# Results — the barrier is analyticity at vanishing strain, not “no \(\mathcal{L}(x)\)”

### Taylor \(W(E=0)\) \(\Rightarrow\) IR Newton · AQUAL is local and aquadratic · quasi-static one-scale continuum is Newton or a step · \(P\) is compactness · prize OPEN

**CPT Framework — FT-WE / FT-CONST-ROUTE / FT-AQUAL-LOCK / FT-LOCK-KEY / FT-PRIZE-PIN**  
**Date:** 2026-09-24  
**Status:** **BANKED** · `PASS_FT_LOCAL_L_BARRIER_ANALYTICITY_NOT_LOCALITY`  
**Stimulus:** User — the real problem is that every local pointwise 4D Lagrangian density \(\mathcal{L}(x)\) fails because local analytic elasticity is always Newtonian at small strain  
**Parents:** `a_geom_ft_w_of_e_results.md` · `a_geom_ft_const_route_results.md` · `a_geom_ft_aqual_lock_results.md` · `a_geom_ft_lock_key_results.md`  
**Script:** `scripts/investigations/s3_geometry/ft_local_l_barrier.py`  
**Data:** `scripts/investigations/s3_geometry/data/ft_local_l_barrier.json`

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

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## 0. One-line

> **The diagnosis is right; the quantifier is too wide.** Any strain energy **analytic at the identity** has a quadratic IR, hence Poisson, hence Newton in the outskirts (FT-WE). Hencky is already that UV. That is why leftover-as-ordinary-elasticity cannot pay the RAR. It does **not** follow that every pointwise density \(\mathcal{L}(x)\) fails. AQUAL \(\mathcal{L}=\rho\Phi+\frac{a_0^2}{8\pi G}F(X)\) *is* local in \(x\); it works as the 3D lock because \(F\sim X^{3/2}\) is **non-analytic** at \(\nabla\Phi=0\) (\(\mu(0)=0\)). Quasi-static *continuum* language with one leftover scale is still only Newton or a step (FT-CONST-ROUTE), which is not SPARC’s fat \(2\,\mathrm{dex}\). Compactness \(P\) is a different book (solvability on \(S^3\)), not this Taylor barrier. The 4D prize remains a constitutive map of the metric that is non-analytic at leftover \(K\), possibly local in \(x\) as AQUAL is, not exhibited. Do not write \(\mathcal{L}(g,K)\). Prize OPEN.

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## 1. Two properties that got fused

| Property | What it kills | What it does not kill |
|----------|---------------|------------------------|
| **Analytic** \(W\) at \(E=0\) | IR Newton (FT-WE, Born-Infeld, quartic, Hencky) | AQUAL / \(\mu(0)=0\) |
| **Local** density \(\mathcal{L}(x)\) | nothing by itself | AQUAL is local; Einstein–Hilbert is local |

Galactic IR is *small* well strain (\(\Phi/c^2\sim 10^{-6}\), \(g\sim a_0\)). Ordinary elasticity expands there and becomes Hooke. The RAR needs the opposite: the modulus to *vanish* as \(g\to 0\). That is non-analyticity at vanishing gradient, not non-locality of the density.

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## 2. Integrity

| Gate | Verdict |
|------|---------|
| Analytic local \(W\) is Newton in the IR | **YES** |
| Every local \(\mathcal{L}(x)\) fails | **NO** |
| AQUAL is a local \(\mathcal{L}(x)\) | **YES** |
| Quasi-static one-scale continuum is Newton or step | **YES** |
| \(P\) forced by this Taylor barrier | **NO** |
| 4D \(\mathcal{L}\) written / prize closed / promote | **NO** |

**Exit:** `PASS_FT_LOCAL_L_BARRIER_ANALYTICITY_NOT_LOCALITY`

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## 3. One-line for agents

> **§FT-LOCAL-L-BARRIER: analytic \(W(E=0)\) \(\Rightarrow\) IR Newton; that is the real local-elasticity kill; AQUAL is local in \(x\) and non-analytic in \(\nabla\Phi\); do not conclude “no \(\mathcal{L}(x)\)”; prize OPEN; no promote.**