# RLMT Companion Paper: Code Bundle

Supporting code for "R-layer Mode Theory: A Representation-Theoretic
Origin for the Mode Amplitude Coefficients, and a Negative Result for
Tensor-Sector Coupling" (companion to the RLMT/Planck 2018 paper,
Tohi 2026).

## Requirements

```
pip install camb numpy scipy --break-system-packages
```
Tested with CAMB 1.6.6, Python 3.12.

## Files and what they reproduce

- **cmbs4_forecast.py** -- Core module. Implements the RLMT primordial
  spectrum (Eq. 5.1-5.5 of the parent paper), a CAMB wrapper, a
  CMB-S4-like noise model, and the base Fisher-matrix builder
  (TT+TE+EE). All other scripts import from this file. Running it
  directly reproduces the full 13-parameter CMB-S4 Fisher forecast at
  the Planck best-fit point (Table 1 of the parent paper) and shows
  the near-singular behaviour discussed in Appendix on the exact
  scaling degeneracy.

- **cmbs4_forecast_extended.py** -- Eigen-decomposition of the Fisher
  matrix (identifies the exact beta_eff/c_n scaling degeneracy),
  external BAO/LSS Fisher-prior builders, and a CMB lensing (phi-phi)
  Fisher block with a flat-sky TT-only N0 reconstruction-noise
  calculation. Reproduces the sigma(kc), sigma(eps3) tables at the
  paper's detectability threshold (kc=0.3, eps3=0.02).

- **tensor_coupling_candidates.py** -- The full tensor-sector search.
  Implements six candidate tensor-coupling constructions (shared-kc
  cutoff, ad hoc -eps3 tilt, standard consistency relation, and three
  Clebsch-Gordan/quadrupole constructions) and the CMB-S4 BB Fisher
  pipeline. Reproduces every number quoted in the "tensor-sector
  coupling" section of the companion paper, including the two
  corrected bugs described there (tensor-only vs total BB; proper
  marginalisation over ns, As when combining with TT/TE/EE).

- **external_ns_prior.py** -- Reproduces the external-Planck-ns-prior
  result (importing sigma(ns)=0.0042 breaks the internal eps3-ns
  degeneracy in a way LSS/21cm/mu-distortion probes at the same k-range
  cannot).

- **mu_distortion.py**, **lss_shape_fisher_v2.py**,
  **future_probes_forecast.py** -- The mu-distortion (PIXIE-like),
  Lyman-alpha forest, and 21cm intensity-mapping Fisher forecasts
  (both the optimistic kmax=2 and the realistic kmax=0.3-0.5 +
  foreground/nonlinear-suppression versions) referenced in the
  "external probes" discussion.

## Typical usage

```bash
# Base Fisher forecast at the Planck best-fit (shows the degeneracy problem)
python3 cmbs4_forecast.py

# Eigen-analysis + external BAO/LSS + lensing, at the detectability threshold
python3 cmbs4_forecast_extended.py

# Full tensor-coupling candidate comparison (takes ~5 minutes: CAMB is
# called ~20-30 times with WantTensors=True)
python3 tensor_coupling_candidates.py

# External ns-prior comparison
python3 external_ns_prior.py
```

Note: `cmbs4_forecast_extended.py` and `tensor_coupling_candidates.py`
expect a cached `F_threshold.npy` (the 8-parameter Fisher matrix at
the kc=0.3, eps3=0.02 fiducial) in the working directory; running
`cmbs4_forecast_extended.py`'s `main()` first will generate it, or see
the inline `build_fisher(...)` calls to regenerate it directly.

## Caveats (see paper for full discussion)

- The 21cm and Lyman-alpha forecasts use a simplified Tegmark (1997)
  effective-volume Fisher formula, not real survey covariances; DESI
  and SKA's actual published forecasts (see paper Sec. on external
  data) generally import an external ns prior rather than
  self-constraining it, which the toy model here reproduces
  qualitatively but not quantitatively.
- The CMB lensing N0 is a flat-sky, TT-only minimum-variance estimator
  (Hu & Okamoto 2002 limit), not the full TT+TE+EE+EB+TB combination a
  real CMB-S4 analysis would use.
- All tensor-coupling constructions in `tensor_coupling_candidates.py`
  are explicitly flagged in-code as postulates, not derivations, with
  the single exception of the linear-response coherence calculation
  (candidate justification in the paper's Sec. on microscopic
  derivation), which is a genuine first-order perturbative result --
  and which is exactly what rules out the earlier postulates.