# Where this constitutive line stands

### Container geometry named · one \(\mu\) unpaid · completion test placed · not an observational closure

**CPT Framework — FT-AXIS-WHERE / FT-AXIS-COMPLETION / FT-AXIS-OBLIGATIONS**  
**Date:** 2026-09-29  
**Status:** **BANKED** · `PASS_FT_AXIS_WHERE_WE_ARE_OPEN`  
**Stimulus:** User — bank where we are, and say what the public site may carry.  
**Parents:** `a_geom_ft_axis_completion_consideration.md` · `a_geom_ft_axis_obligations_results.md` · `a_geom_ft_axis_kappa_willmore_results.md` · `a_geom_ft_axis_close_obligations_consideration.md` · `a_geom_ft_axis_nest_join_consideration.md` · `a_geom_ft_axis_metric_act_results.md`

Tags: **[E honesty]** · **[Programme]** · **[O residual]**

No new sum. Chips unflipped. Not a promote of a paid \(W\).

---

## 0. One-line

> **The container geometry of this line is named. It has not selected one \(\mu\).** Two accelerations stand. One assigned barrier sits inside the Cassini band at the comparison scale and outside it at both framework scales. The test a future action has to pass is written. The page’s law is unpaid.

---

## 1. Geometry that stands

The seat of this line is a round \(S^3\) of radius \(R=3.28\times 10^{26}\,\mathrm{m}\), filled by the Hopf nest \(T_\eta\). The Clifford leaf is \(\eta=\pi/4\). On the unit nest

\[
H=-\cot(2\eta),\qquad W=\frac{2\pi^2}{\sin(2\eta)},
\]

with minimum \(W=2\pi^2\) at that leaf. The excess above the minimum is \(4\pi^2\delta^2\) in the leaf angle \(\delta=\eta-\pi/4\). The axis exchange \(\eta\to\pi/2-\eta\) makes \(H\), \(W\), and \(\tan\eta\) one rate on both sides of the leaf.

The dictionary is \([K]=\Delta\ell/\rho\) and \([g]=c^2\Delta\ell/\rho\). At unit windings, \(\Delta\ell=2-\sqrt{2}\), so

\[
a_{\mathrm{sym}}=(2-\sqrt{2})\,c^2/R=1.605\times 10^{-10}\,\mathrm{m\,s^{-2}},
\]

beside the bare surface \(a_\star=c^2/R=2.740\times 10^{-10}\,\mathrm{m\,s^{-2}}\). Both stand. The comparison value \(1.2\times 10^{-10}\,\mathrm{m\,s^{-2}}\) is an input.

The cap has one index: modes \(r^{n}\) and \(r^{-n}\), log-slopes \(+n\) and \(-n\). The coexact spectrum is every integer \(n\ge 2\). Hopf \(c_1=1\) and the base curvature \(4\) are constants across the Clifford leaf. The static lapse gives acceleration \(|\nabla\Phi|\), so the coefficient on \(a_{\mathrm{sym}}\) is \(1\).

---

## 2. Observational rows

These are the scored rows. They are not a single closure, and they are not a match to five decimal places.

| Row | Score |
|-----|------:|
| Assigned \(\kappa=2\pi^2\) on the published McGaugh north, comparison scale \(1.2\times 10^{-10}\) | \(Q_2=4.40904776\times 10^{-27}\,\mathrm{s^{-2}}\), \(0.470\sigma\), width \(1.419715\,\mathrm{dex}\) |
| Same assignment at \(a_{\mathrm{sym}}\) | \(Q_2=2.27299248\times 10^{-26}\,\mathrm{s^{-2}}\), \(6.5766\sigma\) |
| Same assignment at \(a_\star\) | \(Q_2=3.60829792\times 10^{-26}\,\mathrm{s^{-2}}\), \(11.0277\sigma\) |
| Ground slope \(\kappa=2\) on \(\mu_2\) | \(Q_2=1.803\times 10^{-26}\,\mathrm{s^{-2}}\), \(5.01\sigma\) |
| Power-family equator at \(n=2\) | \(\mu(1)=0.707107\). McGaugh \(0.510590826977\) would need \(n=1.031182\), below the spectrum |
| Factor that would turn \(a_{\mathrm{sym}}\) into \(1.2\times 10^{-10}\) | \(0.747663551402\). No booked pure number equals it. The nearest, \(1/\sqrt{2}\), misses by \(5.42\%\) |

The comparison-scale row uses the published McGaugh inversion and an assigned slope equal to the Clifford minimum. It lies inside the Cassini band \((3\pm 3)\times 10^{-27}\,\mathrm{s^{-2}}\). The same assignment at either framework scale does not. The quoted product \(5.897\times 10^{-27}\) is the comparison quadrupole times \(1.605/1.2\). It is not the surrogate at \(a_{\mathrm{sym}}\).

---

## 3. What remains

Explicit Einstein–Hilbert plus \(F(X)\) is still an infinite convex set of both-end \(\mu\). Raw \(H\) and raw \(W\) are not members of that set. The classical nest energy returns the Clifford leaf and the number \(2\pi^2\).

Completion is the test in `a_geom_ft_axis_completion_consideration.md`. One explicit functional, density displayed before it is varied, no free dimensionless coefficient, one critical \(\mu=F'(X)\) with \(\mu\to 1\) at large \(x\) and \(\mu\sim x\) at small \(x\). Until that density is written, there is nothing to vary.

---

## 4. What the public site may say

The public index may say that this geometry is the named substrate and that the constitutive function is unpaid, with the table in §2 as the observational score. It may link this note. It may not say that a unique \(\mu\) has been derived, that every constraint matches, or that \(W\) is a paid theorem. Doctrine chips stay as they are. Solar and Marklund stay parked. Macro leftover-RAR stays paused.

---

## 5. Integrity

| Gate | Verdict |
|------|---------|
| Fork | **`OPEN`** |
| Container geometry named | **YES** |
| One \(\mu\) selected | **NO** |
| Every constraint matched | **NO** — framework-scale rows are \(6.58\sigma\) and \(11.0\sigma\) out |
| Comparison-scale assigned barrier inside Cassini \(1\sigma\) | **YES** — one ansatz, one input scale |
| Completion test written | **YES** — §0bn1004 |
| \(F\) crowned / chip flipped / promote | **NO** |

**Exit:** `PASS_FT_AXIS_WHERE_WE_ARE_OPEN`

### One-line for agents
> **§FT-AXIS-WHERE: geometry named; \(\mu\) unpaid; comparison-scale ansatz is \(0.47\sigma\); framework scales are out; completion test placed; no promote.**