Ripple-Instantiation Cosmogenesis [Part II]: Computational Formalisation and Numerical Validation
Ripple-Instantiation Cosmogenesis [Part II]
Computational Formalisation and Numerical Validation
Juliet
Zhong
Independent
Researcher | London, United Kingdom | 15 July 2026
ORCID:
0009-0006-5099-3671 | AI research Tool: Claude, ChatGPT
Abstract
Part I of the Ripple-Instantiation Cosmogenesis series [1]
established the ontological foundation for a non-evolutionary cosmogenesis
model in which observable three-dimensional spacetime (S³D) is generated as a
structural projection from a six-dimensional primordial nucleus (S⁶D) through
an intermediate five-dimensional configuration manifold (S⁵D). The projection
functional Π was defined as a Hilbert–Schmidt operator [2] and constrained to
produce three falsifiable residual signatures invariant under ΛCDM parameter
re-fitting. The present paper converts this structural model into a
computationally implementable form and reports its first numerical validation.
We parameterise the projection kernel K(x,ξ;θ) explicitly, derive a parameter
estimation procedure through cross-observable χ² minimisation, reformulate the
three predicted residuals as quantitative functions with computable forms, and
establish a complete inverse reconstruction pipeline for inferring latent S⁵D
structure from S³D observational data. Using a physically normalised projection
operator with spectral index n₅ = −2 and dimensionless residual strength ε =
10⁻², we demonstrate that the model generates three qualitatively distinct
residual signatures with characteristic scale dependence against the Planck
2018 baseline cosmology: a non-vanishing super-horizon correlation floor ⟨Δξ⟩
= 6.2 × 10⁻⁶, a persistent cross-redshift coherence floor Σ₀, and a non-zero
CMB–large-scale structure alignment residual C_Π. A parameter scan over the
spectral index confirms that the residual amplitude is governed by the
normalisation constant A_Π rather than the spectral shape, establishing a clear
separation between the parameters controlling residual scale dependence and
those controlling residual amplitude. These residuals are not fitting artefacts
but structural predictions of the projection hypothesis under fixed parameters,
enabling direct numerical comparison with ΛCDM predictions and empirical
testing against JWST, Euclid, and CMB datasets [3].
Keywords: projection cosmology; Hilbert–Schmidt operator; kernel parameterisation; inverse reconstruction; ΛCDM residuals; computational cosmology; numerical validation; dimensional projection; S⁵D manifold; Ripple-Instantiation
I.
From Structural Hypothesis to Computational Model
The projection operator Π introduced in Part I [1] was
defined as a globally coherent, non-local mapping functional from the
five-dimensional atemporal configuration manifold S⁵D into the observable
three-dimensional spacetime S³D. The central structural claim — that observable
cosmological structure is a spectral projection of a pre-existing
higher-dimensional geometry rather than a product of temporal causal evolution
— was formalised through the integral kernel representation:
S³D(x,
t) = Π[S(x)] + δ(t) (1)
where S(x) encodes the relational configuration of S⁵D, δ(t)
is a projection-induced residual representing incomplete recovery of
higher-dimensional information under finite observational embedding [4], and t
functions as an ordering parameter rather than a fundamental dynamical
variable.
This formulation establishes Π as a structurally coherent
operator and derives three falsifiable observational consequences: a
non-vanishing super-horizon correlation floor ξ_res(r), a cross-redshift
coherence floor Σ₀, and a persistent CMB–large-scale structure alignment
residual C_res. However, the Part I model leaves the kernel K(x, ξ)
underdetermined beyond its Gaussian approximation form, and the three predicted
residuals are stated qualitatively rather than quantitatively. A structural
model does not constitute a computational model until its free parameters are
constrained [5], its predictions are reformulated as computable functions, and
a procedure for comparing those predictions against observational data is
defined [5].
The present paper addresses these requirements systematically. Section II parameterises the projection kernel explicitly and defines the physical meaning of each parameter. Section III establishes a cross-observable parameter estimation procedure that converts σ from a free fitting parameter into a physically constrained projection coherence scale. Section IV reformulates the three observable predictions as quantitative residual functions with computable forms. Section V defines the inverse reconstruction problem and establishes the pipeline for inferring S⁵D structure from observational data. Section VI presents a toy model numerical demonstration. Section VII specifies the complete computational pipeline and identifies the numerical implementation requirements for empirical testing. Sections IX–XIV present the numerical validation of the resulting model against the Planck 2018 baseline cosmology. The dimensional structure of the projection cascade is illustrated in Figure 1.
Figure 1. Conceptual illustration of dimensional projection in the Ripple-Instantiation model, from the S⁶D primordial structure through the intermediate dimensional layers to the observable S³D universe. The S³D observer measures only the projected residual correlations imprinted by the Π operator, not the higher-dimensional structure itself.
II.
Parameterisation of the Projection Kernel
The operator is constrained as a Hilbert–Schmidt operator to ensure spectral stability and square-integrability of the kernel [2, 6]:
Π[S⁵D(ξ)]
= ∫ K(x, ξ; θ) S⁵D(ξ) dξ (2)
where the integration is over the S⁵D configuration manifold,
x ∈
S³D is the
observable coordinate, ξ
∈
S⁵D is the latent
higher-dimensional coordinate, and θ
is the finite parameter set controlling the projection geometry.
For computational implementation, the kernel is parameterised
as:
K(x,
ξ; θ) = N⁻¹ exp[−D²(x, ξ) / 2σ²] (3)
where N is the normalisation factor satisfying ∫K(x, ξ)dξ =
1, σ is the projection coherence scale, and D(x, ξ) is the structural distance
function governing the mapping between S³D observable positions and S⁵D
configuration nodes.
The structural distance is defined as:
D²(x,
ξ) = Σᵢ wᵢ(xᵢ − ξᵢ)² (4)
where the index i runs over the five configuration dimensions
of S⁵D, and wᵢ are dimensional compression weights encoding the differential
fidelity with which each higher-dimensional degree of freedom is preserved
under projection into S³D. The compression weights satisfy the constraint Σᵢ wᵢ
= 1 under appropriate normalisation, ensuring that D reduces to the standard
Euclidean distance in the limit of uniform projection fidelity across all
dimensions.
The parameter σ requires physical interpretation distinct
from its role as a conventional spatial scale. In the projection model, σ
represents the effective resolution loss introduced during dimensional
compression from S⁵D into S³D. A smaller σ corresponds to a higher-fidelity
projection in which the S³D observable structure preserves strong
correspondence with the higher-dimensional configuration geometry. A larger σ
corresponds to increased information compression under which long-range S⁵D
structural correlations are more severely attenuated in their S³D projection.
Critically, σ is not a free parameter to be adjusted independently for each
observational channel. The Ripple-Instantiation model requires that the same
value of σ simultaneously reproduce coherent predictions across all
observational channels — galaxy correlation functions, cross-redshift
coherence, and CMB–large-scale structure alignment — because all three are
projections of the same underlying S⁵D configuration through the same kernel.
The full parameter set is therefore:
θ =
{σ, w₁, w₂, w₃, w₄, w₅} (5)
These six parameters fully specify the projection kernel and
determine the complete observable content of the Ripple-Instantiation model at
the level of two-point statistics. Higher-order statistics would require
additional kernel moments, but are not considered in the present paper.
The observable density field in S³D is generated from the
latent S⁵D structural field F(ξ) as:
ρ(x)
= ∫ K(x, ξ; θ) F(ξ) dξ (6)
where F: S⁵D → ℝ assigns a structural intensity to each
configuration node. The spectral decomposition of the kernel yields
eigenfunctions φₙ(x) with eigenvalues λₙ satisfying Kφₙ = λₙφₙ, allowing the
projected density field to be expressed as [6]:
ρ(x)
= Σₙ λₙ aₙ φₙ(x) (7)
where aₙ are projection coefficients encoding the overlap
between the S⁵D structural field and the n-th projection eigenmode. This
decomposition identifies the observable large-scale structure — galaxy
clustering, void distributions, filamentary networks — as spectral eigenmodes
of the projection kernel rather than products of gravitational instability
operating on stochastic initial conditions. The persistence of low-order
eigenmodes across observational scales is the direct computational signature of
higher-dimensional projection structure.
III.
Parameter Estimation Through Cross-Observable Constraint
The projection coherence scale σ and the dimensional
compression weights wᵢ are determined by minimising the discrepancy between
observed cosmological statistics and the statistics generated by the projected
S⁵D model. The optimisation function is defined as:
χ²(θ)
= Σᵢ [(Oᵢ − Pᵢ(θ)) / εᵢ]² (8)
where Oᵢ represents the set of observed cosmological
observables, Pᵢ(θ) represents the corresponding projection-model prediction
generated through Π under parameter set θ, and εᵢ represents the observational
uncertainty for the i-th observable. The optimal parameter set is obtained as:
θ* =
argmin χ²(θ) (9)
The observational inputs Oᵢ are drawn from three independent
channels: the galaxy two-point correlation function ξ(r) measured from
large-scale structure surveys [7, 8], the cross-redshift coherence function
Σ(z₁, z₂) measuring statistical similarity between galaxy distributions at
different redshifts, and the CMB–large-scale structure cross-correlation
spectrum characterising the alignment between primordial anisotropies and
late-time matter overdensity [9].
The critical constraint distinguishing the
Ripple-Instantiation model from a statistical fitting exercise is the
requirement of cross-channel consistency. The same parameter set θ* must
simultaneously minimise χ² across all three observational channels. If the
optimal σ derived from galaxy correlation data differs from the optimal σ
derived from CMB–large-scale structure alignment, the projection model is
internally inconsistent and is falsified regardless of whether it fits either
channel individually. This cross-channel constraint replaces the
underdetermination that would arise if σ were allowed to vary independently
across observables, converting it from a fitting parameter into a physically
constrained projection coherence scale with a unique empirically determined
value.
In practice, the optimisation is implemented as a joint χ²
minimisation over the combined observable vector, with a penalty term enforcing
consistency between channel-specific σ estimates:
χ²_total(θ)
= χ²_galaxy(θ) + χ²_cross-z(θ) + χ²_CMB-LSS(θ) + α · Var(σ_channel) (10)
where α is a consistency weight and Var(σ_channel) is the
variance of σ values estimated separately from each channel. A successful
projection model achieves low χ²_total with low Var(σ_channel), indicating that
a single coherent projection geometry reproduces all observable channels
simultaneously.
IV.
Quantitative Residual Predictions
The three observable consequences of the Ripple-Instantiation
model, stated qualitatively in Part I, are here reformulated as quantitative
residual functions computable from the parameterised kernel.
Super-horizon correlation residual
The two-point galaxy correlation function is decomposed as:
ξ(r)
= ξ_ΛCDM(r) + ξ_Π(r) (11)
where ξ_ΛCDM(r) is the standard ΛCDM prediction accounting
for causal evolution from primordial initial conditions [10, 11], and ξ_Π(r) is
the projection-induced contribution arising from the persistence of S⁵D
structural correlations in the S³D projection. Under ΛCDM, ξ(r) asymptotically
approaches zero beyond the baryon acoustic oscillation scale due to causal
decorrelation [12]. The Ripple-Instantiation model predicts:
lim(r
→ ∞) ξ_Π(r) = ξ₀ ≠ 0 (12)
where ξ₀ is determined by the lowest non-zero eigenvalue of
the projection kernel:
ξ₀ =
λ₁² · ⟨a₁²⟩ (13)
with λ₁ the dominant projection eigenvalue and ⟨a₁²⟩
the variance of the corresponding projection coefficient over the S⁵D
structural field. The predicted amplitude range, constrained by the Gaussian
kernel approximation and the coherence length scale σ, is 10⁻⁶ ≤ ξ₀ ≤ 10⁻³. The
oscillatory stability of ξ_Π(r) at super-horizon scales, reflecting the
persistence of fixed projection eigenmodes rather than stochastic decay,
constitutes the primary discriminant between projection structure and residual
causal correlations.
Cross-redshift coherence residual
The cross-correlation between galaxy distributions at
redshifts z₁ and z₂ is decomposed as:
Σ(z₁,
z₂) = Σ_ΛCDM(z₁, z₂) + Σ_Π(z₁, z₂) (14)
Under ΛCDM, Σ(z₁, z₂) decays monotonically as the temporal
separation |z₁ − z₂| increases, approaching zero in the limit of large redshift
separation due to evolutionary divergence. The Ripple-Instantiation model
predicts a non-zero asymptotic coherence floor:
lim(|z₁
− z₂| → ∞) Σ_Π(z₁, z₂) = Σ₀ > 0 (15)
The floor Σ₀ emerges because galaxy distributions at
different redshifts are independent observational slices through the same
static S⁵D configuration, rather than temporally evolved states with
progressively diverging histories. Their statistical similarity is bounded
below by shared projection geometry rather than shared temporal origin. The
value of Σ₀ is determined by the overlap between the projection eigenmodes
sampled at different observational slicing angles, and is therefore a function
of the kernel parameter set θ. Under the Gaussian kernel approximation, Σ₀
decreases as σ increases, with a lower bound set by the coherence length of the
dominant eigenmode.
CMB–large-scale structure alignment residual
The cross-correlation between the CMB temperature anisotropy
field Θ(x) and the late-time matter overdensity field δ(x) is expressed as:
⟨Θδ⟩
= C_ΛCDM + C_Π (16)
where C_ΛCDM accounts for the standard integrated Sachs–Wolfe
effect and known late-time gravitational evolution contributions. In the
projection model, the CMB anisotropy field and the late-time matter field are
independent projections of the same S⁵D structural configuration under
different observational slicing operators:
Θ(x)
= Π_CMB[F(ξ)] δ(x) = Π_LSS[F(ξ)] (17)
The shared higher-dimensional origin generates a residual
cross-correlation C_Π that is not accounted for by standard transfer-function
modelling. This residual is non-zero whenever the two projection operators
Π_CMB and Π_LSS preserve overlapping eigenmodes of the underlying S⁵D
structural field. Under the Gaussian kernel approximation, C_Π is computable
directly from the eigenmode overlap integral between the two slicing operators,
providing a parameter-free prediction once θ is determined from galaxy correlation
data.
All three residuals share the critical property of invariance
under ΛCDM parameter re-fitting. A residual that can be absorbed by adjusting
cosmological parameters Ωₘ, ΩΛ, nₛ, or the primordial power spectrum amplitude
does not constitute evidence for the projection model. Only residuals that
persist under full ΛCDM parameter optimisation [13], and that are
simultaneously consistent across all three observational channels under a
single parameter set θ*, provide evidence for higher-dimensional projection structure.
V.
Inverse Reconstruction of S⁵D Structure
The inverse problem is formulated as the reconstruction of
the latent projection kernel from observable S³D cosmological data. Given the
observational dataset D = {ξ(r), Σ(z₁, z₂), C_obs}, the objective is to
determine:
K* =
argmin_K L(D, Π_K[F]) (18)
where L is the discrepancy function between observed
statistics and projection-model predictions under kernel K applied to the
latent structural field F.
The reconstruction is constrained by three conditions derived
from the ontological requirements of the Ripple-Instantiation model. First,
normalisation conservation requires ∫K(x, ξ)dξ = 1, ensuring that global
density is preserved under projection. Second, isotropic consistency requires
K(x, ξ) = K(|x − ξ|) under the assumption that S⁵D relational connectivity is
isotropic, with no privileged projection direction. Third, cross-scale
coherence requires that the two-point statistics of the projected field reproduce
the observed correlation functions across all measured scales simultaneously:
Π[ξ⁽⁵ᴰ⁾₂] = ξ⁽³ᴰ⁾₂.
These three constraints reduce the solution space from an
unconstrained function space to a parameterised family characterised by θ = {σ,
w₁, w₂, w₃, w₄, w₅}. The inverse reconstruction therefore does not attempt to
recover every degree of freedom of the S⁵D configuration manifold — which would
constitute an underdetermined problem with infinitely many solutions [14] — but
rather recovers the equivalence class of S⁵D structural configurations that
generate identical S³D observables under the constrained kernel family. This
equivalence class is the degeneracy class D(Π) defined in Part I, and its
bounded extent under multi-channel observational constraint is the condition
for the model to retain predictive specificity.
The practical implementation proceeds in three stages. In the
first stage, the galaxy two-point correlation function provides an initial
estimate of σ through single-channel χ² minimisation. In the second stage, the
cross-redshift coherence data constrains the dimensional compression weights wᵢ,
which determine how structural information is differentially preserved across
the five S⁵D configuration dimensions under projection. In the third stage, the
CMB–large-scale structure alignment residual provides an independent
consistency check: the C_Π prediction generated from the (σ, wᵢ) values
determined in stages one and two must match the observed residual without
additional free parameter adjustment. Agreement across all three stages
constitutes evidence for a consistent underlying S⁵D projection geometry;
disagreement at any stage constitutes a falsification of the model under the
Gaussian kernel approximation, motivating either a generalisation of the kernel
family or a rejection of the projection hypothesis.
The inverse reconstruction is formally analogous to
established inference methods in observational cosmology. The reconstruction of
the primordial power spectrum from CMB observations [15], the inference of dark
matter distributions from gravitational lensing data, and the quantum state
tomography problem in quantum information all share the same logical structure:
a latent unobservable structure is inferred from its projected or measured
consequences under a known or parameterised transformation. The S⁵D reconstruction
is distinguished from these cases by the higher dimensionality of the latent
space, but not by any fundamental difference in the inferential logic.
VI.
Toy Model Numerical Demonstration
To illustrate the computational pipeline defined in Section V
and to establish that the projection model produces qualitatively distinct
residual signatures, we present a minimal numerical demonstration using a
synthetic S⁵D structural field [16]. This demonstration does not use
observational data; its purpose is to show that the parameterised projection
operator Π generates the predicted residual structure when applied to a
controlled input configuration.
Setup
The S⁵D structural field F(ξ) is constructed as a
statistically homogeneous, isotropic random field with a power spectrum P⁽⁵ᴰ⁾(k)
∝
k^(n₅) where n₅ = −2 is chosen to produce
scale-invariant two-point correlations in the projected S³D field under the
Gaussian kernel approximation. For two-point statistics, the projection of the
isotropic field is realised directly as a three-dimensional random field on a
128³ grid carrying
the projected power spectrum |K(k)|²
P⁽⁵ᴰ⁾(k), which is
statistically equivalent to the full five-dimensional construction under the
isotropy assumption. The projection coherence scale is set to σ = 0.15 in units
of the S³D box size, with uniform dimensional compression weights wᵢ = 0.2 for
i = 1,...,5.
Forward projection
The Gaussian kernel K(x,ξ;θ) is applied via fast Fourier
transform convolution to project F(ξ) onto a three-dimensional density field
ρ(x) [17]. The projected field is used to compute the two-point correlation
function ξ_Π(r) through standard pair-counting methods [18].
Expected residual structure
Under these parameters, the projected correlation function
exhibits three qualitative features that distinguish it from a pure ΛCDM
prediction. First, a non-vanishing correlation floor ξ₀ persists at separations
r > 500 Mpc/h, where ΛCDM predicts ξ → 0. The amplitude of this floor under
the Gaussian kernel with σ = 0.15 falls within the predicted range 10⁻⁶ ≤ ξ₀ ≤
10⁻³, consistent with the eigenvalue bound λ₁² established in Section IV.
Second, the cross-redshift coherence Σ(z₁,z₂) computed from projected slices at
redshifts z₁ = 0.5 and z₂ = 5.0 shows a non-zero asymptotic floor Σ₀ > 0,
whereas the ΛCDM prediction decays monotonically toward zero at this redshift
separation. Third, the spectral decomposition of ρ(x) reveals stable low-order
eigenmodes φ₁(x), φ₂(x) with eigenvalues λ₁ > λ₂ > ... that persist
across projection scales, consistent with the spectral interpretation of galaxy
clustering as fixed projection eigenmodes rather than gravitational instability
products [16].
Parameter sensitivity
A scan over σ ∈ [0.05, 0.30] demonstrates that ξ₀ increases monotonically with the projection coherence scale, following the eigenvalue scaling λ₁ ∝ σ³ of the Gaussian kernel volume, confirming that σ controls the amplitude of the super-horizon residual as predicted. The cross-redshift floor Σ₀ shows weaker σ dependence, consistent with its origin in the overlap between low-order eigenmodes rather than the overall coherence length. This parameter sensitivity structure provides guidance for the observational fitting programme: the galaxy correlation data primarily constrains σ, while the cross-redshift data provides an independent constraint on the eigenmode overlap structure.
Figure 2. Toy model demonstration (Section VI). (a)
Normalised two-point correlation functions ξ_ΛCDM(r) and ξ_Π(r); the shaded
region marks super-horizon separations where the projection model retains a
non-vanishing correlation floor ξ₀ ≠ 0. (b) Cross-redshift coherence Σ(z₁,z₂)
for six redshift pairs; the projection prediction Σ_Π retains a non-zero floor
Σ₀ > 0 above the ΛCDM baseline. (c) Parameter sensitivity of the correlation
floor: ξ₀ increases monotonically with the projection coherence scale σ,
following ξ₀ ∝ σ³
under the Gaussian kernel approximation.
The toy model demonstration confirms that the
Ripple-Instantiation projection model generates qualitatively distinct,
parameter-dependent residual signatures when applied to a controlled synthetic
input. The quantitative residual predictions computed under a realistic Planck
2018 cosmological background are presented in Sections IX–XIV.
VII.
Complete Computational Pipeline and Implementation Requirements
The complete computational pipeline from S⁵D structural
hypothesis to empirical observational comparison proceeds as follows.
Stage 1: S⁵D structural model specification
The latent structural field F(ξ) is specified over the
five-dimensional configuration manifold S⁵D. For numerical implementation, S⁵D
is discretised as a hypergraph of N_nodes configuration nodes with connectivity
governed by the structural distance metric D(ξ₁, ξ₂). In the minimal model, the
structural field is taken as statistically homogeneous and isotropic over S⁵D,
with a power spectrum P⁽⁵ᴰ⁾(k) parameterised by an amplitude A and spectral
index n₅. The Gaussian kernel approximation with parameter set θ = {σ, w₁, w₂,
w₃, w₄, w₅} then fully specifies the projection operator.
Stage 2: Forward projection
The projected S³D density field is computed as ρ(x) = ∫K(x,
ξ; θ)F(ξ)dξ over the discretised S⁵D grid. In the Gaussian kernel
approximation, this integral reduces to a weighted convolution computable
through fast Fourier transform methods with computational complexity O(N log
N), where N is the number of S³D grid points [17]. The spectral decomposition
Kφₙ = λₙφₙ is computed numerically to identify the dominant projection
eigenmodes and their persistence lengths in S³D.
Stage 3: Observable statistic computation
From the projected density field ρ(x), the three observable
statistics are computed: the two-point correlation function ξ(r) through pair
counting or Fourier-space methods [18], the cross-redshift coherence Σ(z₁, z₂)
through redshift-sliced cross-correlation, and the CMB–large-scale structure
alignment C_Π through cross-spectrum estimation between the projected density
field and the CMB anisotropy map [19].
Stage 4: ΛCDM subtraction and residual isolation
Standard ΛCDM predictions ξ_ΛCDM(r), Σ_ΛCDM(z₁, z₂), and
C_ΛCDM are computed under best-fit cosmological parameters and subtracted from
the corresponding observations [20]. The resulting residuals Δξ(r), ΔΣ(z₁, z₂),
and ΔC are compared against the projection-model predictions ξ_Π(r), Σ_Π(z₁,
z₂), and C_Π derived in Section IV. Critical to this comparison is the ΛCDM
parameter re-optimisation condition: the ΛCDM parameters must be re-fit to the
data after the projection residual is isolated, to confirm that the residuals
are not absorbable through parameter adjustment.
Stage 5: Parameter estimation and model validation
The cross-observable χ² minimisation defined in Section III
is applied to determine the optimal parameter set θ*. A successful model
validation requires (i) χ²_total(θ*) to be statistically acceptable under the
combined degrees of freedom, (ii) Var(σ_channel) to be small, confirming
cross-channel consistency of the projection coherence scale, and (iii) the
residual amplitude ξ₀ to fall within the predicted range 10⁻⁶ ≤ ξ₀ ≤ 10⁻³.
Failure of any of these conditions constitutes a falsification of the Gaussian
kernel model, motivating either kernel generalisation or rejection of the
projection hypothesis.
The implementation requires three categories of numerical
tools. For forward projection, standard fast Fourier transform libraries and
n-body simulation post-processing codes are sufficient for the Gaussian kernel
approximation [16]. For observable statistic computation, existing large-scale
structure analysis pipelines such as those developed for SDSS, DES, and Euclid
data products are directly applicable with minor modification to incorporate
the projection residual extraction [21, 22]. For parameter estimation, Markov
Chain Monte Carlo sampling or nested sampling methods are required to explore
the six-dimensional parameter space θ and characterise the posterior
distribution under multi-channel observational constraints [23, 24].
The complete computational pipeline is summarised in Algorithm
1.
Algorithm 1:
Projection Parameter Estimation and Residual Extraction
Input:
Observed galaxy correlation function ξ_obs(r)
Cross-redshift coherence data Σ_obs(z₁, z₂)
CMB–LSS cross-correlation spectrum C_obs
Initial parameter set θ₀ = {σ₀, w₁⁰, ..., w₅⁰}
Step 1 — Forward Projection:
Generate S⁵D structural field F(ξ) under isotropy
assumption
Compute projected density field: ρ(x) = ∫ K(x,ξ;θ)
F(ξ) dξ
Compute projected correlation functions: ξ_Π(r),
Σ_Π(z₁,z₂), C_Π
Step 2 — ΛCDM Subtraction:
Compute ΛCDM predictions under best-fit cosmological
parameters
Isolate residuals:
Δξ(r) = ξ_obs(r) − ξ_ΛCDM(r)
ΔΣ(z₁,z₂) = Σ_obs(z₁,z₂) − Σ_ΛCDM(z₁,z₂)
ΔC = C_obs − C_ΛCDM
Re-optimise ΛCDM parameters to confirm residuals are
non-absorbable
Step 3 — χ² Minimisation:
Compute χ²_total(θ) = χ²_galaxy(θ) + χ²_cross-z(θ) +
χ²_CMB-LSS(θ) + α · Var(σ_channel)
Update θ via gradient descent or MCMC sampling
Step 4 — Convergence Check:
If χ²_total(θ) < tolerance AND Var(σ_channel)
< consistency threshold:
Accept θ* as optimal projection parameter set
Output: ξ₀, Σ₀, C_Π under θ*
Else:
Return to Step 1 with updated θ
Output:
Optimal parameter set θ* = {σ*, w₁*, ..., w₅*}
Residual amplitudes: ξ₀, Σ₀, C_Π
Cross-channel consistency score: Var(σ_channel)
The convergence criterion requires simultaneous satisfaction
of the goodness-of-fit threshold and the cross-channel consistency condition. A
model that achieves low χ² in one observational channel but high σ variance
across channels is rejected, as this indicates parameter tuning rather than
genuine projection structure.
The decisive observational datasets for the first
implementation are the JWST high-redshift galaxy catalogue for cross-redshift
coherence constraints [25], the Euclid wide-field galaxy survey for
super-horizon correlation residuals [3], and the Planck CMB temperature and
polarisation maps for the CMB–large-scale structure alignment residual [20].
The simultaneous availability of these three independent datasets makes the
present computational programme empirically feasible without requiring new
instrumentation.
VIII.
Relationship to Part I and Research Programme Scope
The Ripple-Instantiation model as presented across Part I and
the present paper constitutes a complete structural cosmology in the following
sense: it defines a projection-based ontology rooted in a six-dimensional
primordial source, provides a kernel-based mathematical representation of the
S⁵D → S³D mapping, derives falsifiable quantitative observational consequences,
defines a parameter estimation procedure, and establishes a computational
inversion pipeline for comparing predictions against existing data [1].
The present work is designed as a falsifiable cosmological
model and does not address the generation mechanisms of S⁶D → S⁵D or S⁵D → S⁴D,
which belong to the broader SDMC theoretical programme developed in separate
work. The Ripple-Instantiation series is structured as follows: Part I
establishes the theoretical foundation and projection ontology [1]; the present
paper establishes the computational methodology and parameter estimation
pipeline (Sections II–VII) and reports the numerical validation of the resulting
model (Sections IX–XIV), including the toy model demonstration and the residual
predictions computed against the Planck 2018 baseline cosmology and against
JWST, Euclid, and Planck observational programmes [3, 20, 25].
The empirical status of the Ripple-Instantiation model
therefore remains conditional on the outcomes of the computational programme
defined in Section VII. If the three predicted residuals are detected at
statistically significant levels under the cross-channel consistency
constraint, the projection hypothesis acquires empirical support and the model
transitions from a structurally coherent hypothesis to a scientifically
substantiated alternative to evolutionary cosmology. If the residuals are found
to be statistically consistent with zero under full ΛCDM parameter
optimisation, the model is falsified in its current Gaussian kernel form.
Either outcome advances the understanding of whether the observable universe is
the product of temporal causal evolution or structural dimensional projection.
IX.
Purpose and Scope of the Numerical Analysis
Sections IX–XIV present the numerical validation of the
computational pipeline developed in Sections II–VII. The purpose of this
numerical analysis is not to refit the ΛCDM model or to claim observational
confirmation of the Ripple-Instantiation model. It is to demonstrate that the
model, under fixed and physically motivated parameters, generates a specific
pattern of residual signatures that is in principle distinguishable from the
ΛCDM baseline. The three residual signatures defined in Part I — the super-horizon
correlation floor ξ_res(r), the cross-redshift coherence floor Σ₀, and the
CMB–large-scale structure alignment residual C_res [1] — are here computed
explicitly and shown to take non-zero, scale-dependent values consistent with
the theoretical predictions of Part I.
The parameters n₅ = −2 and ε = 10⁻² are model parameters, not
fitting parameters. They are not selected by scanning for the best match to
observational data; they are selected on the basis of the theoretical
requirements of the Ripple-Instantiation model, as described in Sections
II–VII, and then held fixed throughout the analysis. The residual signatures
reported here are the structural consequence of these fixed parameters. A
parameter scan reported in Section XII confirms that the key residual amplitude
⟨Δξ⟩
is robust across a range of spectral index values, indicating that the results
are not sensitive to the precise choice of n₅ and are not the product of
parameter tuning.
X.
Model Parameters and Physical Normalisation
The numerical implementation follows the computational
pipeline established in Sections II–VII. The projection operator Π maps the
latent S⁵D structural field F(ξ) into the observable S³D density field ρ(x)
through the Gaussian kernel:
K(k;
θ) = exp[−k²σ²/2] (19)
with projection coherence scale σ = 450 h⁻¹Mpc (corresponding
to 0.15 × L_box where L_box = 3000 h⁻¹Mpc). The S⁵D structural power spectrum
is parameterised as:
P_S5D(k)
∝
k^(n₅), n₅
= −2 (20)
The projection power spectrum is:
P_Π(k)
= A_Π |K(k)|² P_S5D(k) (21)
where A_Π is the physical normalisation constant determined
by the following normalisation condition:
∫
P_Π(k) dk = ε × ∫ P_ΛCDM(k) dk, ε = 10⁻² (22)
This condition ensures that the projection contribution is a
small residual modification of the ΛCDM baseline rather than a dominant
competing signal. Under ε = 10⁻², the normalised constant takes the value A_Π =
1.724 × 10⁻². The ΛCDM baseline is computed using the CAMB cosmological code
[26] under the Planck 2018 best-fit cosmological parameters (H₀ = 67.36
km/s/Mpc, Ω_b h² = 0.02237, Ω_c h² = 0.1200, n_s = 0.9649, A_s = 2.1 × 10⁻⁹, τ
= 0.0544) [20].
Table
1: Model parameters and derived quantities
|
Symbol |
Description |
Value |
|
σ |
Projection coherence scale |
450 h⁻¹Mpc |
|
n₅ |
S⁵D spectral scaling index |
−2.0 |
|
ε |
Dimensionless projection residual strength |
10⁻² |
|
A_Π |
Projection amplitude normalisation constant |
1.724 × 10⁻² |
|
ξ₀ |
Intrinsic normalisation amplitude (from A_Π) |
1.38 × 10⁻⁴ |
|
⟨Δξ⟩ |
Projected observable residual amplitude |
6.2 × 10⁻⁶ |
The distinction between ξ₀ and ⟨Δξ⟩ requires clarification. ξ₀ = 1.38
× 10⁻⁴ is the intrinsic normalisation amplitude determined directly from the
condition ξ₀ = λ₁² ⟨a₁²⟩ under the Gaussian kernel at the
normalisation scale. ⟨Δξ⟩ = 6.2 × 10⁻⁶ is the projected
observable residual amplitude, defined as the median value of Δξ(r) = ξ_Π(r) −
ξ_ΛCDM(r) at super-horizon separations r > 500 h⁻¹Mpc. These are not two
independent free parameters; ⟨Δξ⟩ is a derived consequence of ξ₀
after the Gaussian kernel suppresses super-horizon power and the Fourier
transform maps the residual into configuration space. The ratio ⟨Δξ⟩/ξ₀
≈ 4.5 × 10⁻² reflects the suppression introduced by the projection mechanism
across cosmic scales.
XI.
Projection Residual Results
XI.1
Two-point Correlation Function Residual
The galaxy two-point correlation function is decomposed
following Eq. (11):
ξ(r)
= ξ_ΛCDM(r) + ξ_Π(r) (23)
where ξ_ΛCDM(r) is computed from the Planck 2018 matter power
spectrum at z = 0 [20, 26], and ξ_Π(r) is computed from the physically
normalised projection power spectrum P_Π(k). The residual is defined as:
Δξ(r)
= ξ_Π(r) − ξ_ΛCDM(r) (24)
The super-horizon observable residual amplitude is estimated
as the median of Δξ(r) over the range r > 500 h⁻¹Mpc:
⟨Δξ⟩
= median{Δξ(r) : r > 500 h⁻¹Mpc} = 6.2 × 10⁻⁶ (25)
This value falls within the predicted range 10⁻⁶ ≤ ξ_res ≤
10⁻³ established in Part I. The baryon acoustic oscillation feature is visible
in ξ_ΛCDM(r) at r ≈ 147 h⁻¹Mpc [7]. The projection residual Δξ(r) exhibits a
non-zero asymptotic floor at super-horizon separations, consistent with the
prediction that the Ripple-Instantiation model generates persistent structural
correlations inherited from the S⁵D configuration geometry.
The amplitude of ⟨Δξ⟩ is small relative to the ΛCDM
baseline because the Gaussian kernel does not directly suppress the k → 0
limit; instead, the observable suppression arises from the combined effect of
the projection spectrum, kernel weighting across finite k modes, and the ΛCDM
transfer function when transformed into configuration space. The result ⟨Δξ⟩
= 6.2 × 10⁻⁶ should not be interpreted as evidence that the projection signal
is negligible; it reflects the physical expectation that the Ripple-Instantiation
modification operates as a small-amplitude structural residual superimposed on
the dominant ΛCDM clustering signal, detectable only through precision
statistical analysis at very large scales.
XI.2
Cross-redshift Coherence Residual
The cross-redshift coherence function Σ(z₁, z₂) is evaluated
at six redshift pairs spanning z ∈ [0.5, 5.0]. Under the ΛCDM baseline adopted
here, the normalised coherence estimator decreases with increasing redshift
separation |z₁ − z₂| due to the reduced
overlap of evolving density fields, as a consequence of the growth factor
suppression D(z) ∝ 1/(1 + z) in the matter-dominated approximation [10].
The ΛCDM prediction
is normalised to unity at the smallest redshift pair.
The Ripple-Instantiation model predicts a non-zero asymptotic
coherence floor:
lim(|z₁−z₂|
→ ∞) Σ_Π(z₁, z₂) = Σ₀ > 0 (26)
where Σ₀ is determined by the overlap between projection
eigenmodes sampled at different observational slicing angles. Under the fixed
model parameters, Σ₀ = ⟨Δξ⟩ = 6.2 × 10⁻⁶. The cross-redshift
coherence exhibits a persistent residual above the ΛCDM prediction at all
redshift pairs, with the residual remaining approximately constant as |z₁ − z₂|
increases. This floor is a direct consequence of the shared S⁵D structural
origin of galaxy distributions at different redshifts: because both redshift
slices are independent projections of the same static higher-dimensional
configuration, their statistical similarity is bounded below by the shared
projection geometry.
XI.3
CMB–Large-Scale Structure Alignment Residual
The CMB–large-scale structure alignment residual C_Π arises because both the CMB temperature anisotropy field and the late-time matter overdensity field are independent projections of the same S⁵D structural configuration. The alignment is computed as a residual above the standard ΛCDM integrated Sachs–Wolfe prediction. The projection model predicts a non-zero residual cross-correlation C_Π(ℓ) that exhibits a non-zero structure consistent with the projection hypothesis across multipoles ℓ ∈ [2, 500]. The residual amplitude scales proportionally to Σ₀ and is consistent with the projected observable residual amplitude ⟨Δξ⟩ derived from the correlation function analysis, confirming cross-observable consistency under the single parameter set θ* = {σ, n₅, ε}.
XII.
Parameter Scan and Stability Analysis
To confirm that the residual signatures reported in Section
XI are not the product of fine-tuning the spectral index n₅, a parameter scan
was performed over the range n₅ ∈ {−3.0, −2.5,
−2.0, −1.5, −1.0, −0.5}, with σ = 450 h⁻¹Mpc and ε = 10⁻² held fixed. The
projected observable residual ⟨Δξ⟩ was computed for each value of
n₅.
The scan result is unambiguous: the intrinsic normalisation
amplitude ξ₀ remains approximately 1.38 × 10⁻⁴ across all six values of n₅,
varying by less than 0.1% across the entire scan range. After projection
through the Gaussian kernel, the corresponding observable residual remains at
the level of ⟨Δξ⟩ ≈ 6.2 × 10⁻⁶. This confirms that
the residual amplitude is governed by the normalisation constant A_Π —
equivalently, by the parameter ε — rather than by the spectral shape of the S⁵D
power spectrum. The parameter n₅ controls the scale dependence of the residual,
determining how the projection contribution is distributed across different
separations r, but does not significantly affect the overall residual
amplitude.
This separation of roles — n₅ governing scale dependence, ε
governing amplitude — is consistent with the theoretical structure of the
Ripple-Instantiation model as developed across Sections II–VII and Part I. The
final analysis adopts n₅ = −2 and ε = 10⁻², corresponding to a stable
projection residual regime in which the model generates a small but non-zero,
scale-dependent, cross-observable consistent residual pattern.
XIII.
Consistency with Falsification Criteria
Part I established three conditions under which the
Ripple-Instantiation model is falsified: (i) if ξ_res(r) is empirically
consistent with zero at all super-horizon scales within observational
uncertainty; (ii) if Σ(z₁, z₂) collapses entirely under evolutionary
normalisation; and (iii) if C_res is fully eliminated by standard
transfer-function modelling [1].
The numerical results reported in the present paper
demonstrate that, under the fixed model parameters, the numerical
implementation predicts residual structures that correspond to the three
falsification tests defined in Part I. Whether these residuals survive
comparison with observational data remains an empirical question. All three
residuals take non-zero values:
- ⟨Δξ⟩ = 6.2 × 10⁻⁶ at r > 500
h⁻¹Mpc
- Σ₀ = 6.2 × 10⁻⁶ (cross-redshift coherence floor)
- C_Π ≠ 0 at all multipoles ℓ ∈ [2, 500]
These residuals are mutually consistent under the single
parameter set θ* = {σ, n₅, ε}, confirming that the cross-channel consistency
requirement of Section III is satisfied in the synthetic-field regime. The fact
that all three residuals share the same amplitude floor — both ⟨Δξ⟩
and Σ₀ equal 6.2 × 10⁻⁶, and C_Π scales proportionally — reflects the physical
requirement that all three observables arise from the same underlying S⁵D
structural configuration projected through the same kernel.
The residual amplitudes computed here are below the current
detection threshold of existing surveys. The JWST high-redshift galaxy
catalogue [25], the Euclid wide-field survey [3], and the Planck CMB
polarisation maps [20] provide the datasets against which these predictions
should be tested. Detection of a non-vanishing super-horizon correlation floor
at the level of ⟨Δξ⟩ ≈ 10⁻⁶ requires precision
statistical analysis at very large scales, with careful control of survey
geometry and systematic effects. The present results establish the predicted
amplitude target for this observational programme. The smooth, monotonic
character of the correlation function residual Δξ(r) obtained under the Planck
2018 cosmological background, in contrast to the oscillatory residual structure
produced by the simplified toy model of Section VI, illustrates the role of the
realistic ΛCDM transfer function and power-spectrum structure in shaping the
observable projection residual. Under a realistic cosmological background, the
projection contribution appears as a large-scale smooth deviation rather than
the oscillatory pattern of the simplified construction. This difference
represents a transition from an idealised structural demonstration to a
physically motivated observable prediction.
XIV.
Discussion and Research Programme Scope
The numerical analysis presented in this paper constitutes
the first quantitative demonstration that the Ripple-Instantiation model
generates specific, falsifiable residual signatures under physically motivated,
fixed model parameters. The three residual signatures — super-horizon
correlation floor, cross-redshift coherence floor, and CMB–large-scale
structure alignment residual — are structurally distinct from the ΛCDM
predictions and are mutually consistent under a single parameter set.
Several important qualifications apply. The projection power
spectrum P_Π(k) is computed here from a synthetic S⁵D structural field, not
from real observational data. The normalisation condition ε = 10⁻² is adopted
on physical grounds — it ensures that the projection contribution is a small
residual modification rather than a dominant signal — but its precise value is
not yet independently constrained by observations. The growth factor
approximation D(z) ∝ 1/(1 + z) used in the cross-redshift coherence calculation
is a matter-dominated approximation; a more precise calculation would use the
full numerical growth factor from CAMB [26].
The large-scale shape difference between ξ_Π(r) and ξ_ΛCDM(r)
at sub-horizon scales may reflect additional gravitational effects arising from
the dimensional structure of the projection, which in the broader SDMC
theoretical programme corresponds to contributions from higher-dimensional
layers S⁴D and S⁵D that are not directly observable in S³D. Figure 4 compares
the normalised projection prediction with galaxy correlation measurements
reconstructed from published SDSS LRG [7] and BOSS DR11 [27] results at sub-horizon
scales; the quantitative fit at these scales requires the survey-specific
selection functions, bias modelling, and window corrections, and belongs to the
observational fitting programme defined in Section VII.
Figure 4. Comparison of the normalised projection
prediction with galaxy correlation values reconstructed from published SDSS LRG
measurements (Eisenstein et al. 2005 [7]) and BOSS DR11 LOWZ measurements
(Anderson et al. 2014 [27]) at sub-horizon scales (r < 200 h⁻¹Mpc). (a)
Observed correlation functions against the ΛCDM baseline (z = 0.35, Planck
2018) and the projection prediction ξ_Π. (b) Residuals Δξ = ξ_obs − ξ_ΛCDM
(points) against the projection residual prediction (dashed). The data points
are reconstructions of published measurements rather than survey catalogue
analyses; the quantitative sub-horizon comparison requires survey-specific
selection and window corrections (Section XIV).
The next stage of the empirical programme, as outlined in
Section VII, involves fitting the projection model parameters against real
cosmological datasets — specifically the SDSS or DESI galaxy correlation
function [7, 21], the Euclid cross-redshift clustering data [3], and the Planck
CMB–large-scale structure cross-correlation spectrum [9, 20] — using the
cross-observable χ² minimisation procedure defined in Section VII, Algorithm 1.
This will determine whether the projected observable residual ⟨Δξ⟩
= 6.2 × 10⁻⁶ predicted under the current parameters is consistent with or
excluded by current data, and will provide empirical constraints on ε and σ
independent of the normalisation assumption adopted here.
Appendix A:
Non-Gaussian Kernel Extensions
The Gaussian kernel K(x,ξ;σ) = N⁻¹ exp[−D²(x,ξ)/2σ²] employed
throughout the main text represents the minimal parameterisation consistent
with the isotropy and normalisation constraints of the projection model.
However, the physical geometry of S⁵D may impose non-Gaussian correlation
structures that a single-scale Gaussian kernel cannot capture. Two alternative
kernel families are introduced here for completeness and for use in the full
numerical implementation [28, 29].
Matérn Kernel
The Matérn class of kernels is parameterised as:
K_ν(x,ξ)
= N⁻¹ · (2^(1−ν)/Γ(ν)) · (√(2ν) D(x,ξ)/ℓ)^ν · K_ν(√(2ν) D(x,ξ)/ℓ) (A1)
where ν controls the smoothness of the projection (ν = 1/2
gives an exponential kernel, ν → ∞ recovers the Gaussian), ℓ is the
characteristic coherence length, and K_ν denotes the modified Bessel function
of the second kind [30]. The Matérn kernel is preferred when the S⁵D structural
field is expected to have finite differentiability rather than infinite
smoothness, as in the Gaussian case. For ν = 3/2, the kernel reduces to:
K_{3/2}(x,ξ)
= N⁻¹ · (1 + √3 D/ℓ) · exp(−√3 D/ℓ) (A2)
which is computationally tractable and introduces a
characteristic correlation length ℓ distinct from the projection coherence
scale σ.
Lorentzian Kernel
For projection geometries in which S⁵D structural
correlations exhibit heavy-tailed behaviour — consistent with scale-free
relational networks — the Lorentzian kernel provides an alternative [29]:
K_L(x,ξ)
= N⁻¹ · (1 + D²(x,ξ)/γ²)⁻¹ (A3)
where γ is the half-width parameter governing the rate of
correlation decay. Unlike the Gaussian, the Lorentzian kernel decays as a power
law rather than exponentially, predicting stronger residual correlations at
large structural separations. This is directly relevant to the super-horizon
correlation floor ξ₀: under the Lorentzian kernel, ξ₀ is expected to be larger
in amplitude and more extended in scale than under the Gaussian approximation,
providing a sharper observational discriminant between kernel families.
The three kernel families make qualitatively different
predictions for the residual amplitude and scale dependence of ξ_Π(r). A
measurement of the super-horizon correlation floor that constrains both
amplitude and scale dependence simultaneously can therefore discriminate
between kernel families, providing an additional layer of empirical content
beyond the binary test of whether ξ₀ ≠ 0.
Acknowledgements
The author used Claude (Anthropic) and ChatGPT (OpenAI) as
linguistic, structural, and technical assistance in the drafting of this
manuscript, and for the numerical implementation of the projection model in
Python using the CAMB cosmological code, and for the generation of all figures
with author's instruction. All conceptual frameworks, model parameters,
physical interpretations, and final conclusions were developed and verified by
the author, who assumes full responsibility for the integrity of the work.
Declarations
Funding: This research received no external funding.
Conflicts of interest: The author declares no
conflicts of interest.
Data availability: No new observational data were
generated. The ΛCDM baseline was computed using the publicly available CAMB
code with Planck 2018 best-fit parameters. The projection field was generated
from a synthetic S⁵D structural field using the numerical procedures described
in the text. The observational values shown in Figure 4 are reconstructed from
the published SDSS and BOSS measurements cited therein.
Author contributions: Juliet Zhong: conceptualisation,
formal analysis, numerical implementation, writing.
References
[1] Zhong, J. (2026). Ripple-Instantiation Cosmogenesis: The
Six-Dimensional Spherical Cascade as an Alternative to Temporal Assembly.
Research Square preprint. https://doi.org/10.21203/rs.3.rs-9601290/v1
[2] Arfken, G. B., & Weber, H. J. (2013). Mathematical
Methods for Physicists: A Comprehensive Guide (7th ed.). Academic Press.
[3] Euclid Collaboration (2022). Euclid preparation. XIX.
Astronomy & Astrophysics, 662, A112.
https://doi.org/10.1051/0004-6361/202142419
[4] Shannon, C. E. (1948). A mathematical theory of
communication. Bell System Technical Journal, 27(3), 379–423.
https://doi.org/10.1002/j.1538-7305.1948.tb01338.x
[5] Boylan-Kolchin, M. (2023). Stress testing ΛCDM with
high-redshift galaxy candidates. Nature Astronomy, 7(6), 731–735.
https://doi.org/10.1038/s41550-023-01937-7
[6] Reed, M., & Simon, B. (1980). Methods of Modern
Mathematical Physics, Vol. I: Functional Analysis. Academic Press.
[7] Eisenstein, D. J., et al. (2005). Detection of the baryon
acoustic peak in the large-scale correlation function of SDSS luminous red
galaxies. The Astrophysical Journal, 633(2), 560–574.
https://doi.org/10.1086/466512
[8] Hamilton, A. J. S. (1993). Toward better ways to measure
the galaxy correlation function. The Astrophysical Journal, 417, 19.
https://doi.org/10.1086/173284
[9] Hu, W., & Dodelson, S.
(2002). Cosmic microwave background anisotropies. Annual Review of Astronomy
and Astrophysics, 40(1), 171–216.
https://doi.org/10.1146/annurev.astro.40.060401.093926
[10] Peebles, P. J. E. (1993). Principles of Physical
Cosmology. Princeton University Press.
[11] Peebles, P. J. E. (1980). The Large-Scale Structure of
the Universe. Princeton University Press.
[12] Eisenstein, D. J., & Hu, W. (1998). Baryonic
features in the matter power spectrum. The Astrophysical Journal, 496(2),
605–614. https://doi.org/10.1086/305424
[13] Huterer, D., & Starkman, G. (2003). Probing the dark
energy: Methods and strategies. Physical Review Letters, 90(3), 031301.
https://doi.org/10.1103/PhysRevLett.90.031301
[14] Engl, H. W., Hanke, M., & Neubauer, A. (1996).
Regularisation of Inverse Problems. Kluwer Academic Publishers. ISBN:
978-0-7923-4157-4.
[15] Rimes, C. D., & Hamilton, A. J. S. (2005).
Information content of the matter power spectrum. Monthly Notices of the Royal
Astronomical Society, 360(3), L82–L86.
https://doi.org/10.1111/j.1745-3933.2005.00051.x
[16] Springel, V., et al. (2005). Simulations of the
formation, evolution and clustering of galaxies and quasars. Nature, 435(7042),
629–636. https://doi.org/10.1038/nature03597
[17] Cooley, J. W., & Tukey, J. W. (1965). An algorithm
for the machine calculation of complex Fourier series. Mathematics of
Computation, 19(90), 297–301. https://doi.org/10.1090/S0025-5718-1965-0178586-1
[18] Landy, S. D., & Szalay, A. S. (1993). Bias and
variance of angular correlation functions. The Astrophysical Journal, 412(1),
64–71. https://doi.org/10.1086/172900
[19] Tegmark, M. (1997). How to measure CMB power spectra
without losing information. Physical Review D, 55(10), 5895–5907.
https://doi.org/10.1103/PhysRevD.55.5895
[20] Planck Collaboration (2020). Planck 2018 results. VI.
Cosmological parameters. Astronomy & Astrophysics, 641, A6.
https://doi.org/10.1051/0004-6361/201833910
[21] Tegmark, M., et al. (2004). Cosmological parameters from
SDSS luminous red galaxies. Physical Review D, 69(10), 103501.
https://doi.org/10.1103/PhysRevD.69.103501
[22] Cooray, A., & Sheth, R. (2002). Halo models of large
scale structure. Physics Reports, 372(1), 1–129.
https://doi.org/10.1016/S0370-1573(02)00276-4
[23] Foreman-Mackey, D., Hogg, D. W., Lang, D., &
Goodman, J. (2013). emcee: The MCMC Hammer. Publications of the Astronomical
Society of the Pacific, 125(925), 306–312. https://doi.org/10.1086/670067
[24] Speagle, J. S. (2020). dynesty: a dynamic nested
sampling package for estimating Bayesian posteriors and evidences. Monthly
Notices of the Royal Astronomical Society, 493(3), 3132–3158.
https://doi.org/10.1093/mnras/staa278
[25] NASA Webb Mission Team, Goddard Space Flight Center
(2026). NASA Webb Pushes Boundaries of Observable Universe Closer to Big Bang.
NASA Science. Published 28 January 2026.
https://science.nasa.gov/missions/webb/nasa-webb-pushes-boundaries-of-observable-universe-closer-to-big-bang/
[26] Lewis, A., Challinor, A., & Lasenby, A. (2000).
Efficient computation of CMB anisotropies in closed FRW models. The
Astrophysical Journal, 538(2), 473–476. https://doi.org/10.1086/309179
[27] Anderson, L., et al. (2014).
The clustering of galaxies in the SDSS-III Baryon Oscillation Spectroscopic
Survey: baryon acoustic oscillations in the Data Releases 10 and 11 Galaxy
samples. Monthly Notices of the Royal Astronomical Society, 441(1), 24–62.
https://doi.org/10.1093/mnras/stu523
[28] Matérn, B. (1960). Spatial Variation. Meddelanden från
Statens Skogsforskningsinstitut, 49(5). [2nd ed.: Springer, 1986.
https://doi.org/10.1007/978-1-4615-7892-5]
[29] Rasmussen, C. E., & Williams, C. K. I. (2006).
Gaussian Processes for Machine Learning. MIT Press. ISBN: 978-0-262-18253-9.
[30] Abramowitz, M., & Stegun, I. A. (1964). Handbook of
Mathematical Functions. National Bureau of Standards.
COPYRIGHT © 2026 Juliet Zhong. All Rights Reserved.
www.julietzhong.com
Comments
Post a Comment