Cosmic Microwave Background in Pattern Field Theory

Allen, James Johan Sebastian

Cosmic Microwave Background in Pattern Field Theory

This page consolidates Pattern Field Theory’s (PFT™) current thinking on the Cosmic Microwave Background (CMB), the observational signatures we expect from Pi-Particle driven fractal curvature networks, and the reproducible tests performed to compare the Allen Orbital Lattice (AOL / π-matrix) with Planck and large-scale structure data. The material below presents hypothesis, data sources, methods, reproducibility notes and suggested next steps.

                            Executive summary

                            Pattern Field Theory predicts that the CMB preserves low-multipole signatures of an early “breach” emergence. These signatures are expected as structured deviations in low-ℓ modes and as coherent, lensing-like artifacts produced by π-driven curvature networks. The comparison pipeline (AOL → Healpix → Cℓ) is reproducible and available; initial runs show promising low-ℓ structure worthy of deeper statistical testing.

Navigation

Cosmos
Trivergence Sim
CMB
Resolved Paradoxes
Hypothesis
Data & Methods
Analysis Pipeline
Results & Interpretation
Reproducibility
Publication status
References

Universe-scale wedge/hex barrel implied by prime-driven growth — Figure: Macro geometry implied by prime-driven growth on the hex lattice — a wedge/hex-barrel shape consistent with an axial emergence model.
Image © 2025 Generated by PatternFieldTheory.com

Hypothesis

Pattern Field Theory (PFT™) projects the Allen Orbital Lattice (AOL / π-matrix) onto cosmological observables. The central hypothesis for the CMB is:

\[ \textbf{H}_0:\quad \text{Low-multipole structure of the CMB contains measurable components sourced by π-driven lattice modes.} \]

Operationally this predicts:

Excess or deficit power in specific low multipoles (ℓ < 40) aligned with dominant AOL axes.
Localized lensing-like artifacts at the angular scale of π curvature loops (predictive range ≈ 0.05–0.1 arcsec for small scale features when scaled to high-z structure).
Topological fingerprints (non-Gaussianity, persistent homology signatures) consistent with a lattice seeded cosmic web rather than pure Gaussian random fields.

Data & methods

Primary data

AOL simulation output — 3D point cloud (Nx3) exported from the lattice growth runs (runs 001–005). Use of axial q/r mapping for ring projection is described in the notebook.
Planck SMICA — full-sky CMB map (user-supplied FITS). We compare angular power spectra at matched nside and apply the same masks and beam treatment used by Planck where applicable.
Large-scale structure — optional crosschecks with SDSS/2dF/Euclid mock catalogs for filament statistics and connectivity metrics.

Key methods

Projection: radial-projection of AOL points to spherical coordinates (θ,φ) from configurable observer positions (origin or far-offset). Observer placement is a sensitivity knob.
Healpix mapping: counts (or mass-weighted counts) binned to Healpix; optional distance attenuation can be applied to mimic selection functions.
Power spectra: compute alm & Cℓ via healpy; visualisation uses ℓ(ℓ+1)Cℓ/2π convention.
Network analysis: build spatial graphs, compute MST length, degree, betweenness, and persistent homology for topological fingerprints.
Null tests: Monte-Carlo ΛCDM realizations and random rotated-lattice controls to estimate p-values.

Note: methodological choices (observer position, smoothing, mask) strongly affect low-ℓ comparison. All published figures should include the sensitivity table we show in the reproducibility section.

Analysis pipeline (skeleton)

Prepare lattice output — ensure points are in an Nx3 numpy array `aol_points.npy` (units arbitrary). Optionally apply scaling to match comoving distances.
Project to sphere — choose observer position; compute θ, φ for every point.
Make Healpix map — bin counts into pixels (nside default 128); apply Gaussian smoothing to mimic beam.
Compute Cℓ — extract alm via `hp.map2alm` and compute `hp.alm2cl`.
Compare to Planck — degrade Planck map to the same nside or upgrade your map; compute χ² on low-ℓ and perform Monte-Carlo tests.
Topological checks — compute persistent homology (Betti numbers) and filament network metrics; compare distributions against randoms.

                                Practical note

                                We have a ready skeleton script (Healpix + filaments) suitable for immediate runs. If you want, we will add a Jupyter notebook link here (`{{NOTEBOOK_LINK}}`) when uploaded to the repository.

Preliminary results & interpretation

The initial projection runs (AOL runs 001–005) produce low-ℓ structure that overlaps the multipole ranges where Planck reports anomalies. We caution that:

These are preliminary results: statistical significance depends on null-realization procedures, observer placement, and masking choices.
The lattice projection is not a cosmological simulation (no full Boltzmann/transfer modelling) — it is a targeted hypothesis test for global mode imprinting.

Big Bang to present: schematic cone showing cosmic expansion and CMB decoupling — Figure: Big-Bang cone schematic — the CMB is the snapshot near recombination and provides a testbed for global mode hypotheses. Image downloaded thanks to Wikipedia

Interpretive summary: if π-matrix global modes align with Planck low-ℓ features under robust null tests, that supports the claim that the breach event’s lattice modes left a measurable imprint. The next required steps are rigorous Monte-Carlo, blind tests, and cross-validation with independent pipelines.

Reproducibility & data availability

Reproducibility is essential. The following resources / steps are recommended for any public release:

Code: publish the Healpix skeleton, filament metrics and the notebook used for the final figures. Link here: add your repo link or replace `{{NOTEBOOK_LINK}}` when uploaded.
Data: include aol_points.npy (or a lightweight example) plus a small set of null realizations. Planck SMICA is publicly available; link to the Planck archive rather than hosting it.
Parameter table: always publish observer position, nside, smoothing σ, mask used, and the exact statistic (ℓ range) used for χ².
Blind tests: include rotated lattice controls and randomized point sets to exclude alignment by chance.

                                Repro tip:

                                Include a small example dataset (N≈50k points) so reviewers can run the notebook end-to-end within minutes without large downloads.

Publication status & queued tests

STATUS Preliminary results published — further robustness tests queued.

Published on this site

Run PDFs: run001 … run005
AOL → Healpix analysis description and method summary (this page)
Representative figures and schematic images (see figures/ folder)

Queued tests (in queue)

Monte-Carlo null ensemble — ≥1000 ΛCDM realizations to quantify low-ℓ significance.
Blind rotated-lattice controls — randomized rotations of the lattice to measure chance alignment rates.
Observer sensitivity sweep — project with multiple observer positions and selection functions.
Planck lensing & weak-lensing overlays — cross-correlation with lensing mass maps.
Filament cross-validation — compare AOL filament metrics with SDSS / Euclid mock catalogs.
Topological analysis — persistent homology and Betti number comparisons for lattice vs randoms.

When completed, each test will be accompanied by: (1) the notebook and analysis script, (2) a small example dataset for reproduction, and (3) a short results note linked from this page. Repository link and DOI will be provided once the release bundle is published.

Pattern Field Theory