Results — constitutive prize: a realistic Lagrangian route exists, and it is not another \(W(E)\)
Local analytic elasticity is closed · quasi-static rheology is Newton or a step · thin GE-J skin too sharp for SPARC’s \(2\,\mathrm{dex}\) · live: non-graph 4-dent, fat yield zone \(\sim r_y\) · \(P\) on \(S^3\) is necessary and not the interpolator · prize OPEN
CPT Framework — FT-CONST-PRIZE / FT-WE / FT-BEND / FT-RAR-PROFILE / FT-S3-SPECTRUM / GE-J gated / FT-GW170817 Date: 2026-09-22 Status: BANKED ·PASS_FT_CONST_ROUTE_LIVE_NOT_EXHIBITED
Stimulus: User — the transition profile must emerge from the 4D bulk elasticity/deformation Lagrangian; is there a realistic route to model or determine it?
Parents: FT-CONST-PRIZE · FT-WE · FT-BEND · FT-3COMBO · FT-RAR-PROFILE · FT-S3-SPECTRUM · FT-GEJ-A0-DESIGN
Script: scripts/investigations/s3_geometry/ft_const_route.py
Data: scripts/investigations/s3_geometry/data/ft_const_route.json
Tags: [E honesty] · [E algebra] · [Programme] · [O residual]
0. One-line
Yes: two live routes, neither exhibited, both gated. A local strain energy analytic at the identity cannot mint the RAR bend (FT-WE: IR Newton, UV wrecked). Quasi-static local rheology with one leftover scale is Newton or a step (viscoelastic relaxes; Bingham at \(\dot\gamma\to 0\) is a floor). SPARC’s \(\sim 2\,\mathrm{dex}\) in \(g\) is \(\Delta r/r\simeq 2.3\) on a \(1/r^2\) well — a fat yield zone of thickness \(\sim r_y\), not a pudding skin and not an Israel wall. The leftover \(S^3\) projector \(P\) is forced by compactness and still does not pay kpc RAR (Green is Coulomb at \(r\ll R\)). What remains: (1) drop the graph ansatz of FT-BEND and solve the 3-surface in leftover \(B^4\) with a named 4-energy already in the lab; (2) if that interface is thick, a dimensionless ODE in \(y=g/a_0\) with leftover coefficients only. Pre-register UV Newton, IR boost, no centre wreck, \(c_{\mathrm{gw}}=c\). Do not write \(\mu(x)\). Do not pick \(p\). Do not unpark GE-J as \(m_e\). Prize OPEN.
1. Why “just write a 4D \(W\)” is not a route
| Attempt | End | |---------|-----| | Analytic \(W(E)\) at \(E=0\) | IR Poisson (FT-WE) | | More Hencky | already Newton+1PN (BP-1) | | Strain cap \(\lvert\nabla U\rvert\le 1/R\) | SS kills; wrong end | | Graph \(u=(R/c^2)\Phi\) | vertical at the centre; outskirts Newton (FT-BEND) | | Willmore \(\int H^2\) | \(k^4\) face book; O-WV parked | | Thin GE-J / Israel skin | global wall, or local \(p=\infty\) (FT-RAR-PROFILE inverted width) | | Extra metric (TeVeS-class) | \(c_{\mathrm{gw}}\) FIRE | | Pick \(p\) / Padé / simple \(\mu\) | shop (inner SPARC vs Chae tension) |
Finding R1 [E algebra]. In the quasi-static limit a local continuum with one scale \(R\) has two IR fates: a relaxed elastic modulus (Newton) or a yield stress (step/floor). SPARC’s smooth two-decade bend is not that dichotomy. Continuum language that stays local and quasi-static cannot be the prize. Finding R2 [E]. \(\Delta\log_{10} g\sim 2\) on \(g\propto 1/r^2\) is \(\Delta\log_{10} r\sim 1\), \(\Delta r/r\simeq 2.3\). The yield region has to be as fat as \(r_y\) itself. A rice-skin / Israel film is the wrong thickness, even when it is the right kind of threshold (FT-RICE, FT-GEJ-A0).2. What is still live (not exhibited)
Route A — non-graph embedding. FT-BEND failed as a graph. A 3-surface in leftover \(B^4\) may fold; a level-set can stay regular where the graph goes vertical. Named energies already in the lab: 4-volume constraint, induced 3-Hilbert, Hencky of the 4-metric. Not a new modulus. Pre-register: high-\(g\) dimple = Newton; low-\(g\) not Newton; centres exist; photons see the 3-geometry. If the IR is still Newton or the centre still wrecks, CLOSE this branch. Relativistic lift must keep \(c_{\mathrm{gw}}=c\). Route B — fat yield zone, ODE in \(y\). If A produces a diffuse interface of thickness \(\sim r_y\), the profile is a 1-D kink in \(y=g/a_0\). Leftover supplies only \(a_0=c^2/R\). A unique geometric functional (no \(V(\varphi)\) fitted to SPARC) would mint \(\nu(y)\). STOP if a potential is chosen to hit McGaugh. \(p=\infty\) is the thin limit of this route and is already in tension at the join. Not a third route: \(P\) alone. Compact Poisson requires \(P\rho=\rho-\bar\rho\) (FT-S3-SPECTRUM). That is why a pointwise analytic \(W(E(x))\) is the wrong book. It does not replace A or B: the leftover Green at \(r\ll R\) is Coulomb (FT-S3-PHI). \(P\) is the reason the prize is non-local, not the interpolator.\(W_{12}\) stays gated (FT-5). Retry only as the Green function of a named operator from A.
3. Licensed next (one mill, then stop)
1. FT-BEND sequel: DONE / KILL (PASS_FT_BEND_SEQ_KILL_NECK). Parametric \(3\)-volume and leftover CMC still neck at \(\sqrt{mR}\); \(0.1\,\mathrm{kpc}\) has no \(2\)-sphere; \(50\,\mathrm{kpc}\) Kepler. Do not retry \(3\)-volume in another coordinate. 2. Fat-zone ODE: DONE as formulation (PASS_FT_FAT_ODE_CANONICAL_WIDTH_PREDICTED). \(\lambda=1\) frozen; observable \(\nu\)-width \(0.48\,\mathrm{dex}\) vs McGaugh \(2.19\) — do not retune. Still not a 4D \(W\). Prize OPEN.
Do not mill another analytic \(W\). Do not assume \(\sigma\). Do not SPARC \(I\).
4. Integrity
| Gate | Verdict | |------|---------| | Realistic route exists | YES (A, B; not exhibited) | | Local analytic \(W\) as the route | NO | | Quasi-static local rheology as SPARC bend | NO (Newton or step) | | Thin GE-J skin as SPARC width | NO (\(\Delta r/r\simeq 2.3\)) | | \(P\) mints kpc RAR | NO | | Embedding selects \(p\) | NO | | Prize closed | NO | | \(\mu(x)\) / CDM / GE-J \(m_e\) | NO |
Exit:PASS_FT_CONST_ROUTE_LIVE_NOT_EXHIBITED
5. One-line for agents
§FT-CONST-ROUTE: prize still OPEN; analytic \(W\) and thin walls are not the route; live is non-graph 4-dent and/or a fat \(\sim r_y\) yield zone with leftover-only coefficients; \(P\) is necessary not sufficient; next is FT-BEND sequel; no \(\mu\) shop.