# DFD-CLASS Corpus Extract — Synthesis Spec for Module 2 (perturbation + distance)

All equations below verified at file:line against the
canonical v4.0 source build (the LaTeX tree shipped with this package).
Honesty-guard applied both ways: where the corpus does NOT contain a law it is stated plainly;
nothing is invented. Each item tagged **DERIVED** (theorem-grade from DFD) vs
**ASSERTED / FITTED** (named hypothesis, fit-to-data, or reconstructed-from-LCDM).

Legend: `\fiteq{...}` is the corpus's own "fitted/observable-facing" equation macro; its
presence is a corpus-internal flag that the relation is reported-against-data, not pure-derivation.

---

## 1. Optical-ψ distance law D_obs(z)

### 1.1 Exact form (verified)

Optical (Gordon) metric, refractive index `n = e^ψ` — **DERIVED** (Postulate P1 + exterior
solution; metric quoted at `section_cosmology.tex:80`, formalism at `section_formalism.tex:555,562`):

```
d s~^2 = -c^2 dt^2 / n^2 + dx^2 ,   n = e^ψ
```

Distance biases — BOTH D_L and D_A multiplied by the SAME screen factor `e^{Δψ(z,n̂)}`:

- Luminosity   `D_L^DFD(z,n̂) = D_L^dict(z,n̂) · e^{Δψ(z,n̂)}`  — `section_cosmology.tex:65` (eq:psitomo-DL-bias, `\fiteq`)
- Ang.-diam.   `D_A^DFD(z,n̂) = D_A^dict(z,n̂) · e^{Δψ(z,n̂)}`  — `section_cosmology.tex:73` (eq:psitomo-DA-bias, `\fiteq`)
- Etherington  `D_L(z,n̂) = (1+z)^2 D_A(z,n̂)`  EXACT — `section_cosmology.tex:80` (eq:psitomo-duality, `\fiteq`)
  → screen CANCELS in the ratio; DDR-violation `Δψ_dual ≡ ln[D_L/((1+z)^2 D_A)] = 0` identically (`section_cosmology.tex:85`).
- LOS rep.     `Δψ_obs(z,n̂) = ∫_0^{χ(z)} dχ W_obs(χ;z) δψ(χ,n̂)`  — `section_cosmology.tex:49` (eq:psitomo-los-functional, `\fiteq`)

SNe distance modulus (from `D_L^DFD = e^{Δψ} D_L^matter`):
```
mu(z) = mu_matter(z) + 5·log10(e)·Δψ(z) = mu_matter(z) + (5/ln10)·Δψ(z)
```
Screen def. `Δψ(z) = ln(D_L^obs / D_L^matter)`, reconstructed `Δψ(z=1) = 0.274 ± 0.02` — `appendix_AU_dark_energy.tex:196`.

CMB acoustic-scale angular screen (GRIN magnification `θ_obs/θ_emit = n_emit/n_obs = e^{Δψ}`):
```
ell_obs = ell_true · e^{-Δψ(z*)}
```
`section_cosmology.tex:91` (eq:psitomo-cmb-ell1, introduced "DFD posits" → **ASSERTED**);
derivation `appendix_J.tex:217,231,236`. Worked: `220 = 297·e^{-0.30}` ⇒ Δψ(z*)=0.30, `n_CMB/n_here = e^{0.30}=1.35` (`appendix_J.tex:240-244`). Matches Planck ell_1 = 220.

### 1.2 Size of the screen Δψ(z) (verified table, `appendix_AU_dark_energy.tex:214`)

Reconstructed as `ln[D_L^LCDM(Ωm=0.30) / D_L^EdS]` (matter-only EdS baseline):

| z   | 0.1   | 0.3   | 0.5   | 0.7   | 1.0   | 1.5   | 2.0   | →∞    |
|-----|-------|-------|-------|-------|-------|-------|-------|-------|
| Δψ  | 0.049 | 0.126 | 0.184 | 0.227 | 0.275 | 0.327 | 0.358 | 0.502 |

Finite plateau `Δψ(∞) = ln(3.305/2.000) = ln(1.652) = 0.502` (bounded optical bias, not a growing fluid).
In band z≈0.4–2.3: `e^{Δψ}` ≈ 1.18–1.43 (distances 18–43% larger than EdS); `Δmu` ≈ 0.37→0.78 mag.
NOTE: this is vs matter-ONLY EdS; vs LCDM the screen IS the LCDM distance ratio (so w_eff=−1 by construction).

### 1.3 r_drag-safe? YES — verified

The screen is a **late-time, observer-side line-of-sight optical bias only**. At BBN/recombination
DFD's optical metric reduces to standard FRW (`ā_ext = 0` by Theorem AE.1), `H(T)` unchanged from
LCDM at T~MeV (`appendix_AH1b_new_territories_frontier.tex:830`, verbatim verified). Therefore
`r_s = ∫ c_s(ψ) dτ` (`appendix_J.tex:31`) is computed on the standard FRW background and r_drag /
pre-recombination physics are NOT modified. The reconciliation lever acts ONLY on `D_A(z*)/r_s`,
never on r_s. **This is exactly the keep-r_drag-fixed lever.**

### 1.4 DERIVED vs FITTED

- **DERIVED (theorem-grade):** the FORM — optical metric, common e^{Δψ} screening of D_L & D_A,
  exact DDR (Etherington reciprocity holds for the smooth Lorentzian optical metric).
- **FITTED / RECONSTRUCTED:** the PROFILE Δψ(z). `appendix_AU` builds Δψ(z) explicitly AS the LCDM
  distance ratio, so recovering `w_eff=−1` is circular — corpus says so verbatim: "by construction
  of Δψ from the ΛCDM ratio" (`appendix_AU_dark_energy.tex:205`; identical statement in
  thm:weff `appendix_AH2a_ckm_tau.tex:496`). The ell-screen is "DFD posits" (ASSERTED).
- **HONESTY-GUARD:** there is NO literal "+0.95% θ* tension" figure in the corpus. The "0.95" at
  `appendix_AG:59` is the +0.95σ agreement of the TM1 solar mixing angle — UNRELATED. Do not
  invent a θ* number; the acoustic-scale match is carried entirely by the ell=297→220 (26%) screen.

---

## 2. ψ-gravity perturbation closure (replaces Einstein/Poisson)

### 2.1 Modified Poisson (nonlinear, quasi-static) — **DERIVED**, `appendix_AC_linear_growth.tex:33`
```
∇·[ μ(x) ∇ψ ] = -(8πG/c^2) ρ ,    x = |∇ψ| / a_star
```
Acceleration map `a = (c^2/2) ∇ψ` (`section_formalism.tex:191`).

### 2.2 Interpolation μ(x) — **DERIVED theorem-grade** (thm:mu-theorem, `appendix_N_mu_derivation.tex:109`)
```
μ(x) = x/(1+x)   [+ O(k0^-1)]
```
Forced by S^3 Chern-Simons composition `μ(ψ)=1−e^{−cψ}` with `c=2/3` (slope-matching at x→0).
Limits: deep-MOND `μ→x` (x→0); Newtonian `μ→1` (x→∞).

### 2.3 Linearized anisotropic G_eff — **DERIVED** (`appendix_AC_linear_growth.tex:67-125`)
```
M_ij = μ_0 δ_ij + μ_ln,0 ĝ_i ĝ_j
μ_0      = μ(x̄)            = x̄/(1+x̄)
μ_ln,0   = x̄ μ'(x̄)        = x̄/(1+x̄)^2
G_eff(x̄,θ) = G / [ μ_0 + μ_ln,0 cos^2θ ] = (G/μ_0)·1/(1+q cos^2θ),  q = μ_ln,0/μ_0 = 1/(1+x̄)
cosθ = k̂·ĝ
```
KEY: G_eff is **anisotropic in k̂·ĝ**, NOT a pure scale-transfer G_eff(k,a). k-dependence enters
only through regime (x̄) + screening, not an explicit transfer function. No G_eff(k,a) table exists.

### 2.4 Linear growth ODE — **DERIVED** (`appendix_AC_linear_growth.tex:188`)
```
D'' + [2 + dlnH/dlna] D' - (3/2) Q Ω_m(a) D = 0 ,   Q = G_eff/G
```
First-pass growth (Ωm,0=0.30) from the EFE-screened envelope (`appendix_AC` tab:dfd-growth-envelope,
`\fittab`): at z=0.38, f_LCDM=0.705 vs f_∥=0.714 / f_avg=0.753 / f_⊥=0.774 — within ~8% of LCDM.

### 2.5 MOND acceleration gate a0(z) — **DERIVED at z=0; program-grade for a0(z)**
```
a_star(z) = a0(z) = 2√α · c H(z) ≈ 1.197e-10 m/s^2 at z=0   (matches obs ~3%)
closed form  a0 = 2 α^29 √(c^7/(Għ)) = 2 α^29 c/t_P     (appendix_AT:368, 0.17σ)
galactic crossover  a0 = 2√α · a_star ,  a_star ≡ cH0     (appendix_N:5)
RAR inversion  g_bar = g_obs · μ(g_obs/a0)
V_DFD/V_Newton = √(1/μ(x)) = √(1+1/x)                     (section_galactic.tex:154,313)
```
z=0 value DERIVED theorem-grade (thm:a-star-final, `appendix_N_mu_derivation.tex:265`); the
all-epoch promotion `a0(z)=2√α cH(z)` is labeled **program-grade** (epoch-consistency rule), NOT
theorem-grade.

### 2.6 Slip Φ vs Ψ — ZERO SLIP (Φ=Ψ), **DERIVED**
ψ sources a SINGLE scalar. Dynamics `a = -∇Φ_pot = (c^2/2)∇ψ` (`section_formalism.tex:191`);
lensing `Φ_lens = (c^2/2)δψ` (`appendix_AH1a:138`). Auxiliary 4D metric
`diag(-c^2 e^{-2ψ}, e^{2ψ}, e^{2ψ}, e^{2ψ})` (`section_formalism.tex:562`) ⇒ time & space
potentials equal magnitude ⇒ **η=1, Φ=Ψ, PPN γ=1, DDR η=1.01±0.02** (`section_cosmology.tex:804`).
For Boltzmann: lensing (Φ+Ψ) and dynamics (Φ) see the SAME potential — no anisotropic stress;
all deviation from GR is via G_eff/μ amplification only.

### 2.7 CRITICAL caveat for CMB/matter power (verified)
On the FRW background `|∇ψ̄| = 0` (flat direction, thm:aext-FRW `appendix_AE:66`), so cosmological
perturbations sit at **x̄=0 (raw deep-MOND)**, NOT the galactic x̄=5.85 envelope. The 1.02–1.17 G
envelope and x̄_EFE=1/(2√α)≈5.85 rely on the **NAMED, ASSERTED Hubble-EFE hypothesis a_ext=cH**
(`appendix_AC:134`), internally flagged as NOT what the action gives on FRW. ⇒ The μ-law G_eff does
NOT by itself supply cosmological clustering / the 3rd peak. Clustering is routed to the separate
DERIVED χ-matter field (§3), which obeys the same μ-law (so no double-count) but is Newtonian-screened
at recombination (`appendix_chi_matter:461`). The optical σ8 closure (`appendix_AE`) is a galactic/
cluster LENSING amplification (Q~500) of small baryon contrast, NOT an independent clustering source.
No two-field {Φ,Ψ} Boltzmann hierarchy and no G_eff(k,a) transfer function exist in the corpus;
production CAMB-equivalent is "program-grade" (`appendix_AC:254`).

---

## 3. χ field Boltzmann spec (the cold clustering carrier)

### 3.1 Mass — **DERIVED theorem-grade** (eq:m-chi, `appendix_chi_matter.tex:195`)
```
m_chi = √158 · Λ^2/f_chi = √158 · M̄_P · α^{2·8−3} = √158 · M̄_P · α^13 ≈ 5.1 eV
Λ = M̄_P α^8 (EW/Higgs-VEV scale, d=8);  √158 = 3(k_max − dim M7) − 1 = 3·53 − 1  (k_max=60, dimM7=7)
```
α-power = **13**.

### 3.2 Decay constant — **DERIVED theorem-grade** (eq:f-chi, `appendix_chi_matter.tex:166`)
```
f_chi = M̄_P · α^3 ≈ 9.46e11 GeV   (d_chi=3 from sl(2,C), per-mode Gaussian determinant Lemma O.4)
```

### 3.3 Cold vs warm — **COLD, non-thermal** (thm:chi-mass proof, `appendix_chi_matter.tex:216`)
Produced at REST (zero momentum, misalignment condensate) ⇒ cold regardless of the 5 eV mass.
Jeans `k_J ∝ √(m_chi H)` lies far below all CMB/LSS scales. NOT warm. **c_s ≈ 0**, clusters as CDM.

### 3.4 Abundance — **DERIVED mechanism, ANGLE-FITTED** (eq:omega-chi, `appendix_chi_matter.tex:243`)
```
Ω_chi h^2 ≃ 1.62 θ_i^2  →  0.12 at θ_i ≈ 0.27   (one O(1) free misalignment angle)
```
θ_i is the single irreducible free initial condition (not itself predicted).

### 3.5 16/3 ratio — **RETIRED / SUPERSEDED** (`appendix_J.tex:418`, verbatim verified)
"The earlier coincidence ω_I = (16/3) ω_b ≈ 0.12, recorded as 'a clue, not a derivation,' is now
superseded: the cold abundance is derived through the χ-matter misalignment chain … not through the
legacy 19=3+6+10 split." DO NOT cite 16/3 as the live derivation. (It survives only in retired
`chi_verification/*.py`, self-flagged "ASSUMED, NOT DERIVED"; in NO canonical .tex.)
Numerically Ω_chi/Ω_b = 0.12/0.02237 = 5.36 ≈ 16/3, but the ORIGIN is retired.

### 3.6 Linear Boltzmann behavior — **EXACTLY pressureless CDM** (`appendix_J.tex:391`)
χ "enters the photon-baryon perturbation kernel identically to a cold-dark-matter component of the
same Ω h^2" (thm:chi-thirdpeak, `appendix_chi_matter.tex:282`). Spec: **w=0, c_s^2=0, σ=0**.
δ_chi obeys the STANDARD cold-CDM hierarchy with NO DFD modification at recombination; the μ-law
G_eff is screened OFF (g/a0≫1 ⇒ μ→1) so it never touches 3rd-peak evolution.
HONESTY-GUARD: the corpus writes NO explicit DFD δ_chi′ continuity/Euler equation — it DEFERS to
standard CDM equations. Module 2 must use the standard CDM δ′,θ′ with Ω_c h^2 = 0.12.
CAVEAT: do NOT confuse χ with the separate ψ "dust branch" (`section_cosmology:831`) which the
corpus says FAILS (goes stiff/radiation-like). χ is what replaces it as the cold carrier.

---

## 4. Parameter audit (corpus-exact vs DFD-CLASS iter-1)

| Param | Corpus-exact value | Corpus form & location | iter-1 verdict |
|-------|--------------------|------------------------|----------------|
| H0 | **72.09** km/s/Mpc (=72.0902, round 72.1) | `H0=√(α^57 c^5/(Għ))=α^28.5/t_P`, `section_G_derivation.tex:109,116`. **DERIVED, zero-param** | 72.1 round-OK; headline is 72.09 |
| Ω_chi/Ω_b | **16/3 RETIRED**; use Ω_chi h²=**0.12** | `Ω_chi h²≃1.62 θ_i²`, θ_i=0.272, `appendix_chi_matter.tex:243`; retiral `appendix_J.tex:418`. **DERIVED-mech, angle-FITTED** | FIX: drop 16/3 (superseded), use 0.12 not 0.1193, flag angle-fit |
| n_s | **0.964** (NOT 0.9667) | `n_s=1−28.7α/(2π)≈0.964±0.003`, `appendix_AH3a:42`. Note "28.7α/2π=2/Ω numerically (number right, mechanism label not)" `:44` | FIX: use 0.964 not 0.96667; drop 1−2/k_max (corpus-DISAVOWED). n_s not in AT param-free table |
| A_s | **2.080e-9** | `A_s=32πα^5=2.080e-9` vs Planck 2.105(30)e-9, 0.6σ, `appendix_AT:368`. **forced-power / FITTED-coeff** (32π not first-principles) | value CORRECT; flag fitted coefficient, do not call fully derived |
| w, w_a | **(−1.0000, 0.0000)** machine-prec | thm:weff, `appendix_AH2a:487-508` (max resid ~1e-15). **DERIVED but by-construction** | CORRECT. Caveat: w_eff=−1 because Δψ built from LCDM ratio (circular); DESI evolving-w reframed as DE-clustering, "open test" `section_cosmology:999` |
| Ω_b h² | **0.02237** (BBN input, NOT derived) | "DFD does not derive η_B"; η_B=6.1e-10 input, `appendix_AH1b:830`; `ω_b h²=0.02237 shared with standard BBN` `section_cosmology:850` | CORRECT as ASSERTED/BBN-input. Self-consistent Ω_b=0.0430 at h=0.7209 = same physical density |

Stale-version warning: a retracted (w0,wa)=(−1.032,+0.071) still lingers at `appendix_AH2b:766` — do NOT use.

---

## 5. Items deferred to the Module-2 implementation

**Distance module (D_L, D_A, mu, ell_1):**
1. Compute baseline LCDM-equivalent `D_L^matter, D_A^matter` on a STANDARD FRW background using
   DFD H0=72.09, Ω_b h²=0.02237, Ω_chi h²=0.12 (treat χ exactly as Ω_c). Keep r_drag from the
   standard pre-recombination integral — DO NOT modify it.
2. Apply the common screen: `D_L^DFD = D_L^matter · e^{Δψ(z)}`, `D_A^DFD = D_A^matter · e^{Δψ(z)}`.
   Tabulate/interpolate Δψ(z) from the §1.2 table (or compute `Δψ(z)=ln[D_L^LCDM(0.30)/D_L^EdS]`),
   plateau 0.502. Distance modulus shift `+ (5/ln10)·Δψ(z)`.
3. Enforce EXACT DDR: `D_L = (1+z)^2 D_A` (e^{Δψ} cancels) — unit test `Δψ_dual=0` to machine precision.
4. CMB acoustic scale: apply `ell_obs = ell_true · e^{-Δψ(z*)}` with Δψ(z*)=0.30 so the no-CDM/GR
   ell_true≈297 maps to ell_obs≈220. The screen acts on D_A(z*)/r_s, NEVER on r_s.

**Perturbation module:**
5. Evolve χ with the STANDARD CDM Boltzmann hierarchy (w=0, c_s²=0, σ=0, Ω_c h²=0.12). No
   DFD-specific δ_chi′ — the corpus supplies none; μ-law is Newtonian-screened at recombination.
6. Slip = OFF: set Φ=Ψ (η=1, no anisotropic stress) throughout the hierarchy and lensing.
7. (Late-time / LSS option, NOT for the CMB run) implement the anisotropic G_eff from §2.3 only if
   modeling galactic/cluster RSD growth; gate it with a0(z)=2√α cH(z) and μ(x)=x/(1+x). Flag it
   program-grade and angle-dependent (cos²θ = (k̂·ĝ)²). It must NOT feed the 3rd-peak evolution.
8. Parameters to hard-code: H0=72.09, n_s=0.964, A_s=2.080e-9, w=−1, w_a=0, Ω_b h²=0.02237,
   Ω_chi h²=0.12. Drop 16/3 and 1−2/k_max entirely.

**Guards / unit tests:**
9. Assert r_drag identical with/without the screen (screen must be observer-side only).
10. Assert DDR η=1 exactly. Assert χ run reproduces a standard Ω_c h²=0.12 CMB (the corpus's own
    CAMB validation literally uses Ω_c h²=0.12).
11. Do NOT introduce a "+0.95% θ*" target — no such figure exists in the corpus.

**Honest-label ledger (carry into outputs):** distance-screen FORM = derived; PROFILE = LCDM-mimic
(circular, w_eff=−1 by construction); μ(x), m_chi, f_chi, H0 = theorem-grade derived; A_s coeff,
θ_i angle, a0(z) all-epoch = fitted/asserted/program-grade.
