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]
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 ina_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 |
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.
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
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.