# 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]**

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## 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.

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## 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.

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## 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.

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## 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.

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## 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`

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## 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.**