i) As in Lemma 2.7(i), dt=−∫(1+logρ~)∂tρ~ ddt S=- (1+ ρ)\, _t ρ. The transport contribution is ∫ω(J)∂θρ~(1+logρ~)dθdJ=∫ω(J)(∫−π∂θ[ρ~logρ~]dθ)dJ=0 ω(J)\, _θ ρ\,(1+ ρ)\,dθ\,dJ= ω(J) ( _-π^π _θ [ ρ ρ ]dθ )dJ=0 by periodicity (note ∂θ[ρ~logρ~]=(1+logρ~)∂θρ~ _θ[ ρ ρ]=(1+ ρ) _θ ρ). The diffusive contribution is computed exactly as in Lemma 2.7(i), fiberwise in J and integrated dJdJ, using rapid decay to justify Fubini. Equality holds iff ∂θρ~t≡0 _θ ρ_t≡ 0. (i) The bound is Lemma 3.2 on 1×S^1×J plus [Φ(⋅∣J)]≤log2π S[ (· J)]≤ 2π fiberwise (Jensen), i.e., conditional entropy is maximized by the Haar law on each fiber; equality iff Φ(⋅∣J) (· J) is uniform for a.e. J. Stationarity and uniqueness with fixed marginal: ∂tρ~=0 _t ρ=0 with (i) forces ∂θρ~=0 _θ ρ=0, and (i) fixes the J-profile. Moment decay: the nnth fiber moment αn(t∣J) _n(t J) obeys α˙n=(inω(J)−εn2)αn α_n=(in\,ω(J)- n^2) _n, so |αn(t∣J)|=e−εn2t|αn(0∣J)|| _n(t J)|=e^- n^2t| _n(0 J)|; convergence of ρ~t→ρ~∞ ρ_t→ ρ_∞ in L2L^2 follows by Parseval fiberwise and dominated convergence in J. (iv) For ε=0 =0, (11) transports along θ˙=ω(J) θ=ω(J), J˙=0 J=0. A θ-independent density is a fixed point, and any functional of ρJ _J is conserved by (i). ∎ Summary of what is proven and what remains open. Theorem 3.21 proves, within the kime representation (a) equipartition over the phase is the unique entropy-maximal state compatible with the conserved action statistics, the rigorous form of “equipartition of entropy is the bound,” as an upper bound on entropy at fixed action marginal attained exactly at equilibrium; (b) Hamiltonian flow is the entropy-neutral boundary ε=0 =0 of the kime-deformed family and preserves all equilibrium relationships exactly, which is the precise (and here, global rather than merely local) version of the conjecture that Hamiltonian dynamics preserves equilibrium entropic relationships. The correspondingly open statements are the multi-DOF per-DOF refinements (Problem 3.17) and the reconciliation of the upper-bound role of equipartition at fixed actions with the lower-bound role of the equipartitioned uncertainty product in Theorem 3.16; the two are dual faces (max-entropy at fixed invariants vs. min-uncertainty at fixed entropy) of one variational principle, whose sharp joint statement for n≥2n≥ 2 is presented in Conjecture 3.22. Conjecture 3.22 (Equipartition duality). Fix n≥2n≥ 2, an entropy value s, and an action marginal ρ _ J on (0,∞)n(0,∞)^n. Among all states on n×(0,∞)nT^n×(0,∞)^n with entropy ≥s≥ s and action marginal ρ _ J, the phase-equipartitioned product state (unique when it exists) simultaneously (i) maximizes the entropy, (i) minimizes every within-DOF uncertainty uju_j, and (i) is the unique state at which the per-DOF conjectured bound of Problem 3.17(b) is saturated for all j; moreover it is the unique fixed point, with the given marginal, of the multi-DOF kime-deformed semigroup ∂tρ~=∑j(−ωj∂θj+ε∂θj2)ρ~ _t ρ= _j(- _j _ _j+ \,∂^2_ _j) ρ. Statistical formulation of Problem (I). Under Assumption 1.1, every quantity in this section is an attribute of the ensemble of repeated experiments: r, αn _n, ℐ[Φ]I[ ], and RjkR_jk are estimable by kime-phase tomography with quantified error [4]. The terms [ρ] S[ρ] and uju_j are estimable from calibrated observables via the action–angle dictionary. Theorem 3.7, Theorem 3.8, and Theorem 3.16 are thus experimentally checkable inequalities on reproducibility statistics, and Problems 3.17–3.19 come with built-in numerical surrogates (Problem 3.18). This is the distinctive contribution of the kime formulation, the open problems of [1] are re-expressed in estimable coordinates without loss of mathematical content. 4 Problem I (invariant entropy): why continuous quantities pair, and the kime chart as normal form Problem (I) asks for a general, assumption-minimal account of the following phenomenon. A coordinate-invariant notion of “count of states” (hence of entropy) over a continuum appears to exist only when continuous quantities organize into conjugate pairs, and the resulting structure is symplectic. The problem statement further suggests that the natural home of the construction is a complex (or generalized complex) structure, one real dimension pairing with another inside ℂC. The kime coordinate κ=teiθκ=te^iθ is precisely such a complex pairing, and Lemma 2.3 shows the pairing is symplectically exact. In this section we prove the pairing phenomenon as a theorem about invariant measures and entropies (Theorems 4.1–4.3 and Corollary 4.4), identify the kime chart as a Kähler normal form (Proposition 4.5), and state the open remainder (Problems 4.8–4.11). 4.1 Invariant entropy forces an invariant measure Throughout, X is a smooth σ-compact manifold, densities are with respect to a fixed smooth positive reference measure λ, and f∗ρf_*ρ denotes the pushforward density of ρ under a diffeomorphism f (Jacobian Jf>0J_f>0 of f w.r.t. λ, assumed orientation-compatible for simplicity). Theorem 4.1 (Invariant entropy ⇔ invariant measure). Let f∈Diff(X)f (X) with continuous Jacobian JfJ_f. The following statements are equivalent (a) λ[f∗ρ]=λ[ρ] S_λ[f_*ρ]= S_λ[ρ] for every compactly supported continuous density ρ with finite entropy; (b) f preserves λ (i.e., Jf≡1J_f≡ 1). Proof. (b)⇒ (a) is the change-of-variables identity (the proof of Lemma 3.10 with unit Jacobian, valid on any X). For (a)⇒ (b), the same computation gives, for every admissible ρ, λ[f∗ρ]=λ[ρ]+ρ[logJf], S_λ[f_*ρ]= S_λ[ρ]+E_ρ [ J_f ], so (a) forces ∫ρlogJfdλ=0 ρ\, J_f\,dλ=0 for all such ρ. Taking ρ to run through approximate identities concentrated at an arbitrary point x yields logJf(x)=0 J_f(x)=0 by continuity; hence Jf≡1J_f≡ 1. ∎ Theorem 4.1 converts Problem (I) into a question about invariant measures under the physically mandated transformation group. The physical mandate, following [1], is that different experimenters may label the same continuous quantity by arbitrary smooth reparametrizations (units, gauges, monotone recalibrations), so the group must contain all diffeomorphisms of the configuration quantities; entropy must be the same for all of them. Theorem 4.2 (No invariant measure on unpaired quantities). Let Q=ℝmQ=R^m, m≥1m≥ 1, regarded as the value space of m continuous quantities, and let Diff(Q)Diff(Q) act naturally. There is no nonzero σ-finite Borel measure on Q, absolutely continuous with a locally integrable density g, invariant under all of Diff(Q)Diff(Q). Consequently, by Theorem 4.1, no reparametrization-invariant entropy exists for unpaired continuous quantities. Proof. Invariance under f means g(f(x))|detDf(x)|=g(x)g(f(x))\,|\! Df(x)|=g(x) for a.e. x. Taking f to be all translations gives g(x+a)=g(x)g(x+a)=g(x) for a.e. x, for every a, hence g≡c≥0g≡ c≥ 0 a.e. (mollifying, a translation-invariant locally integrable function agrees a.e. with its smooth translation-invariant mollification, which is constant). Taking f(x)=λxf(x)=λ x, λ>1λ>1, gives cλm=c\,λ^m=c, so c=0c=0. ∎ 4.2 Pairing: cotangent lifts and the uniqueness of the Liouville measure The classical repair is to adjoin to each quantity q a conjugate p transforming contragradiently, i.e., to pass from Q to T∗QT^*Q with the tautological lift of Diff(Q)Diff(Q), T∗f:(q,p)⟼(f(q),Df(q)−⊤p),f∈Diff(Q).T^*\!f:(q,p) (f(q),\,Df(q)^-\! p ), f (Q). (12) Theorem 4.3 (Existence and uniqueness of the invariant measure on pairs). Let Q=ℝnQ=R^n and let G=T∗f:f∈Diff(Q)G=\T^*\!f:f (Q)\ act on T∗Q=ℝ2nT^*Q=R^2n by (12). Then, (i) every T∗fT^*\!f has Jacobian identically 11; hence the Liouville measure dnqdnpd^nq\,d^np is G-invariant, and the associated entropy is reparametrization-invariant (Theorem 4.1); (i) conversely, any G-invariant measure with continuous positive density on T∗QT^*Q is a constant multiple of the Liouville measure; equivalently, the invariant entropy is unique up to an additive constant. Proof. (i) The differential of (12) in block form is lower block-triangular with diagonal blocks Df(q)Df(q) and Df(q)−⊤Df(q)^-\! , so its determinant is detDf⋅detDf−⊤=1 Df· Df^-\! =1. (i) Let m>0m>0 be a continuous invariant density: m(T∗f(z))⋅1=m(z)m(T^*\!f(z))· 1=m(z) for all z, f, by (i). We claim G acts transitively on (q,p):p≠0\(q,p):p≠ 0\. Given (q0,p0),(q1,p1)(q_0,p_0),(q_1,p_1) with p0,p1≠0p_0,p_1≠ 0, choose B∈GL(n,ℝ)B (n,R) with Bp0=p1B\,p_0=p_1, set A=B−⊤A=B^-\! (so that A−⊤p0=p1A^-\! p_0=p_1), and let f(x)=q1+A(x−q0)∈Diff(Q)f(x)=q_1+A(x-q_0) (Q). Then T∗f(q0,p0)=(q1,p1)T^*\!f(q_0,p_0)=(q_1,p_1). Hence m is constant on the dense open orbit p≠0\p≠ 0\, and by continuity constant everywhere: m≡c>0m≡ c>0. For the entropy statement, rescaling λ↦cλ c\,λ shifts λ S_λ by the constant logc c. ∎ Corollary 4.4 (The pairing theorem). A theory of m continuous quantities that (a) admits arbitrary smooth relabelings of the quantities themselves and (b) possesses a relabeling-invariant entropy functional on states, cannot realize the quantities as coordinates on the bare value space (Theorem 4.2); it can realize them as the base coordinates of a cotangent bundle with contragradient conjugates, and then the invariant entropy exists and is the Liouville entropy, unique up to an additive constant (Theorem 4.3). In this precise sense, continuous quantities must come in conjugate pairs for entropy to be well defined, and the count of states is fixed (up to units) to be the symplectic volume. Note that Corollary 4.4 is deliberately stated with the group generated by base reparametrizations only. Enlarging G to all symplectomorphisms of T∗QT^*Q preserves the conclusion (they too have unit Jacobian); enlarging it to all volume-preserving diffeomorphisms destroys the symplectic structure while retaining the measure, the gap between these two groups for n≥2n≥ 2 is the content of Problem 4.11 below. 4.3 The kime chart as complex normal form Proposition 4.5 (Kähler triple on the kime plane). On the punctured kime plane ℂ∗∋κ=teiθC^* κ=te^iθ carry the cone metric g0g_0 of (1), the complex structure JκJ_κ of the coordinate κ (i.e., multiplication by i: Jκ∂t=t−1∂θJ_κ _t=t^-1 _θ, Jκ(t−1∂θ)=−∂tJ_κ(t^-1 _θ)=- _t), and the Kähler form ωK=i2dκ∧dκ¯=tdt∧dθ=dJ∧dθ. _K\;=\; i2\,dκ κ\;=\;t\,dt θ\;=\;dJ θ. Then (g0,ωK,Jκ)(g_0, _K,J_κ) is a compatible Kähler triple ωK(⋅,⋅)=g0(Jκ⋅,⋅),g0(Jκ⋅,Jκ⋅)=g0(⋅,⋅),dωK=0. _K(·,·)=g_0(J_κ\,·,·), g_0(J_κ\,·,J_κ\,·)=g_0(·,·), _K=0. Consequently the kime pairing of the two real quantities (t,θ)(t,θ) into one complex quantity κ realizes, in one chart, all three structures whose interplay Problem (I) asks about: the invariant count of states (ωK _K, by Theorem 4.3 and Lemma 2.3), the metric geometry of the time cone (g0g_0, [4]), and the complex pairing (JκJ_κ). Proof. First, dκ=eiθ(dt+itdθ)dκ=e^iθ(dt+it\,dθ) and dκ¯=e−iθ(dt−itdθ)d κ=e^-iθ(dt-it\,dθ), so dκ∧dκ¯=−2itdt∧dθdκ κ=-2it\,dt θ and i2dκ∧dκ¯=tdt∧dθ=dJ∧dθ i2dκ κ=t\,dt θ=dJ θ. In the frame (e1,e2)=(∂t,t−1∂θ)(e_1,e_2)=( _t,t^-1 _θ), which is g0g_0-orthonormal by (1), JκJ_κ acts as the standard rotation e1↦e2↦−e1e_1 e_2 -e_1 (this is multiplication by i in the coordinate κ, since ∂tκ=eiθ _tκ=e^iθ and t−1∂θκ=ieiθt^-1 _θκ=ie^iθ). Then g0(Jκe1,e1)=g0(e2,e1)=0=ωK(e1,e1)g_0(J_κe_1,e_1)=g_0(e_2,e_1)=0= _K(e_1,e_1) and g0(Jκe1,e2)=g0(e2,e2)=1=tdt∧dθ(e1,e2)=ωK(e1,e2)g_0(J_κe_1,e_2)=g_0(e_2,e_2)=1=t\,dt θ(e_1,e_2)= _K(e_1,e_2). Bilinearity and antisymmetry give ωK=g0(Jκ⋅,⋅) _K=g_0(J_κ·,·). Orthogonality of JκJ_κ in an orthonormal frame is clear, and dωK=0d _K=0 in two dimensions is automatic (ωK=dJ∧dθ _K=dJ θ is even exact away from the apex, with primitive JdθJ\,dθ). ∎ Revisiting Orientation. As anticipated in original orientation convention remark 2.2, ωK=dJ∧dθ=−Ψ∗(dq∧dp) _K=dJ θ=- ^*(dq ). The Kähler form of the holomorphic coordinate κ and the action–angle Darboux form of Lemma 2.3 agree up to orientation, being exchanged by κ↦κ¯κ κ (equivalently, by swapping the roles of the conjugate pair, since J,θ=+1\J,θ\=+1 with respect to ωK _K while θ,J=+1\θ,J\=+1 with respect to Ψ∗ω0 ^* _0). Both induce the same unsigned Liouville measure tdtdθt\,dt\,dθ, so every entropy and measure statement in this paper is insensitive to the choice. We keep θ,J=+1\θ,J\=+1 as the mechanical convention and ωK _K as the complex-analytic one. Proposition 4.6 (Kime-ray factorization and Wick rotation; [4]). Let H be self-adjoint and bounded below on a Hilbert space ℋH, and for κ=teiϑκ=te^i with t≥0t≥ 0 define the kime propagator Uϑ(t)=e−iℏκHU_ (t)=e^- i κ H. Then for every ϑ the two factors in Uϑ(t)=exp(sinϑℏtH)⏟self-adjointexp(−icosϑℏtH)⏟unitaryU_ (t)= \! ( \,tH )_self-adjoint\; \! (- i \,tH )_unitary commute, and if H≥0H≥ 0 then ‖Uϑ(t)‖≤1\|U_ (t)\|≤ 1 for all ϑ∈[−π,0] ∈[-π,0]: on the lower half of the kime circle the self-adjoint factor is a contraction semigroup. The special cases are: ϑ=0 =0, the real-time Schrödinger group e−itH/ℏe^-itH/ ; ϑ=−π/2 =-π/2, the Euclidean (Wick-rotated) heat semigroup e−tH/ℏe^-tH/ ; and −π<ϑ<0-π< <0, damped oscillatory propagators with contraction rate |sinϑ|| | and phase rate cosϑ . Proof. Functional calculus for the self-adjoint operator H: the scalar identity −ieiϑ=sinϑ−icosϑ-ie^i = -i gives, for every spectral value λ, e−iℏλteiϑ=esinϑℏλte−icosϑℏλte^- i λ te^i =e λ t\,e^- i λ t, and both factors are functions of the single operator H, hence commute. If H≥0H≥ 0 and ϑ∈[−π,0] ∈[-π,0] then sinϑ≤0 ≤ 0, so the first factor is a contraction and the second unitary, whence the norm bound. The special cases are read off directly. ∎ Reflection on Problem (I) in the kime normal form. Proposition 4.5 exhibits the pairing demanded by Corollary 4.4 as literally complex-analytic: one compact quantity (θ, Haar-uniform at equilibrium by Theorem 3.21) pairs with one noncompact quantity (J=t2/2J=t^2/2), and the invariant count of states is the Kähler area. Proposition 4.6 shows the same complex structure organizes dynamics. Rotating κ interpolates between the entropy-conserving (unitary/Hamiltonian, Theorem 3.21(i) with ε=0 =0) and entropy-producing (contractive/diffusive) sectors. Thus, the Problem (I) suggestion that “complex structures may be central to why quantities pair”, holds exactly in the kime chart. What remains open is whether it is forced in general, which we now state. 4.4 Remaining Problem I Open Problems Proposition 4.7 (Symplectic maps preserve all symplectic spectra). If S∈Sp(2n,ℝ)S (2n,R) then for every Σ≻0 0 the symplectic spectrum of SΣS⊤S S^\! equals that of Σ (proof in Lemma 3.14). Antisymplectic maps (SΩS⊤=−ΩS S^\! =- ) do likewise. Problem 4.8 (Spectral characterization of the symplectic group). Prove or disprove the converse: if S∈GL(2n,ℝ)S (2n,R) preserves the symplectic spectrum of every Σ≻0 0, then S is symplectic or antisymplectic up to the scaling S↦λS λ S forced by ν(λ2Σ)=λ2ν(Σ)ν(λ^2 )=λ^2ν( ) (so, for normalized S with |detS|=1| S|=1). A proof would characterize Sp(2n,ℝ)Sp(2n,R) purely by an estimable statistical invariant (symplectic spectra of covariance matrices), replacing the geometric definition by an information-theoretic one, the sharpest available answer to Problem (I)’s request for an entropy-first derivation of the symplectic structure. Problem 4.9 (Entropy-only rigidity). Theorem 4.1 assumed entropy invariance for all states. Determine the minimal state classes ℱF for which “[f∗ρ]=[ρ] S[f_*ρ]= S[ρ] for all ρ∈ℱρ ” still forces Jf≡1J_f≡ 1 (e.g., Gaussians only; kime states with von Mises phase laws only), and, dually, characterize the group of transformations preserving the entropy of every equilibrium (phase-equipartitioned) kime state. The latter group is strictly larger than the measure-preserving group (it contains all fiberwise rotations θ↦θ+c(J)θ θ+c(J) trivially, but also non-measure-preserving maps acting only on null sets of equilibria); its computation quantifies exactly how much of the symplectic structure is visible to equilibrium thermodynamics alone. Problem 4.10 (Generalized complex rigidity of the Wick interpolation). Proposition 4.5 produces a Kähler triple in one kime DOF; Proposition 4.6 deforms the dynamics between its symplectic and metric legs. Formulate and prove (or refute) the following rigidity statement: if a 2n2n-dimensional state continuum carries (a) a reparametrization-invariant entropy (hence, by Theorems 4.1 and 4.3, a distinguished volume), (b) a one-parameter interpolation of evolutions that is entropy-conserving at one end and satisfies a de Bruijn identity dt=εℐ≥0 ddt S= \,I≥ 0 elsewhere (Theorem 3.21(i)), then the infinitesimal generators assemble into a generalized complex (indeed generalized Kähler) structure in the sense of [23], whose pure-symplectic locus is the Hamiltonian sector and whose type jumps encode the diffusive sector. A positive answer would derive “quantities pair inside ℂC” from entropy axioms alone, completing the program of Problem (I). Problem 4.11 (Volume-preserving vs. symplectic for n≥2n≥ 2). Corollary 4.4 pins down the measure but, for n≥2n≥ 2, not the finer symplectic structure: SDiff(ℝ2n)⊋Symp(ℝ2n)SDiff(R^2n) (R^2n). Identify the weakest statistical requirement that reduces the invariance group from volume-preserving to symplectic. Candidates, in increasing strength: (i) invariance of all within-DOF uncertainties uju_j at equilibrium; (i) invariance of the linear symplectic capacity of covariance ellipsoids (see the capacity remark 3.4; by [9, 10] this fails for generic volume-preserving linear maps once n≥2n≥ 2); (i) invariance of the whole symplectic spectrum (Problem 4.8). Determine which of (i)–(i) are equivalent, and which are estimable from kime-tomographic data in the sense of Problem 3.18. 5 Problem I (directional DOF): classical spin, coadjoint orbits, and the kime circle Problem (I) asks for a classical relativistic directional degree of freedom whether, nonrelativistically, the direction phase space is the sphere with conjugate pair θxy,Sz=1\θ^xy,S_z\=1. The relativistic construction, the correct conjugate variables, and the choice between “spin as four-vector” and “spin as two-form” are open. The kime contribution is threefold: (a) the conjugate variable θxyθ^xy is a kime-type circular phase, so the entire statistical machinery of Sections 2–3 applies fiberwise and yields a compact-compact uncertainty theorem (Theorem 5.3); (b) the null decomposition nR,Ln_R,L suggested in [1] is exactly the classical shadow of the D=5D=5 kime Dirac structure of [6], whose chirality obstruction (Proposition 5.6) constrains the admissible answers; (c) the vector/two-form dichotomy becomes a precise moment-map question on Poincaré coadjoint orbits (Problem 5.11). 5.1 The nonrelativistic sector: the sphere as a kime cylinder Definition 5.1 (Spin phase space). For s>0s>0, the spin-s phase space is the sphere s2=∈ℝ3:||=sS^2_s=\ S ^3:| S|=s\ with symplectic form ωs=ssinΘdΘ∧dφ _s=s \,d in spherical coordinates, where =s(sinΘcosφ,sinΘsinφ,cosΘ) S=s\,( ,\; ,\; ), normalized so that the components Sx,Sy,SzS_x,S_y,S_z satisfy Si,Sj=ϵijkSk\S_i,S_j\= _ijkS_k. Here, φ is the rotation angle about the z-axis, i.e., the θxyθ^xy of Problem 1.20 in [1]. Lemma 5.2 (Darboux/kime chart on the sphere; Archimedes). On s2∖polesS^2_s \poles\, the pair (φ,Sz)=(φ,scosΘ)( ,S_z)=( ,s ) satisfies ωs=dφ∧dSz _s=d _z, hence φ,Sz=1\ ,S_z\=1, and the chart s2∖±sz^→≅1×(−s,s),⟼(φ,Sz),S^2_s \± s z\\; \; \;\;S^1×(-s,s), S ( ,S_z), is a symplectomorphism onto the finite kime cylinder (1×(−s,s),dφ∧dSz) (S^1×(-s,s),\,d _z ). In particular the invariant count of states is Vol(s2,ωs)=4πsVol(S^2_s, _s)=4π s. Proof. dSz=−ssinΘdΘdS_z=-s \,d , so dφ∧dSz=ssinΘdΘ∧dφ=ωsd _z=s \,d = _s; bijectivity onto the open cylinder is clear. (This is Archimedes’ hat-box theorem in symplectic form.) The bracket normalization matches Definition 5.1: e.g., Sx,Sy=Sz\S_x,S_y\=S_z is verified in the chart by direct computation with Sx=s2−Sz2cosφS_x= s^2-S_z^2 , Sy=s2−Sz2sinφS_y= s^2-S_z^2 . Thus, the total volume is ∫dφdSz=2π⋅2s \,dS_z=2π· 2s. ∎ Lemma 5.2 says the directional DOF is a kime DOF with compact conjugate momentum: the latent phase φ∈1 ^1 carries the trial-to-trial variability of repeated orientation measurements (Assumption 1.1), and its conjugate SzS_z ranges over a finite interval. Both marginals are now compact, so the uncertainty principle acquires floors on both sides. Theorem 5.3 (Compact–compact entropic uncertainty relation for a directional DOF). Let ρ be a probability density on (1×(−s,s),dφdSz) (S^1×(-s,s),\;d \,dS_z ) with [ρ]>−∞ S[ρ]>-∞. Let r∈[0,1)r∈[0,1) be the mean resultant length of the φ -marginal and ρSz _S_z the SzS_z-marginal. Then Λ(r)⋅e[ρSz]≥e[ρ],withΛ(r)≤2π,e[ρSz]≤2s, (r)· e S[ _S_z]\;≥\;e S[ρ], (r)≤ 2π, e S[ _S_z]≤ 2s, (13) with equality in (13) iff ρ is a product of a von Mises law in φ and an arbitrary SzS_z-marginal density achieving its entropy (and equality in the two ceilings iff the respective marginals are uniform). Under any Hamiltonian flow on (s2,ωs)(S^2_s, _s), e.g., Larmor precession H=γ⋅H=γ\, B· S, the right-hand side e[ρ]e S[ρ] is a constant of motion, while r and [ρSz] S[ _S_z] evolve; (13) caps their joint concentration at all times, and [ρ]≤log(4πs) S[ρ]≤ (4π s) is the absolute ceiling set by the count of states of Lemma 5.2. Proof. Subadditivity (Lemma 3.2) gives [ρ]≤[Φφ]+[ρSz] S[ρ]≤ S[ _ ]+ S[ _S_z], and Lemma 3.4 bounds [Φφ]≤hc(r)=logΛ(r) S[ _ ]≤ h_c(r)= (r). After exponentiating, direct equality analysis is as in Theorem 3.7. The ceilings are Jensen’s inequality on the two compact ranges. Hamiltonian flows preserve ωs _s, hence the measure dφdSzd \,dS_z (Lemma 5.2), hence [ρ] S[ρ] and the global ceiling is Jensen on the total space. ∎ Note that for the nonrelativistic directional DOF, Theorem 5.3 answers the question posed in Problem (I)(a) as it recurs inside Problem (I). The correct “uncertainty product” for the non-canonical, doubly compact pair (φ,Sz)( ,S_z) is the product of entropy widths, its floor is the invariant e[ρ]e S[ρ], and the von Mises family is again extremal on the circular leg. All quantities are estimable from repeated orientation measurements via circular tomography [4, 11]. 5.2 Relativistic Poincaré coadjoint orbits and the null pair In this subsection and the next we work on four-dimensional Minkowski space with signature (+,−,−,−)(+,-,-,-); uμu^μ is the four-velocity (u⋅u=c2u· u=c^2) and SμS^μ the spin four-vector, spacelike with S⋅S=−s2S· S=-s^2 and S⋅u=0S· u=0. Definition 5.4 (Classical spinning particle; [20, 21]). The phase space of a free classical particle of mass m>0m>0 and spin s>0s>0 is the coadjoint orbit m,sO_m,s of the (connected) Poincaré group through an element with Casimirs P⋅P=m2c2P· P=m^2c^2 and W⋅W=−m2c2s2W· W=-m^2c^2s^2, where Wμ=12ϵμνρσPνSρσW^μ= 12ε^μνρσP_νS_ρσ is the Pauli–Lubański vector and SρσS_ρσ the spin two-form (intrinsic angular momentum). m,sO_m,s carries the canonical Kirillov–Kostant–Souriau symplectic structure; it is 88-dimensional, fibering over the mass shell with fiber the sphere s2S^2_s of Definition 5.1 (the little-group orbit). Proposition 5.5 (The null pair nR,nLn_R,n_L). On m,sO_m,s define, along each state, the four-vectors nR=uc+Ss,nL=uc−Ss.n_R= uc+ Ss, n_L= uc- Ss. Then nR,nLn_R,n_L are future-directed null vectors with nR⋅nL=2n_R· n_L=2, and the map (u,S)↦(nR,nL)(u,S) (n_R,n_L) is a bijection onto ordered pairs of future-directed null vectors with inner product 22, with inverse u=c2(nR+nL)u= c2(n_R+n_L), S=s2(nR−nL)S= s2(n_R-n_L). Consequently the directional content of a classical spinning particle is exactly a pair of null directions, the classical counterpart of the right/left Weyl decomposition of a Dirac spinor in D=4D=4 [6]. Proof. nR⋅nR=u⋅u/c2+2u⋅S/(cs)+S⋅S/s2=1+0−1=0n_R· n_R=u· u/c^2+2\,u· S/(cs)+S· S/s^2=1+0-1=0, and similarly nL⋅nL=0n_L· n_L=0; nR⋅nL=u⋅u/c2−S⋅S/s2=1−(−1)=2n_R· n_L=u· u/c^2-S· S/s^2=1-(-1)=2. Future-directedness: since S⋅u=0S· u=0, u⋅nR=u⋅nL=u⋅uc=c>0,u· n_R=u· n_L= u· uc=c>0, and a nonzero null vector whose Minkowski product with a future-directed timelike vector is positive is itself future-directed (in signature (+,−,−,−)(+,-,-,-), u⋅n=u0n0−⋅u· n=u^0n^0- u· n with ||=n0| n|=n^0 forces sgnn0=sgn(u⋅n)sgnn^0=sgn(u· n) by Cauchy–Schwarz, ||<u0| u|<u^0). The displayed inverse is linear algebra; it maps the stated pair set back into u⋅u=c2,S⋅S=−s2,S⋅u=0\u· u=c^2,\ S· S=-s^2,\ S· u=0\ by reversing the three computations. ∎ 5.3 The chirality obstruction from the kime compactification Kime representation realizes complex time via a second, compact temporal direction. The natural relativistic arena is then the (3+2)(3+2)-signature Clifford algebra Cl(3,2)Cl(3,2) of [6], with metric η=diag(−1,−1,−1,+1,+1)η=diag(-1,-1,-1,+1,+1) in the conventions of that paper. The following algebraic fact, proved there in the spinorial setting (the “no chirality in five dimensions” theorem of [6]), constrains every classical construction that descends from it; note that the argument uses only that the spacetime dimension D=5D=5 is odd, not the specific signature. Proposition 5.6 (No chirality in D=5D=5; [6]). Let γ1,…,γ5γ^1,…,γ^5 generate an irreducible complex representation of Cl(3,2)Cl(3,2) (γM,γN=2ηMN\γ^M,γ^N\=2η^MN, η=diag(−1,−1,−1,+1,+1)η=diag(-1,-1,-1,+1,+1)). Then there is no operator χ with χ2=χ^2=1 and χ,γM=0\χ,γ^M\=0 for all M. Proof. Set Γ=γ1γ2γ3γ4γ5 =γ^1γ^2γ^3γ^4γ^5. For each fixed M, moving γMγ^M through the four other factors of Γ produces four sign flips, and through itself none, so γMΓ=ΓγMγ^M = γ^M, Γ is central. By Schur’s lemma, Γ=c 1 =c\,1 with c≠0c≠ 0 (Γ is invertible, each γMγ^M being so). If χ anticommuted with every γMγ^M, then moving χ through the five factors of Γ gives χΓ=(−1)5Γχ=−Γχ =(-1)^5 χ=- χ; but Γ=c 1 =c\,1 commutes with everything, so cχ=−cχc\,χ=-c\,χ, i.e., χ=0χ=0, contradicting χ2=χ^2=1. ∎ Corollary 5.7 (Constraint on the classical construction). In any kime-compactified (D=5D=5) relativistic extension of the directional DOF whose classical limit descends from an irreducible Cl(3,2)Cl(3,2) structure, the null pair (nR,nL)(n_R,n_L) of Proposition 5.5 is a change of basis on one irreducible phase space, not a decomposition into two independent invariant subsystems. There is no invariant that separates a right-handed from a left-handed sector. Any proposed relativistic phase space for Problem (I) that splits into decoupled nRn_R- and nLn_L-sectors is therefore incompatible with the kime compactification; compatible proposals must realize (nR,nL)(n_R,n_L) as coupled coordinates on a single orbit, as m,sO_m,s indeed does. Proof. Immediate from Propositions 5.5 and 5.6, a decoupling invariant would be the classical limit of a chirality operator, whose nonexistence in irreducible Cl(3,2)Cl(3,2) representations is Proposition 5.6; the coupling on m,sO_m,s is visible in nR⋅nL=2n_R· n_L=2 and in the Kirillov–Kostant–Souriau form, which does not split. ∎ Lemma 5.8 (Vector and two-form determine each other on shell). On m,sO_m,s (so P2=m2c2P^2=m^2c^2 and the Tulczyjew condition SμνPν=0S_μνP^ν=0 holds), the Pauli–Lubański vector and the spin two-form are mutually inverse data, Wμ=12ϵμνρσPνSρσ,Sρσ=1m2c2ϵρσμνPμWν,W^μ= 12\,ε^μνρσP_νS_ρσ, S_ρσ= 1m^2c^2\, _ρσμνP^μW^ν, up to the overall sign fixed by the convention ϵ0123=+1ε^0123=+1 (conventions as in [24, Ch. 2]); moreover Sμ:=Wμ/(mc)S^μ:=W^μ/(mc) satisfies S⋅P=0S· P=0, S⋅S=−s2S· S=-s^2, recovering the four-vector of Proposition 5.5 in the rest frame. Proof. Contract the definition of W with ϵρσμνPμ _ρσμνP^μ and use the ϵϵ=−det(δ)ε=- (δ) identity together with SμνPν=0S_μνP^ν=0 and P2=m2c2P^2=m^2c^2. The orthogonality W⋅P=0W· P=0 is immediate from antisymmetry. ∎ 5.4 Remaining Problem I Open Problems Problem 5.9 (Kime action–angle atlas on m,sO_m,s). Construct an atlas of Darboux charts on m,sO_m,s adapted to the kime fibration, i.e., charts of the form (xi,pi;φ,Sz′)(x^i,p_i; ,S_z ) in which the directional factor is the kime cylinder of Lemma 5.2 for the little-group sphere, and quantify the obstruction to a single global chart. The fiber s2S^2_s has ∫s2ωs=4πs≠0 _S^2_s _s=4π s≠ 0, so no global Darboux chart exists, and the minimal atlas is governed by the class [ωs]/2πℏ[ _s]/2π , which is integral iff 2s/ℏ∈ℤ2s/ (Weil integrality; [22, 21]). Make precise, within Assumption 1.1, the resulting statement that a consistent single-valued kime phase law on the directional fiber exists iff the spin is (half-)integer in units of ℏ , the sharpest available classical bridge to spin-12 12, and the direct analogue for Problem (I) of the 2π2π-holonomy correction anticipated in Problem 3.12. Problem 5.10 (Sharp uncertainty relation on m,sO_m,s). Prove the relativistic extension of Theorem 5.3: an entropic inequality on m,sO_m,s, invariant under the Poincaré group and under all Hamiltonian flows, that reduces to (13) on the little-group fiber in the rest frame and to (4) on the translational factor in the nonrelativistic limit. Identify the extremal family (conjecturally: relativistic Gaussian in (x,p)(x,p) ⊗ von Mises–Fisher in the direction, coupled only through the Tulczyjew constraint) and the role of the two Casimirs as the invariant scales, with W⋅W=−m2c2s2W· W=-m^2c^2s^2 playing the part of ∏jνj _j _j in Theorem 3.16. Problem 5.11 (Four-vector vs. two-form as a moment-map criterion). Lemma 5.8 shows the two candidate generalizations of spin carry the same information on shell, so the dichotomy of Problem (I) cannot be about information content; we propose it is about conjugacy. Specifically, on m,sO_m,s, determine which object is the moment map of the kime phase circle action, i.e., find the function F with φ,F=1\ ,F\=1 for the globally defined little-group phase φ of Problem 5.9, and decide whether F is (a) a frame component of the Pauli–Lubański vector W, or (b) a flux pairing 12Sμνζμν 12S_μν\,ζ^μν of the spin two-form against a fixed bivector ζ (the infinitesimal rotation plane). Conjecture: (b), with ζ the generator of the little-group rotation defining φ . Equivalently, spin generalizes as the two-form (a rotation plane, whose conjugate is the angle in that plane), and the four-vector W is the derived, non-conjugate repackaging, consistent with the nonrelativistic normal form φ,Sz=1\ ,S_z\=1 (where SzS_z is the moment map of rotation about z z), with the D=5D=5 kime picture in which the extra dimension supplies the compact conjugate phase [6], and with Corollary 5.7, which forbids resolving the dichotomy by splitting into chiral vector sectors. 6 Interpretation, scope, and discussion 6.1 What the kime representation contributes The formulations above rest on one exact identification (Lemma 2.3: kime cone == action–angle chart, kime measure == Liouville measure) and one standing modeling postulate (Assumption 1.1, the kime phase is a latent circular variable whose law encodes the intrinsic variability of repeated, identically controlled experiments). Given these, the three open problems of [1] decompose as follows. Problem (I), uncertainty. The compact-phase sector is solved in sharp form, the kime-cylinder entropic uncertainty principle (Theorem 3.7) with von Mises ⊗ Gaussian extremals, the sharp circular Fisher inequality (Theorem 3.8) with von Mises extremals, and their kinetic consequence for the reconstructed phase Hamiltonian of [5] (Corollary 3.9). The non-canonical sector is settled for diffeomorphic pairs with the geometric-mean Poisson-bracket correction (Theorem 3.11), which clarifies and corrects the conjectured expected-bracket form (Problem 3.12). The multi-DOF sector is settled in aggregate (Theorem 3.16: Fischer defect == cross-DOF correlation. Equi-partition value as orbit minimum of the uncertainty product) and open per-DOF, where it is now a concrete matrix-analytic question of symplectic Schur–Horn type (Problem 3.17) with an estimable torus surrogate (Problem 3.18). The thermodynamic conjecture is proved in the form available to one kime DOF (Theorem 3.21: equipartition is the unique max-entropy attractor of phase diffusion. Hamiltonian flow is the entropy-neutral member of the Wick-interpolated family and preserves all equilibrium relationships), with the joint multi-DOF statement recorded as Conjecture 3.22. Problem (I), invariant entropy. The pairing phenomenon is proved as a rigidity theorem: invariant entropy ⟺ invariant measure (Theorem 4.1). No invariant measure exists on unpaired quantities (Theorem 4.2); on contragradient pairs the invariant measure exists and is uniquely Liouville, so the invariant entropy is unique up to an additive constant (Theorem 4.3, Corollary 4.4). The kime chart realizes the pairing as a Kähler triple with the complex coordinate κ (Proposition 4.5), and the kime-ray factorization of [4] shows the same complex structure grades dynamics into entropy-conserving and entropy-producing sectors (Proposition 4.6, Theorem 3.21). Whether this complex organization is forced by entropy axioms, via generalized complex geometry, and whether symplectic (rather than merely volume-preserving) structure is forced by estimable statistical invariants, is stated as Problems 4.10, 4.8, and 4.11. Problem (I), directional DOF. Nonrelativistically the directional DOF is a kime DOF: the sphere is symplectically a finite kime cylinder (Lemma 5.2), and the compact–compact uncertainty relation with its Larmor-invariant floor is Theorem 5.3. Relativistically the correct arena is the Poincaré coadjoint orbit m,sO_m,s (Definition 5.4); the suggested null pair is a bijective repackaging of (u,S)(u,S) (Proposition 5.5); the kime compactification’s chirality obstruction rules out any answer that decouples right- and left-handed sectors (Proposition 5.6, Corollary 5.7); and the four-vector/two-form dichotomy, informationally empty by Lemma 5.8, is reformulated as the sharp question of which object is conjugate to the kime phase (Problem 5.11), with the Weil-integrality bridge to spin-12 12 isolated in Problem 5.9. 6.2 Epistemic status and falsifiability In the spirit of the caveats of [5], there are 3 specific points that delimit the claims. First, every theorem above is a statement about the kime representation. The identification of the latent statistical phase with a mechanical angle variable (remark 2.2) is a modeling postulate, adopted because it is the unique identification under which the kime measure is the invariant count of states (Theorem 4.3). Second, nothing here derives quantum mechanics or new physics. The inequalities are classical information-geometric statements, and their quantum-facing appearances (Corollary 3.9, Problem 5.9) are mediated by explicitly flagged postulates of [5, 6]. Third, the interpretation of the phase law as intrinsic experimental variability (Assumption 1.1) is falsifiable in the concrete sense of [4]. The phase law, its resultant r, its Fisher information, and the relative-phase matrix R are identifiable from repeated-measurement data up to the stated gauges, with Cramér–Rao–controlled error; hence Theorems 3.7, 3.8, 3.16, and 5.3 are empirically checkable inequalities, and violations would falsify the identification, not merely the estimation procedure. 6.3 Summary table of problem status Component Proved here Open (stated as) (I) canonical, 1 DOF, compact phase Thms. 3.7, 3.8; Cor. 3.9 (I) non-canonical pairs Thm. 3.11 (geometric-mean bracket) Prob. 3.12 (I) multi-DOF Thm. 3.16 (aggregate floor) Probs. 3.17, 3.18, 3.19; Conj. 3.22 (I) thermodynamic conjecture Thm. 3.21 Conj. 3.22 (I) pairing of quantities Thms. 4.1–4.3; Cor. 4.4; Props. 4.5, 4.6, 4.7 Probs. 4.8, 4.9, 4.10, 4.11 (I) nonrelativistic direction Lem. 5.2; Thm. 5.3 (I) relativistic direction Props. 5.5, 5.6; Cor. 5.7; Lem. 5.8 Probs. 5.9, 5.10, 5.11 Acknowledgments This study is motivated by the novel work of Gabriele Carcassi and Christine Aidala at the University of Michigan who developed the Assumptions of Physics book [1]. 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