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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]

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.

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.

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

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.
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