Results — the constitutive slot is isolated; the action is not written
Named leftover 4-energies closed · unique \(y\)-map for isolated wells · interpolator family unpaid · \(P\) allowed · \(a_{\mathrm{ext}}\) not eliminated · writing \(\mathcal{L}\) now is a shop · prize OPEN
CPT Framework — FT-RAR-ELIM / FT-COV-YIELD-EQ / FT-METRIC-ELAS / FT-CONST-ROUTE / FT-POISSON-EMBED / FT-FAT-FORCE / FT-CONST-PRIZE Date: 2026-09-23 Status: BANKED ·PASS_FT_PRIZE_PIN_SLOT_NOT_L
Stimulus: User — constitutive prize looks isolated; can we pinpoint the action now?
Parents: a_geom_ft_rar_elim_results.md · a_geom_ft_cov_yield_eq_results.md · a_geom_ft_metric_elas_results.md · a_geom_ft_poisson_embed_results.md · a_geom_ft_const_route_results.md · a_geom_ft_constitutive_prize_freeze.md
Script: scripts/investigations/s3_geometry/ft_prize_pin.py
Data: scripts/investigations/s3_geometry/data/ft_prize_pin.json
Tags: [E honesty] · [Programme] · [O residual]
0. One-line
The slot is pinned. The Lagrangian is not. After elimination, leftover RAR is extra metric \(\Phi\) organised by \(y=g_N/a_0\) for an isolated well, non-analytic at leftover \(K\sim 1/R\), UV Einstein/Hencky, \(c_{\mathrm{gw}}=c\), IR rail \(g_{\mathrm{obs}}^2=g_N a_0\) kind. Every named 4-energy already in the lab fails as that action (analytic \(W\), Hencky, 3-volume/CMC, 3-Hilbert, \(\mathrm{Vol}_4\), thin Israel, Poisson-embed of those, extra \(\phi\), second metric). Quasi-static local rheology is Newton or a step, which is not SPARC’s fat \(2\,\mathrm{dex}\) bend — so the prize may be non-local (\(P\)). That still does not select McGaugh versus Pythagorean versus a \(p\)-family, and it does not kill EFE \(a_{\mathrm{ext}}\). Writing \(\mathcal{L}(g,K)\) or \(\mu(x)\) here would be a shop. Skordis–Złośnik-class existence is CAT-EVID that a covariant MOND with \(c_{\mathrm{gw}}=c\) can exist; they still shop the interpolator. \(\sqrt{6}\) still [M]. P5 stands. Prize OPEN.
1. What is isolated (the slot)
| Gate | Forced | |------|--------| | Operand | 4-metric (photons see \(\Phi\)); not \(m(a)\), not extra \(\phi\), not a second metric | | Threshold | \(\lvert\nabla\Phi\rvert\sim 1/R\), not Ricci | | Isolated reduction | universal \(y\mapsto g_{\mathrm{obs}}\); no leftover \(M\) in the EL | | UV | Newton + scored 1PN | | IR kind | \(g_{\mathrm{obs}}^2=g_N a_0\) (\(k=1\) at join) | | Waves | \(c_{\mathrm{gw}}=c\) in the UV principal symbol | | Scale | leftover \(K\sim 1/R\) only; \(R,c\) invariant | | Allowed | non-local \(P=\mathrm{id}-\mathrm{mean}\) | | Not forced | a unique interpolator; elimination of \(a_{\mathrm{ext}}\) |
FT-CONST-ROUTE’s “live named embedding energy” is closed as a derivation (FT-BEND-SEQ, FT-POISSON-EMBED). The fat \(y\)-ODE is a formulation of this slot, not the 4D action.
2. Why the action is not pinpointed
A Lagrangian is a density plus a function. The density’s named leftover candidates are dead. The function along \(y\) is a family (FT-RAR-CURVE, FT-RAR-PROFILE). Local analytic \(W\) cannot supply that function (FT-WE). Local non-analytic rheology supplies a step, not two decades (FT-CONST-ROUTE). Compactness forces \(P\) on Poisson but does not mint \(\nu(y)\) (FT-S3-PHI).
So the unpaid object is exactly:
a (possibly non-local) constitutive response of the metric, non-analytic at leftover \(K\), whose EL is the \(y\)-map, with UV Einstein and \(c_{\mathrm{gw}}=c\).
That sentence is the pin. It is not \(\mathcal{L}=\ldots\)
Finding P1 [E honesty]. Picking Pythagorean \(g=\sqrt{g_N(g_N+a_0)}\) as “the” action because it is distinguished as a two-term inverse-square is still a choice of interpolator (SPARC does not award it). Picking AQUAL/QUMOND \(\mu\) is a shop. Picking Skordis–Złośnik and then leftover \(a_0=c^2/R\) still leaves their free function. Finding P2 [E]. Quasi-static local continuum with one scale cannot be the prize (Newton or step). If the action is local, it is not quasi-static yield language. If it is quasi-static, it is not pointwise. Those are constraints on the slot, not a density.3. Integrity
| Gate | Verdict | |------|---------| | Slot isolated | YES | | Action written | NO | | Named lab 4-energy is the action | NO | | Unique interpolator | NO | | EFE eliminated | NO | | \(\mathcal{L}(g,K)\) / \(\mu(x)\) invented | NO | | Prize closed / promote | NO |
Exit:PASS_FT_PRIZE_PIN_SLOT_NOT_L
4. One-line for agents
§FT-PRIZE-PIN: constitutive slot isolated (metric \(\Phi\), non-analytic at leftover \(K\), universal \(y\)-map, UV Einstein, \(c_{\mathrm{gw}}=c\), \(P\) allowed); named 4-energies closed; interpolator and \(a_{\mathrm{ext}}\) unpaid; do not write \(\mathcal{L}\); prize OPEN; no promote.