# Results — continuous \(W(E)\) with leftover \(R\), no fitted \(\mu(x)\)

### Analytic \(W\) at \(E=0\) \(\Rightarrow\) IR **Newton** (wrong end) · high-\(g\) is where it modifies · simple \(\mu\) is a choice · leftover OPEN

**CPT Framework — FT-A0-KIND / BP-1 / FT-W12-LOCAL / GE-J**  
**Date:** 2026-09-22  
**Status:** **BANKED** · `PASS_FT_WE_ANALYTIC_IR_NEWTON`  
**Stimulus:** User — authorize \(W(E)\) with background \(R\); test if a smooth tail emerges without fitting \(\mu(x)\)  
**Parents:** FT-A0-KIND · BP-1 Hencky=Newton · FT-RICE  
**Script:** `scripts/investigations/s3_geometry/ft_w_of_e.py`  
**Data:** `scripts/investigations/s3_geometry/data/ft_w_of_e.json`

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

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## 0. One-line

> **A continuous strain energy that is analytic at zero strain cannot produce the observed low-\(a\) tail.** Spherical vacuum is \(\mu(g)\,g=g_N\). Analytic \(W(E)\) \(\Rightarrow\) \(\mu(0)\) finite \(\Rightarrow\) \(g\to g_N\) in the outskirts (Newton, \(1/r\) *potential*). Quartic and Born-Infeld-style laws with leftover \(a_\star=c^2/R\) **saturate the centre** (\(g/g_N\sim 0.005\) at 0.1 kpc) and are Kepler at 50 kpc (\(g/g_N\to 1\)). That is the **wrong end** of the RAR. Hencky quadratic is already Newton (BP-1). A “simple” \(\mu=x/(1+x)\) with the same \(a_\star\) *does* boost the outskirts (\(g/g_N\sim 10\) at 50 kpc) — because \(\mu(0)=0\), which is **non-analytic** and a **choice**, not a derivation. No SPARC fit was used for \(a_\star\). Leftover OPEN.

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## 1. Spec

\(W=W(E;R)\), one scale \(R=\sqrt{6}\,c/H_0\). Spherical well, Book G mass. No Book C \(\mathbf{J}\). No fitted interpolator.

If \(W\) is Taylor-expandable at \(E=0\), the IR (dwarfs, outskirts) is the quadratic piece \(\Rightarrow\) Poisson.

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## 2. Board (\(M=5\times 10^{10}M_\odot\), \(a_\star=c^2/R\simeq 2.3\,a_0\))

\(g/g_N\):

| \(r\) (kpc) | Analytic quartic | Strain-cap \(\sqrt{1+(g/a_\star)^2}\) | Simple \(\mu=x/(1+x)\) **choice** |
|-------------|------------------|----------------------------------------|-----------------------------------|
| 0.1 | 0.005 | 0.020 | 1.00 |
| 1 | 0.11 | 0.20 | 1.04 |
| 8.5 | 0.91 | 0.95 | 2.27 |
| 20 | 1.00 | 1.00 | 4.52 |
| 50 | **1.00** | **1.00** | **10.5** |

**Finding W1 [E algebra].** Analytic \(W\): IR Newton. UV wrecked (centres suppressed). Opposite to FT-A0-KIND.

**Finding W2 [E honesty].** The simple interpolator “works” in the outskirts only because it was built to have \(\mu\to 0\) as \(g\to 0\). That is not an emergent tail from \(W(E)\).

**Finding W3 [E process].** A pudding-skin / low-\(|g|\) phase that matches the data is **non-analytic at vanishing strain**. That is allowed as a question; it is GE-J/yield language, not a smooth hyperelastic \(W(E)\).

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## 3. Integrity

| Gate | Verdict |
|------|---------|
| Dedicated \(W(E)\) test | **PASS** |
| Smooth MOND tail from analytic \(W\) | **NO** |
| \(\mu(x)\) fitted | **NO** |
| Chip / CDM / SPARC \(I\) | **NO** |

**Looked into (22 Sep, combinatory doctrine):** Floor + Bingham + CGM is a coherent *picture*. Tuning \(\sigma_{\mathrm{yield}}\) or a CGM profile to SPARC is a **parameter trap**. A derivation still demands: yield uniquely from leftover 4-ball \([K]\) (already computed \(\sigma\), but a **wall**); transition from well \(\times E_{\mathrm{cosmic}}\) without choosing \(\mu(x)\). That lock is **not exhibited**. `a_geom_ft_three_combo_doctrine.md`.

**Exit:** `PASS_FT_WE_ANALYTIC_IR_NEWTON`

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## 4. One-line for agents

> **§FT-WE: analytic \(W(E)\) \(\Rightarrow\) Newton in the IR and saturation in the UV (wrong end); simple \(\mu\) is a choice; leftover OPEN.**