# Opinion — A-geom river/PG model: UV is GR in other clothes; shear \(\neq a_0\)

### Painlevé–Gullstrand dual KIND · \(|\mathrm{d}v/\mathrm{d}r|\sim c/R\) needs \(v\sim c\) · galactic \(v_{\mathrm{PG}}/c\simeq 7.5\times 10^{-4}\) · Door-B exhaust not RAR · IR-PG not a \(c_{\mathrm{gw}}\) theorem · prize OPEN

**CPT Framework — FT-AQUAL-LOCK / FT-SHEET-VS-MEDIUM / FT-H0FREQ / FT-GW170817 / FT-P4-BULK-LEAK / TF-ADM / Book G**  
**Date:** 2026-09-24  
**Status:** **BANKED** · `PASS_FT_VEL_MODEL_SCORE_PG_DUAL_SHEAR_NOT_A0`  
**Stimulus:** User — consider Gemini A-geom (typed “Albenian”) geometry velocity model as a GR replacement  
**Parents:** `a_geom_ft_vel_model_consideration.md` · `a_geom_ft_aqual_lock_results.md` · `a_geom_ft_hubble_freq_results.md` · `a_geom_ft_gw170817_results.md` · `a_geom_tf_adm_path_cones_consideration.md`  
**Script:** `scripts/investigations/s3_geometry/ft_vel_model_score.py`  
**Data:** `scripts/investigations/s3_geometry/data/ft_vel_model_score.json`

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

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## 0. One-line

> **Keep the river as ADM-shift language for the metric. Do not replace GR, and do not identify leftover shear with \(a_0\).** Painlevé–Gullstrand is an exact coordinate dual of Schwarzschild: flat slices, shift \(\beta^i=-v^i\), \(v=\sqrt{2GM/r}\). In the UV that *is* GR, so SS tests and the factor-of-2 light deflection are not new proofs — they are GR in river clothes (TF-ADM already). Convective \((\mathbf{v}\cdot\nabla)\mathbf{v}=-GM/r^2\) is the same dual. The proposed IR, \(|\mathrm{d}v/\mathrm{d}r|\sim c/R\ \Leftrightarrow\ a\sim c^2/R\), replaces \(v\) by \(c\). At the MW solar circle \(v_{\mathrm{PG}}/c\simeq 7.5\times 10^{-4}\); the shear is \(\lvert\mathrm{d}v/\mathrm{d}r\rvert\simeq 460\,c/R\), the \(\Omega\) clock, not \(a/c\) (FT-H0FREQ). Matching leftover frequency would saturate at \(g\sim v\,c/R\simeq 0.0017\,a_0\), not \(a_0\). Dumping \(\nabla\cdot\mathbf{v}\neq 0\) into leftover-normal flux is P4 / Door B, not galactic RAR. An IR-modified PG metric is no longer Einstein; \(c_{\mathrm{gw}}=c\) at \(10^{-15}\) is then a gate, not a theorem. Putting Pythagorean \(g\) into \(v\,\mathrm{d}v/\mathrm{d}r\) is an interpolator shop. Photons seeing \(\Phi_{\mathrm{eff}}=-v^2/2\) is the constitutive prize in PG coordinates, not a derivation. Prize OPEN. No promote.

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## 1. What is right

| Claim | Score |
|-------|--------|
| PG river is a legitimate writing of spherical vacuum GR | **KIND** (Painlevé 1921, Gullstrand 1922) |
| UV SS / light / Shapiro if the metric *is* Schwarzschild | **YES** (GR dual, not a new mill) |
| Matter and light share one metric | **YES** (KiDS vs \(m(a)\) already) |
| Not a 3-area membrane (\(\cos\theta\) neck) | **YES** (tine B language) |
| Gaps named (PDE, 3-body, core) | **honest** |

TF-ADM already put cones on the path of the 3-geometry via \((N,\beta^i,K_{ij})\). PG is \(N=1\), \(\gamma_{ij}=\delta_{ij}\), \(\beta=-\mathbf{v}\). Useful dictionary. Not Einstein’s Hamiltonian.

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## 2. The illicit step (MW \(M=5\times 10^{10}M_\odot\), \(8.5\,\mathrm{kpc}\))

PG inflow \(v=\sqrt{2GM/r}\). Convective \(a=v\,\lvert\mathrm{d}v/\mathrm{d}r\rvert=GM/r^2\). The briefing sets \(\lvert\mathrm{d}v/\mathrm{d}r\rvert\sim c/R\) and writes \(a\sim c^2/R\). That needs \(v\sim c\).

| Quantity | Value |
|----------|--------|
| \(v_{\mathrm{PG}}/c\) | \(7.5\times 10^{-4}\) |
| \(\lvert\mathrm{d}v/\mathrm{d}r\rvert/(c/R)\) | \(460\) |
| \(\Omega/H_0\) (\(v=220\,\mathrm{km\,s^{-1}}\)) | \(370\) |
| \((a/c)/H_0\) | \(0.27\) |
| \(v_{\mathrm{PG}}(c/R)/a_0\) if shear were leftover | \(0.0017\) |

**Finding V1 [E algebra].** River shear is a *frequency*. Leftover \(c/R\) is a frequency. Matching them is the orbital/\(\Omega\) clock, already eliminated as a RAR organiser (FT-RAR-ELIM, FT-H0FREQ). The RAR clock is acceleration \(a/c\) versus \(c/R\).

**Finding V2 [E honesty].** \(v_{\mathrm{flat}}=(GMa_0)^{1/4}\) is orbital BTFR, not PG inflow \(\sqrt{2GM/r}\). Saturating the river speed is not the same object as a flat rotation curve.

The Euler line with Pythagorean \(\sqrt{g_N^2+g_N a_0}\) is FT-RAR-CURVE inserted by hand. Not a flow derivation.

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## 3. Other gates

**Light / KiDS.** If \(v(r)\) is exactly PG-Schwarzschild, null geodesics are GR. The heuristic \(n=1+v^2/c^2\) is isotropic-coordinate packaging, not an independent mill. If \(v(r)\) is *changed* in the IR, photons see that metric — extra \(\Phi\) *kind*, but only after the interpolator is put in.

**\(c_{\mathrm{gw}}\).** Acoustic \((\omega-\mathbf{v}\cdot\mathbf{k})^2=c^2k^2\) is the PG null cone while the metric is Einstein. An IR-saturated shift is a preferred-frame / aether-class metric. GW170817 remains a **hard gate** on the UV principal symbol (FT-GW170817), not a free lunch from \(v_{\mathrm{bg}}=0\).

**\(\nabla\cdot\mathbf{v}\neq 0\).** In GR that is \(K=-\nabla\cdot\mathbf{v}\) of the slice, not fluid disappearing. Sending the flux down leftover-normal \(n\) is 4-volume leak / Door B. P4: product \(T^{n\alpha}=0\); origin line PARK. Constitutive tine B does not need a 4th-direction drain.

**LLI / 3-body.** UV exact GR is safe. A global river PDE is not automatically 1PN n-body. LLR is a real unpaid gate, as the briefing said.

**Replace GR.** Overclaim. UV is GR. IR is the constitutive prize in shift language. No chip.

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## 4. Integrity

| Gate | Verdict |
|------|---------|
| PG UV = GR dual | **YES** |
| Replaces GR | **NO** |
| \(\lvert\mathrm{d}v/\mathrm{d}r\rvert\sim c/R\) is \(a_0\) | **NO** |
| KiDS / \(c_{\mathrm{gw}}\) derived beyond GR dual | **NO** |
| Door-B exhaust is leftover RAR | **NO** |
| Pythagorean Euler derived | **NO** |
| Useful ADM-shift language | **YES** |
| Prize closed / promote | **NO** |

**Exit:** `PASS_FT_VEL_MODEL_SCORE_PG_DUAL_SHEAR_NOT_A0`

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## 5. One-line for agents

> **§FT-VEL-MODEL: PG river is Schwarzschild in shift clothes, not a GR replacement; shear \(\lvert\mathrm{d}v/\mathrm{d}r\rvert\sim c/R\) is the \(\Omega\) clock (\(v\not\sim c\)); IR-PG is not a \(c_{\mathrm{gw}}\) theorem; Door-B flux not RAR; Pythagorean-in-Euler is a shop; prize OPEN; no promote.**