# Results — Pythagorean join FIRE in the Solar System; standard \(\mu\) isolated tail is safe, AQUAL-EFE quadrupole is not

### \(n=1\) tail \(\delta g\sim a_0/2\) · Mercury \(0.82''/\mathrm{cy}\) vs \(\sim 0.003\) · standard \(n=2\) perihelion tiny · Cassini \(Q_2\) wants \(n\ge 6\) · SPARC fat vs SS sharp · prize OPEN

**CPT Framework — FT-STOP-REVISIT / BP-1 / FT-LOCK-KEY / FT-AQUAL-EFE / FT-AQUAL-LOCK / Book G**  
**Date:** 2026-09-24  
**Status:** **BANKED** · `PASS_FT_SS_JOIN_PYTH_FIRE_STD_Q2_TENSION`  
**Stimulus:** User — mill SS leakage of the two named joins first  
**Parents:** `a_geom_ft_stop_revisit_results.md` · `a_geom_bp1_two_well_hencky_overlap_results.md` · `a_geom_ft_lock_key_results.md` · `a_geom_ft_aqual_efe_results.md`  
**Script:** `scripts/investigations/s3_geometry/ft_ss_join.py`  
**Data:** `scripts/investigations/s3_geometry/data/ft_ss_join.json`

Tags: **[E algebra]** · **[E honesty]** · **[Programme]** · **[O residual]** · **[CAT-EVID]**

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## 0. One-line

> **Pythagorean is dead in the Solar System. Standard \(\mu=x/\sqrt{1+x^2}\) survives isolated perihelion and dies as AQUAL-EFE quadrupole (CAT-EVID).** Large-\(x\) tails: Pythagorean \(\mu=1-1/(2x)+\cdots\) \(\Rightarrow\) extra acceleration \(\delta g\simeq a_0/2\) (constant). That is \(0.82''/\mathrm{cy}\) on Mercury, \(1.3''/\mathrm{cy}\) Earth, \(4.0''/\mathrm{cy}\) Saturn, against MESSENGER/INPOP extras \(\sim 0.003''/\mathrm{cy}\) and Book G’s scored \(42.98''/\mathrm{cy}\) 1PN. Standard \(\mu=1-1/(2x^2)+\cdots\) gives Mercury \(\sim 10^{-9}''/\mathrm{cy}\) — harmless in isolation. Cassini \(Q_2=(3\pm 3)\times 10^{-27}\,\mathrm{s^{-2}}\) (Hees+2014) vs Blanchet–Novak AQUAL-EFE quadrupole \(2.1\times 10^{-27}\)–\(4.1\times 10^{-26}\) for slow-to-standard \(\mu\); literature needs \(n\ge 6\) (Fienga even \(n\ge 20\)). SPARC’s fat \(2\,\mathrm{dex}\) join fights that sharp UV (Desmond+2024 \(\sim 8.7\sigma\) CAT-EVID). Leftover-locked AQUAL cannot keep both 0-parameter named joins and Book G. Do not pick a sharp \(\mu_n\) from SPARC. Sciama parked. Prize OPEN.

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## 1. Tails (no SPARC fit)

| Join | \(\mu(x)\) as \(x\to\infty\) | Isolated extra \(\delta g\) |
|------|------------------------------|------------------------------|
| Pythagorean \(g=\sqrt{g_N(g_N+a_0)}\) | \(1-1/(2x)+\cdots\) | \(\simeq a_0/2\) (**constant**) |
| Standard \(x/\sqrt{1+x^2}\) | \(1-1/(2x^2)+\cdots\) | \(\simeq a_0^2/(2g)\) |

Hees+2014 even writes the Pythagorean \(\mu=(\sqrt{1+4x^2}-1)/(2x)\) as a named interpolator.

Perihelion mill: \(\Delta\varpi/\mathrm{rev}\simeq 2\pi\,(\delta g/g_N)\), \(a_0=1.2\times 10^{-10}\,\mathrm{m\,s^{-2}}\).

| Planet | \(x=g_N/a_0\) | Pythagorean \(''/\mathrm{cy}\) | Standard \(''/\mathrm{cy}\) |
|--------|---------------|--------------------------------|------------------------------|
| Mercury | \(3.3\times 10^8\) | **\(0.82\)** | \(\sim 10^{-9}\) |
| Earth | \(4.9\times 10^7\) | **\(1.3\)** | \(\sim 2\times 10^{-8}\) |
| Saturn | \(5.4\times 10^5\) | **\(4.0\)** | \(\sim 7\times 10^{-6}\) |

GR Mercury is \(42.98''/\mathrm{cy}\) (BP-1 / Book G). Extra at \(0.8''/\mathrm{cy}\) is not a 1PN remainder.

**Finding J1 [E algebra].** Pythagorean’s distinguished Gauss join has an \(n=1\) UV tail. It is **FIRE** in the Solar System, leftover \(a_\star=2.31\,a_0\) would make \(\delta g\) worse.

**Finding J2 [E].** Standard isolated tail is Book-G safe. The remaining SS gate is AQUAL’s **EFE quadrupole** in the Galactic field (same nonlinearity as FT-AQUAL-EFE).

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## 2. Cassini \(Q_2\) (CAT-EVID, not re-derived)

Hees, Folkner, Jacobson, Park 2014, PRD **89** 102002: \(Q_2=(3\pm 3)\times 10^{-27}\,\mathrm{s^{-2}}\) from Cassini Saturn ranging.

Blanchet & Novak 2011: AQUAL-EFE quadrupole for standard \(\mu\)-choices \(2.1\times 10^{-27}\)–\(4.1\times 10^{-26}\,\mathrm{s^{-2}}\). Slow joins sit high; \(n=2\) (standard) is not in the Cassini core. Later work: \(n\ge 6\) (Blanchet–Novak / Hees), even \(n\ge 20\) (Fienga+2018). Desmond et al. 2024: RAR-favoured interpolator vs SS quadrupole \(\sim 8.7\sigma\).

**Finding J3 [E honesty].** This mill did **not** recompute \(Q_2\). It is CAT-EVID that AQUAL with a fat RAR join fails Saturn ranging, while a join sharp enough for Cassini is not the SPARC mean. Leftover does not yet force a join that does both.

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## 3. Integrity

| Gate | Verdict |
|------|---------|
| Pythagorean \(n=1\) tail FIRE perihelion | **YES** |
| Standard \(n=2\) isolated perihelion OK | **YES** |
| Standard AQUAL-EFE \(Q_2\) tension | **CAT-EVID** (not recomputed) |
| SPARC \(I\) / \(\mu_n\) shop / \(\mathcal{L}\) written | **NO** |
| Book G 1PN stands | **YES** |
| Sciama unparked / prize closed / promote | **NO** |

**Exit:** `PASS_FT_SS_JOIN_PYTH_FIRE_STD_Q2_TENSION`

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## 4. One-line for agents

> **§FT-SS-JOIN: Pythagorean FIRE SS (\(\delta g\sim a_0/2\), Mercury \(0.82''/\mathrm{cy}\)); standard isolated tail OK; AQUAL-EFE Cassini \(Q_2\) wants \(n\ge 6\) (CAT-EVID) vs SPARC fat join; leftover does not yet force a join that does both; prize OPEN; no promote.**