PFT vs ΛCDM — Predictions & Tests (v0.1)
Side-by-side predictions using PFT’s canonical language. This page lists the observable differences, the statistic to measure them, and the current test status. (Geometry appears at Emergence; π emerges at Emergence as the invariant of the first curvature.)
| Phenomenon | PFT (your terms) | ΛCDM baseline | Measurement / Statistic | Status |
|---|---|---|---|---|
| Origin mechanism | Emergence (geometry on) → π emerges with first curvature → Bang = 2D→3D supercritical instability (bamboo-like staccato chain reaction). | Past boundary (FLRW); no physical mechanism at \(t\!=\!0\) inside GR. | — | Canonical |
| First geometric object | Π-particle: minimal closed constant-curvature loop with \(\oint \kappa\,ds = 2\pi\). | No “first shape” specified. | Loop detection & rim metrics (see Π-locking below). | Defined |
| Π-locking on rims | Rims circularize under tension; **π emerges at Emergence**; closed loops satisfy \(\oint \kappa ds = 2\pi\); curvature variance drops. | No special π-locking expectation. |
\(R_\pi=\Big|\sum_i \kappa_i \Delta s_i - 2\pi\Big|\), curvature variance \( \mathrm{var}_w(\kappa) \), Hessian smallest eigenvalue \( \lambda_{\min}(\mathbf H_\pi) \). |
Field test |
| CMB TT/TE/EE spectra (shape) | Single-amplitude fit over \(30 \le \ell \le 1500\) matches Planck PR3 binned shapes; differences to emerge at fine scales / non-Gaussian stats. | ΛCDM fits full spectra with 6 parameters (plus nuisances). | \(\chi^2/\mathrm{DoF}\) for TT, TE, EE (binned); AIC/BIC vs ΛCDM (amplitude-only PFT). | Round-2: TT ≈ 1.00; TE ≈ 1.19; EE ≈ 0.79 (pipeline) |
| High-\(\ell\) micro-asymmetry | Tiny prime-seeded ripple band (order \(10^{-3}\) fractional) expected at high \(\ell\). | No specific ripple beyond noise/foregrounds. | Residual power vs ΛCDM; narrow-band search; look for stable peaks under mask/beam changes. | Search |
| Non-Gaussianity (staccato) | Chain-reaction rupture leaves weak bispectrum/tri-spectrum fingerprints. | Nearly Gaussian primary; small fNL. | Bispectrum templates aligned to rupture modes; nulls via phase randomization. | Design |
| Lensing micro-offsets | Minor systematic offsets (\(\sim 0.05''\text{–}0.1''\)) from sheet-to-shell nucleation history (hypothesis). | No such offset predicted. | Stacked strong-lens residual maps; centroid shift statistics. | Hypothesis |
| Topology / matched rings | Excess of near-π closed rims relative to controls. | No excess beyond chance. | Ring-finder → \(R_\pi\) distribution; KS against randomized skies. | Run |
Π-locking metrics
Definitions. For a sampled loop \((x_i,y_i)\) with segments \(\Delta s_i\) and discrete curvature \(\kappa_i\):
- Closure residual \(R_\pi=\big|\sum_i \kappa_i \Delta s_i - 2\pi\big|\).
- Curvature variance (length-weighted) \( \mathrm{var}_w(\kappa) \).
- Π-matrix stability via smallest eigenvalue \( \lambda_{\min}(\mathbf H_\pi) \) of the π-locking Hessian.
Acceptance bands (tunable): Π-locked candidates typically have \(R_\pi \lesssim 10^{-2}\), low variance, and small positive \( \lambda_{\min} \).
Breach instability (2D→3D)
Not a “controlled rupture”: when size–tension–potential cross a threshold, long-wave modes go unstable together (bamboo-like, staccato chain reaction):
\[
\mathcal{F}=\int \Big[\tfrac{\kappa}{2}(\nabla^2 h)^2+\tfrac{\sigma}{2}\lvert\nabla h\rvert^2-\tfrac{\alpha}{2}h^2\Big]\,dA,\quad
\kappa k^{4}+\sigma k^{2}-\alpha<0,\quad
L_{\text{crit}}=\frac{2\pi}{k_c}.
\]
Test protocol (Round-3 — CMB, map-level)
- Data: Planck 2018 PR3 maps (e.g., SMICA); apply official mask/beam \(W_\ell\).
- Covariance: use full bandpower covariances for \(\chi^2\) (no scalar σ shortcuts).
- Fit: single-amplitude to TT/TE/EE over \(30\!\le\!\ell\!\le\!1500\); record \(\chi^2/\mathrm{DoF}\), AIC/BIC vs ΛCDM.
- Residuals: inspect high-\(\ell\) narrow-band excess; run bispectrum template tests.
- Robustness: vary masks, recalibrate beam, recheck stability.
Status (internal pipeline)
- Round-2 shapes: TT \(\chi^2/\mathrm{DoF}\approx 1.00\), TE \(\approx 1.19\), EE \(\approx 0.79\) (amplitude-only fit).
- Π-locking toolkit prepared: outputs \(R_\pi\), curvature variance, and \( \lambda_{\min}(\mathbf H_\pi) \).
- Next: ring statistics on maps, high-\(\ell\) ripple search, bispectrum pass.