Results — 4-medium constitutive keyhole: tensors from \((g,u,a)\), not a written \(W\)
Algebraic basis \(g_{\mu\nu},\;u_\mu u_\nu,\;a_\mu a_\nu,\;u_{(\mu}a_{\nu)}\) · scalar \(\lvert a\rvert\) is AQUAL · \(Q_2\) is elliptic leak · Pythagorean out · UV Einstein · prize OPEN
CPT Framework — FT-SCALAR-SS / FT-DERIV-ORDER / FT-LOCK-KEY / FT-SS-JOIN / FT-GR-CURV / FT-AQUAL-LOCK / FT-PRIZE-PIN Date: 2026-09-24 Status: BANKED ·PASS_FT_KEYHOLE_4D_BASIS_NOT_W
Stimulus: User — look for the fundamental constitutive tensor of the 4-medium; shape of the keyhole as a key, not a shop
Parents: a_geom_ft_scalar_ss_results.md · a_geom_ft_deriv_order_results.md · a_geom_ft_lock_key_results.md · a_geom_ft_ss_join_results.md · a_geom_ft_gr_curv_results.md
Script: scripts/investigations/s3_geometry/ft_keyhole_4d.py
Data: scripts/investigations/s3_geometry/data/ft_keyhole_4d.json
Tags: [E algebra] · [E honesty] · [Programme] · [O residual]
0. One-line
The keyhole is a symmetric 4-tensor built from leftover rest frame \(u^\mu\) and its 4-acceleration \(a_\mu\), with leftover scale \(1/R\). At a point, with \(u\cdot a=0\) static, the algebraic basis is \(g_{\mu\nu}\), \(u_\mu u_\nu\), \(a_\mu a_\nu\), \(u_{(\mu}a_{\nu)}\). Ricci is the wrong order. ADM \(K\) misses static wells. Scalar \(\mu(\lvert a\rvert)\) is the isotropic reduction: AQUAL, elliptic, Cassini \(Q_2\) from the outer MOND bubble. Pythagorean (\(n=1\) tail) is FIRE. UV must be Einstein (\(c_{\mathrm{gw}}=c\), Book G 1PN, tail \(n\ge 2\) isolated, \(n\ge 6\) if the static EL stays AQUAL-divergence). Coefficients of that basis are not written: that would be a shop. AeST is CAT-EVID that a vector-tensor host exists, not a paste-in. Prize OPEN.
1. Shape of the keyhole (what any key must pass)
| Gate | Forced | |------|--------| | Photons | same 4-metric (KiDS; no \(m(a)\), no second metric) | | Operand | \(a_\mu=u^\nu\nabla_\nu u_\mu\), \(x=\lvert a\rvert R\) — not \(R_{\mu\nu}\), not static ADM \(K\) | | Vacuum of \(W\) | leftover umbilic \(a=0\) (FT-PRESTRAIN) | | Isolated spherical IR | AQUAL ends: \(F\sim X^{3/2}\) | | Isolated UV tail | \(n\ge 2\) (Pythagorean FIRE); Book G \(\gamma=\beta=1\) | | Waves | Einstein principal symbol, \(c_{\mathrm{gw}}=c\) at \(10^{-15}\) | | Conservation | Bianchi / \(\nabla_\mu T^{\mu\nu}=0\) — cannot be an arbitrary \(\mu_{ij}\) | | Scale | leftover \(K=1/R\) only; \(\sqrt{6}\) still [M] | | Not | extra \(\phi\), empty-\(B^4\) area, leftover \(P\) as \(\nu\), SPARC \(I\), Sciama delay |
Cassini \(Q_2\) is not a local \(1/x\) tail at Mercury (\(x\sim 10^8\), \(\mu=1\)). It is an elliptic leak: AQUAL’s outer solar MOND bubble in the Galactic field back-reacts as an inner quadrupole. A purely local UV-Einstein constitutive does not automatically make \(Q_2\); a static AQUAL-divergence EL does.
2. Named tensors vs the hole
| Tensor | Fit? | |--------|------| | Einstein \(G_{\mu\nu}\) | UV only; IR Laplace | | Hencky stress of 4-metric strain | analytic at leftover vacuum; IR Newton | | Leftover embedding \(K_{ij}\) / area EL | wrong interpolator end; BEND | | Scalar \(\mu(\lvert a\rvert)\delta_{ij}\) | AQUAL 3D key; \(Q_2\) elliptic leak; fat vs sharp join strained | | Algebraic \(\mu_{\mu\nu}[g,u,a]\) | keyhole — four structures, unpaid functions of \(a^2 R^2\) | | Disformal \(g_{\mu\nu}+B\,a_\mu a_\nu\) | photon/matter split risk (KiDS) | | Non-metricity / \(f(Q)\) analytic | GR-equivalent; IR Newton | | AeST / Einstein-aether | host kind (\(c_{\mathrm{gw}}=c\) possible); still shops \(F\) and often a kinetic term for \(u\) |
Finding K1 [E algebra]. With leftover \(u^\mu\) and \(a_\mu\perp u\), a symmetric 4-tensor at a point has four algebraic slots. Isotropic AQUAL uses one (a function of \(\lvert a\rvert\) times \(h_{ij}\)). The extra slots (\(a_\mu a_\nu\), mixed \(u_{(\mu}a_{\nu)}\)) are the only local tensor handles that are not Ricci. They do not by themselves cancel \(Q_2\). Finding K2 [E honesty]. Writing \(A,B,C,D(\lvert a\rvert R)\) is the \(\mu\) shop with more indices. Leftover would have to force those functions. It has not. The lock-key is the basis, not the coefficients.3. What looking at the hole does teach
The 3D lock was AQUAL aquadratic, not leftover area. The 4D lock is not a curvature density and not a unique scalar \(\mu(\lvert a\rvert)\) if one takes Cassini vs fat RAR seriously (CAT-EVID). The remaining geometric object is leftover \(u^\mu\) and \(a_\mu\) as constitutive arguments of the same 4-metric Einstein already uses in the UV.
Static spherical isolation still has to recover AQUAL ends with a legal UV tail. Galactic EFE can remain (Crater II kind). SS quadrupole is the extra demand on how \(a_\mu\) of the Sun and leftover \(u\) contract — elliptic AQUAL vs a hyperbolic Einstein-UV with IR constitutive only. That split is unpaid.
4. Integrity
| Gate | Verdict | |------|---------| | Keyhole is algebraic \((g,u,a)\) tensor | YES | | Coefficients / \(\mathcal{L}\) written | NO | | Scalar \(\mu(\lvert a\rvert)\) uniquely killed | NO (strained; AQUAL reduction) | | \(Q_2\) cancelled by the basis alone | NO | | AeST pasted in | NO | | Pythagorean reused | NO | | Prize closed / promote | NO |
Exit:PASS_FT_KEYHOLE_4D_BASIS_NOT_W
5. One-line for agents
§FT-KEYHOLE-4D: constitutive tensor keyhole is algebraic in leftover \(u^\mu\) and \(a_\mu\) (\(g,uu,aa,ua\)); Ricci/ADM \(K\)/area out; scalar \(\lvert a\rvert\) is AQUAL (\(Q_2\) elliptic); Pythagorean FIRE; coefficients not written; prize OPEN; no promote.