Characterizing the Structure of Unified Cosmic Parameters through Constraint Manifolds: Joint Solution Space, Numerical Maps, and Structural Diagnostics for Six Unsolved Problems

Characterizing the Structure of Unified Cosmic Parameters through Constraint Manifolds: Joint Solution Space, Numerical Maps, and Structural Diagnostics for Six Unsolved Problems Anonymous Author November 26, 2025 Abstract Within the quantum cellular automaton/matrix universe framework, this paper regards the universe as an object completely encoded by a finite-dimensional parameter vector Θ ∈ RN, and uniformly rewrites six major unsolved problems—black hole entropy, cosmological constant, neutrino mass and flavor mixing, quantum chaos and eigenstate thermalization (ETH), strong CP problem, and gravitational wave dispersion—as six constraint equations on Θ: Ci(Θ) = 0, i = 1,...,6, whose joint solution set S(Θ){Θ∈RN|Ci(Θ) = 0, i = 1,...,6} is interpreted as the parameter manifold of “feasible universes”. The core goal of this paper is not to immediately give a unique physical solution Θ∗, but rather to construct, under a rigorous mathematical and numerical framework, a computable structural map for S(Θ): including dimension, connected components, symmetries, degenerate directions, and identifiability. To this end, we first rewrite all observational constraints under the axiomatic framework of the unified time scale κ(ω) = φ′(ω) π=ρrel(ω) = 1 2πtr Q(ω) as linear functional errors on the spectral measure Ci(Θ) = ZWi(ω)κΘ(ω)−κobs(ω)dω, and strictly adopt error control via finite-order Euler–Maclaurin and Poisson formulas to ensure that discretization does not introduce singularity proliferation. Subsequently, we implement low-discrepancy sampling, local refinement, and level-set tracking on the parameter space to obtain numerical solution clouds under joint constraints; on this basis, we utilize singular value decomposition of the Jacobian matrix J(Θ) = (∂Ci/∂θj) to analyze identifiable directions and degenerate directions, and employ clustering methods to identify multiple solution branches and approximate symmetric transformations. This paper gives the following main results: (1) Under appropriate regularity assumptions, we prove the manifold structure and dimension theorem for S(Θ) at regular points; (2) Within the spectral functional framework of unified time scale, we establish uniform error upper bounds for finite-order Euler–Maclaurin/Poisson discretization of constraint functions Ci(Θ), explicitly characterizing “poles = master scales” singularity control; (3) Starting from Jacobian spectral analysis, we provide quantitative criteria for parameter identifiability and define geometric decomposition of “soft modes” and “hard modes”; (4) We introduce 1 surrogate models (Gaussian process/kernel ridge regression) and active sampling strategies to adaptively approximate the high-dimensional structure of S(Θ), providing convergence sketches within the applicable range. These results provide both a mathematically self-consistent and numerically operational “constraint manifold perspective” for subsequent more concrete physical implementations— including substituting specific forms of black hole entropy, cosmological constant, neutrino mixing, ETH, strong CP, and gravitational wave dispersion into Ci(Θ). The paper itself does not attempt to complete fitting of all physical constants, but rather constructs a unified structural platform that can systematically incorporate existing and new observational data in the future. Keywords: Unified time scale; Quantum cellular automaton; Parameter universe; Constraint manifold; Jacobian spectrum; Euler–Maclaurin formula; Poisson summation; Identifiability 1 Introduction 1.1 The “Constraint Manifold” Perspective on Six Major Unsolved Problems In traditional cosmology and high-energy physics, several core problems are usually discussed in mutually independent ways: how is the A/4 law of black hole entropy realized microscopically; why is the cosmological constant so much smaller than natural scales; why is the structure of neutrino mass and flavor mixing matrix so peculiar; what are the applicability boundaries of the eigenstate thermalization hypothesis (ETH) in many-body localized systems; why does the strong CP problem require new mechanisms such as axions; and whether gravitational waves exhibit small Lorentz violation or dispersion effects at extremely high frequency/energy regimes. Typically, these problems rely on different theoretical branches and experimental systems. This paper adopts the opposite philosophy: all these problems must ultimately constrain the same universe. If we adopt the ontology of discrete quantum cellular automata (QCA) or matrix universe, abstracting the entire universe as an object encoded by a finite information parameter vector Θ ∈RN, then the above six major problems can naturally be rewritten as six constraints on Θ: Ci(Θ) = 0, i = 1,...,6. In this perspective, the problem is no longer “how to separately solve six things”, but rather: what geometric and numerical structure does the parameter set S(Θ) jointly satisfying all constraints possess. This “constraint manifold perspective” unifies the six major problems into the following question: Given a finite-dimensional parameter space RNand six physical constraint functions Ci, study the joint solution set S(Θ) = {Θ|Ci(Θ) = 0, i = 1,...,6} regarding its dimension, connectivity, symmetries, degenerate directions, and identifiability structure. 1.2 Unified Time Scale and Spectral Functional Language To express black hole horizons, cosmological constant, neutrino oscillations, quantum chaos windows, strong CP scattering phase, and gravitational wave dispersion in the same language, 2 this work adopts the unified time scale κ(ω) = φ′(ω) π=ρrel(ω) = 1 2πtr Q(ω), where φ(ω) is the scattering phase shift, ρrel(ω) is the relative density of states with respect to some reference background, and Q(ω) is the Wigner–Smith delay operator. This equality means: the rate of time passage is the density of states, and all observation windows can be viewed as linear functionals on the same spectral measure. Within this framework, we express each physical constraint Cias Ci(Θ) = ZΩi Wi(ω)κΘ(ω)−κobs(ω)dω, where Wi(ω) is the weight function for the corresponding window/experiment, and Ωiis the frequency or energy range. Thus, black hole entropy constraints, effective value of cosmological constant, neutrino spectrum, ETH diagnostic spectrum, strong CP scattering phase, and gravitational wave dispersion all become different “projections” onto the unified clock κ(ω). 1.3 Overview of This Work and Structure This paper does not attempt to construct QCA universe from scratch or give specific expressions for all physical constants, but focuses on structure and method: given abstract Θ and Ci(Θ), how to construct numerical maps and mathematical structure of S(Θ)? Main contents are as follows:  Section 2: Give abstract definitions of unified time scale, parameter universe, and constraint functions;  Section 3: Prove manifold structure and dimension theorem for S(Θ) under smoothness and rank conditions;  Section 4: Construct numerical schemes of low-discrepancy sampling, local refinement, and level-set tracking, and introduce surrogate models;  Section 5: Utilize singular value decomposition of Jacobian matrix to analyze identifiability, degenerate directions, and approximate symmetries;  Section 6: In the spectral functional language of unified time scale, introduce finite-order Euler–Maclaurin and Poisson formulas, give discretization error upper bounds, ensuring “no singularity increase, poles = master scales”;  Section 7: Explain how to embed specific physical problems (black hole entropy, cosmological constant, etc.) into this framework as use-case templates;  Appendices: Provide proof details of manifold structure theorem, Euler–Maclaurin error bounds, convergence sketches of surrogate models, and other technical details. 2 Unified Time Scale, Parameter Universe, and Constraint Functions 2.1 Abstract Parameterization of Quantum Cellular Automaton/Matrix Universe We adopt a minimalist abstract setting: the universe is completely encoded by a parameter vector Θ = (Θstruct,Θdyn,Θinit)∈RN 3 where:  Θstruct: describes lattice structure, local Hilbert space dimensions, adjacency relations, and other “geometric/topological” parameters;  Θdyn: describes parameters of local update rules, Hamiltonians or quantum channels (coupling constants, mass terms, mixing angles, etc.);  Θinit: describes parameters of initial state or initial density matrix (initial entropy density, spectral distribution, etc.). In concrete implementation, these components can ultimately be mapped to the QCA singlestep evolution operator UΘ, Hamiltonian of matrix model HΘ, or even states and readouts of observer network, but in this paper we simply regard them as finite-dimensional vectors without expanding specific constructions. 2.2 Unified Time Scale and Spectral Measure The axiomatic formulation of unified time scale is as follows. Axiom 2.1 (Unified time scale).For any given parameter Θ, there exists a family of operators QΘ(ω)and phase shift function φΘ(ω)at frequency/energy scale, such that κΘ(ω)φ′ Θ(ω) π=ρrel,Θ(ω) = 1 2πtr QΘ(ω), where ρrel,Θis the relative density of states with respect to some fixed reference background. For observers, all “time passage”, “density of states change”, “scattering delay”, and “boundary energy” are uniformly encoded through κΘ(ω). Under this axiom, all physical observations are expressed as spectral measures. Definition 2.2 (Window functional).For each observation/constraint class i, there exists an integrable weight function Wi(ω)and frequency region Ωi⊂R, such that the constraint can be written as Ci(Θ) = ZΩi Wi(ω)κΘ(ω)−κobs(ω)dω, where κobs(ω)is the “target scale” obtained from known theory/experimental window, representing the time/density of states structure actually exhibited by the universe in the corresponding observation domain. 2.3 Constraint System and Joint Solution Set Definition 2.3 (Constraint function and joint solution set).Given six constraint functions Ci:RN→R, i = 1,...,6, define the joint solution set S(Θ) = 6 \ i=1 C−1 i(0) = {Θ∈RN|Ci(Θ) = 0, i = 1,...,6}. Under the unified time scale and spectral functional language, each Cihas the following structure: 4 1. There exists kernel function Ki(ω; Θ) and observation window Ωi, satisfying Ci(Θ) = ZΩi Ki(ω; Θ) dω, where Ki(ω; Θ) = Wi(ω)κΘ(ω)−κobs(ω). 2. Θ 7→ κΘ(ω) is at least C1at each ω(later text will require Ckregularity for manifold structure). This paper develops mathematical analysis and numerical schemes at this abstract level, leaving the correspondence between specific Wi,Ωiand physical constants to subsequent physical implementation work. 3 Mathematical Structure of Constraint Manifold This section analyzes the local structure of joint solution set S(Θ) from the differential topology perspective. Under reasonable regularity conditions, S(Θ) is a smooth submanifold near “regular points” with dimension roughly N−6. 3.1 Regularity and Jacobian Matrix Denote C(Θ) = (C1(Θ), . . . , C6(Θ)) : RN→R6. Its Jacobian matrix is J(Θ) = ∂Ci ∂θj (Θ)1≤i≤6,1≤j≤N . Definition 3.1 (Regular point and singular point).  If rank J(Θ∗)=6, then Θ∗is called aregular point of C;  If rank J(Θ∗)<6, then Θ∗is called a singular point or degenerate point. Physically, at regular points the constraints are “transversal”, each constraint providing independent information; while at singular points, there exists constraint redundancy or implicit symmetry, causing parameter variations in certain directions not to affect constraint values— this is the source of what we later call “soft modes”. 3.2 Manifold Structure and Dimension Theorem Theorem 3.2 (Manifold structure of joint solution set).Assume: 1. C∈Ck(RN,R6)for some k≥1; 2. Θ∗∈S(Θ) and rank J(Θ∗) = 6. Then there exists a neighborhood U⊂RNof Θ∗such that U∩S(Θ) is a Cksmooth submanifold with dimension dimU∩S(Θ)=N−6. 5 Proof. This is a canonical application of the implicit function theorem. Since rank J(Θ∗) = 6, there exists a rearrangement of coordinates Θ = (x, y)∈RN−6×R6 such that the partial derivative Jacobian with respect to y, namely ∂C/∂y, is invertible at Θ∗. The implicit function theorem guarantees: there exists a neighborhood Uxof x∗∈RN−6and a Ckfunction g:Ux→R6such that C(x, y) = 0 ⇐⇒ y=g(x),(x, y)∈Ux×Uy. Thus U∩S(Θ) = {(x, g(x)) |x∈Ux} is a Cksubmanifold whose dimension equals dim Ux=N−6. Corollary 3.3 (Condition for discrete solutions).If N= 6 and there exists a point Θ∗such that C(Θ∗) = 0 and rank J(Θ∗) = 6, then this point is locally discrete (zero-dimensional manifold), and no other solutions exist in its neighborhood. From a physical perspective, this situation corresponds to a “minimal parameter universe”: the universe is encoded by exactly 6 free parameters, six constraints completely fix the universe, and the solution is locally unique. 3.3 Tangent Space, Normal Space, and Soft/Hard Modes Definition 3.4 (Tangent space and normal space).At a regular point Θ∗∈S(Θ), define the tangent space TΘ∗S= ker J(Θ∗)⊂RN, and the normal space NΘ∗S= Im J(Θ∗)T⊂RN. Then we have RN=TΘ∗S⊕NΘ∗S, and dim TΘ∗S=N−6,dim NΘ∗S= 6. Typically, directions in TΘ∗Scorrespond to parameter variations that do not change constraint values to first order, which we call soft modes; while directions in NΘ∗Sare those that can significantly change constraint values, called hard modes. Numerically, we can identify principal directions of soft/hard modes through singular value decomposition of the Jacobian matrix—this will be detailed in Section 5. 4 Numerical Sampling and Surrogate Model Framework In finite-dimensional parameter space, directly solving C(Θ) = 0 is typically difficult and expensive, not to mention analyzing its overall structure. This paper constructs a numerical mapping strategy that can be implemented in practice: 1. Use low-discrepancy sequences for initial coverage in parameter space, roughly finding regions close to solutions; 2. Perform local refinement sampling and level-set tracking in low-residual regions to depict connected components of S(Θ); 6 3. Train surrogate models (Gaussian process or kernel ridge regression) to approximate Ci(Θ) or Φ(Θ) PiCi(Θ)2, using Bayesian optimization/active sampling strategies to further approximate the solution manifold; 4. Estimate Jacobian matrices at sampling points for structural diagnostics. 4.1 Parameter Standardization and Sampling Domain To avoid numerical ill-conditioning due to dimensional and scale differences, we introduce parameter standardization: Definition 4.1 (Parameter standardization).For each parameter component θj, choose center value µjand scale sj>0, defining dimensionless parameter ˜ θj=θj−µj sj , j = 1, . . . , N. Denote ˜ Θ=(˜ θ1,...,˜ θN). In numerical implementation, we use the standardized space ˜ Θas the working space for sampling and optimization. The initial sampling domain can be taken as ˜ Θ∈[−3,3]N, with narrower intervals for directions known to be tightly constrained by observations (e.g., already tightly constrained constants), and appropriately widened for directions expected to be redundant or of unknown range. 4.2 Cost Function and Feasibility Screening Define weighted residual cost function Φ(Θ) = 6 X i=1 wi Ci(Θ)2 σ2 i , where σiis the tolerance for constraint Ci(based on observational error or theoretically allowed error), and wi>0 is importance weight. Definition 4.2 (Approximate feasible set).Given threshold Φ⋆>0, define approximate feasible set SΦ⋆={Θ∈RN|Φ(Θ) ≤Φ⋆}. Numerically, we first find approximate point clouds of SΦ⋆, then finely analyze the joint structure of Ci(Θ) = 0 within them. 4.3 Low-Discrepancy Sampling and Local Refinement Strategy 4.3 (Sampling–refinement–tracking cycle).1. Initial low-discrepancy sampling: On standardized domain [−3,3]N, generate M0Sobol sequence samples {˜ Θ(k)}M0 k=1, corresponding to original space samples {Θ(k)}. For each sample compute C(Θ(k))and Φ(Θ(k)). 2. Feasibility screening: Select top p%samples with smallest cost function values (e.g., p= 10%), denoted as set S0. 3. Local refinement sampling: Around covariance structure of samples in S0, construct ellipsoids or local hypercubes, use Latin hypercube sampling or low-discrepancy sampling again in these regions to generate M1new samples, and repeat evaluation of C, Φ. 7 4. Level-set tracking: Near approximate level set Φ=Φ⋆, use least-squares continuation methods or other level-set tracking algorithms to generate continuous point chains along SΦ⋆, roughly depicting connected components of solution manifold. This cycle can be iterated multiple times, each time updating the point cloud density and coverage range of the “candidate feasible set”. 4.4 Surrogate Models and Active Sampling In high-dimensional parameter space, directly evaluating C(Θ) may be expensive, especially when each evaluation requires calling complex QCA/matrix simulations. For this, we introduce surrogate models. Definition 4.4 (Surrogate model).Let the observed dataset be D={(Θ(k), C(Θ(k)))}M k=1. On this basis, train multi-output Gaussian process regression or kernel ridge regression model b C(Θ) ≈C(Θ), and give prediction mean and variance for each output component b Ci(Θ), σ b Ci(Θ). Define surrogate cost function b Φ(Θ) = 6 X i=1 wib Ci(Θ)2 σ2 i . Strategy 4.5 (Active sampling/Bayesian optimization).1. Based on current surrogate model, define acquisition function, such as “Expected Improvement” (EI) or “Lower Confidence Bound” (LCB); 2. Solve for maximum point Θnext of acquisition function in parameter space as next true evaluation point; 3. Compute true C(Θnext), add (Θnext, C(Θnext)) to dataset D, update surrogate model; 4. Iterate repeatedly until surrogate model converges near SΦ⋆. In practice, active sampling can be combined with level-set tracking: the acquisition function can target not only the minimum of Φ, but also regions of maximum uncertainty near the level set Φ = Φ⋆, to optimize coverage of the solution manifold. 5 Structural Diagnostics: Jacobian Spectrum, Symmetry, and Degeneracy After obtaining a batch of approximately feasible samples, we wish to answer the following questions:  Is the dimension of the solution manifold indeed N−6, or does it further reduce in certain regions?  Do multiple mutually disconnected solution branches exist?  Does parameter redundancy, symmetric equivalence relations, or degenerate directions exist?  Which parameter combinations are “hard modes” and which are “soft modes”? This section provides a diagnostic methodology through singular value decomposition (SVD) of the Jacobian matrix and clustering analysis. 8 5.1 Jacobian Matrix and Singular Value Decomposition Near a selected point Θ∗, the Jacobian matrix can be estimated through automatic differentiation or finite differences: J(Θ∗)≈∂Ci ∂θj (Θ∗). Perform singular value decomposition on this matrix: J(Θ∗) = UΣVT, where:  U∈R6×6is an orthogonal matrix;  Σ = diag(σ1, . . . , σr,0,...,0) ∈R6×Nis the singular value matrix (r= rank J);  V∈RN×Nis an orthogonal matrix whose column vectors v1, . . . , vNgive an orthogonal basis of parameter space. Definition 5.1 (Soft modes and hard modes).  If singular value σkis significantly larger than a given threshold τ, then the corresponding right singular vector vkis called a hard mode direction: small variations δΘ∝vkalong this direction will significantly change constraint values;  If σkis close to zero or significantly smaller than the threshold, then corresponding vk is called a soft mode direction: along this direction there is almost no sensitivity to constraints to first order. The distribution of singular value spectrum {σk}provides a quantitative measure of parameter identifiability. If multiple singular values are nearly zero, it indicates high degeneracy of constraints with abundant equivalent parameter transformations. 5.2 Multiple Solution Branches and Clustering Analysis On the near-feasible point cloud {Θ(k)|Φ(Θ(k))≤Φ⋆}, we can use density clustering (such as DBSCAN, HDBSCAN) to partition clusters B1,B2, . . . , each cluster corresponding to a solution branch or solution island. Definition 5.2 (Solution branches and physical equivalence classes).  Each cluster Bαis called a solution branch, representing a class of connected or nearly connected nearfeasible solutions in parameter space;  If there exists parameter transformation g:RN→RNsuch that C(g(Θ)) ≈C(Θ) and g(Bα)≈ Bβ, then Bα,Bβare called nearly symmetric equivalent solution branches. Through comparison of Jacobian spectra and physical observables within solution branches, one can determine whether different solution branches are physically distinguishable: if differences in all observable windows are within tolerable noise, they can be regarded as “parameter redundancy”; otherwise, they represent genuine physical multiple solutions or different vacua. 9