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Information Geometry of Bounded Rationality: Entropy--Regularised Choice with Hyperbolic and Elliptic Quantum Geometries
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Models of bounded rationality often describe behaviour through probabilistic rather than deterministic choice. In discrete choice, such behaviour is represented by a probability vector \(q=(q_1,\dots,q_n)\in\Delta^{n-1}\), interpreted either as a stochastic choice rule in the sense of Luce1959 or as a von Neumann--Morgenstern lottery over \(n\) outcomes vonNeumannMorgenstern1944. Two influential paradigms build on this representation in markedly different styles. Quantum--like cognitive (QL) models represent behaviour through amplitudes and contextual projections, whereas entropy--regularised (ER) models derive smooth probabilistic choice rules directly on the simplex. The aim of this paper is to show that information geometry provides a natural common framework for these approaches. Moreover, we show that the same geometric lift underlying ER models also canonically induces a hyperbolic (split--signature) quantum--like structure.
Quantum--like cognitive (QL) models describe the cognitive state as an amplitude---either a state vector or a density operator---in a (typically complex) Hilbert space. Deliberation and contextual change are represented by unitary (or more generally completely positive) maps, and choice probabilities follow from the Born rule (e.g.\ Aerts1995,BusemeyerEtAl2006,Aerts2009,BusemeyerBruza2012,YearsleyBusemeyer2016,Yearsley2017). Because contextual projections and observables need not commute, QL models naturally generate several empirical signatures of bounded rationality, such as context effects, order effects, and deviations from the classical law of total probability (e.g.\ Khrennikov2005,KhrennikovHaven2009,BruzaKittoNelsonMcEvoy2009,PothosBusemeyer2013,HavenKhrennikov2016). In this perspective bounded rationality reflects a contextual, noncommutative calculus, rather than noisy perturbations of an underlying classical utility model.\footnote{The “quantum” label here refers to the use of Hilbert--space contextuality and interference, rather than to any claim of microscopic quantum computation in neural tissue deBarrosSuppes2009.}
Entropy--regularised (ER) models provide smooth probabilistic choice rules directly on the simplex. A canonical example is Rational Inattention, in which a decision maker trades off deterministic utility against an information cost. The Luce--logit (softmax) rule arises when this cost is Shannon entropy, while more general variational formulations allow broader classes of information costs Sims2003,MatejkaMcKay2015. More broadly, information--theoretic ideas relating probabilistic choice, entropy, and minimum information have appeared in the econometric and psychological literatures in several forms.\footnote{See, e.g., McFadden1974,SnickarsWeibull1977,Luce2003.} In all cases, the resulting choice rules reflect a balance between preference and information cost.
Although QL and ER models are usually presented as distinct paradigms, they admit a common interpretation in information geometry. On the ER side, entropy penalties and exponential--family structure arise from convex duality on the Fisher--Rao statistical manifold \((\Delta^{n-1}_{>0},g_F)\).\footnote{See Ly2017 for an exposition of Fisher--Rao geometry in mathematical psychology.} On the QL side, a closely related geometric structure appears after the amplitude reparametrisation \(q\mapsto\rho\) given by \(\rho_k=\sqrt{q_k}\). This map identifies the simplex interior with the positive orthant of the unit sphere, pulls back \(g_F\) (up to scale) to the round metric, and places the amplitude description in the standard symplectic/Hamiltonian framework familiar from geometric formulations of quantum mechanics Kibble1979,Heslot1985,BrodyHughston2001,Goyal2010,ReginattoHall2012,Reginatto2013,Molitor2012, Caticha2021. Under this reparametrisation, expected utilities become quadratic forms in amplitudes AertsHavenSozzo2018, and the dynamics of probabilities may be analysed via an induced dynamics on amplitudes.
Our starting point is this shared information--geometric foundation. We formulate a local least--action rationality (LAR) principle for decision dynamics, in the spirit of the differential approach to preferences Machina1982,Russell1991 and related least--action proposals in cognitive modelling (cf.\ BettiGori2016LeastCognitiveAction,FoxKotelba2018PLPA), but applied at the level of amplitudes rather than directly on the simplex. At each state, the realised velocity is the metric projection (in the Fisher--Rao geometry induced by the amplitude embedding) of a preferred direction specified by a preference covector field. In the linear case this yields an evolution for unnormalised amplitudes \(\dot{\tilde\rho}=\widehat V\tilde\rho\), where the generator decomposes canonically as \(\widehat V=\widehat S+\widehat F\) with \(\widehat S^{\!\top}=\widehat S\) (evaluative channel) and \(\widehat F^{\!\top}=-\widehat F\) (co--utility/rotational channel).
Passing from amplitudes to their cotangent lift yields an ambient first--order Hamiltonian system on \(\mathbb R^{2n}\cong T^*\mathbb R^n\). The lifted Fisher--Rao structure canonically induces a flat para--K\"ahler geometry on this space (neutral metric, symplectic form, and product structure). On a distinguished zero--residual sector (where the least--action projection constraint is exactly met), the dynamics admit a split--complex (hyperbolic) Schr\"odinger--type representation.
This para--Hermitian lift is rich enough to reproduce several signatures typically attributed to quantum--like models. It yields a hyperbolic Born--type readout, permits noncommutative observables, and generates phase sensitivity, interference patterns, and contextual effects, all without positing a complex Hilbert space. Moreover, the hyperbolic geometry provides a canonical neutral quadratic index whose evolution obeys an exact balance law with a nonnegative source term, thereby organising episodes by the cumulative accumulation of least--action residual.
Although QL models are often formulated in complex Hilbert space, it is frequently acknowledged---and sometimes adopted---that real--valued formulations already capture the essential contextual--projection structure without appealing to complex phases (e.g.\ BruzaKittoNelsonMcEvoy2009). The para--K\"ahler framework developed here provides a principled foundation for such real formulations by deriving a genuinely hyperbolic (split--signature) quantum geometry directly from classical Fisher--Rao decision dynamics, connecting also to earlier work on hyperbolic quantum mechanics Khrennikov2000hyperbolicquantummechanics. In this view, the split--complex notation used later is a convenient packaging of an underlying real geometry, not an approximation to an elliptic (complex) quantum model.
QL models with coherent (unitary) evolution form a different and equally important strand of the QL literature (e.g.\ BusemeyerWangLambertMogiliansky2009,BusemeyerZhangBalakrishnanWang2020). We show in (ref) how such elliptic (Riemannian/complex) quantum dynamics arises in our framework only after an additional coherence input: one must impose an admissibility restriction that selects a holomorphic Lagrangian sector (a polarisation) compatible with the symplectic structure, so that the projected dynamics become unitary on the corresponding complex state space.
The rest of the paper is organised as follows. (ref) formulates least--action rationality on the probability simplex from three geometric ingredients: Fisher--Rao distinguishability, a preference covector field, and a least--action variational principle. (ref) lifts this structure to an ambient amplitude model on \(\mathbb R^{2n}\): we derive the quadratic LAR Lagrangian, the associated Hamiltonian system, its canonical para--K\"ahler geometry, and the split--complex (hyperbolic) Schr\"odinger representation on the distinguished zero--residual sector. (ref) then shows how standard (elliptic) quantum geometry arises by imposing a polarised admissibility restriction, making precise in what sense coherence--preserving unitary dynamics is a special case of the lifted theory. (ref) returns to behavioural implications, analysing how the para--K\"ahler lift organises systematic deviations from expected utility. (ref) concludes and discusses directions for future work.
In this section we begin with making explicit the three foundational assumptions that constitute a principle of rationality used in this paper. Each assumption isolates a specific geometric ingredient: (i) a notion of distinguishability between nearby redistributions of a lottery, (ii) an evaluative (preference) structure defined on such redistributions, and (iii) a variational principle.
We take as primitive observable objects lotteries over the finite outcome set $\{1,\dots,n\}$, represented by probability vectors. Accordingly, we consider the open probability simplex \[ \mathcal M:=\Delta^{n-1}_{>0} =\Bigl\{\, q=(q_1,\dots,q_n)\in\mathbb{R}^n \ \Big|\ q_i>0,\ \sum_{i=1}^n q_i=1 \,\Bigr\}, \] which is a smooth embedded submanifold of the affine hyperplane $\{q\in\mathbb{R}^n:\sum_i q_i=1\}$ of dimension $n-1$. Its closure is the closed simplex \[ \Delta^{n-1} =\Bigl\{\, q\in\mathbb{R}^n \ \Big|\ q_i\ge 0,\ \sum_{i=1}^n q_i=1 \,\Bigr\}, \] so that $\mathcal M$ is the interior of $\Delta^{n-1}$. For each $q\in\mathcal M$ the tangent space is canonically identified with the codimension--one subspace \[ T_q\mathcal M=\Bigl\{\, v\in\mathbb{R}^n \ \Big|\ \sum_{i=1}^n v_i=0 \,\Bigr\}. \]
To formalise “distinguishability” we endow lotteries with a perceptual metric. We obtain it by lifting lotteries to the unit sphere, where the round geometry provides a canonical notion of distance, and then pulling this distance back to the simplex. Let $\langle\cdot,\cdot\rangle$ be the Euclidean inner product on $\mathbb{R}^n$ and $\|x\|:=\sqrt{\langle x,x\rangle}$ its norm. Given any nonzero unnormalised amplitude $\tilde\rho\in\mathbb{R}^n\setminus\{0\}$, set \[ \rho:=\frac{\tilde\rho}{\|\tilde\rho\|}\in\mathbb S^{n-1}, \qquad \mathbb S^{n-1}:=\{\,\rho\in\mathbb{R}^n:\|\rho\|=1\,\}. \] We read out an observable lottery by componentwise squaring, \[ q_i=\rho_i^2=\frac{\tilde\rho_i^{\,2}}{\|\tilde\rho\|^2}\qquad(i=1,\dots,n), \] equivalently via the smooth surjection \[ \pi:\mathbb S^{n-1}\to\Delta^{n-1}, \qquad \pi(\rho)=(\rho_1^2,\dots,\rho_n^2)=\rho^{\odot 2}, \] where $x^{\odot 2}$ denotes componentwise squaring. The map $\pi$ is sign--blind: for $q\in\mathcal M$ the fibre $\pi^{-1}(q)$ consists of the $2^n$ sign choices $\rho\mapsto s\rho$ with $s=\mathrm{diag}(\pm1)$, whereas on faces the same action produces degeneracies when some components vanish. For geometric constructions on lotteries it is convenient to choose the canonical square--root section into the nonnegative orthant \[ \overline{\mathbb S}^{\,n-1}_+:=\{\,\rho\in\mathbb S^{n-1}:\rho_i\ge 0\ \forall i\,\}, \qquad \iota:\Delta^{n-1}\to\overline{\mathbb S}^{\,n-1}_+, \qquad \iota(q)=\sqrt q:=(\sqrt{q_1},\dots,\sqrt{q_n}), \] so that $\pi\circ\iota=\mathrm{id}_{\Delta^{n-1}}$. On the interior $\mathcal M$ the restriction of $\iota$ is smooth and maps into the open orthant \[ \mathbb S^{n-1}_+:=\{\,\rho\in\mathbb S^{n-1}:\rho_i>0\ \forall i\,\}. \] We emphasise that $\iota$ serves only as a smooth lift (equivalently, a diffeomorphism $\iota:\mathcal M\to\mathbb S^{n-1}_+$ with inverse $\pi\!\mid_{\mathbb S^{n-1}_+}$) for expressing simplex--level geometric objects; it does not constrain the sign pattern of the underlying normalised amplitude $\tilde\rho/\|\tilde\rho\|$, which remains unconstrained.
Let $g_{\mathbb S}$ denote the round metric on $\mathbb S^{n-1}$ induced by $\langle\cdot,\cdot\rangle$, and let $d_{\mathbb S}$ be the associated geodesic distance. Diagonal sign flips $s=\mathrm{diag}(\pm 1)\in O(n)$ act by isometries on $(\mathbb S^{n-1},g_{\mathbb S})$, hence any sign section $q\mapsto s\sqrt q$ induces the same pullback metric on $\mathcal M$.
We now postulate that perceptual distance between lotteries is the angular distance between their square--root lifts on the sphere, up to an overall scale.
Since $\iota$ is injective, $d_{\mathrm{perc}}$ is the restriction of the spherical geodesic distance to the embedded set $\iota(\Delta^{n-1})\subset\mathbb S^{n-1}$, hence a metric on $\Delta^{n-1}$. Moreover, because $\iota(\Delta^{n-1})\subset\overline{\mathbb S}^{\,n-1}_+$, one has $0\le d_{\mathrm{perc}}(q,q')\le c\pi/2$.
The definition above provides a global perceptual metric on $\Delta^{n-1}$. On the interior $\mathcal M$ we refine it infinitesimally to obtain a local distinguishability tensor by differentiating the lift $\iota$. Since the square--root map becomes singular as some coordinates approach zero, this refinement is asserted on $\mathcal M$ and must be interpreted piecewise if a trajectory contacts the boundary.
For $q\in\mathcal M$ and $v,w\in T_q\mathcal M$, writing $\rho=\iota(q)=\sqrt q$, differentiation of $q_i=\rho_i^2$ gives $dq_i=2\rho_i\,d\rho_i$, hence \[ d\rho_i(v)=\frac{v_i}{2\rho_i}, \qquad \bigl(\iota^*g_{\mathbb S}\bigr)(q)[v,w] =\frac14\sum_{i=1}^n\frac{v_iw_i}{q_i}. \] It follows that \[ \iota^*g_{\mathbb S}=\tfrac14\,g_F, \qquad g_F(q)[v,w]:=\sum_{i=1}^n\frac{v_iw_i}{q_i}\quad\text{for all }v,w\in T_q\mathcal M, \] where $g_F$ is the Fisher--Rao metric on $\mathcal M$. Thus the induced local distinguishability metric on lotteries is (up to the constant factor $1/4$) the Fisher--Rao metric; we take $g_F$ as the canonical simplex--level tensor associated with A1.
The constant $c$ in Assumption (ref) sets the overall units of distance; infinitesimally, the induced quadratic form is proportional to $\iota^*g_{\mathbb S}$, namely $c^2\,\iota^*g_{\mathbb S}=\tfrac{c^2}{4}\,g_F$ on $T_q\mathcal M$. In particular, $g_F$ weights perturbations in rare outcomes more heavily via the factors $1/q_i$, and becomes singular at the boundary of $\Delta^{n-1}$, motivating our use of the interior $\mathcal M$ for smooth Riemannian geometry.
Finally, for any smooth trajectory $t\mapsto q(t)\in\mathcal M$, the lifted curve $\rho(t):=\iota(q(t))\in\mathbb S^{n-1}_+$ is an auxiliary geometric lift and satisfies \[ \|\dot q(t)\|_{g_F}^2=4\,\|\dot\rho(t)\|_{g_{\mathbb S}}^2. \]
As emphasised already by Russell1991, differential geometry provides a natural language for preferences: marginal evaluation is encoded by a covector field (a $1$--form). In our framework, this covector field is induced by a fixed preference operator $\widehat V\in\mathrm{End}(\mathbb{R}^n)$ acting on latent amplitudes.
Let $g_{\mathbb S}$ be the round metric on the amplitude sphere $\mathbb S^{n-1}\subset\mathbb{R}^n$ induced by the Euclidean inner product $\langle\cdot,\cdot\rangle$. For $\rho\in\mathbb S^{n-1}$ denote by \[ Q_\rho:=I-\rho\rho^\top \] the orthogonal projector onto the tangent space \[ T_\rho\mathbb S^{n-1}=\{\eta\in\mathbb{R}^n:\langle\rho,\eta\rangle=0\}. \] A (spherical) preference covector is a $1$--form $\bar\alpha^{\mathbb S}\in\Omega^1(\mathbb S^{n-1})$ assigning to each infinitesimal adjustment $\eta\in T_\rho\mathbb S^{n-1}$ an instantaneous evaluative score $\bar\alpha^{\mathbb S}_\rho(\eta)\in\mathbb{R}$. We restrict attention to the ambient--operator class: fix $\widehat V\in\mathrm{End}(\mathbb{R}^n)$ and define
Since $\eta\perp\rho$, we have $\langle \eta,Q_\rho \widehat V\rho\rangle=\langle \eta,\widehat V\rho\rangle$. Thus $\bar\alpha^{\mathbb S}$ is the $g_{\mathbb S}$--musical dual of the tangent vector field
Within this class, $\bar\alpha^{\mathbb S}$ is invariant under the shift $\widehat V\mapsto \widehat V+\lambda I$, since $Q_\rho(\lambda\rho)=0$. Hence only the class $[\widehat V]\in\mathrm{End}(\mathbb{R}^n)/(\mathbb{R}\cdot I)$ is observable at the spherical level; when convenient we fix a representative by imposing $\tr(\widehat V)=0$.
Observable lotteries live on the interior simplex \[ \mathcal M=\Delta^{n-1}_{>0} =\Bigl\{q\in\mathbb{R}^n_{>0}:\sum_{i=1}^n q_i=1\Bigr\}. \] We represent $q\in\mathcal M$ by its positive square root \[ \iota:\mathcal M\to\mathbb S^{n-1}_{+}, \qquad \iota(q)=\rho=\sqrt q, \] so that $\rho_i=\sqrt{q_i}$ and $\sum_i\rho_i^2=1$. Restricting $\bar\alpha^{\mathbb S}$ to the open set $\mathbb S^{n-1}_{+}$ and pulling back along $\iota$ yields the marginal preference covector on lotteries: \[ \mathcal T:\mathrm{End}(\mathbb{R}^n)\to\Omega^1(\mathcal M), \qquad \mathcal T(\widehat V):=\beta:=\iota^*\!\bigl(\bar\alpha^{\mathbb S}\!\mid_{\mathbb S^{n-1}_{+}}\bigr). \] Here $\iota$ is used only as a smooth lift (equivalently, a diffeomorphic identification of $\mathcal M$ with $\mathbb S^{n-1}_+$); it does not constrain the sign pattern of any latent amplitude trajectory.
Decompose \[ \widehat V=\widehat S+\widehat F, \qquad \widehat S^\top=\widehat S, \qquad \widehat F^\top=-\widehat F. \] The central structural fact proved below is that $\widehat S$ induces an exact (utility) component on $\mathcal M$, while $\widehat F$ induces a purely circulatory component which is co--exact in Fisher--Rao geometry.
The equivalence of (A2) and (A2$'$) is immediate from tangency: if $\eta\in T_\rho\mathbb S^{n-1}$ then $Q_\rho\eta=\eta$, hence \[ \bar\alpha^{\mathbb S}_\rho(\eta) =\langle \eta,\,Q_\rho\widehat V\rho\rangle =\langle Q_\rho\eta,\,\widehat V\rho\rangle =\langle \eta,\,\widehat V\rho\rangle, \] so $\bar\alpha^{\mathbb S}=(\widehat V\rho)^\top d\rho=(Q_\rho\widehat V\rho)^\top d\rho$.
Before concluding this discussion of preference structure, it is useful to separate the symmetric and skew--symmetric parts of the latent preference operator, $\widehat V=\widehat S+\widehat F$, and to discuss the distinct behavioural roles they play when the induced marginal covector is expressed on $\mathcal M$ via the square--root chart.
Starting with the symmetric component $\widehat S^\top=\widehat S$, the induced simplex preference field is exact within the linear lift class: it admits the potential \[ U(q)=\tfrac12\,(\sqrt q)^\top \widehat S\,\sqrt q =\tfrac12\sum_{i,j}\widehat S_{ij}\sqrt{q_i q_j}. \] This expression is quadratic in amplitudes (square--root probabilities). When $\widehat S$ is diagonal, it reduces to the standard additive expected--utility form $U(q)=\sum_i u_i q_i$ with $u_i=\tfrac12\widehat S_{ii}$, recovering classical utility. More generally, off--diagonal entries encode pairwise evaluative couplings between alternatives, weighted by belief amplitudes rather than probabilities. In this sense the symmetric potential provides a canonical pairwise--interaction extension of additive utility; within the square--root chart, these are exactly the operator--induced utility potentials (unique up to an additive constant, reflecting the $\mathbb{R}\cdot I$ gauge): by Proposition (ref)(1), $\mathcal T(\widehat S)=dU$.
By contrast, the skew--symmetric component $\widehat F^\top=-\widehat F$ does not contribute to the exact (utility) channel and instead generates circulation. On the amplitude sphere it induces the divergence--free Killing field $X_{\widehat F}(\rho)=\widehat F\rho$, whose dual $1$--form $\bar\alpha^{\widehat F}(\rho)=(\widehat F\rho)^\top d\rho$ is co--closed. Moreover, for any pair of amplitude states $\rho,\rho'\in\mathbb S^{n-1}$ it defines the skew bilinear form \[ R(\rho,\rho'):=\rho^\top \widehat F\,\rho',\qquad R(\rho,\rho')=-R(\rho',\rho). \] When restricted to the principal section $\rho=\sqrt q$ this yields the corresponding skew bilinear comparison $R(\sqrt q,\sqrt{q'})$ on lotteries, matching the structure of Fishburn's skew--symmetric bilinear (SSB) preference form in these coordinates Fishburn1984. Accordingly, $\widehat F$ may be interpreted as a continuous regret--type field, encoding non--potential components of preference intensity that generate circulation rather than gradient flow. This aligns with Russell's differential--geometric view of regret as the nonconservative part of a preference field Russell1991, where non--closed components can generate holonomy around loops. In summary, $\widehat S$ governs the conservative evaluative channel through a utility potential, whereas $\widehat F$ governs an intrinsic pairwise comparison (co--utility) channel that can generate path dependence.
Given Assumption A2 specifying a spherical preference covector field $\bar\alpha^{\mathbb S}\in\Omega^1(\mathbb S^{n-1})$, we consider infinitesimal adjustments of an agent's state on $\mathbb S^{n-1}$. When relating spherical objects to lotteries $q\in\mathcal M$, we evaluate them on the principal section $\iota(q)=\sqrt q\in\mathbb S^{n-1}_+$, i.e.\ we use the restriction $\alpha^{\mathbb S}:=\bar\alpha^{\mathbb S}\!\mid_{\mathbb S^{n-1}_+}$; this is a representative choice and does not constrain the latent amplitude trajectory.
At $\rho\in\mathbb S^{n-1}$ admissible instantaneous displacements are tangent vectors $\dot\rho\in T_\rho\mathbb S^{n-1}$. To compare such displacements with a (generally non-tangent) preferred direction in the ambient space, fix any smooth ambient lift $\widetilde X:\mathbb S^{n-1}\to\mathbb{R}^n$ of the covector field, meaning \[ \bar\alpha^{\mathbb S}_\rho(\delta\rho)=\langle \delta\rho,\,\widetilde X(\rho)\rangle \qquad\text{for all }\rho\in\mathbb S^{n-1},\ \delta\rho\in T_\rho\mathbb S^{n-1}. \] (Such $\widetilde X$ always exists, and is unique up to addition of a radial term $\lambda(\rho)\rho$.) Let $Q_\rho:=I-\rho\rho^\top$ denote the orthogonal projector onto $T_\rho\mathbb S^{n-1}$. Least--action (least--effort) rationality postulates that the realised displacement is the closest admissible direction to $\widetilde X(\rho)$: \[ \dot\rho =\arg\min_{\xi\in T_\rho\mathbb S^{n-1}}\|\xi-\widetilde X(\rho)\|^2 =Q_\rho\,\widetilde X(\rho). \] Equivalently, since $Q_\rho\widetilde X(\rho)$ is the unique tangent vector satisfying $\langle\delta\rho,Q_\rho\widetilde X(\rho)\rangle=\bar\alpha^{\mathbb S}_\rho(\delta\rho)$ for all $\delta\rho\in T_\rho\mathbb S^{n-1}$, one has \[ \dot\rho = g_{\mathbb S}^{-1}\bar\alpha^{\mathbb S}_\rho \in T_\rho\mathbb S^{n-1}. \]
When $\bar\alpha^{\mathbb S}$ is induced by a linear evaluator in the ambient space, \[ \bar\alpha^{\mathbb S}_\rho(\delta\rho)=\langle \delta\rho,\,Q_\rho\,\widehat V\rho\rangle =\langle \delta\rho,\,\widehat V\rho\rangle, \qquad \delta\rho\in T_\rho\mathbb S^{n-1}, \] we may take $\widetilde X(\rho)=\widehat V\rho$, and the least--action direction becomes \[ \dot\rho = Q_\rho\,\widehat V\rho. \] This defines the smooth vector field
which we refer to as the Least--Action Rationality (LAR) vector field.
Restricting to the principal section $\iota(\mathcal M)\subset\mathbb S^{n-1}_+$, write $\alpha^{\mathbb S}:=\bar\alpha^{\mathbb S}\!\mid_{\mathbb S^{n-1}_+}$ and recall that the induced simplex preference form is $\beta=\iota^*\alpha^{\mathbb S}\in\Omega^1(\mathcal M)$. On $\mathcal M$ define the Fisher--Rao natural--gradient field $X_{\mathcal M}$ by \[ g_F\bigl(X_{\mathcal M},\cdot\bigr)=\beta, \qquad\text{where}\qquad g_F=4\,\iota^*g_{\mathbb S}. \] Then for each $q\in\mathcal M$ one has the pushforward relation \[ d\iota_q\bigl(X_{\mathcal M}(q)\bigr) =\tfrac14\,g_{\mathbb S}^{-1}\alpha^{\mathbb S}_{\iota(q)}\ \in\ T_{\iota(q)}\mathbb S^{n-1}, \] i.e.\ the simplex Fisher--Rao gradient flow is the $\iota^{-1}$--image of the spherical least--action direction field, up to the constant time rescaling by a factor $4$.
The previous section formulated Least--Action Rationality (LAR) as a variational principle on the information--geometric manifold of amplitudes. In the present section we recast it as an equivalent first--order system on the doubled ambient space \(\mathbb{R}^{2n}\cong\mathbb{R}^n\times\mathbb{R}^n\) of amplitudes and their residuals (momenta), remaining entirely within the classical information--geometric framework. The same equations can be represented as a hyperbolic Schr\"odinger--type equation in the split--complex algebra \(\mathbb D\) ((ref)).
To pass from the constrained, pointwise projection dynamics on the unit sphere to an unconstrained system on a linear space, we encode the LAR least--squares condition by a quadratic residual Lagrangian. Given a preferred ambient direction field $\widehat V\tilde\rho$, define the instantaneous residual $u:=\dot{\tilde\rho}-\widehat V\tilde\rho$ and penalise its Euclidean norm. The resulting first--order cotangent lift on $T^*\mathbb{R}^n\simeq\mathbb{R}^{2n}$ will be shown below to reproduce the spherical LAR flow for the normalised amplitudes $\rho=\tilde\rho/\|\tilde\rho\|$ on the distinguished zero--residual sector.
To do so, we define the least--action Lagrangian
defined on the ambient space of unnormalised amplitudes $\tilde\rho\in\widetilde{\mathcal M}=\mathbb{R}^n$ equipped with the Euclidean metric. Throughout Section (ref), unless otherwise stated, we adopt the trace--free convention $\tr(\widehat V)=0$ (as in Section (ref)).
The simplex $\mathcal M=\Delta^{n-1}_{>0}$ enters only through the sign--blind readout map $q(\tilde\rho)$ defined for $\tilde\rho\in\mathbb{R}^n\setminus\{0\}$ by \[ q_i=\frac{\tilde\rho_i^2}{\|\tilde\rho\|^2}, \] and, when expressing simplex--level geometric objects, through the square--root section $\iota:\mathcal M\to\mathbb S^{n-1}_+$, $\iota(q)=\sqrt q$. No sign restriction is imposed on the latent amplitude $\tilde\rho$: trajectories may have any sign pattern and may cross coordinate hyperplanes without affecting the readout. Accordingly, Fisher--Rao geometry (the metric $g_F$ and its derived operators) is invoked only on intervals where the readout satisfies $q(t)\in\mathcal M$ (and otherwise is interpreted in the stratified sense of Remark (ref)), whereas the amplitude/cotangent dynamics are global on $\widetilde{\mathcal M}=\mathbb{R}^n$ and $T^*\widetilde{\mathcal M}\simeq\mathbb{R}^{2n}$.
The Legendre transform of (ref) \[ \mathbb FL:\;(\tilde\rho,\dot{\tilde\rho})\longmapsto(\tilde\rho,y), \qquad y=\partial_{\dot{\tilde\rho}}L=\dot{\tilde\rho}-\widehat V\tilde\rho, \] is hyperregular and therefore a diffeomorphism. Moreover, the Poincaré--Cartan form $\theta_L=y^\top d\tilde\rho$ satisfies $\omega_L=-d\theta_L$, and the pullback identity \[ (\mathbb FL)^*\Omega=\omega_L, \qquad \Omega=d\tilde\rho^i\wedge dy_i, \] where $\Omega$ is the canonical symplectic $2$--form on $T^*\widetilde{\mathcal M}$, shows that $\mathbb FL:(T\widetilde{\mathcal M},\omega_L)\to (T^*\widetilde{\mathcal M},\Omega)$ is a symplectomorphism. Thus the residual variable \[ u=\dot{\tilde\rho}-\widehat V\tilde\rho \] is the canonical momentum $y$ expressed in velocity coordinates, and $(\tilde\rho,y)$ are Darboux coordinates for the ambient phase space $T^*\widetilde{\mathcal M}\simeq\mathbb{R}^{2n}$.
The corresponding Hamiltonian
generates the linear Hamiltonian system
or in vector notation
Hence $\Phi_t=\exp(t\mathsf A)\in\mathrm{Sp}(2n,\mathbb{R})$ preserves the canonical symplectic form.
The geometric structure underlying this Hamiltonian system arises from the information--geometric origin of the model. In information geometry, exponential and mixture families are dually flat and hence carry a Hessian structure Shima2007,Nielsen2020Entropy. In particular, the probability simplex $\mathcal M=\Delta^{n-1}_{>0}$ may be viewed as a mixture family in the coordinates $q$, so it comes equipped with the flat mixture connection $\nabla^{(m)}$ and the Fisher--Rao metric $g_F$, which is locally Hessian. Concretely, with the (negative) entropy potential $ \Phi(q):=\sum_{i=1}^n q_i\log q_i, $ one has $g_F=(\nabla^{(m)} d\Phi)\!\mid_{T\mathcal M}$.
The square--root lift \[ \iota:\mathcal M\longrightarrow\mathbb S^{n-1}_+\subset \mathbb S^{n-1}\subset\widetilde{\mathcal M}\simeq\mathbb{R}^n, \qquad \iota(q)=\sqrt q, \] is a smooth embedding that transports the metric geometry to the ambient amplitude space: it satisfies $\iota^*g_{\mathbb S}=\tfrac14\,g_F$, where $g_{\mathbb S}=g_{\mathrm{Euc}}|_{\mathbb S^{n-1}}$ is the round metric. We view $\mathbb S^{n-1}_+$ as a submanifold of the Euclidean space $(\widetilde{\mathcal M},g_{\mathrm{Euc}})$ and endow $\widetilde{\mathcal M}$ with the standard flat (hence Levi--Civita) connection $\nabla^{\mathrm{Euc}}$. Thus $(\widetilde{\mathcal M},g_{\mathrm{Euc}},\nabla^{\mathrm{Euc}})$ provides a canonical ambient affine structure for the cotangent/Hamiltonian formulation, naturally associated with the underlying statistical geometry of $(\mathcal M,g_F,\nabla^{(m)})$.
A crucial geometric consequence of the Hessian structure is that the flat affine connection is part of the given data. In the present setting this distinguished connection is $\nabla^{(m)}$ on $\mathcal M$. In general, Sasaki--type lifts on tangent and cotangent bundles depend on a chosen connection; for a Hessian manifold $(M,g,\nabla)$ the torsion--free flat connection $\nabla$ is fixed in advance. On the tangent side, this underlies the classical Sasaki/Dombrowski construction and its Hessian refinement: the tangent bundle \(TM\) of a dually flat or Hessian manifold carries a canonical K\"ahler structure induced by \((g,\nabla)\), see Dombrowski1962 and Section 2.2 in Shima2007, and its information--geometric applications to quantum mechanics developed by Molitor2013JGP. Here, we will instead follow the corresponding approach on the cotangent side, following Reginatto2013,Caticha2021, where the chosen flat connection provides a distinguished way of lifting curves and covectors: parallel transport of covectors with respect to the dual connection induced by $\nabla$ defines, at each point $\alpha\in T^*M$, a horizontal subspace $H_\alpha\subset T_\alpha(T^*M)$ complementary to the vertical subspace $V_\alpha:=\ker d\nu_\alpha$, where $\nu:T^*M\to M$ is the bundle projection. These subspaces vary smoothly and give a canonical splitting \[ T(T^*M)=H\oplus V \] into \(\nabla\)--horizontal and vertical subbundles. The next lemma summarise their basic symplectic properties.
Lemma (ref) provides the symplectic input: for any torsion--free affine connection $\nabla$ the induced horizontal distribution $H$ is Lagrangian, complementary to the vertical Lagrangian $V=\ker d\nu$, and hence determines a canonical Lagrangian splitting $T(T^*M)=H\oplus V$. This splitting already defines an almost para--Hermitian package by setting $K|_H=+\mathrm{id}$, $K|_V=-\mathrm{id}$ and $G:=\Omega(\,\cdot\,,K\,\cdot\,)$; however, to obtain a para--K\"ahler structure in the strict sense one must also have integrability of $H$ (equivalently, of $K$), which is ensured by flatness of $\nabla$. We therefore specialise to the Hessian case:
Lemma (ref) requires only torsion--freeness to obtain a transversal Lagrangian splitting $T(T^*M)=H\oplus V$ and hence an almost para--Hermitian package $(G,K,\Omega)$. The additional “Hessian” hypothesis (flatness of $\nabla$) is used only to guarantee integrability of $H$ (and therefore of $K$), so that the structure is para--K\"ahler in the strict sense. Then Proposition (ref) shows that, once a torsion--free flat connection \(\nabla\) is fixed (in particular, when \((M,g,\nabla)\) is a Hessian manifold), the cotangent bundle \(T^*M\) carries a canonical para--K\"ahler structure \((G,K,\Omega)\). The construction depends only on the canonical symplectic form \(\Omega=-d\theta\) on \(T^*M\) and on the \(\nabla\)--horizontal/vertical splitting \(T(T^*M)=H\oplus V\). In this sense the para--complex structure \(K\) is intrinsically determined by this splitting, and \(G\) is then uniquely determined by the relation \(\Omega(u,v)=G(Ku,v)\).
By contrast, there is no complex structure \(J\) on \(T^*M\) that is canonically determined by the affine/Hessian data \((M,g,\nabla)\) alone and compatible with the canonical symplectic form \(\Omega\). Of course, once one chooses an \(\Omega\)--compatible integrable almost complex structure \(J\) on \(T^*M\), the associated metric \(g_J(\cdot,\cdot):=\Omega(\cdot,J\cdot)\) makes \((T^*M,g_J,J,\Omega)\) K\"ahler. The point is that \(J\) is extra structure: it is not canonically selected by \(\nabla\) or by the Hessian metric on \(M\), and different choices of \(J\) lead to genuinely different K\"ahler lifts. In this sense, the para--K\"ahler structure of Proposition (ref) is canonical relative to the Hessian data, whereas an elliptic K\"ahler structure on \(T^*M\) is not. We return to this in (ref) and (ref).
Next, specialising Proposition (ref) to the lifted Hessian manifold $(\widetilde{\mathcal M},g_{\mathrm{Euc}},\nabla^{\mathrm{Euc}})$ yields a canonical para--K\"ahler structure $(G,K,\Omega)$ on $T^*\widetilde{\mathcal M}\simeq\mathbb{R}^{2n}$. The least--action Hamiltonian system (ref) is Hamiltonian; hence its flow preserves the canonical symplectic form $\Omega$. In particular, in the present linear setting on $\mathbb{R}^{2n}$ its generator $\mathsf A$ is an element of $\mathfrak{sp}(2n,\mathbb{R})$.
With the canonical $K$--splitting $T(T^*\widetilde{\mathcal M})=E_+\oplus E_-$ from Corollary (ref) at hand, we rewrite the least--action Hamiltonian system in the corresponding $K$--adapted (Witt) coordinates $(z_+,z_-)=(\tilde\rho,y)$ on $T^*\widetilde{\mathcal M}\simeq\mathbb{R}^{2n}$, where (under the translation identification) the distributions $E_+$ and $E_-$ are spanned by $\partial/\partial z_+^i$ and $\partial/\partial z_-{}_i$, respectively. This exposes the dynamics as a block flow perturbed by a shear mixing $E_-$ into $E_+$, and it yields the monotonicity identity for the neutral coordinate $\Lambda(t)=2\langle z_+(t),z_-(t)\rangle$:
In the following, we use the terminology on/off-shell relative to the invariant zero section $ \Sigma_0=\{y=0\}. $ A trajectory is on-shell if it is entirely contained in \(\Sigma_0\), equivalently if \(y(t)\equiv 0\) on the interval under consideration. It is off-shell if it is not confined to \(\Sigma_0\), equivalently if \(y(t)\neq 0\) at least at some time. Thus off-shell trajectories are not, in general, confined to a fixed leaf of the \(E_+\)-foliation, since the residual variable \(y\) typically evolves.
We are now in the position where we can define a para--qudit model on the zero--residual leaf:
Next we can identify rational inattention (RI) and Projective Expected Utilities (PEU) as special cases in the following two examples:
For a symmetric but non--diagonal $\widehat S$, the map $q(T)$ in (ref) is still given by the same amplitude evolution $\tilde\rho_T=e^{T\widehat S}\tilde\rho_0$, but now depends on the full operator $\widehat S$ (and hence on all components of $\tilde\rho_0$ in the choice basis). In particular, in the choice basis this generically cannot be reduced to a coordinatewise exponential tilt \[ q_k(T)\propto q_k(0)\exp\{\eta v_k\} \qquad\text{(for full--support $q(0)\in\mathcal M$).} \] for any single vector $v\in\mathbb{R}^n$. Equivalently, the diagonal ER/RI form is recovered if and only if $\widehat S$ is simultaneously diagonal in the choice basis; otherwise the off--diagonal entries mix coordinates and act as a rotation of the evaluative basis, inducing genuine contextual couplings between alternatives. Thus, within the same LAR model, the diagonal (choice--separable) regime reproduces classical entropy--regularised / RI posteriors, whereas non--diagonal symmetric generators produce the contextual, interference--like, and order--sensitive distortions of choice probabilities characteristic of quantum--like (QL) models. In this sense ER/RI and QL appear as two regimes of a single information--geometric dynamical framework.
To summarise, we have now obtained the dynamical equation (ref) in the real doubled space \(\mathbb{R}^{2n}\). This real formulation is the most direct, since it keeps the linear--algebraic dual spaces explicit and makes the canonical split metric transparent, but it is often advantageous to repackage the same system in different algebraic languages. Because the doubled space carries a natural split signature, it is convenient to introduce the split--complex \(\mathbb D^n\) representation in (ref), deriving a (split) para--Schr\"odinger equation. This will also allow a clean comparison, in (ref), with the standard complex packaging in \(\mathbb{C}^n\).
Having derived the lifted least--action equations of motion on \(N=T^*\widetilde{\mathcal M}\simeq\mathbb{R}^{2n}\) (cf.\ (ref)), we now show how the shear--free dynamics on the zero--residual leaf admit an intrinsic split--complex packaging. In this representation the canonical Witt splitting is encoded by the idempotents of \(\mathbb D\), and the on--shell flow takes the compact hyperbolic Schr\"odinger form.
We stay in the global Darboux chart \((\tilde\rho,y)\in\mathbb{R}^n\oplus\mathbb{R}^n\) on \(N=T^*\widetilde{\mathcal M}\), with canonical symplectic form \[ \Omega=d\tilde\rho^i\wedge dy_i. \] By Proposition (ref), the flat Euclidean para--K\"ahler structure determines a para--complex endomorphism \(K\) with Witt decomposition \(T(T^*\widetilde{\mathcal M})=E_+\oplus E_-\), \(E_\pm=\ker(K\mp\mathrm{id})\), and the lifted least--action flow is linear with generator
where \(\mathsf A_{\mathrm{pu}}\) preserves the Witt splitting (it commutes with \(K\)) and \(\mathsf A_{\mathrm{sh}}\) is a nilpotent shear mapping \(E_-\) into \(E_+\). In the \(K\)--adapted coordinates \((z_+,z_-)=(\tilde\rho,y)\), the zero--residual leaf \(\Sigma_0=\{z_-=0\}\) is invariant, and on \(\Sigma_0\) the dynamics reduce to the decoupled Witt evolution
which is the regime naturally encoded by the split--complex (hyperbolic) Schr\"odinger form.
To see this and to package (ref) intrinsically, we use the split--complex algebra \(\mathbb D=\{a+bj\mid a,b\in\mathbb{R},\ j^2=+1\}\) and its idempotents \(e_\pm=\tfrac12(1\pm j)\), which satisfy \(j e_\pm=\pm e_\pm\) and provide a complete pair of orthogonal projectors.\footnote{Terminology varies by field. The scalar algebra generated by a unit \(j\) with \(j^{2}=+1\) is known as the split--complex, hyperbolic, or double numbers. In differential geometry, the corresponding endomorphism \(K\) with \(K^{2}=\mathrm{id}\) and equal--rank eigenspaces is a para--complex structure; together with a neutral metric \(G\) and compatible symplectic form \(\Omega\), the triple \((G,K,\Omega)\) is para--K\"ahler.} These algebraic projectors mirror the Witt projectors \(P_\pm=\tfrac12(\mathrm{id}\pm K)\) onto \(E_\pm\), and thus identify the real direct sum \(E_+\oplus E_-\cong\mathbb{R}^n\oplus\mathbb{R}^n\) with the free \(\mathbb D\)--module \(\mathbb D^n\). Accordingly, any Witt state \((z_+,z_-)\) can be encoded as the split--complex state \[ \Psi_{\mathbb D} := z_+ e_+ + z_- e_- \in \mathbb D^n. \] Viewed as a real vector space, \(\mathbb{R}^{2n}\) equipped with the neutral metric \(G\) is a Krein space; the split--complex packaging simply recasts the same para--Hermitian geometry in module form.
On the zero--residual leaf, it is convenient to introduce the para--Hermitian Hamiltonian operator \[ \widehat H_{\mathbb D} = \widehat S + j\,\widehat F \in \mathrm{End}(\mathbb D^n), \qquad \widehat H_{\mathbb D}^{\#} = \widehat H_{\mathbb D}, \] where \((\cdot)^\#\) denotes split--conjugate transpose, i.e.\ \((a+bj)^\#:=a-bj\) and \(\widehat A^\#:=(\widehat A^{\#\textnormal{(entrywise)}})^{\!\top}\). The two real equations in (ref) then combine into the compact para--Schr\"odinger equation
Projecting (ref) onto \(e_\pm\) recovers the Witt components, since \(j e_\pm=\pm e_\pm\) implies \(\dot z_\pm=(\widehat F\pm\widehat S)z_\pm\). Away from the zero--residual leaf, the full lifted dynamics (ref) can be viewed as the para--Schr\"odinger evolution supplemented by the nilpotent shear term \(\mathsf A_{\mathrm{sh}}\), i.e.\ a forcing from \(E_-\) into \(E_+\) that vanishes exactly on \(\Sigma_0\). In this formulation \(\widehat H_{\mathbb D}\) is para--Hermitian, \((\widehat H_{\mathbb D})^\#=\widehat H_{\mathbb D}\), hence \((j\widehat H_{\mathbb D})^\#=-\,j\widehat H_{\mathbb D}\). Therefore the propagator \[ U(t):=\exp(j\widehat H_{\mathbb D}\,t) \] is para--unitary, \(U(t)^\#U(t)=I\), and preserves the induced para--Hermitian (Krein) pairing \(\langle\Psi,\Phi\rangle_{\mathbb D}:=\Psi^\#\Phi\) (equivalently its real part \(G=\Re\langle\cdot,\cdot\rangle_{\mathbb D}\)) along the shear--free on--shell flow on \(\Sigma_0\). The split--complex packaging thus provides a compact description in which the hyperbolic Schr\"odinger form holds precisely on the zero--residual leaf.
The split--complex representation in (ref) is intrinsic to the Hessian para--K\"ahler lift: the para--complex structure \(K\) and its Witt splitting are part of the geometric data of the model, and the shear--free flow on the zero--residual leaf is naturally expressed as a para--Schr\"odinger equation. One may nevertheless repackage the same real lifted equations in complex coordinates by choosing an auxiliary almost complex structure \(J\) with \(J^2=-\mathrm{id}\). Such choices always exist (e.g.\ \(\Omega\)--compatible ones on a symplectic manifold), but they are noncanonical and affect only how one represents the fixed symplectic form \(\Omega\) (for instance via \(\Omega(\cdot,J\cdot)\)), not \(\Omega\) itself. Consequently, the resulting complex form of the equations need not be unitary and, in general, does not reflect the underlying para--Hermitian geometry.
To make this concrete, in the flat Darboux chart \((\tilde\rho,y)\in\mathbb{R}^n\times\mathbb{R}^n\) on \(\mathbb{R}^{2n}\simeq T^*\widetilde{\mathcal M}\) we choose the standard almost complex structure that pairs base and fibre coordinates. Define \[ J(\tilde\rho,y)=(-y,\tilde\rho), \qquad J^2=-\mathrm{id}, \] and identify \(\mathbb{R}^{2n}\) with \(\mathbb C^n\) via
Under this identification, the real lifted Hamiltonian system (ref), \[ \dot{\tilde\rho}=\widehat V\,\tilde\rho+y, \qquad \dot y=-\,\widehat V^{\!\top}y, \qquad \widehat V=\widehat S+\widehat F,\ \ \widehat S^{\!\top}=\widehat S,\ \ \widehat F^{\!\top}=-\widehat F, \] becomes, after substituting (ref) and collecting the \(z\) and \(\bar z\) components,
Equivalently, \[ \dot z = \widehat F z + \widehat S \bar z \;-\;\frac{i}{2}(z-\bar z) = \widehat F z + \widehat S \bar z + y, \qquad y=\frac{1}{2i}(z-\bar z), \] so the scalar terms \(\pm\tfrac{i}{2}I\) in (ref) are simply the canonical \(+y\) coupling expressed in the complex coordinates \(z=\tilde\rho+iy\). In particular, (ref) is not complex--linear off the zero--residual leaf: it has (the Bogoliubov) form that mixes \(z\) and \(\bar z\).
On the zero--residual leaf \(y=0\) the subspace \(y\equiv 0\) is invariant (\(\dot y=-\widehat V^{\!\top}y\)), hence \(z=\bar z\) and (ref) reduces to \[ \dot z=(\widehat S+\widehat F)z=\widehat V z. \] Thus the on--shell dynamics become complex--linear and can be written in a (generally nonunitary) Schr\"odinger form
However, \(\widehat H_{\mathbb C}\) is non--Hermitian whenever \(\widehat S\neq 0\): indeed \(\widehat H_{\mathbb C}^\dagger=-\,i\,\widehat V^{\!\top}\neq \widehat H_{\mathbb C}\) in general. Off the zero--residual leaf (\(y\neq 0\)), the coupling to \(\bar z\) in (ref) precludes any reduction to a unitary complex Schr\"odinger evolution.
This extrinsic complex packaging also makes clear why the usual Hermitian norm need not be conserved. Differentiating \(\|z\|^2=z^\dagger z\) along solutions of (ref) yields \[ \frac{d}{dt}\|z\|^2 = z^\dagger\!\Bigl(\widehat S+\tfrac{i}{2}I\Bigr)\bar z + \bar z^{\!\dagger}\!\Bigl(\widehat S-\tfrac{i}{2}I\Bigr) z =2\,\Re\!\Bigl(z^\dagger\!\Bigl(\widehat S+\tfrac{i}{2}I\Bigr)\bar z\Bigr), \] which is generically nonzero. In summary, identifying \(\mathbb{R}^{2n}\simeq\mathbb C^n\) provides an elegant coordinate description of the same lifted real dynamics, but it obscures the para--K\"ahler geometry: unless the real generator preserves the chosen almost complex structure \(J\), the resulting representation is not a unitary Schr\"odinger evolution, and even on the zero--residual leaf the apparent Hamiltonian \(\widehat H_{\mathbb C}\) is non--Hermitian whenever \(\widehat S\neq 0\). This contrast will be further elaborated in (ref).
The preceding sections show that the least--action rationality (LAR) lift admits a natural hyperbolic quantum--geometric reading. Starting from the information--geometric (Hessian) data, the cotangent lift $ N:=T^*\widetilde{\mathcal M} $ comes with its canonical symplectic form \(\Omega\) and a canonical para--K\"ahler (para--Hermitian) way of organising that symplectic space: a product structure \(K\) with \(K^2=\mathrm{id}\) splits directions into two complementary sectors, and a neutral metric \(G\) relates the splitting to the symplectic form via \[ \Omega(u,v)=G(Ku,v). \] This gives an intrinsic Witt decomposition $ TN=E_+\oplus E_-, $ so the hyperbolic lift does not merely say that the dynamics are Hamiltonian on \((N,\Omega)\); it also supplies a canonical “two--sector” geometry in which the split--complex Schr\"odinger picture becomes a convenient shorthand (most transparently on the zero--residual sheet).
By contrast, much of the quantum cognition literature (e.g.\ BusemeyerBruza2012) assumes an explicitly elliptic quantum geometry: amplitudes live in a complex Hilbert space and evolution is unitary. From the present perspective, the key change is not the symplectic data (since \((N,\Omega)\) is the same) but the choice of coherence structure used to interpret the lifted geometry (symplectic space). In the hyperbolic theory, coherence is organised by the canonical para--complex split (\(K^2=+\mathrm{id}\)), leading at the linear level to the noncompact symmetry group \[ \mathrm{Sp}(2n,\mathbb{R})\cap\mathrm{O}(n,n)\cong \mathrm{GL}(n,\mathbb{R}). \] An elliptic model instead requires a K\"ahler organisation \((g,J,\Omega)\) with \(J^2=-\mathrm{id}\), \(g\) positive definite, and \[ \Omega(u,v)=g(Ju,v), \] whose linear symmetry group is the compact unitary group \[ \mathrm{Sp}(2n,\mathbb{R})\cap\mathrm{O}(2n)\cong \mathrm{U}(n). \] The intuitive point is simple: hyperbolic structure is delivered by the lift, whereas elliptic structure amounts to an extra modelling choice: a way of selecting a complex notion of orthogonality/coherence on the same symplectic space.
The purpose of this section is to make that additional choice explicit, and to show how standard complex (unitary) quantum dynamics can be recovered from the classical Hamiltonian framework. Rather than postulating a complex structure ad hoc, we proceed variationally. Starting from the Hamilton action on \((N,\Omega)\), we impose an eigenbundle admissibility restriction: admissible trajectories are required to evolve within a distinguished half--dimensional Lagrangian subbundle of the (complexified) tangent bundle, while admissible variations are confined to the complementary Lagrangian directions. As shown below, this replaces the full Hamilton equation by a projected evolution and is equivalent to choosing a K\"ahler polarisation. In this sense, polarisation is not an independent geometric postulate, but the symplectic encoding of an admissibility constraint imposed at the level of the variational principle.
We keep the same lifted manifold \(N:=T^*\widetilde{\mathcal M}\simeq\mathbb{R}^{2n}\) and its canonical symplectic form \(\Omega\). Since \(\Omega\) is exact, there exists a globally defined one--form \(\theta\) on \(N\) with \(\Omega=-\,d\theta\). Any smooth function \(H:N\to\mathbb{R}\) then determines a Hamiltonian vector field \(X_H\in\mathfrak X(N)\) by
equivalently, the integral curves \(z(t)\) satisfy \(\dot z(t)=X_H(z(t))\). These same curves are also characterised variationally as stationary points (with fixed endpoints) of the Hamilton action
The purpose of the present discussion is to explain how an elliptic (Hilbert--space) quantum dynamics can be obtained from this same real Hamiltonian structure \((N,\Omega,H)\) by adding an admissibility structure that selects a complex Lagrangian sector. This additional structure is not canonically fixed by \(\Omega\) (nor by the information--geometric lift) and therefore constitutes genuine modelling input.
We consider the global Darboux chart $(\tilde\rho,y)\in\mathbb{R}^n\oplus\mathbb{R}^n$ on $N$, so that $\Omega=d\tilde\rho^i\wedge dy_i$. In these coordinates we consider the (formal) complexification $N_\mathbb{C}\simeq\mathbb{C}^{2n}$ and its complexified tangent bundle $TN_\mathbb{C}:=TN\otimes_\mathbb{R}\mathbb{C}$, and we extend $\Omega$ $\mathbb{C}$--bilinearly to $\Omega_\mathbb{C}:=\Omega\otimes 1$. Fix a complex symmetric matrix $M\in\operatorname{Sym}(n,\mathbb{C})$ with $\Im M$ nondegenerate and define constant complex subbundles of $TN_\mathbb{C}$ (by identifying each fibre with $\mathbb{C}^{2n}$ in Darboux coordinates) as graphs
The following elementary statement shows that this choice provides a complex bi--Lagrangian splitting of $TN_\mathbb{C}$.
For later use, it is convenient to record an explicit formula for $\Pi_M^{(+)}$ in Darboux coordinates. Given $(a,b)\in\mathbb{C}^n\oplus\mathbb{C}^n$, the unique decomposition $(a,b)=(u,Mu)+(v,\bar M v)$ is determined by $u+v=a$ and $Mu+\bar M v=b$, equivalently $(M-\bar M)u=b-\bar M a$. Since $M-\bar M=2i\,\Im M$ is invertible, one finds
We now impose a complex eigenbundle admissibility restriction directly at the level of the variational principle on $N_\mathbb{C}$. A $C^1$ curve $z:[t_0,t_1]\to N_\mathbb{C}$ is called admissible if
and an admissible variation along $z(\cdot)$ is a $C^1$ vector field $\delta z(t)\in T_{z(t)}N_\mathbb{C}$ with fixed endpoints and values constrained by
Stationarity of the Hamilton action under the restricted variations replaces the full Hamilton equation by its projection onto the admissible Lagrangian sector.
In the Darboux chart (ref), the kinematic constraint (ref) is the graph relation
Thus Proposition (ref) provides the general reduction principle: the variational admissibility restriction converts Hamilton's equation into a projected evolution tangent to the chosen complex Lagrangian sector. The remainder of the elliptic construction is then an explicit analysis of this projected dynamics in coordinates adapted to the graph constraint.
When $\Im M$ is definite, one may (after a constant real symplectic change of Darboux coordinates, i.e.\ an $\mathrm{Sp}(2n,\mathbb{R})$--change of linear coordinates preserving $\Omega$) normalise $\Im M=-I$ and write $M=R-iI$ with $R^\top=R$. In this normalised class, define the complex linear combinations
Differentiating and substituting (ref) gives
so admissible kinematics foliate the constrained dynamics by affine leaves $\{\phi=\phi_0\}$, and on each leaf the projected equation (ref) closes as an evolution equation for the holomorphic coordinate $\psi$.
In the symplectically normalised coordinates (ref), the admissibility restriction can be stated equivalently as follows: \(\dot z\in L_M^{(+)}\) is equivalent to the graph relation (ref), and in the \((\psi,\phi)\) variables this is precisely the conservation law \(\dot\phi=0\). Thus admissible kinematics foliate the constrained dynamics by affine leaves \(\{\phi=\phi_0\}\), and on each leaf the projected equation (ref) closes as a first--order evolution equation for the holomorphic coordinate \(\psi\).
To connect this general reduction principle with standard complex quantum dynamics, we now specialise to a distinguished polarisation. For the lifted least--action Hamiltonian (ref), the choice $ M_\ast=(1-i)I $ (i.e.\ \(R=I\) in (ref)) yields, on the invariant holomorphic leaf \(\{\phi=0\}\), an elliptic Schr\"odinger equation:
To summarise, in the present elliptic construction one supplies, as additional modelling input, an \(\Omega\)--compatible complex Lagrangian sector \(L_M^{(+)}\subset TN_\mathbb{C}\). In the adapted coordinates \((\psi,\phi)\) this admissibility requirement is equivalent to the conservation law \(\dot\phi=0\). Accordingly, the projected dynamics is foliated by invariant affine leaves \(\{\phi=\phi_0\}\). For the distinguished choice \(M_\ast=(1-i)I\), the leaf \(\{\phi=0\}\) (vanishing anti--holomorphic coordinate) carries a Schr\"odinger evolution with generator \(\widehat S+i\widehat F\), up to an additive scalar multiple of the identity (equivalently, a global phase / projective gauge).
It is useful to contrast this with the hyperbolic (para--K\"ahler) lift of (ref). There the canonical Witt splitting is fixed by the Fisher--Rao/Hessian data, and the deviation from para--unitarity is carried entirely by the nilpotent shear. On the {zero--residual leaf} $ \Sigma_0=\{(\tilde\rho,y):y=0\}, $ the shear is inactive, the dynamics closes as a para--unitary block flow, and one obtains a para--holomorphic (split--complex, \(\mathbb D\)) Schr\"odinger representation (the “para--Schr\"odinger” equation ((ref)). Thus both sectors admit a natural “zero leaf”: \(\{\phi=0\}\) in the elliptic case and \(\Sigma_0\) in the hyperbolic case. The distinction is that \(\Sigma_0\) is canonical from the lift itself, whereas \(\{\phi=0\}\) depends on the chosen complex sector \(L_M^{(+)}\).
Consequently, elliptic coherence does not follow from least--action rationality alone: it requires an additional admissibility restriction selecting a complex Lagrangian sector. This motivates a modified least--action postulate within the selected holomorphic sector, which we formulate next as coherent least--action rationality (CLAR).
The behavioural assumptions in (ref), in particular the assumption on LAR in (ref), determine a canonical para--K\"ahler lift on \(N:=T^*\widetilde{\mathcal M}\) and yield the lifted hyperbolic (Krein--space) model as derived in (ref). By contrast, an elliptic (Hilbert--space) quantum model requires additional structure not fixed by the Fisher--Rao data: as we saw above in (ref), one must specify an \(\Omega\)--compatible complex sector, equivalently a complex Lagrangian polarisation, and then restrict admissible evolutions to that sector. In behavioural terms, elliptic coherence is therefore not an implication of Assumptions A1--A3, but an additional admissibility requirement.
We formulate this additional requirement as a modified least--action postulate on the same underlying real symplectic phase space. Let \(\theta\) be the tautological one--form on \(N\) with \(\Omega=-\,d\theta\), let \(H:N\to\mathbb{R}\) be the lifted Hamiltonian, and extend \((N,\Omega,\theta,H)\) to the complexification \(N_\mathbb{C}\). Fix a complex bi--Lagrangian splitting of the complexified tangent bundle, \[ TN_\mathbb{C}=L^{(+)}\oplus L^{(-)}, \qquad \Omega_\mathbb{C}|_{L^{(\pm)}}\equiv 0, \qquad L^{(-)}=\overline{L^{(+)}}. \] In the constant Darboux setting used in (ref), such a splitting is equivalently specified by a matrix \(M\in\operatorname{Sym}(n,\mathbb{C})\) with \(\Im M\) nondegenerate via \(L^{(+)}=L_M^{(+)}\), \(L^{(-)}=L_M^{(-)}\) as in (ref), with associated projector \(\Pi^{(+)}=\Pi_M^{(+)}\) onto \(L^{(+)}\) along \(L^{(-)}\).
Consider the Hamilton action on \(C^1\) curves \(z:[t_0,t_1]\to N_\mathbb{C}\), \[ \mathcal S[z]:=\int_{t_0}^{t_1}\bigl(\theta(\dot z(t)) - H(z(t))\bigr)\,dt. \] A curve \(z(\cdot)\) is called coherent admissible (for the chosen polarisation) if
and an admissible variation along \(z(\cdot)\) is a \(C^1\) field \(\delta z(t)\in T_{z(t)}N_\mathbb{C}\) with fixed endpoints and values constrained by
Assumption 3' [Coherent least--action rationality (CLAR)] Given a choice of complex polarisation \(TN_\mathbb{C}=L^{(+)}\oplus L^{(-)}\), the decision maker realises coherent admissible episodes \(z(\cdot)\) that are stationary for the Hamilton action \(\mathcal S[z]\) under all admissible variations (ref).
By Proposition (ref), Assumption 3' is equivalent to the projected Hamilton equation
where \(X_H\) is the (complex) Hamiltonian vector field defined by \(\iota_{X_H}\Omega_\mathbb{C}=dH\). Thus CLAR replaces the full Hamilton evolution by its polarisation component: the complementary \(L^{(-)}\) component of \(X_H\) is not followed, but excluded by admissibility.
When \(\Im M\) is definite (after normalization \(\Im M<0\)), the polarisation induces a positive definite sesquilinear form on \(L^{(+)}\) via \(\langle u,v\rangle_M:=-\,i\,\Omega_\mathbb{C}(u,\bar v)\), so that (ref) becomes a unitary evolution on the corresponding Hilbert space. For the lifted least--action Hamiltonian (ref), the distinguished choice \(M_\ast=(1-i)I\) yields, on the invariant holomorphic leaf \(\{\phi=0\}\), the standard Schr\"odinger dynamics with Hermitian generator \(\widehat S+i\widehat F\) (Proposition (ref)). In particular, coherence in the elliptic model is an explicit behavioural restriction: it is imposed by selecting a complex polarisation and applying the least--action principle within its holomorphic sector, rather than emerging from the hyperbolic LAR lift alone.
This section reviews the least--action rationality (LAR) model through the lens of bounded rationality. Our guiding theme is the separation between the latent information--geometric dynamics, formulated on amplitudes (and their lift), and the epistemic readout on the simplex that produces observable choice frequencies. We will primarily consider the hyperbolic (split--complex/para--K\"ahler) regime developed above, where the lifted phase space \[ N:=T^*\widetilde{\mathcal M}\subset T^*\mathbb{R}^n \] carries its canonical symplectic form \(\Omega\) together with the para--K\"ahler structure induced by the Hessian data, and the Hamiltonian LAR dynamics on \(N\) restricts on the zero--residual leaf to the para--unitary (split--complex Schr\"odinger--type) sector. A final subsection ((ref)) returns briefly to the elliptic (complex, unitary) coherent sector to contrast which bounded--rationality mechanisms persist under CLAR and which are specific to the hyperbolic lift.
As in other quantum--like approaches, two levels of state description coexist. The latent state is an amplitude (or lifted amplitude--momentum pair) evolving on the information--geometric lift, while the epistemic state is a probability distribution on the simplex $\mathcal M=\Delta^{n-1}_{>0}$ obtained by geometric projection (normalisation and squaring in a chosen readout basis). The latent variables need not be endowed with an ontological interpretation; what matters for our purposes is that they support a canonical differential geometry and a preferred LAR dynamics, whereas the epistemic variables are what is observable as choice frequencies in a fixed context. Bounded rationality is then understood as the mismatch between these two descriptions: even when the latent layer obeys a clean variational principle, the induced epistemic behaviour can be path dependent, history dependent, and context dependent because of structural constraints in the latent generator and because of the geometric reductions required to pass to observables.
We separate these mechanisms according to where they arise: intrinsically on the epistemic simplex \(\mathcal M\), intrinsically in the latent on--shell flow on the zero--residual leaf, intrinsically in the full lifted (off--shell) phase--space dynamics, or from the geometric readout map that reduces latent amplitudes to observable probabilities. Accordingly, we proceed as follows. (ref) studies epistemic non--integrability of the preference form \(\beta\) on \(\mathcal M\). (ref) studies the latent on--shell split \(\widehat V=\widehat S+\widehat F\) on the zero--residual leaf and introduces a monotone entropic clock on rays. (ref) studies off--shell deviation using the neutral form \(Q\) and the associated index \(\Lambda(t)\). (ref) studies context dependence and hyperbolic interference arising from the readout family \(\pi_B\). We conclude in (ref) by returning briefly to the elliptic coherent sector (CLAR), to contrast which bounded--rationality mechanisms persist under unitary Hilbert--space evolution and which are specific to the hyperbolic lift.
Assume $n\ge3$, so $\mathcal M=\Delta^{n-1}_{>0}$ has $\dim\mathcal M\ge2$ and admits nonconstant smooth loops and embedded disks. Recall from (ref) that the induced epistemic preference field on lotteries is the $1$--form $\beta=\mathcal T(\widehat V)\in\Omega^1(\mathcal M)$ obtained by restricting the spherical form $\bar\alpha^{\mathbb S}$ to $\mathbb S^{n-1}_{+}$ and pulling it back along the square--root chart $\iota:\mathcal M\to\mathbb S^{n-1}_{+}$, $\iota(q)=\sqrt q$. All simplex--level differential identities involving $\beta$ (and hence $d\beta$) are understood on $\mathcal M$ and, if a trajectory touches the boundary of $\Delta^{n-1}$, are to be read piecewise on each time interval of constant support as in (ref). By Proposition (ref) this field splits canonically as \[ \beta=\mathcal U+\mathcal R, \qquad \mathcal U=dU, \qquad \mathcal R=\beta-\mathcal U, \qquad d\beta=d\mathcal R. \] The conservative part $\mathcal U$ admits a global potential, while $\mathcal R$ captures the obstruction to global integrability. We package this obstruction as follows.
Following the geometric viewpoint of Russell1991, we therefore interpret the $2$--form \[ \mathcal K:=d\beta \] as a preference curvature: it measures the failure of path independence, and its flux controls the accumulated loop holonomy (regret) generated by sequential changes. In our operator class the conservative part is precisely the utility channel, so the obstruction is carried entirely by the co--utility channel: $d\beta=d\mathcal R$ by Proposition (ref). Thus expected--utility behaviour corresponds to $\mathcal K\equiv0$, whereas nonzero curvature predicts path dependence and order effects in sequential choice. This packaging is consistent with classical regret/disappointment motivations LoomesSugden1982,Bell1982. The nonconservative component arises intrinsically as the co--utility component induced by the same latent operator that also generates the utility potential.
Recall that in our operator representation \(\widehat V=\widehat S+\widehat F\) the symmetric channel \(\widehat S\) encodes evaluative norm change, while the skew channel \(\widehat F\) generates circulatory (regret--like) drift. The purpose of the present subsection is to make this separation explicit in polar variables and to introduce a canonical scalar clock on rays, anchored by a spectral gauge choice that will later serve as a reference when we leave the zero--residual leaf. The clock is called entropic in the sense that it is the log--partition normaliser in the quadratic readout (hence paired with Shannon entropy by convex duality), and in the sense that its on--shell production rate is generated by the evaluative channel \(\widehat S\), not by the circulatory channel \(\widehat F\).
Recall that on the zero--residual leaf we set \(y\equiv 0\), so the lifted amplitude satisfies the linear on--shell dynamics (ref), i.e.\ \(\dot{\tilde\rho}=\widehat V\,\tilde\rho\) with \(\widehat V=\widehat S+\widehat F\). We write the polar decomposition \[ r(t):=\|\tilde\rho(t)\|>0, \qquad \rho(t):=\frac{\tilde\rho(t)}{\|\tilde\rho(t)\|}\in\mathbb S^{n-1}, \qquad \tilde\rho(t)=r(t)\rho(t). \] Differentiating \(\|\rho(t)\|^2\equiv 1\) yields \(\ip{\rho(t)}{\dot\rho(t)}=0\). Substituting \(\dot{\tilde\rho}=\dot r\,\rho+r\,\dot\rho\) into the on--shell evolution \(\dot{\tilde\rho}=\widehat V\tilde\rho\) and separating radial and tangential components gives the polar identities
and
The channel separation is already visible at the level of norm change: by skew--symmetry, \(\ip{\rho}{\widehat F \rho}=0\) for all \(\rho\), so \(\widehat F\) contributes no direct radial component, whereas \(\widehat S\) completely determines the radial rate \(\dot r/r\) via (ref). At the same time, \(\widehat S\) also affects the tangential drift through \((\widehat S-\ip{\rho}{\widehat S\rho}I)\rho\), and the instantaneous direction \(\rho(t)\) influences the radial rate through the Rayleigh quotient \(\ip{\rho(t)}{\widehat S\rho(t)}\). Circulation may therefore reshape the path \(\rho(t)\), but it cannot generate clock production when \(\widehat S=0\).
The epistemic state is obtained by squaring coordinates, \[ q_i(t):=\rho_i(t)^2, \qquad q(t)\in\Delta^{n-1}, \] so simplex observables depend only on the ray \(\mathbb{R}_{>0}\tilde\rho(t)\), not on the scale \(r(t)\). A convenient way to encode this ray--invariance is the identity--shift invariance of the simplex projection: for any \(c\in\mathbb{R}\), \[ \widehat S\mapsto \widehat S+cI \quad\Longrightarrow\quad \tilde\rho(t)\mapsto e^{ct}\tilde\rho(t) \quad\Longrightarrow\quad q_i(t)=\frac{\tilde\rho_i(t)^2}{\|\tilde\rho(t)\|^2}\ \text{ unchanged}, \] and the direction equation (ref) is unchanged as well, since the shift cancels in \(\widehat S-\ip{\rho}{\widehat S \rho}I\). We therefore regard \(\widehat S\mapsto \widehat S+cI\) as a projective gauge.
We now introduce the scale variable that will serve as an intrinsic entropic (equivalently, welfare) clock. Define the normaliser \[ \mathcal Z(t):=\|\tilde\rho(t)\|^{2}=\sum_{i=1}^n \tilde\rho_i(t)^2, \qquad \sigma(t):=\log\|\tilde\rho(t)\|=\tfrac12\log \mathcal Z(t), \] and the nonnegative weights \(w_i(t):=\tilde\rho_i(t)^2\), so that \(\mathcal Z=\sum_i w_i\) and \(q_i=w_i/\sum_j w_j\). When \(w_i(t)>0\) we may write \(w_i=e^{\varphi_i}\) with \(\varphi_i(t):=\log w_i(t)\), giving \[ \mathcal Z(t)=\sum_{i=1}^n e^{\varphi_i(t)}, \qquad q_i(t)=\frac{e^{\varphi_i(t)}}{\sum_{j=1}^n e^{\varphi_j(t)}}. \] (These identities extend to the boundary by continuity, with the convention \(0\log 0:=0\).) Thus \(\log \mathcal Z=\log\sum_i e^{\varphi_i}\) is the log--sum--exp normaliser (the “inclusive value” in discrete choice), and it is canonically paired with Shannon entropy by the variational identity \[ \log\Bigl(\sum_{i=1}^n e^{\varphi_i}\Bigr) = \sup_{q\in\Delta^{n-1}} \Bigl\{\sum_{i=1}^n q_i\varphi_i + \mathcal H(q)\Bigr\}, \qquad \mathcal H(q):=-\sum_{i=1}^n q_i\log q_i, \] (with the convention \(0\log 0:=0\)). Since \(q(t)\) is precisely the normalised weight vector, the supremum is attained along the episode, yielding the free--energy identity
This is the sense in which \(\sigma=\tfrac12\log\mathcal Z\) is simultaneously “welfare” (a log--partition potential) and “entropic” (paired with \(\mathcal H\) by convex duality), independently of whether the tangential drift contains circulation (\(\widehat F\neq 0\)).
For dynamical purposes, it is convenient to anchor \(\sigma\) in a canonical way that respects the projective gauge above. Let \[ \lambda_{\min}:=\lambda_{\min}(\widehat S), \qquad \widehat S_{+}:=\widehat S-\lambda_{\min}I\succeq 0, \] and define the spectrally normalised entropic welfare clock
This anchoring is projectively invariant: under \(\widehat S\mapsto\widehat S+cI\) one has \(\log\|\tilde\rho(t)\|\mapsto \log\|\tilde\rho(t)\|+ct\) and \(\lambda_{\min}\mapsto\lambda_{\min}+c\), so \(\sigma_+\) is unchanged. Along any zero--residual trajectory we have
so \(\sigma_{+}\) is nondecreasing and strictly increasing unless \(\rho(t)\) remains in the ground eigenspace \(\ker(\widehat S_{+})\). In particular, the clock production is driven by the evaluative channel: \(\widehat F\) contributes no direct radial component, and when \(\widehat S=0\) one has \(\dot\sigma_{+}\equiv 0\). Moreover, (ref) depends only on the ray \(\rho(t)\), so up to an additive constant the clock is determined by the ray trajectory: \[ \sigma_{+}(t)-\sigma_{+}(t_0)=\int_{t_0}^t \ip{\rho(s)}{\widehat S_{+}\rho(s)}\,ds. \] This is consistent with the auxiliary complex repackaging in (ref), in which the packaged generator \(\widehat H_{\mathbb{C}}\) is defined. If \(\widehat H_{\mathbb{C}}\) is hermitian with respect to the standard complex inner product induced by the Euclidean metric implies unitary evolution and hence norm preservation. In particular, in the purely circulatory case (\(\widehat S=0\)) the evolution is unitary and preserves the Euclidean norm.
Finally, mapping (ref) through \(q_i=\rho_i^2\) yields the induced simplex drift. At the level of the operator decomposition, this drift separates into an evaluative contribution driven by \(\widehat S\) and a circulatory contribution driven by \(\widehat F\). When \(n\ge3\), Proposition (ref) identifies this separation on the simplex by decomposing the induced preference $1$--form \(\beta=\mathcal T(\widehat V)\) into its potential part \(dU\) (utility) and its complementary part (co--utility) under the regularity assumptions stated there.
The polar variables also anticipate what changes off--shell. Along a general lifted episode one has \(\dot{\tilde\rho}=y+\widehat V\tilde\rho\), so the radial rate acquires precisely the radial component of the residual, \[ \dot r(t)=\ip{\rho(t)}{y(t)}+r(t)\ip{\rho(t)}{\widehat S\rho(t)}. \] Equivalently, at the level of the normaliser one finds \[ \dot{\mathcal Z}(t)=2\,\ip{\tilde\rho(t)}{y(t)}+2\,\ip{\tilde\rho(t)}{\widehat S\,\tilde\rho(t)}, \qquad \dot\sigma(t)=\ip{\rho(t)}{\widehat S\rho(t)}+\frac{\ip{\tilde\rho(t)}{y(t)}}{\mathcal Z(t)}. \] Thus the unique new scalar contribution is measured by the neutral pairing \(\ip{\tilde\rho(t)}{y(t)}\). In the next subsection we make this phase--space observable explicit via the null--cone index \(\Lambda(t)=2\ip{\tilde\rho(t)}{y(t)}\) (so that \(\dot\sigma=\ip{\rho}{\widehat S\rho}+\Lambda/(2\mathcal Z)\)), and show how it records action accumulation and enforces a one--way cone--crossing constraint for off--shell episodes.
The preceding subsections isolated bounded rationality mechanisms visible on the epistemic manifold \(\mathcal M\): non--integrability is detected by the preference curvature \(d\beta\), and on the zero--residual leaf the latent split \(\widehat V=\widehat S+\widehat F\) separates evaluative norm change from circulatory drift. We now focus on a distinct, specifically hyperbolic mechanism that is only visible in the full lifted dynamics: off--shell episodes with nonzero residual are organised by an orthogonality relation in phase space, encoded by the canonical neutral form of the para--K\"ahler lift.
Recall that we consider the Darboux chart \(Z=(\tilde\rho,y)\in T^*\widetilde{\mathcal M}\simeq\mathbb{R}^{2n}\) of the lifted model, where \(y\) denotes the residual (momentum) and \(t\mapsto Z(t)=(\tilde\rho(t),y(t))\) is a lifted episode satisfying (ref). By (ref) and (ref), the Euclidean Hessian lift carries the canonical neutral quadratic form \[ Q(\tilde\rho,y)=2\langle \tilde\rho,y\rangle . \] Along an episode \(Z(t)=(\tilde\rho(t),y(t))\) we therefore consider the associated scalar observable
The level set \[ \mathcal N:=\{Z\in T^*\widetilde{\mathcal M}:Q(Z)=0\} =\{(\tilde\rho,y):\langle\tilde\rho,y\rangle=0\} \] is the decisional light cone\footnote{Levin2019 define a cognitive light cone, which is a different concept in another context.}, or simply the null cone. It contains the zero--residual leaf \(\{y=0\}\), but is strictly larger: the null constraint \(\Lambda=0\) imposes only the orthogonality \(\langle\tilde\rho,y\rangle=0\) and does not force \(y=0\).
The interpretation of the null constraint is an orthogonal splitting of bounded rationality deviation. For \(\tilde\rho\neq0\), write \(r:=\|\tilde\rho\|>0\) and \(\rho:=\tilde\rho/r\in\mathbb S^{n-1}\). Decompose the residual into its component parallel to the instantaneous amplitude direction and its orthogonal complement, \[ y_{\parallel}:=\langle\rho,y\rangle\,\rho, \qquad y_{\perp}:=y-y_{\parallel}, \qquad \langle \rho,y_{\perp}\rangle=0. \] Then \[ \Lambda=2\langle \tilde\rho,y\rangle=2r\,\langle\rho,y\rangle, \] so \(\Lambda\) measures exactly the radial component \(y_{\parallel}\). In particular, \[ Z\in\mathcal N \quad\Longleftrightarrow\quad \langle\tilde\rho,y\rangle=0 \quad\Longleftrightarrow\quad y_{\parallel}=0, \] i.e.\ on the null cone the instantaneous least--action deviation is purely tangential to the sphere of radius \(r\).
This distinction is behaviourally meaningful because simplex observables depend only on the ray \(\mathbb{R}_{>0}\tilde\rho\) via the quadratic readout \[ q_i=\frac{\tilde\rho_i^2}{\|\tilde\rho\|^2}=\rho_i^2, \] whereas the lifted model shows how deviation accumulates in phase space. The quantity \(\Lambda\) isolates the component of the residual that is aligned with the current amplitude direction and therefore separates “radial” deviation (\(\Lambda\neq0\)) from “purely redistributive” deviation at fixed radius (\(\Lambda=0\)).
A central feature of the hyperbolic lift is that the corresponding episode--level index \(\Lambda\) satisfies an exact balance law with a nonnegative source term. Differentiating (ref) along a solution and substituting \(\dot{\tilde\rho}=y+\widehat V\tilde\rho\) and \(\dot y=-\widehat V^{\!\top}y\) from (ref) yields \[ \dot\Lambda(t) =2\langle \dot{\tilde\rho}(t),y(t)\rangle+2\langle \tilde\rho(t),\dot y(t)\rangle =2\|y(t)\|^2, \] hence the monotonicity identity
Integrating (ref) from \(t_0\) to \(t\) gives
It is therefore natural to interpret \(\Lambda\) as a null--cone index whose increment records the cumulative deviation from the zero--residual leaf. We name the associated nonnegative accumulation functional
Since \(y=\dot{\tilde\rho}-\widehat V\tilde\rho\) along any episode, one has \(L(\tilde\rho,\dot{\tilde\rho})=\tfrac12\|y\|^2\), hence the action accumulated over \([t_0,t]\) is \[ \int_{t_0}^t L\,d\tau=\tfrac12\,\mathcal A(t;t_0)=\tfrac14\bigl(\Lambda(t)-\Lambda(t_0)\bigr). \] Thus \(\Lambda\) is simultaneously a geometric phase--space index and a quantitative record of accumulated least--action deviation.
The monotonicity of \(\Lambda\) imposes a one--way constraint relative to the null cone. Fix \(t_0\). If \(\Lambda(t_0)>0\), then \(\Lambda(t)>0\) for all \(t\ge t_0\). If \(\Lambda(t_0)=0\) and \(y(t_0)\neq 0\), then \(\dot\Lambda(t_0)=2\|y(t_0)\|^2>0\), hence \(\Lambda(t)>0\) for all \(t>t_0\). If \(\Lambda(t_0)<0\), then \(\Lambda\) can cross \(0\) at most once for \(t\ge t_0\), and any crossing time \(t_*>t_0\) (if it exists) is uniquely characterised by \[ \Lambda(t_*)=0 \quad\Longleftrightarrow\quad \mathcal A(t_*;t_0)=\int_{t_0}^{t_*}\|y(\tau)\|^2\,d\tau=-\frac{\Lambda(t_0)}{2}. \] Finally, remaining on the null cone for all forward time is equivalent to vanishing residual: by (ref), the condition \(\Lambda(t)\equiv \Lambda(t_0)\) for all \(t\ge t_0\) holds if and only if \(\|y(t)\|\equiv 0\), i.e.\ \(y(t)\equiv 0\), for all \(t\ge t_0\). In particular, any episode that touches \(\mathcal N\) with nonzero residual departs immediately into the region \(\{Q>0\}\), and cannot return to \(\{Q<0\}\) for forward time.
It is useful to contrast this lifted, episode--level constraint with the intrinsic simplex decompositions discussed earlier. On \(\mathcal M\) the preference field \(\beta=\mathcal T(\widehat V)\) admits the canonical splitting \(\beta=\mathcal U+\mathcal R\) with \(\mathcal U=dU\) and \(d\beta=d\mathcal R\) (Proposition (ref)), so co--utility bounded rationality is detected by the curvature \(d\beta\) and can be probed by loop holonomy. By contrast, \(\Lambda(t)=2\langle\tilde\rho(t),y(t)\rangle\) depends on the residual variable \(y\) and is therefore not determined by the epistemic state \(q(t)\) alone. Co--utility bounded rationality is thus a property of the epistemic preference field on \(\mathcal M\), whereas the null--cone index \(\Lambda\) is a trajectory diagnostic of off--shell episodes in the lifted model: it quantifies accumulated deviation and enforces the one--way cone--crossing constraint that is characteristic of the hyperbolic least--action lift.
The mechanisms analysed so far are intrinsic either to the epistemic simplex dynamics (e.g.\ non--integrability of the induced preference form) or to the lifted least--action dynamics (e.g.\ the on--shell channel split and the off--shell null--cone index). We now isolate a distinct source of bounded rationality that is neither a property of the latent generator alone nor a property of the simplex geometry alone, but of the geometric reduction that connects them: observable choice frequencies are obtained from amplitudes by a nonlinear readout map defined with respect to a fixed orthonormal readout basis of the latent amplitude space. Consequently, two latent episodes that are close (or even identical up to a latent gauge) can induce different observable drifts if the readout basis is changed, and conversely a fixed latent episode can display context dependence solely because the projection to probabilities is nonlinear. Since this effect is already present in the real (hyperbolic) LAR formulation, with linear amplitude dynamics together with a quadratic normalisation/squaring readout in an orthonormal basis, it can be analysed without invoking elliptic unitarity or any additional measurement postulates.
We therefore proceed in two steps. First we record the probability readout in a form that is compatible with both the elliptic and hyperbolic coherent descriptions by introducing a uniform phase--group notation. We then study how varying the readout map across contexts induces distinct observable drifts and produces the familiar cross--term mechanism usually called (interference) when multiple amplitude contributions are coherently combined prior to quadratic normalisation.
The epistemic readout is defined directly from the real amplitude variable \(\tilde\rho\). For \(\tilde\rho\neq 0\), set
Thus \(q(\tilde\rho,y)\) depends on \((\tilde\rho,y)\in N\) only through \(\tilde\rho\in\mathbb{R}^n\); the residual variable \(y\) affects \(q\) only through the induced trajectory \(t\mapsto(\tilde\rho(t),y(t))\). The map (ref) is homogeneous of degree \(0\), hence invariant under \(\tilde\rho\mapsto\lambda\tilde\rho\) for \(\lambda\neq 0\), and it is sign--blind in each component: \(\tilde\rho_i\mapsto-\tilde\rho_i\) leaves \(q\) unchanged. Boundary events \(q_i=0\) are equivalent to \(\tilde\rho_i=0\).
For notational comparison between the elliptic and hyperbolic sectors, let \[ U(\mathbb{C}):=\{u\in\mathbb{C}^\times:\overline u\,u=1\}\cong S^1, \qquad U(\mathbb{D}):=\{u\in\mathbb{D}^\times:u^\#u=1\}. \] Here \(U(\mathbb{D})\) is the unit hyperbola, with two connected components; any choice of component is irrelevant for the readout, since only the relation \(u^\#u=1\) enters below. Given \(\tilde\rho\in\mathbb{R}^n\setminus\{0\}\) and \(u=(u_1,\dots,u_n)\in U(\mathbb K)^n\), define the associated \(\mathbb K\)--valued vector
Then, componentwise, \[ \psi_i^*\psi_i=(\tilde\rho_i)^2 \quad (\mathbb K=\mathbb{C}), \qquad \psi_i^\#\psi_i=(\tilde\rho_i)^2 \quad (\mathbb K=\mathbb{D}), \] and therefore \[ \sum_{k=1}^n\psi_k^*\psi_k=\mathcal Z(\tilde\rho) \quad (\mathbb K=\mathbb{C}), \qquad \sum_{k=1}^n\psi_k^\#\psi_k=\mathcal Z(\tilde\rho) \quad (\mathbb K=\mathbb{D}). \]
Hence the quadratic readout (ref) admits the the following Born-form identity
Equation (ref) is an identity, not an additional postulate. In the elliptic sector, Caticha2021 likewise derives the Born-form expression rather than postulating it. Here the probabilities are defined by (ref), and (ref) merely expresses this quadratic readout as normalised component intensities for the vectors \(\psi\) of the form (ref). In particular, (ref) is not asserted for arbitrary \(\psi\in\mathbb K^n\).
On the chosen holomorphic eigenbundle one may take \(\psi=\Psi\in\mathbb{C}^n\), and on the chosen para--holomorphic eigenbundle one may take \(\psi=\Psi_{\mathbb{D}}\in\mathbb{D}^n\). In both cases the probability rule is still (ref); only its notation changes. The elliptic/hyperbolic distinction therefore lies not in the quadratic readout (ref)--(ref), but in the phase geometry and in the corresponding \(\mathbb K\)--Hermitian packaging of the state evolution (cf.\ (ref), (ref)). For \(n=2\), this is the Lorentzian \(\mathrm{SO}(2,1)\) carrier of Remark (ref).
Having fixed the quadratic readout and its phase--group notation above, we now allow the readout itself to vary. A context is specified by a choice of readout basis (equivalently, by a choice of readout map), and the same latent amplitude state can therefore produce different observable probability assignments in different contexts. Because the readout is nonlinear, its differential depends on the chosen context, so even when the latent dynamics is context--independent the induced simplex drift need not be. This readout--geometric dependence is the basic contextuality mechanism in our framework.
Recall that latent states are represented by normalised amplitudes \(\rho\in\mathbb S^{n-1}\subset\mathbb{R}^n\), and that observable probabilities arise from the quadratic readout. The canonical readout corresponds to the standard basis, i.e.\ \(q_k=\rho_k^2\). To model contextual readout, we allow the readout basis to vary: a context is an orthogonal matrix \(B\in O(n)\) with columns \(b_1,\dots,b_n\). The associated readout map is
so that \(\sum_k \pi_B(\rho)_k=\|\rho\|^2=1\). The canonical context is \(B=I\), giving \(\pi_I(\rho)_k=\rho_k^2\). Equivalently, writing \(\rho^{(B)}:=B^\top\rho\), one has \(\pi_B(\rho)_k=(\rho^{(B)}_k)^2\), i.e.\ the same quadratic readout applied in rotated coordinates; in particular, for real contexts \(B\in O(n)\) the split--complex form (ref) remains available by setting \(c^{(B)}_k:=\rho^{(B)}_k u_k\) with \(u_k\in U(\mathbb{D})\), so that \[ (c^{(B)}_k)^*c^{(B)}_k=(\rho^{(B)}_k)^2=\pi_B(\rho)_k. \]
Now let \(X\) denote the latent LAR vector field (ref) on \(\mathbb S^{n-1}\). In context \(B\), write \(q:=\pi_B(\rho)\in\Delta^{n-1}\). The induced instantaneous observable drift at \(\rho\) is the pushforward \(d\pi_{B,\rho}(X(\rho))\), which satisfies \(\sum_k \dot q_k=0\) and hence defines an admissible simplex drift whenever \(q\in\mathcal M\). Since \(\pi_B\) is nonlinear, its differential depends on \(B\), so the observable simplex trajectory depends on which readout basis \(B\) is used to read out the same latent amplitude episode.
This does not exhaust the effect of changing basis. Even for a fixed context \(B\), the same quadratic readout can produce cross terms whenever the \(B\)--coordinates of the latent state contain several modal contributions before readout. In that case squaring generates off--diagonal terms. In particular, this occurs whenever the chosen readout basis is not aligned with the spectral basis of the on--shell generator. In the split--complex extension \(\mathbb{D}\), the same algebraic mechanism yields the hyperbolic interference laws studied in the hyperbolic quantum literature (e.g.\ Khrennikov2000hyperbolicquantummechanics).
To make this explicit, on the zero--residual leaf one may use the auxiliary complex packaging (ref). The corresponding effective complex Hamiltonian \(\widehat H_{\mathbb C}=i\,\widehat V\) is generically non--Hermitian whenever \(\widehat S\neq 0\). Assuming \(\widehat V\) is diagonalizable over \(\mathbb C\), write \[ \widehat V=P\Lambda P^{-1}. \] Then \[ z_i(t)=\sum_a P_{ia}e^{\lambda_a t}c_a, \qquad w_i(t):=z_i(t)^2 = \sum_{a,b}P_{ia}P_{ib}\,c_a c_b\,e^{(\lambda_a+\lambda_b)t}. \] The terms with \(a\neq b\) are precisely the interference terms. Since \(\widehat V\) and the on--shell initial data are real, the complex modal contributions recombine to a real trajectory \(z(t)=\tilde\rho(t)\in\mathbb{R}^n\). The observable probabilities are therefore still computed by the same readout, written equivalently in the form (ref) for representatives of the type (ref).
Thus, a misaligned context can produce interference, but the two notions should not be identified: context refers to the chosen readout basis, whereas interference refers to the off--diagonal cross terms generated within that fixed basis by the quadratic readout. In the real hyperbolic LAR framework, both are intrinsic consequences of linear amplitude dynamics combined with the quadratic readout map to the simplex, and neither requires complex Hilbert space, elliptic unitarity, or any additional measurement postulate.
So far in (ref) we have worked in the hyperbolic information--geometric regime: a real para--K\"ahler LAR lift on the doubled phase space \(N=T^*\widetilde{\mathcal M}\), and bounded rationality arising when the latent flow is reduced to the epistemic simplex. In that setting the latent split \(\widehat V=\widehat S+\widehat F\) separates evaluative and co--utility channels, and a specifically hyperbolic, lifted diagnostic is available off--shell: the neutral form \(Q\) (and its index \(\Lambda\)) controls residual accumulation through the doubled variable \(y\).
By contrast, the elliptic (complex, Hilbert--space) quantum model arises in our framework by imposing the coherent admissibility restriction of Coherent Least--Action Rationality (CLAR). One passes to the complexification \(N_\mathbb{C}\) and restricts to a chosen complex Lagrangian splitting \(TN_\mathbb{C}=L^{(+)}\oplus L^{(-)}\) (cf.\ (ref) and (ref)). On the resulting holomorphic sector the reduced dynamics closes as the standard Schr\"odinger evolution with Hermitian packaged generator in ((ref)) \[ \widehat H\;:=\;\widehat S+i\widehat F, \] hence is unitary with respect to the induced Hilbert inner product. In particular, the Hilbert norm \(\|\psi(t)\|\) is conserved along CLAR trajectories. From the viewpoint of the present section, this means that the lifted, off--shell hyperbolic mechanism of (ref) does not play the same role in the elliptic regime: the neutral index \(\Lambda\) is a trajectory diagnostic tied to the real para--K\"ahler lift on the doubled variables \((\tilde\rho,y)\), whereas CLAR replaces the full lifted flow by a polarization--projected, coherence--preserving evolution on \(N_\mathbb{C}\). Nevertheless, bounded rationality in the elliptic regime still persists through epistemic non--integrability (curvature of the induced preference form) and through readout--induced contextuality, since both arise from the reduction from latent amplitudes to observable probabilities rather than from the off--shell hyperbolic lift.
To make this persistence precise, we now pass from the latent unitary CLAR evolution to its induced epistemic dynamics in a fixed readout context. Since the off--shell lifted diagnostic \(\Lambda\) is absent in the elliptic regime, the relevant bounded--rationality mechanisms are those that survive the reduction to observable probabilities: the induced simplex drift, its Fisher--Rao dual preference form, and their dependence on the chosen readout context.
As in (ref), fix an admissible readout context \(B=\{b_1,\dots,b_n\}\subset\mathbb{R}^n\), i.e.\ the orthonormal basis used in the quadratic readout. In the elliptic model the latent state is a complex amplitude \(\psi(t)\in\mathbb{C}^n\setminus\{0\}\) evolving unitarily on a coherent sector, and the associated epistemic state is obtained by the modulus--squared readout in the chosen basis, \[ q_B(t)=\pi_B(\psi(t))\in\mathcal M, \qquad \pi_B(\psi)_k:=\frac{|\langle b_k,\psi\rangle|^2}{\|\psi\|^2}, \] where we restrict attention to episodes for which \(q_B(t)\in\mathcal M=\Delta^{n-1}_{>0}\) (so Fisher--Rao geometry applies without boundary issues). For unitary CLAR trajectories \(\|\psi(t)\|\) is constant, so the normalisation is inessential, but we keep it explicit to emphasise ray--invariance.
The induced simplex drift in context \(B\) is obtained by pushing forward the latent velocity, \[ \dot q_B(t)=d\pi_{B,\psi(t)}\bigl(\dot\psi(t)\bigr), \] which is automatically tangent to the simplex hyperplane \(\sum_k q_k=1\). Since \(\pi_B\) is nonlinear and not injective, \(\dot q_B(t)\) generally depends on latent phase information in \(\psi(t)\). Accordingly, the reduced drift is naturally viewed as an object defined along a trajectory rather than as an autonomous vector field on \(\mathcal M\).
To relate back to the epistemic analysis of (ref), one may associate to the reduced drift a (preference) covector along the trajectory by Fisher--Rao duality, \[ \beta_B(t)\;:=\;g_F\!\bigl(\dot q_B(t),\,\cdot\,\bigr)\in T^*_{q_B(t)}\mathcal M. \] Whenever the reduced drift admits a smooth representative one--form field \(\beta_B\in\Omega^1(\mathcal M)\) in a fixed context (e.g.\ after choosing a coherent gauge/section on the holomorphic sector), the same integrability question arises as in the hyperbolic case: the curvature \(d\beta_B\) detects epistemic path dependence in that context, while variation of \(B\) probes the readout--geometric contextuality mechanism of (ref).
Thus, in the elliptic regime, CLAR provides coherence--preserving (unitary) latent motion generated by the Hermitian operator \(\widehat H=\widehat S+i\widehat F\), while epistemic bounded rationality is determined by how this motion is reduced to observable probabilities by the nonlinear readout family \(\pi_B\). In fact, this is precisely the setting of the complex Hilbert--space quantum--cognition literature: bounded rationality is compatible with coherence--preserving (unitary) latent evolution, and arises through geometric reduction, namely the nonlinear, context--dependent readout family \(\pi_B\) that maps latent states to observable probabilities on the simplex. What our framework adds in the hyperbolic (para--K\"ahler) regime is an additional, genuinely lifted mechanism that is invisible under coherent elliptic restriction: the full least--action dynamics on the doubled variables \((\tilde\rho,y)\) admits the neutral index \(\Lambda(t)=2\langle\tilde\rho(t),y(t)\rangle\), which provides an intrinsic accumulation diagnostic and a one--way cone--crossing constraint for residual deviation. This \(\Lambda\)--mechanism has no analogue in the unitary CLAR sector, where one restricts attention to coherent holomorphic evolution and does not track episodes outside the non-zero residual leaf.
The role of information theory in behavioural choice modelling has been mixed. Luce2003 argued that, despite its early appeal, information theory offered limited explanatory value for psychological theories of choice. His critique was structural: standard information--theoretic formulations treat alternatives as unstructured elements of a finite set, whereas behavioural stimuli typically come with internal relations (similarity, salience, or metric structure) that shape observed choice patterns. In this view, entropy--based approaches risk capturing only a generic “softening” of choice, without explaining how context and internal structure enter.
Interestingly, the same year, Shannon entropy was reintroduced into models of bounded rationality in a different role: Sims2003 treated entropy not as a description of stimuli but as a resource cost governing internal representation. This reinterpretation sidesteps Luce's objection about unstructured alternatives, yet it still operates directly on the simplex and does not by itself supply an intrinsic geometry linking the latent layer of representation to the observable layer of choice.
\paragraph{Main contribution.} We develop a unified information--geometric theory of choice in which observable probabilities on the simplex are obtained by a quadratic (Born--type) readout from a latent amplitude dynamics governed by a single preference operator \(\widehat V\). The operator admits the canonical decomposition \(\widehat V=\widehat S+\widehat F\), and this split has direct behavioural content. The symmetric sector \(\widehat S\) is the evaluative channel: it drives welfare/utility production through norm change on rays, induces gradient--type drift, and generates a canonical monotone entropic welfare clock given by the log--partition normaliser. This is the same log--sum--exp (inclusive--value) potential that underlies entropy--regularised choice, free--energy formulations, and rational--inattention objective functions, and it is canonically paired with Shannon entropy by convex duality. The skew sector \(\widehat F\) is the circulatory channel: it generates redistributive drift without welfare production and induces co--utility/regret via non--integrable (holonomic) components of the epistemic preference field.
Crucially, \(\widehat F\) identifies a genuine split--complex (para--complex) sector of the lifted dynamics. On the cotangent lift \(N=T^*\widetilde{\mathcal M}\), equipped with its canonical para--K\"ahler structure, we derive the Hamiltonian least--action equations and show that on the distinguished zero--residual leaf \(y=0\) the dynamics closes as a linear para--Schr\"odinger evolution. This provides a principled motivation for working with real amplitudes in quantum--like cognition models: the canonical hyperbolic lift already contains a coherent Schr\"odinger--type sector with a transparent welfare--vs--circulation interpretation, without assuming complex Hilbert space.
The framework extends beyond the coherent leaf. Away from \(y=0\), bounded rationality is organised by intrinsic phase--space diagnostics of the hyperbolic lift, including the neutral pairing and its null--cone index, which record accumulated least--action deviation and enforce one--way cone--crossing constraints for non-zero residual episodes. Taken together, the theory delivers an operator--geometric foundation in which welfare production, regret/co--utility, coherent split--complex dynamics, arise from a single least--action principle, even outside the non-zero residual leaf.
These constructions also clarify the relationship between entropy--regularised choice and quantum--like cognition models. The approach is explicitly probabilities--first, in the spirit of information--geometric reconstructions, but it does not begin by postulating a complex Hilbert space. Instead, the Fisher--Rao/Hessian data determine a canonical para--K\"ahler structure on the lifted manifold, and observable probabilities are obtained directly on the real leaf via the same quadratic normalisation.
At the same time, the familiar elliptic (Hilbert--space) regime can be accommodated within the same information--geometric origin. What changes is not the underlying symplectic structure \((N,\Omega)\) but the class of admissible variations: an elliptic model is obtained only after supplying additional admissibility input that selects an \(\Omega\)--compatible complex Lagrangian sector \(L^{(+)}\subset TN_\mathbb{C}\) and restricts trajectories accordingly. Within such a sector the restricted dynamics closes as a unitary Schr\"odinger--type evolution, and Born--type probabilities again arise from the same quadratic readout. In this precise sense, elliptic coherence is not a primitive consequence of least--action rationality; it is a coherent restriction of the canonical hyperbolic lift, motivating coherent least--action rationality (CLAR) as a least--action postulate formulated within the selected complex sector.
Several directions for further research follow naturally. On the applied side, the operator--based (pairwise) structure suggests new estimators and model classes that interpolate between additive utility, pairwise contextual couplings, and regret--driven circulation, with clear links to existing entropy--regularised and context--dependent choice models. On the geometric side, it would be valuable to characterise when (and in what sense) admissible complex sectors can be selected in a data--driven way, and to understand how the hyperbolic null--cone and intrinsic clocks interact with projection, aggregation and measurement schemes on the simplex. A related direction is to study hybrid and open--system perturbations that interpolate between coherence--restricted and unrestricted lifted dynamics (e.g.\ channel/Lindblad--type effects in quantum--like decision dynamics; AsanoEtAl2012).
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