Curvature Replication Cosmology

Allen, James Johan Sebastian

In Pattern Field Theory (PFT), large scale structure does not arise from the expansion of a pre-existing metric space. Instead, space itself emerges from curvature replication on the Allen Orbital Lattice (AOL). Curvature replication cosmology replaces scale factor evolution with the growth and interaction of discrete curvature shells. The observable universe is the integrated record of these replication events.

Curvature replication shells on the Allen Orbital Lattice — Conceptual diagram: discrete curvature shells replicating outward on the Allen Orbital Lattice, with density peaks along constructive interference nodes that later become filaments and clusters.

1. From metric expansion to curvature replication

Standard cosmology describes the universe with a Friedmann–Lemaître–Robertson–Walker (FLRW) metric and a time-dependent scale factor \( a(t) \). Distances stretch in proportion to \( a(t) \), and redshift traces this stretch. PFT replaces this picture with a discrete curvature field \( R(x,t) \) defined on the AOL. There is no global metric that stretches. Instead, new curvature is replicated outward from earlier curvature.

Let \( \mathcal{L}_{\text{AOL}} \) denote the curvature operator on the Allen Orbital Lattice. For a lattice site \( a_{i,j,\ell} \) with curvature \( \kappa_{i,j,\ell} \), replication is expressed as

\[ \kappa_{i,j,\ell}(t + \Delta t) = \kappa_{i,j,\ell}(t) + \mathcal{R}\big[\kappa_{\text{neigh}}(t)\big] \]

where \( \mathcal{R} \) is the replication kernel, acting on a local neighborhood of the lattice. Instead of scaling all distances, the universe grows by emitting new curvature shells that satisfy PFT coherence constraints, including the Crystalline Coherence Equation (CCE) and Phase Alignment Lock (PAL).

2. Curvature replication field

On large scales, the discrete replication process can be approximated by a continuous curvature replication field \( R(r,t) \), where \( r \) is comoving radial distance measured along AOL geodesics. Define \( R(r,t) \ge 0 \) as the local curvature replication density. Then the replication integral out to distance \( d \) is

\[ \chi(d) = \int_{0}^{d} R(s,t)\,\eta(s)\,\mathrm{d}s, \]

where \( \eta(s) \) is a scale dependent efficiency factor that incorporates layer depth, prime index density, and local CCE cost. The quantity \( \chi(d) \) measures the accumulated curvature replication along the path from the observer to distance \( d \).

In PFT, observables such as redshift, time dilation, and lensing are functions of \( \chi(d) \) rather than the FLRW scale factor. This allows curvature replication cosmology to reproduce ΛCDM-like behaviour where appropriate while predicting distinct deviations at specific scales and redshift ranges.

3. Trishift field and distance–signal relations

PFT replaces the single-valued distance–redshift function with a Trishift field \[ T(d) = \big(T_1(d), T_2(d), T_3(d)\big), \] where each component tracks one contribution to the observed signal:

\( T_1(d) \): energy and wavelength effects from curvature replication,
\( T_2(d) \): path geometry effects, including AOL geodesic detours,
\( T_3(d) \): phase and coherence effects affecting interference and polarization.

For a given line of sight, the primary component couples to the replication integral by

\[ \frac{\mathrm{d}T_1}{\mathrm{d}d} = H_{\text{pft}}\,\eta(d), \]

where \( H_{\text{pft}} \) is an effective PFT Hubble parameter, defined as an emergent summary of replication intensity and lattice depth, rather than a fundamental constant governing metric expansion.

Effective redshift \( z_{\text{pft}}(d) \) is then given by a deformation of the standard law

\[ 1 + z_{\text{pft}}(d) = \exp\big( \alpha_1 T_1(d) + \alpha_2 T_2(d) + \alpha_3 T_3(d) \big), \]

with coefficients \( \alpha_i \) constrained by CMBR data, baryon acoustic features, and galaxy survey anisotropies. This gives curvature replication cosmology a clean route to fit empirical data while retaining its structural foundations.

4. Curvature replication and structure formation

In ΛCDM, structure formation is driven by the growth of density perturbations in an expanding fluid. In curvature replication cosmology, filaments and voids are direct consequences of how curvature shells replicate and interfere on the AOL. Let \( \rho(r,t) \) denote effective matter density and \( \Phi(r,t) \) the PFT gravitational potential. PFT links them via a curvature sourced field equation

\[ \nabla^2 \Phi(r,t) = 4\pi G\,\rho(r,t) + \beta\,R(r,t), \]

where the additional source term \( \beta\,R(r,t) \) encodes the contribution of curvature replication itself to gravitational potential, without introducing a separate dark energy fluid. Regions with strong replication gradients naturally produce overdensities along AOL-compatible directions, matching filamentary structures and the observed fractal dimensionality around \( D \approx 2 \).

5. CMBR anisotropies and replication imprint

Curvature replication cosmology predicts that the cosmic microwave background radiation (CMBR) is an edge snapshot of a specific replication epoch. The power spectrum \( C_\ell \) is not only a record of acoustic oscillations, but also of how curvature shells replicated and interfered over AOL depth.

PFT expresses the anisotropy field as

\[ \Delta T(\hat{n}) = \sum_{\ell,m} a_{\ell m} Y_{\ell m}(\hat{n}), \]

with coefficients

\[ a_{\ell m} = \int \mathcal{K}_\ell(r)\,R(r,t_\ast)\,\Psi_{\ell m}(r,\hat{n})\,\mathrm{d}r, \]

where \( \mathcal{K}_\ell(r) \) is a replication kernel at decoupling time \( t_\ast \), and \( \Psi_{\ell m} \) encodes AOL geodesic and phase contributions. This structure allows PFT to explain:

low multipole anomalies as replication bias rather than primordial randomness,
hemispherical asymmetry as a directionally biased replication sector,
preferred axes and cold spots as persistent AOL-aligned features.

Curvature replication cosmology therefore treats the CMBR as a direct observational window into the replication history, rather than a simple projection of early fluid dynamics.

6. Curvature replication visualisation

The following schematic visualises curvature replication as expanding shells on a 2D slice of the AOL. Each ring corresponds to a replication front. Intensity encodes \( R(r,t) \); interference regions between fronts illustrate where filaments and nodes of the cosmic web are seeded.

7. Comparison with ΛCDM

Curvature replication cosmology and ΛCDM share several empirical targets while differing structurally:

Both describe an early hot, dense phase and a later dilute phase.
Both aim to reproduce the observed CMBR spectrum, large scale structure, and supernova distance moduli.
PFT removes the need for a separate dark energy component by attributing apparent acceleration to evolving \( R(r,t) \) and Trishift structure.

In PFT, the effective Hubble parameter and equation of state are emergent summaries of AOL-based replication, rather than fundamental dynamical drivers. This allows the same data to be fit under a different ontology, giving new leverage on anomalies such as Hubble tension, lensing discrepancies, and large scale alignments.

8. Predictions and falsifiability

Curvature replication cosmology makes several concrete, testable predictions:

Specific scale dependent departures from ΛCDM in distance–signal relations, especially at intermediate redshifts where Trishift components change dominance.
Preferred directions and axes in CMBR statistics that align with AOL-compatible great circles.
A fractal dimension of the cosmic web that stabilises near \( D \approx 2 \) over a wide range of scales, due to curvature carrier surfaces.
Systematic offsets in strong lensing reconstructions consistent with AOL shell geometry rather than a smooth metric background.

Data that violates these predictions in a robust, scale stable way would falsify the curvature replication model. Agreement within the predicted bands strengthens the case that replication, not metric stretch, is the correct structural description of cosmic evolution.

Pattern Field Theory