# SP-COS-OBS — redshift, Tolman SB, CMB acoustic horizon

### Named slice geometry · spectroscopic Hubble **FAIL** · Tolman **FAIL** · acoustic **not accounted** · G-17 not a theorem · chips OPEN

**CPT Framework — absolute-scale / origin book**  
**Date:** 2026-08-22  
**Status:** **SCORED_NO** · `PASS_SP_COS_OBS_SCORED_NO` · not a cosmology close · public chips **unchanged**  
**Parents:** SP-COS §6 bar · SP-TILT · SP-TIME · R-EFF · SFH-1 K4 · CL closeout · G-17 **[M]** unpromoted  
**Census:** `scripts/investigations/s3_geometry/s3_sp_cos_obs_redshift_tolman_acoustic.py`  
**Not:** FLRW \(a(t)\) as SP · tired light as already proved · G-17 as \(\ell_1\) theorem · naming \(R\) · \(\alpha\) mint · recycling \(H_0=69\)

Tags: **[E algebra]** · **[E honesty]** · **[O residual]** · **[Programme]** · **[F]**

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## 0. One-line

> **The named slice-projection geometry does not account for spectroscopic redshift, Tolman \((1+z)^{-4}\) surface brightness, or the CMB acoustic horizon.** Einstein product plus global \(\pi_\beta\) conserves Killing frequency (\(z_{\mathrm{spec}}=0\)). Tired light can linearise \(z\propto d\) only as a **new** law, and then fails Tolman and SNe dilation. A Hubble-time sound horizon on the Mach length gives \(\ell=\pi\sqrt{18}\approx 13.3\), about \(17\times\) too small versus \(\sim 220\). G-17’s exact \(220.50\) inserts \(\Lambda\)CDM \(z_*\) and a fitted bracket — **[M]**, not an SP theorem. Bangless SEL has no recombination history. Hold OPEN. α INPUT. No promote.

---

## 1. What is being tested

SP-COS named these probes as the bar that killed historical static / steady-state / tired-light alternatives. SP-TILT filled \(\Pi_{\mathrm{cosmic}}=\pi_\beta\) with SEL (tilt is a slice, not \(\beta(t)\)). This slice asks whether **that** named geometry, without FLRW \(a(t)\), carries:

| Probe | What “accounted for” would mean |
|-------|----------------------------------|
| **Redshift law** | Spectroscopic \(z(d)\) linear at small \(d\), from the named \(\Pi\) or the Einstein product, **not** a Hubble dial |
| **Tolman SB** | Bolometric surface brightness \(\sim(1+z)^{-4}\) (or a scored different exponent with dilation) |
| **Acoustic horizon** | First-peak scale \(\ell_1\sim 220\) from a named \(r_s\) and \(D_A\) of this geometry, without \(\Lambda\)CDM \(z_*\) as an input |

Etherington reciprocity \(d_L=(1+z)^2 d_A\) is the GR/FLRW partner of Tolman; it is not assumed as a theorem here. It is the bar.

---

## 2. Redshift

**Finding O1 [E algebra].** On the Einstein universe \(S^3\times\mathbb{R}\) (SP-SLICE / DYN-G), \(g_{00}=-1\) is constant. The Killing field \(\partial_t\) conserves angular frequency for static observers. **Spectroscopic \(z=0\) at every distance.** This is the *empty-product* channel. Static bulk-mass gravity is a different channel, scored **SCORED_NO** for Hubble in `a_geom_sp_z_grav_bulk_mass_results.md` (homogeneous \(S^3\) still \(z=0\); a mass ball blueshifts and is quadratic).

**Finding O2 [E algebra].** Global \(\pi_\beta\) is a time-independent linear map \(\mathbb{H}\to\mathbb{R}^3\). It can collapse leak-channel *cross-sections* (SP-TILT contact). It does **not** shift the \(e^{-i\omega t}\) factor of a Maxwell mode. **\(z_{\mathrm{spec}}=0\) still.**

Tired light \(z=\chi=d/R_{\mathrm{eff}}\) is **not** \(\pi_\beta\) and **not** the product metric. It is a new dissipative law. Census, using the published Mach length \(R_{\mathrm{eff}}\approx 3.285\times 10^{26}\,\mathrm{m}\) (ontology **unnamed**):

| Construction | \(H_0\) / km s\(^{-1}\) Mpc\(^{-1}\) |
|--------------|------------------------------------:|
| \(z=d/R_{\mathrm{eff}}\) | \(28.16\) |
| \(\sqrt{6}\,c/R_{\mathrm{eff}}\) | \(68.98\) (M13 dual) |
| M13 invert (published book) | \(68.97\) |

**Finding O3 [E honesty].** Matching \(\sim 69\) from \(R_{\mathrm{eff}}\) **is** the M13 dual, not a redshift theorem. Putting \(H_0\) into \(z=H_0 d/c\) by hand is the Hubble dial SP-COS forbids. Linear Hubble is **FAIL** on the named geometry; **possible** only by adding tired light, which is a different book.

---

## 3. Tolman surface brightness and dilation

Bolometric extra factor \((1+z)^{-n}\) *on top of* geometric dilution:

| Law | \(n\) | SNe time dilation |
|-----|------:|-------------------|
| FLRW (energy + rate + aberration) | **4** | \(1+z\) |
| Tired light (energy loss only) | **1** | **none** |
| SP-SLICE / global \(\pi_\beta\) (no \(z_{\mathrm{spec}}\)) | **0** | **none** |

**Finding O4 [E honesty].** Tolman \(n=4\) is **FAIL** for both the named slice geometry and tired light. SNe waveform stretching is **FAIL** for both. Those two probes are exactly why “expansion is propagation” was posture, not proof (four insights). Compact volume does not replace them.

---

## 4. Acoustic horizon

Bangless SEL has **no** recombination time, **no** baryon-photon sound speed history, and **no** native \(z_*\). So there is no SP-derived sound horizon \(r_s(z_*)\).

A *naive* contact that pretends a Hubble-time sound travel \(r_s=c/(\sqrt{3}\,H)\) with \(H=\sqrt{6}\,c/R_{\mathrm{eff}}\) (dual) and angular diameter \(D_A=R_{\mathrm{eff}}\) (\(\chi=\pi/2\)):

\[
\frac{r_s}{R}=\frac{1}{\sqrt{18}},
\qquad
\ell \;=\; \frac{\pi}{\theta} \;=\; \pi\sqrt{18} \;\approx\; 13.3.
\]

Observed first peak \(\ell_1\sim 220\). Naive ℓ is **too small by \(\approx 16.5\times\)**.

**G-17** writes \(\ell_1=\pi\sqrt{6}\,[(\pi/4)\sqrt{z_*}+2.7248]=220.50\) exactly. That formula **inserts** \(\Lambda\)CDM \(z_*\approx 1090\) and a fitted additive. Lab status is already **[M] unpromoted**. Using it as “SP accounts for the acoustic horizon” is **[F]**.

**Finding O5 [E honesty].** Slice geometry plus Mach length does **not** account for \(\ell_1\sim 220\). SFH-1 already stressed \(\lvert\Omega_k\rvert=1/6\) under standard recombination (\(\sim 167\times\) typical CMB \(\lvert\Omega_k\rvert\)). CL already failed to fix the quadrupole with \(S^3\) modes. This slice does not reopen those as theorems.

---

## 5. Board

| Probe | Named SP-SLICE / \(\pi_\beta\) | Tired light add-on | G-17 / FLRW recycle |
|-------|:------------------------------:|:------------------:|:-------------------:|
| Spectroscopic Hubble | **FAIL** (\(z=0\)) | Linear possible; **new law** | Not SP |
| Tolman \(n=4\) | **FAIL** | **FAIL** | FLRW, not SP |
| SNe dilation | **FAIL** | **FAIL** | FLRW, not SP |
| Acoustic \(\ell_1\sim 220\) | **FAIL** (\(\pi\sqrt{18}\approx 13\)) | No \(r_s\) | **[M]** fit, **FIRE** as theorem |
| CMB blackbody | OPEN as a cavity *possibility* | — | Not scored here (CL closed) |

**Finding O6 [E honesty].** Emergence as slice geometry can stay structured (INV vs SHADOW, SEL vs HIST) **and** fail these three cosmological observables. That is not a kill of the 3-layer picture. It is a kill of “therefore Hubble, Tolman, and \(\ell_1\) come for free.”

---

## 6. Still not

| Claim | Status |
|-------|--------|
| Slice cosmogony closed | **No** |
| Tired light proved | **No** |
| G-17 is an SP acoustic theorem | **FIRE** |
| \(R_{\mathrm{eff}}\) named as FLRW \(R_k\) / horizon | **No** |
| \(\alpha\) or \(m_e\) minted | **FIRE** if claimed |
| Public chips | **unchanged** |
| Promote | **NOT APPROVED** |

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## 7. Licensed next (not opened)

A later account would need a **new named law** for spectroscopic \(z(d)\) that is not \(a(t)\), not \(\tan\beta\) as \(H_0\), and not energy-loss-only, **and** a photon-SHADOW theorem that faces dilation + Tolman + a recombination (or replacement) history. That is a new principle. Do not restart SFH or CL. Do not open SP-TILT-HIST as a bang substitute.

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## 8. Integrity

| Gate | Verdict |
|------|---------|
| Three probes scored on named geometry | **PASS** |
| \(z_{\mathrm{spec}}=0\) on product / \(\pi_\beta\) | **PASS** |
| Tolman / dilation not sloganned | **PASS** |
| Acoustic miss \(\pi\sqrt{18}\) vs 220 named | **PASS** |
| G-17 not sold as theorem | **PASS** |
| \(R\) unnamed; α INPUT | **PASS** |
| Hold / public chips | **unchanged** |

**Exit:** `PASS_SP_COS_OBS_SCORED_NO`

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## 9. One-line for agents

> **§SP-COS-OBS: named slice geometry \(z_{\mathrm{spec}}=0\); Tolman \(n=4\) FAIL; acoustic \(\pi\sqrt{18}\approx 13\) not 220; G-17 [M] not theorem; not accounted; chips OPEN; no promote.**