# Results — leftover foam residual rhymes with the DE *scale*, not with \(q_0<0\)

### Clothes \(cH_0\sim a_0\sim c^2/R\) · wall energy density \(O(1)\times\rho_\Lambda c^2\) · spring-back is \(\dot R\), FIRE against lock · frozen residual is GE-J wall · closed \(S^3\) without \(\Lambda\) decelerates · sky \(z\) TO BED

**CPT Framework — FT-FOAM-RESIDUAL / FT-VOL4-MULTIPLIER / FT-R-LOCK / FT-A0-Z / FT-S3-PHI / FT-GEJ-A0-COMPUTE / FT-S3ROT / SFH-1**  
**Date:** 2026-09-23  
**Status:** **BANKED** · `PASS_FT_FOAM_DE_SCALE_KIND_NOT_Q0`  
**Stimulus:** User — might moving leftover / memory-foam residual explain dark-energy attributed observations, in full or in part?  
**Parents:** `a_geom_ft_foam_residual_results.md` · `a_geom_ft_r_lock_freeze.md` · `a_geom_ft_a0_z_results.md` · `a_geom_ft_s3_phi_results.md` · `a_geom_ft_s3_rotation_ga_de_consideration.md` · `a_geom_ft_gej_a0_compute_results.md`  
**Script:** `scripts/investigations/s3_geometry/ft_foam_de.py`  
**Data:** `scripts/investigations/s3_geometry/data/ft_foam_de.json`

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

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## 0. One-line

> **In part, the *scale*. Not in full, and not as \(q_0<0\).** Dark-energy *attribution* is an FLRW kinematic (\(q_0<0\), \(\Omega_\Lambda\sim 0.7\)). Leftover \(R\) is spectrally locked, not an FLRW \(a(t)\) (FT-R-LOCK). A memory-foam *spring-back* of leftover is \(\dot R\neq 0\), which is that FIRE, and \(a_0(z)\) from expanding \(R\) has the wrong sign (FT-A0-Z). What *does* rhyme is the acceleration board: \(cH_0\simeq 5.67\,a_0\), leftover \(c^2/R\simeq 2.31\,a_0\), de Sitter acceleration at \(r=c/H_0\) is \(\Omega_\Lambda cH_0\simeq 3.97\,a_0\). Umbilic-wall energy density \(\sigma_B/R\) is \(\simeq 0.32\,\rho_\Lambda c^2\) on the two-ball glue (same \(O(1)\) convention class as GE-J). That is clothes, not a supernova theorem. Frozen residual = leftover wall, acceleration along the leftover *normal*, not an isotropic 3-acceleration. In-slice \(g_\Lambda=\Omega_\Lambda H_0^2 r\) is \(8\times 10^{-6}\,a_0\) at \(8.5\,\mathrm{kpc}\) (FT-S3-PHI) — not the RAR, not a Hubble diagram. Closed \(S^3\) without \(\Lambda\) *decelerates* (\(q_0\approx +0.58\), SFH-1). Sky \(z\) stays TO BED. Do not mint \(\Lambda\). Prize OPEN.

---

## 1. Split the claim

| Object | What it is | This mill |
|--------|------------|-----------|
| DE-attributed *observations* | SN dimming, BAO, CMB acoustic scale, late ISW, growth | **not paid** (sky \(z\) TO BED; no Hubble diagram mill) |
| ΛCDM *attribution* | \(q_0<0\), \(\Omega_\Lambda\simeq 0.7\) fluid | **not identified** with leftover foam |
| Scale coincidence | \(a_0\sim cH_0\sim c^2/R\) | **KIND / clothes** (\(\sqrt{6}\) still **[M]**) |
| Foam still recovering | \(\tau\sim R/c\), leftover \(K(t)\) | **FIRE** if it is \(\dot R\) |
| Foam frozen | leftover has not relaxed | **GE-J wall rename** |

“In part” is allowed only for the scale rhyme. “In full” would be a SN/BAO/CMB theorem. That is not exhibited.

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## 2. Scale board (clothes \(R=\sqrt{6}\,c/H_0\), \(\Omega_\Lambda=0.7\) input not derived)

| Quantity | / \(a_0\) |
|----------|-----------|
| \(cH_0\) | \(5.67\) |
| leftover \(c^2/R\) | \(2.31\) |
| de Sitter \(g_\Lambda(r=c/H_0)=\Omega_\Lambda cH_0\) | \(3.97\) |
| \(g_\Lambda\) at \(8.5\,\mathrm{kpc}\) | \(7.9\times 10^{-6}\) |
| \(g_\Lambda\) at \(300\,\mathrm{kpc}\) | \(2.8\times 10^{-4}\) |
| umbilic \(\sigma_B/R\) vs \(\rho_\Lambda c^2\) | \(0.32\) (ratio of energy densities) |

**Finding D1 [E algebra].** The cosmic acceleration *scale* and leftover \(K\) are the same \(O(1)\) board already used for \(a_0\sim c^2/R\). Wall energy density vs \(\rho_\Lambda c^2\) is the same convention class as GE-J’s \(0.18\)–\(2.3\,a_0\). Do not select the cell that hits \(\Omega_\Lambda\) after the fact.

**Finding D2 [E honesty].** In-slice de Sitter is linear in \(r\) and is negligible on RAR radii (FT-S3-PHI). It is not extra galactic \(\Phi\), and it is not a substitute SN diagram.

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## 3. Why foam does not pay \(q_0<0\)

**Spring-back.** Memory foam recovering is leftover trying to change shape. The only leftover length is \(R\). \(\dot R\neq 0\) is expanding (or shrinking) leftover, FT-R-LOCK **FIRE**. If \(a_0=c^2/R\) ran, high-\(z\) would see *more* yield, not baryon Newton (FT-A0-Z, wrong sign). Hubble expansion, if real, is **not** \(\dot R/R\) of this container.

**Frozen residual.** \(\tau\gg t_H\) so leftover has not relaxed: that is a *constant* leftover \(K\), i.e. the umbilic wall. Acceleration is along the leftover **normal** (GE-J), not an isotropic 3-acceleration \(q_0\). Tensor structure does not match (same class as FT-S3ROT: vorticity \(\neq\Lambda\)).

**Closed \(S^3\) kinematics.** SFH-1: no-\(\Lambda\) closed distances give \(q_0\approx +0.58\) (**deceleration**). Curvature alone does not flip to acceleration. Weber did not free-derive DE (SFH-W). A CPT \(\Lambda\) sector remains shopping.

**Redshift.** SN “acceleration” is a luminosity-distance vs \(z\) fit. Sky \(z\) stays TO BED. This mill does not write a Hubble diagram.

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## 4. Integrity

| Gate | Verdict |
|------|---------|
| Scale rhyme leftover \(K\) / \(cH_0\) / \(\rho_\Lambda\) | **KIND**, not a \(W\) |
| Foam spring-back pays \(q_0<0\) | **NO** (\(R\)-lock FIRE) |
| Frozen foam pays SN Hubble diagram | **NO** |
| In-slice \(g_\Lambda\) pays RAR | **NO** |
| Closed \(S^3\) without \(\Lambda\) accelerates | **NO** (decelerates) |
| DE-attributed observations paid | **NO** |
| \(\Lambda\) minted as a fix | **NO** |
| Sky \(z\) / Sciama unparked | **NO** |
| Prize closed / promote | **NO** |

**Exit:** `PASS_FT_FOAM_DE_SCALE_KIND_NOT_Q0`

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## 5. One-line for agents

> **§FT-FOAM-DE: leftover foam/wall rhymes with the DE *scale* (\(cH_0\sim c^2/R\), \(\sigma/R\sim 0.3\,\rho_\Lambda c^2\)); spring-back is \(\dot R\) FIRE; frozen residual is GE-J wall not \(q_0\); closed \(S^3\) decelerates; SN/BAO/CMB not paid; sky \(z\) TO BED; no \(\Lambda\) fix; prize OPEN; no promote.**