\documentclass[11pt]{article} \usepackage[margin=1in]{geometry} \usepackage{amsmath,amssymb,mathtools,booktabs,hyperref,enumitem,microtype} \hypersetup{colorlinks=true,linkcolor=blue,citecolor=blue,urlcolor=blue} \title{Intrinsic Spectral Cosmology} \author{Rajesh Dachiraju\\ Hyderabad, India\\ rajesh.dachiraju@gmail.com} \date{20 October 2026} \newcommand{\RI}{R_{\mathrm I}} \newcommand{\HI}{H_{\mathrm I}} \newcommand{\dd}{\mathrm d} \newcommand{\kk}{\mathbf k} \newcommand{\JJ}{\mathbf J_k} \newcommand{\nablaH}{\nabla_{\!H}} \newcommand{\dtH}{\partial_{H,t}} \begin{document} \maketitle \begin{abstract} We propose an intrinsic spectral cosmology based on intrinsic geometric wave mechanics, spectral dynamics, localized resonances, HJ source flow, and Hamilton--Jacobi gravity for point masses. The intrinsic field is represented through its time-local spectrum \(P(\mathbf k,t)\), whose second moment defines \(M_2\), an intrinsic scale \(R_I=M_2^{-1/2}\), and \(H_I=\dot R_I/R_I=-\dot M_2/(2M_2)\). A canonical spectral Hamiltonian gives a conditional continuity equation and current \(\mathbf J_k=\partial h/\partial(\nabla_k\theta)\); the actual Hamiltonian needed to determine that current is not fixed by the intrinsic space-time weak equation. We exhibit a minimal scale-drift Hamiltonian as a clearly labeled closure ansatz, not as a derived result. Gravity acts only between finite point masses through their Hamilton--Jacobi dynamics. Redshift, distance, structure formation, and the relation of localized resonances to masses remain hypotheses or open derivations. The framework makes no claim of empirical validation. \end{abstract} \noindent\textbf{Scope and status.} The paper separates results established within the cited frameworks, direct mathematical consequences, and conjectural closure assumptions introduced here. The model excludes general relativity, quantum mechanics, and special relativity as foundations; it is a research proposal, not an empirically established cosmology. \section{Introduction} The intrinsic geometric wave-mechanics preprint develops a Hilbert-transform geometry, intrinsic weak derivatives, and a normalized space-time wave equation \cite{Wave}. The spectral-dynamics preprint proposes a hierarchy from intrinsic state to spectral evolution, propagation, resonances, and observables \cite{Mass}. The Haar-wavelet preprint gives a localization result for jump discontinuities \cite{Haar}. The revised HJ-charge preprint proposes an action-generated source flow and candidate charge-like fields \cite{ChargeHJ}. The HJ-gravity preprint formulates pairwise gravity for finite sets of point masses \cite{HJG}. The cosmological proposal developed here has a deliberately narrow foundation: \begin{enumerate}[leftmargin=2em] \item intrinsic wave mechanics supplies the deterministic state and spectral representation; \item spectral moments define a characteristic intrinsic scale; \item localized, stable resonances are candidate material objects; \item only after a resonance is assigned point-mass parameters does the finite-body Hamilton--Jacobi gravity sector apply; \item an HJ source-flow sector may be associated with localized states, while its field interpretation remains conjectural. \end{enumerate} No continuous gravitating mass density or gravitational field equation is introduced. The intrinsic scale evolution and the point-mass mechanics remain distinct until an explicit coupling is derived. In particular, this proposal does not yet give a closed predictive history for an expanding cosmos. \section{Literature survey} \label{sec:literature} \subsection{Hamilton--Jacobi mechanics and point-mass gravity} Hamilton's characteristic and principal functions provide a classical formulation of finite-dimensional dynamics in terms of an action function and its first-order Hamilton--Jacobi equation \cite{Hamilton1834}. The finite-body sector studied here belongs to that tradition: a chosen N-body Hamiltonian generates particle trajectories and a Hamilton--Jacobi equation for the principal function. The cosmological proposal does not claim a new Hamilton--Jacobi formalism. Its modeling choice is to restrict gravity to pair interactions among point masses and to leave the continuous spectral background outside that gravitational sector. This restriction distinguishes the present construction from continuum gravitational field models. \subsection{Spectral transfer and the problem of closure} In classical wave systems, Fourier modes can exchange energy through nonlinear interactions. Wave-turbulence theory derives statistical kinetic descriptions under explicit assumptions about the interaction regime and statistical state; Newell and Rumpf review the construction, its domain of validity, and open closure problems \cite{NewellRumpf}. This literature is relevant methodologically rather than as a cosmological model: spectral normalization and a formal continuity equation do not determine a transfer current. The ISEC proposal similarly needs a specified spectral Hamiltonian with phase-gradient dependence before its conditional current becomes predictive. The assumptions used in wave-turbulence closures are not imported here. \subsection{Wavelets as localization tools} Haar introduced an orthogonal function system that became the basis for the Haar wavelet \cite{Haar1910}. Subsequent wavelet analysis developed methods for locating singularities and estimating local regularity from scale-dependent transform behavior; Mallat and Hwang provide a foundational treatment \cite{MallatHwang}. These results motivate using Haar-scale signatures to locate candidate discontinuities in intrinsic fields. They establish a signal-analysis diagnostic, not the physical interpretation of a discontinuity as stable matter. That interpretation would additionally require a solution of the intrinsic dynamics, stability under perturbations, and a justified mass assignment. \subsection{Relation to the cited intrinsic-framework preprints} The cited preprints develop the intrinsic Hilbert geometry, spectral hierarchy and moment-defined scale, jump-localization construction, HJ source-flow proposal, and point-mass gravity sector \cite{Wave,Mass,Haar,ChargeHJ,HJG}. This paper's contribution is to organize those elements as a cosmological hypothesis and make their interfaces explicit. The phase-space formula for spectral current is conditional on a Hamiltonian that the intrinsic space-time weak equation does not yet specify as a time-local evolution law. The scale-drift term later in the paper is therefore an illustrative closure ansatz, not a result of the cited dynamics. The literature reviewed here supplies mathematical precedents; it does not validate the cosmological interpretation or its observational hypotheses. \section{Intrinsic geometry and wave mechanics} Let \(\mathbb T^m\) denote the periodic domain used in the wave-mechanics preprint, and let \(f^h\) denote the Hilbert transform. The cited framework defines the embedding and inner product schematically by \begin{equation} T(f)=f+i f^h,\qquad \langle f,g\rangle_{\mathcal H}=\langle T(f),T(g)\rangle_{L^2(\mathbb T^m)}. \label{eq:geometry} \end{equation} The induced norm is \(\|f\|_{\mathcal H}=\|T(f)\|_{L^2}\). Intrinsic derivatives \(\partial_j^H\) are defined weakly by duality using this inner product. The wave-mechanics preprint states that on their natural domain these derivatives agree with classical weak derivatives, are skew-adjoint, and have Fourier eigenmodes with eigenvalues \(2\pi i k_j\). Those operator claims belong to the preprint's mathematical construction \cite{Wave}. The wave-mechanics source uses the space-time domain \(\Omega=(0,1)^3\times(0,1)\), a normalized intrinsic field \(\psi\in\mathcal H(\Omega)\), and a wave-number parameter \(\gamma>0\). Its constrained variational construction yields a weak space-time equation of the form \begin{equation} \frac{1}{2\gamma}\langle\Delta_H\psi,\varphi\rangle_{\mathcal H} +\langle\bar V\psi,\varphi\rangle_{\mathcal H} +\sum_{j=1}^{3}\langle A_j\partial_j^H\psi,\varphi\rangle_{\mathcal H} +\lambda\langle\psi,\varphi\rangle_{\mathcal H}=0, \quad \|\psi\|_{\mathcal H}=1, \label{eq:wave} \end{equation} for admissible test functions \(\varphi\). The cited framework defines \(\alpha=1/(2\gamma)\), the Hilbert-transformed vector potential \(\vec A^h\), and \(\bar V=V+\partial^H_t\nabla_H^s\cdot\vec A^h\). It writes the constrained energy as \(E(\psi)=\alpha\|\nabla_H^{st}\psi\|_{\mathcal H}^2+\|\sqrt{\bar V}\psi\|_{\mathcal H}^2+\|\vec A\cdot\nabla_H^s\psi\|_{\mathcal H}^2\), and states that the weak equation follows by constrained variation. Equation \eqref{eq:wave} is a space-time weak equation; it is not yet a closed first-order initial-value law for a time-local spectrum. The wave-mechanics preprint reports a Fourier-matrix solution and existence/uniqueness argument. For cosmological prediction, the boundary/initial data, the phase convention for a complex normalized solution, and continuous dependence of the time-local observables must be made explicit. Here \(\psi\) is the intrinsic deterministic field defined by the framework. Its Hilbert-space norm and spectral weights are not given a quantum-mechanical measurement or Born-rule interpretation. The model uses neither quantum postulates nor special-relativistic kinematics; these are exclusions of its proposed foundations, not experimentally established conclusions. \section{Spectral state and intrinsic cosmological scale} Following the spectral-dynamics preprint's four-dimensional Fourier representation, write \begin{equation} \psi(\mathbf x,t)=\sum_{\kk\in\mathbb Z^3}\sum_{\ell\in\mathbb Z} \widehat\psi(\kk,\ell)e^{2\pi i(\kk\cdot\mathbf x+\ell t)},\qquad P(\kk,t)=\sum_{\ell\in\mathbb Z}|\widehat\psi(\kk,\ell)|^2, \qquad \sum_{\kk}P(\kk,t)=1. \label{eq:sourceP} \end{equation} The spectral-dynamics preprint defines \(P\) as an intrinsic spectral density, not a quantum probability distribution \cite{Mass}. Its broader overview also uses an ensemble power spectrum and a continuum integral. We retain both conventions as source statements but do not identify them without a specified normalization and continuum limit. There is a time-indexing issue to resolve: as printed, the full temporal Fourier coefficients \(\widehat\psi(\kk,\ell)\) in \eqref{eq:sourceP} do not themselves depend on the displayed time \(t\). A cosmological evolution calculation therefore needs a time-local spatial spectrum. One possible new closure is to define spatial Fourier amplitudes \(\widetilde\psi_{\kk}(t)\) on each time slice and set \begin{equation} P_{\kk}(t)=\frac{|\widetilde\psi_{\kk}(t)|^2}{\sum_{\mathbf q}|\widetilde\psi_{\mathbf q}(t)|^2}, \qquad \sum_{\kk}P_{\kk}(t)=1. \label{eq:proposedP} \end{equation} Equation \eqref{eq:proposedP} is our explicit time-slice definition, introduced to resolve the temporal-indexing ambiguity in the spectral preprint. Other ensemble or windowed-time definitions are possible and must be chosen before dynamics are computed. With the discrete convention, define \begin{equation} M_2(t)=\sum_{\kk}|\kk|^2P_{\kk}(t),\qquad \RI(t)=M_2(t)^{-1/2},\qquad \HI(t)=\frac{\dot\RI}{\RI}. \label{eq:scale} \end{equation} The spectral-dynamics preprint also writes the continuum version \(M_2=\int k^2P(\kk,t)\,\dd^3k\). In either convention, when \(M_2>0\) and differentiable, \begin{equation} \boxed{\HI=-\frac{1}{2}\frac{\dot M_2}{M_2}}. \label{eq:Hidentity} \end{equation} This is an identity following from the definition. It defines the fractional growth rate of \(\RI\); it is not by itself an observed expansion rate. With dimensionless torus modes, \(\RI\) is a mode-space scale and requires a calibration length before comparison with physical distances. The spectral-dynamics preprint interprets cosmological expansion as spectral evolution rather than expansion of a dynamical spacetime metric \cite{Mass}. \section{Hamiltonian derivation of the spectral current and the closure gap} The spectral-dynamics preprint sets out a formal canonical construction that sharpens the current discussion. Write a time-local spectral amplitude as \begin{equation} a(\kk,t)=\sqrt{P(\kk,t)}\,e^{i\theta(\kk,t)},\qquad \mathcal H_{\rm spec}=\mathcal H_{\rm spec}[P,\theta]. \label{eq:polar} \end{equation} The proposed canonical variables obey \(\{P(\kk),\theta(\kk')\}=\delta^{(3)}(\kk-\kk')\), with evolution convention \begin{equation} \partial_tP=\frac{\delta\mathcal H_{\rm spec}}{\delta\theta},\qquad \partial_t\theta=-\frac{\delta\mathcal H_{\rm spec}}{\delta P}. \label{eq:canonicalSpec} \end{equation} Suppose the Hamiltonian depends on \(\theta\) only through \(\mathbf q=\nabla_{\kk}\theta\), and has a local functional representation \(\mathcal H_{\rm spec}=\int h(P,\mathbf q,\kk,t)\,\dd^3k\). For variations vanishing at the spectral boundary, \begin{equation} \delta\mathcal H_{\rm spec} =\int \frac{\partial h}{\partial\mathbf q}\cdot\nabla_{\kk}\delta\theta\,\dd^3k =-\int\nabla_{\kk}\cdot\left(\frac{\partial h}{\partial\mathbf q}\right)\delta\theta\,\dd^3k, \qquad \frac{\delta\mathcal H_{\rm spec}}{\delta\theta} =-\nabla_{\kk}\cdot\frac{\partial h}{\partial\mathbf q}. \label{eq:variationSpec} \end{equation} Combining \eqref{eq:canonicalSpec} and \eqref{eq:variationSpec} gives the continuity law and current \begin{equation} \partial_tP+\nabla_{\kk}\cdot\mathbf J_k=0, \qquad \boxed{\mathbf J_k=\frac{\partial h}{\partial(\nabla_{\kk}\theta)}}. \label{eq:derivedJ} \end{equation} For a nonlocal Hamiltonian, replace the partial derivative by the functional derivative with respect to \(\nabla_{\kk}\theta\); a local vector current exists only if that variational derivative can be represented as such. A divergence-free addition to \(\mathbf J_k\) leaves the continuity equation unchanged, so a unique current convention also requires boundary or constitutive information. There is a sign inconsistency in the spectral preprint at this point \cite{Mass}. Its displayed canonical equations \(\partial_tP=\delta\mathcal H/\delta\theta\), \(\partial_t\theta=-\delta\mathcal H/\delta P\), together with \(\partial_tP+\nabla_{\kk}\cdot\mathbf J_k=0\), imply the plus sign in \eqref{eq:derivedJ}; the manuscript's printed formula for \(\mathbf J_k\) has a minus sign. Reversing the canonical bracket/evolution convention reverses both Hamilton equations and can yield that minus sign. This paper retains the explicitly displayed evolution convention and uses the mathematically corresponding plus sign. The sign choice should be reconciled with the source's symplectic convention before treating the current as fixed. This is a structural derivation, not yet the requested physical closure. The wave-mechanics weak equation is a constrained weak equation on space-time involving \(\Delta_H\), potentials, and a norm multiplier. As written, it is not a first-order time-local Hamilton equation and does not specify the canonical spectral Hamiltonian \(\mathcal H_{\rm spec}[P,\theta]\), its phase-gradient dependence, or an initial-value evolution for \(a(\kk,t)\). The quadratic example in the spectral manuscript, \(\mathcal H_{\rm quad}=\int\omega(\kk)P(\kk)\,\dd^3k\), has no phase-gradient dependence and therefore yields \(\mathbf J_k=0\). Nonzero spectral transfer requires additional phase-dependent terms, nonlinear couplings, or a derivation from the underlying field action. Thus the framework determines the form of the current conditionally, but the cited preprints do not determine its value or predict expansion. \subsection{A minimal Hamiltonian ansatz realizing scale drift} To exhibit exactly what extra assumption closes a simple background model, consider the following explicit ansatz: \begin{equation} h(P,\theta,\kk,t)=\omega(\kk)P-\nu(t)P\,\kk\cdot\nabla_{\kk}\theta, \qquad \nu(t)\geq0. \label{eq:ansatzH} \end{equation} It is not derived from the intrinsic weak equation in the wave-mechanics preprint \cite{Wave}. Applying \eqref{eq:derivedJ} and \eqref{eq:canonicalSpec} gives \begin{align} \mathbf J_k&=-\nu(t)\kk P,\label{eq:ansatzJ}\\ \partial_tP-\nu(t)\nabla_{\kk}\cdot(\kk P)&=0, \label{eq:ansatzP}\\ \partial_t\theta&=-\omega(\kk)+\nu(t)\kk\cdot\nabla_{\kk}\theta. \label{eq:ansatzTheta} \end{align} With sufficient decay at large \(|\kk|\), the continuity equation preserves \(\int P\,\dd^3k=1\) and positivity under its characteristic flow. Define \(N(t)=\int_{t_*}^{t}\nu(s)\,\dd s\). Its solution and second moment are \begin{equation} P(\kk,t)=e^{3N(t)}P\!\left(e^{N(t)}\kk,t_*\right),\qquad M_2(t)=e^{-2N(t)}M_2(t_*). \label{eq:ansatzSolution} \end{equation} Consequently, \begin{equation} \RI(t)=\RI(t_*)e^{N(t)},\qquad \HI(t)=\nu(t),\qquad \frac{\ddot\RI}{\RI}=\dot\nu+\nu^2. \label{eq:ansatzCosmo} \end{equation} These equations follow exactly from the ansatz. A constant \(\nu\) gives exponential growth of the defined intrinsic scale; this does not follow from the intrinsic wave equation. The ansatz Hamiltonian is explicitly time-dependent when \(\nu\) varies, and its admissible function space, lower-boundedness, physical units, and compatibility with the original wave action have not been established. A first-principles closure must derive \(\mathcal H_{\rm spec}\), including its phase-gradient terms, from the intrinsic dynamics and thereby determine \(\nu\) or replace this ansatz. Until then, \(\nu(t)\) is a free closure input rather than a prediction. For the discrete torus spectrum, the functional derivatives above must be replaced by a defined lattice Hamiltonian and discrete summation-by-parts identity. The continuum derivation cannot be transferred to the discrete modes without that step. In either representation, normalization preservation, spectral boundary conditions, and continuous dependence on initial data must be proved for the selected Hamiltonian. \section{Localized resonances and Haar-wavelet diagnostics} The spectral-dynamics preprint proposes that matter consists of localized, dynamically stable intrinsic spectral resonances. It discusses Fourier jump localization and gives a Haar-wavelet theorem. For a locally bounded-variation function with jump set \(\{x_m\}\) and jump amounts \(c_m\), the Haar-wavelet preprint defines a scale-averaged transform \(R_y(b_0)\) and states, with its normalization constant \(K\), \begin{equation} \lim_{y\to\infty}R_y(b_0)= \begin{cases} Kc_m,&b_0=x_m,\\ 0,&b_0\notin\{x_m\}. \end{cases} \label{eq:haar} \end{equation} This can serve as a proposed diagnostic for jump-like localization in candidate intrinsic structures \cite{Haar,Mass}. The theorem concerns a mathematical signal class; by itself it does not establish physical resonances, their stability, or a particle spectrum. A candidate material object is thus a solution of the intrinsic wave system that is localized, regular in its allowed function space, and stable under the specified intrinsic evolution. Its position may be identified by a localization center. A mass assignment is not derived in the wave-mechanics or HJ-gravity preprints: they do not derive \(m_a\) from a resonance energy. For the gravity sector below, \(m_a\) must remain an independent point-mass parameter until a derivation is provided. \section{Gravity for point masses: Hamilton--Jacobi sector} \label{sec:gravity} Gravity in the proposed model is restricted to finite sets of point masses, following the HJ-gravity preprint \cite{HJG}. For \(N\) masses \(m_i\) at \(\mathbf x_i\), define \(r_{ij}=|\mathbf x_i-\mathbf x_j|\). The pairwise potential and Newton equations in the HJ-gravity preprint are \begin{align} V_G&=-G\sum_{i