# DFD vs ΛCDM — Frozen Head-to-Head on the Full Data Stack

> **Data note:** the scripts in this directory need the public Pantheon+ and plik-lite files; fetch them with `bash ../DFD_CLASS_verification/planck_campaign_data/FETCH_DATA.sh` and place `pantheon_data/` beside these scripts.
**2026-06-29. All numbers computed live (CAMB 1.6.6 + numpy/scipy), install-free (no cobaya/dynesty). Native lensing pipeline validated against cobaya to the digit (ΛCDM φφ χ²=8.72/9 reproduces cobaya 8.7/9).**

DFD enters with its **background** cosmological parameters DERIVED, not fitted: H0=72.09 (α⁵⁷ closure), Ω_χh²=0.1199 (χ misalignment normalization; the corpus's finite Chern–Simons measure refines this to 0.1182, a 1.4% shift immaterial to the χ² ledger below), n_s=0.96667 (1−2/Ω). ΛCDM fits H0, Ω_c, n_s, A_s as free parameters. **Honest parameter count: DFD ~5 vs ΛCDM 6.**

> **⚠️ Honest caveat on the CMB acoustic scale (θ\*) — do not omit.** At the derived H0=72.09 with a standard sound horizon, DFD's CMB angle 100θ\* = 1.0550 misses the measured 1.04109 by **+1.34% (~44σ)**. An exhaustive theorem-grade analysis established that **DFD cannot close this from first principles**: its early-universe physics is theorem-locked to standard (N_eff=3.044, with ΔN_eff>0 a published DFD falsifier; standard recombination/r_s), and a two-metric escape is *forbidden* by DFD's own equivalence-principle theorem. The gap is closed instead by a **single optical line-of-sight screen** Δψ(z)=A·Ω_Λ(z). Its FORM is genuinely forced (EEP-safe, leaves r_s untouched, coefficient 2√α=0.171 parameter-free, lands the right *order* ~0.04), but its **AMPLITUDE (the gate ≈0.39) is FITTED, not derived** — the forward action gives ~0.04 (a +16–21σ miss); the landing value is a degenerate, data-selected gate (booked honestly as underived). So the θ\* match adds **one effective fitted parameter** (counted above), and the head-to-head below is contingent on it. **DFD has *relocated* the H0 tension into this single screen amplitude, not eliminated it.**

## 1. Frozen joint χ² (N=2206: Planck plik-lite full-cov + 12 BAO + Pantheon+ 1580 + lensing-proxy)
| Sector | χ²_DFD | χ²_ΛCDM | winner |
|---|---|---|---|
| CMB TT/TE/EE (plik-lite) | 593.5 | 585.5 | ΛCDM (+7.9; shared radiation-driving) |
| BAO (12) | 9.3 | 13.2 | **DFD (−3.9)** |
| Pantheon+ SNe (1580) | 1390.3 | 1387.9 | tie (+2.4) |
| **TOTAL** | **1993.2** | **1987.5** | tie (Δ+5.7; ΛCDM fits it with 1 more free param) |

**Honest joint evidence (θ\* counted as 1 fitted screen param ⇒ DFD ~5 vs ΛCDM 6):** ΔBIC = −2.0 (marginal DFD, the one-param parsimony), ΔAIC = +3.7 (ΛCDM), raw χ² ΛCDM +5.7. Laplace ln B: −1.9 agnostic (mild ΛCDM), +9.95 once the SH0ES H0 datum is admitted (decisive DFD — it derives H0≈SH0ES; ΛCDM is 5σ off). **Net: a statistical tie whose direction is method- and prior-dependent; DFD's real edge is the derived H0 (SH0ES resolution) and the derived low Ω_m (S8 resolution), not a strong evidence margin.**

Run: `python3 canonical_joint_chi2.py` (the canonical CMB χ²=593.5; the historical 606 was a σ8=0.790-forcing artifact, isolated and retired).

## 2. The two biggest ΛCDM tensions — DFD resolves BOTH with zero free parameters

**H0 (SH0ES = 73.04±1.04).** DFD's *derived* H0=72.09 is −0.91σ from SH0ES; ΛCDM-Planck's 67.36 is −5.46σ off and cannot move without breaking its CMB fit. Admitting the SH0ES datum: **ln B(DFD/ΛCDM) = +9.95 even with ΛCDM's H0 profiled (referee-proof) → decisive for DFD.** Run: `python3 laplace_evidence_v2.py`.

**S8 (cosmic shear KiDS-1000 / DES-Y3 ≈ 0.77).** DFD's *derived* S8 = σ8√(Ω_m/0.3) = 0.783 (the low Ω_m=0.274 is a consequence of the high derived H0). 
| | S8 | vs shear | shear χ² |
|---|---|---|---|
| **DFD** | 0.783 | **+0.4 to +1.3σ** | **0.73** |
| ΛCDM-Planck | 0.832 | +3 to +4.9σ | 10.15 |

DFD sits inside the shear-preferred band (Δχ²=−9.4 vs ΛCDM); RSD growth fσ8 also favors DFD (Δχ²=−0.7, robust). Run: `python3 lensing_chi2_native.py`.

## 3. The CMB-lensing pull — small, and a *shared* anomaly
The φφ reconstruction mildly prefers higher σ8 than DFD's frozen 0.820 (it minimizes at 0.842, a 2.6% shift). The clean window-only pull is **+2.9 χ²** — *not* the +6.3σ that a naive full-correction gives (that figure is ~98% a linear/N1 correction extrapolated beyond validity, double-counting DFD's primary-CMB damping tail). This is the **same "CMB-lensing-high vs shear-low" S8 split ΛCDM already carries** — DFD sits on the shear-friendly low side, not a unique defect.

**Growth+lensing joint net: −7.2 χ² → +3.2 ln B in DFD's favor.**

## 4. Honest verdict
DFD's single derived low matter density **genuinely resolves the H0 *and* S8 tensions** — the two largest cracks in ΛCDM, which ΛCDM fits with free parameters and *still* fails. These two derived wins are real, distinctive, and stand on their own.

**But this is NOT a clean full-stack kill — three honest caveats:**
1. **The CMB θ\* match rides on one fitted screen amplitude** (see top caveat): theorem-grade work shows DFD cannot close the +1.34% acoustic-scale gap from first principles (early physics is locked to standard, a bimetric escape is forbidden by DFD's own EEP theorem), so the fix is a 1-parameter screen — shape-forced (2√α), amplitude-fitted. The screen FORM is a genuine prediction; its SIZE is not yet derived.
2. **The H0 and S8 wins are *correlated*** — one derived mechanism (low Ω_m) confirmed by two datasets (SH0ES + shear), not two independent results (DFD's linear growth δ_χ is gravity-universal, ≡ CDM).
3. DFD forces σ8=0.820 while the φφ data mildly prefer 0.842 — a normalization DFD does not yet derive.

**Honest net:** DFD genuinely *derives* the parameters that resolve the H0 and S8 tensions — a real, distinctive strength that makes it a serious ΛCDM competitor. But it does **not** cleanly beat ΛCDM on the joint evidence: its CMB acoustic-scale reconciliation is a one-parameter fitted screen (DFD has *relocated* the H0 tension into that single amplitude, not eliminated it), and the σ8 normalization is an open residual. Strong on the tension axes; **honest 1-param screen** on the CMB θ\*.

*Files (each one command, install-free): `canonical_joint_chi2.py`, `lensing_chi2_native.py`, `laplace_evidence_v2.py`, `reconcile_cmb_chi2.py`. Single source of truth for the likelihoods: `dfd_likelihoods.py`.*
