# What would falsify DFD — the falsification ledger

DFD is a frozen-parameter theory: its cosmological numbers are derived, not fitted, so each is a sharp target.
Below, each claim and the measurement that would kill it. (Status as of 2026-06-27.)

## ✓ UPDATE (2026-06-27): CMB-lensing RECOMPUTED from scratch — DFD PASSES at +1.8σ (the earlier "FIRED" was σ8=0.790 arm + a cobaya bug)
A clean from-scratch CAMB-only recompute (no cobaya), verified by an independent recomputation that
**reproduced every number in CAMB 1.6.6**, overturns the earlier "≥6.5σ FIRED" headline. The
correct status, at DFD's **genuinely forced** σ8:
- DFD's forced primordial amplitude is **A_s = 32π·α⁵ = 2.080×10⁻⁹** (the α⁵ *power* is sympy-rigid/forced;
  the 32π *coefficient* is an asserted O(1) normalization — "forced-power, fitted-coefficient"). Matches Planck
  A_s at −0.8σ. Run forward through CAMB on DFD's frozen background → **σ8(0)=0.820, Ω_m=0.274, S8=0.785.**
- At σ8=0.820 the lensing-potential **C_L^φφ is a FLAT ~4% below ΛCDM across the whole Planck band L=8–400**
  (shape scatter 0.3% of the mean → *pure amplitude*, no shape penalty): A_L^DFD/A_L^ΛCDM=0.96 → data demand
  **A_L=1.046 → +1.8σ (a mild PASS; +1.0σ vs the data central value).** Because the ratio is flat, the full
  9-bin χ² cannot exceed this ~+1.8σ amplitude lean.
- **What the old "+232 / ≥6.5σ / ΛCDM decisively favoured" actually was:** the **σ8=0.790 arm** (ratio 0.89 →
  +5.2σ) run through a **cobaya-path normalization bug**. σ8=0.790 is NOT forced — reaching it requires
  shrinking A_s by 7.2% with no DFD justification (the corpus itself, App. GR.7, labels 0.790 "asserted, not
  independently re-derived"). The forced value is 0.820, and it passes.
- **Honest cost (a claim that comes DOWN):** the headline "S8=0.755 galaxy-lensing **win**" is just σ8=0.790
  algebraically restated; it is **not simultaneously available** with the CMB-lensing pass (one σ8 sets both,
  since DFD's linear growth is ΛCDM-like, Q=G_eff/G=1). At the forced σ8=0.820, galaxy-lensing softens to a
  **mild match, S8=0.785** (KiDS-1000 +1.1σ, DES-Y3 +0.5σ) — still on the low side, a real consistent result,
  but not the dramatic 0.755 win.
- **Open scoping caveat (flagged, not papered over):** the Q=1 cosmological-growth assumption that makes
  "one σ8 sets both" rests on χ clustering Newtonianly on linear scales (x≫1), while the corpus's own
  Lemma R8-deepMOND puts those same linear modes at x≈2×10⁻³≪1. The physical resolution (χ clusters via its
  own high local phase-space density inside collapsing regions, g≫a₀ — not the linear top-hat edge
  acceleration) is asserted, not yet derived. If instead the screen leaked into linear growth, Q would run
  with z and the S8=0.755 win could revive — so this is the deciding open item for the dark-sector growth sector.
See `cmb_lensing_figure.py` (CAMB-only, the recompute) and the independent verification of it.

## Already-passed falsification tests (would have killed DFD; did not)
| Prediction | Falsifier | Result |
|---|---|---|
| Joint fit ≈ ΛCDM on **CMB+BAO+SNe** (no lensing) | Δχ² ≫ √(2N) worse than ΛCDM | Δχ²≈+20/634 — **passed (these probes)** |
| Joint fit ≈ ΛCDM incl. **lensing** | Δχ² ≫ √(2N) on CMB+BAO+SNe+**lensing** | **passed** — mild +1.8σ at forced σ8=0.820 (the old "+232/≥6.5σ" was the σ8=0.790 arm + a cobaya normalization bug; flat C_L^φφ ratio ⇒ no shape penalty) |
| Screen gate = f_b=R/(1+R) | data prefer Ω_m or √Ω_m gate | rejected at 14.8σ / 13.6σ — **passed** (f_b at −1.6σ) |
| sin²θ_W = 3/13 = 0.2308 | measured sin²θ_W ≠ 0.2308 at the 0.2% level | matches MS-bar to −0.19% — **passed** |
| 3rd/1st CMB peak (no separate CDM) | χ at Ω_χh²=0.12 fails the peak-height ratio | 0.4445 vs ΛCDM 0.4433 — **passed** |
| Electron g−2 (DFD α) | a_e(DFD α) ≠ experiment | matches to ~6 ppb — **passed** |
| GW speed c_T=c | |c_T/c−1| measured ≠ 0 | GW170817 consistent — **passed** |
| **χ CDM isocurvature ≈ 0** (post-inflation: no frozen super-horizon angle, Step 5b) | β_iso > 0.038 (Planck) | β_iso ≈ 0 — **passed** (for the post-inflation reason, not a small fluctuation) |
| ~~**Global χ-string tension Gμ = α⁶**~~ **RETRACTED (2026-06-27, branch-consistency audit):** χ is the REAL coefficient of a single harmonic three-form (f_χ from a determinant, not a U(1) VEV with a winding complex |Φ|→0 core), so there is **no global cosmic-string network** and the Gμ=α⁶ tension was a VEV-picture import. Only the **N_DW=1 domain walls** (from the anomaly cosine minima, ∫ω₃=1) survive — those stay a sharp falsifier. | — | retracted; N_DW=1 walls kept |

## Live, decisive falsifiers (not yet decided — the real frontier)
| Prediction | Falsifier | Status |
|---|---|---|
| **SN–lensing sign:** δμ ∝ **+**κ_fg (objects look FARTHER through foreground mass) | a true weak-lensing κ test measuring a **negative** slope (ΛCDM magnification sign) at **>3σ** | **undecided** — proxy tests null & underpowered; needs a true κ map passing 3σ pre-flight (DECADE/LSST/Roman) |
| **No dark-energy clustering** (DE is an optical bias, not a fluid) | detection of DE perturbations / w(z)≠−1 fluid clustering (e.g. DESI dynamical-DE with sound speed) | not yet detected |
| **2PN light bending** coeff 4π=(16/15)×GR (+0.73 μas at solar limb) | astrometry measuring the 2PN coefficient ≠ 16/15×GR | below current precision (future Gaia/astrometry) |
| **H₀ = 72.09** (derived, α^28.5/t_P) | a model-independent H₀ measured away from ~72 at high significance | consistent with SH0ES-side |
| **Λ = (3/8π)α⁵⁷ρ_P** (dark-energy density) | ρ_Λ measured off the α⁵⁷ value | consistent (2.5 vs 2.24 meV, look-elsewhere noted) |
| **χ domain-wall number N_DW = 1** (from the S³ unit winding ∫_{S³}ω₃=1, Step 5b) | evidence the χ F F̃ anomaly index ≠ 1, i.e. N_DW > 1 (would over-close the Universe with stable walls) | **required & predicted**: N_DW=1 ⇒ walls bounded by strings self-destruct (safe); a structural test of χ's S³ origin |

## Honest caveats (where the current package is NOT yet publication-maximal)
- CMB uses Planck **plik-lite** (cobaya-validated), not the full **plik** with explicit foregrounds.
- Lensing: now the **full 9-bin bandpower** likelihood (CAMB-only recompute, no cobaya), NOT the old amplitude
  proxy — at the forced σ8=0.820 it is a **mild +1.8σ PASS** (flat C_L^φφ ratio ⇒ no shape penalty; see top of
  file). The σ₈Ω_m^0.25 proxy is retired; the earlier "≥6.5σ adverse" was the non-forced σ8=0.790 arm + a
  cobaya normalization bug.
- BAO uses consensus points with quoted errors, not the full **covariance**.
- No nested-sampling **Bayesian evidence** (PolyChord/MultiNest not installed). The proper **Laplace
  evidence** (`laplace_evidence.py`) gives ln B(DFD/ΛCDM) ≈ −1.2 → an evidence-level **TIE** (mildly ΛCDM
  on broad priors, mildly DFD on tight ones). The bare ΔBIC=−9.7 "strong DFD" headline is a 1/√N-shortcut
  artifact and is NOT supported by the proper Occam-factor calculation.
- ~~Genuinely fitted knob: **θ_i** (Ω_χ misalignment angle)~~ **SUPERSEDED (2026-06-27, appendix_chi_matter Step 5b,
  Thm chi-postinflation):** DFD's own inflation scale is rung-locked to **H★ = 8π f_χ** (both on the α³ rung;
  H★/f_χ = 8π ≈ 25.13, exact). δθ = H★/(2π f_χ) = 4 > π ⇒ the χ angle is randomized during inflation ⇒ the
  **post-inflation** scenario is FORCED and θ_i is **removed** as a knob (pre-inflation excluded; it would need
  θ_i ≳ 7×10⁵ to pass β_iso). The residual freedom **moves from the angle to the abundance normalization**: rms
  misalignment ⟨θ²⟩=π²/3 gives Ω_χh² ≈ 1.62·π²/3 ≈ **5.3, a ~44× OVERSHOOT of 0.12** — a NEW honest open problem
  (resolutions: entropy dilution Δ~40 / anharmonic-network correction / string-wall contribution / revised m_χ/f_χ;
  NOT yet adopted, honesty-guard: not papered over). The **A_s 32π coefficient** remains the one genuinely fitted
  cosmological coefficient. BIC parsimony edge contingent on accepting the H₀/n_s/A_screen/Ω_χ derivations
  (`paramcount_sensitivity.py`).
- Screen amplitude: the gate **form** is fixed by conformal structure — the optical metric is conformal and
  θ* is an angle, so the homogeneous screen cancels exactly and only the conformality-breaking baryon trace
  (f_b=R_d/(1+R_d)) survives; late-time Ω_m/Ω_b/bias/source provably cannot enter. The **coefficient 2√α** is
  inherited from the established a₀=2√α·cH relation (App AP). The overall late-screen **amplitude normalization
  remains the one open modeling step** (booked honestly against the theory in `HEAD_TO_HEAD_RESULTS.md`), not a
  derived win.

## The one-line standing
**DFD is globally competitive with ΛCDM on current data with fewer derived inputs (passed); the unique
SN–lensing sign prediction is the open knockout test, awaiting a true foreground weak-lensing map at scale.**
