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]
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.
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.
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 \(W(E)\).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
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.