# Freeze — Absolute-Scale Theory · M13 Three-Gate Hypothesis · \(R\) Residual

### Geometry \(\to\) spectrum · \(R\) sets scale · stop minting \(m_e\) · no promote

**CPT Framework — absolute scale**  
**Date:** 2026-08-16  
**Status:** **THEORY FREEZE** · `PASS_M13_THREE_GATE_FREEZE` · local \(m_e\) mint **CLOSED-NULL / PARK**  
**Parents:** Scale wall · Lemma L1 · GS stance · Mach M7–M16 · SC4 · GE-C · P3 · MCL consideration  
**Census:** `scripts/investigations/s3_geometry/m13_three_gate_hypothesis_census.py` · `data/m13_three_gate_hypothesis_census.json`

Tags: **[E process]** · **[E honesty]** · **[Programme]** · **[M packaging]** · **[O residual]** · **[P prediction]**

---

## 0. One-line

> **Absolute scale is frozen as a theory of the cosmic radius \(R\) (Mach / Sciama / mixed-mass closure), not as a local electron factory: geometry supplies dimensionless structure and mass *ratios*; M13 is a **falsifiable three-gate hypothesis** (Hubble invert, independent-\(H_0\) contact, \(R\)-ontology); theoretical effort concentrates on **what \(R\) is and how it does or does not move**, not on reproducing \(511\,\mathrm{keV}\); GE linear kernel and Compton-pixel stacks stay parked; no promote.**

---

## 1. Why this freeze now

| Banked fact | Consequence |
|-------------|-------------|
| L1 / scale wall | Local \(1/a\) cannot mint \(a_*\) or \(m_e\) |
| GE-0…5 / GE-C | Linear Sciama / ZM / self-energy **informative negative** |
| MCL-1 | Compton cell \(4\pi\bar\lambda_e^{2}\) **circular** |
| P3 | **SOFT_SCALE_OK**; **HARD_SCALE_PASS false** |
| SC4 / M13–M14 | Soft absolute *relation* already exists; \(H_0\) is the live prediction |
| M12 | Naive expanding \(R\) **FIRE** vs LLR |

Continuing to invent local \(m_e\) theorems repeats a closed-null. The remaining *theory* is the ontology and dynamics of \(R\).

---

## 2. Absolute-scale theory (frozen)

\[
\boxed{
\begin{aligned}
&\text{geometry on }S^3(R)\text{ acts on the spectrum (modes, }C_i\text{, ratios);}\\
&\text{the cosmic radius }R\text{ (Mach/Sciama state) sets the absolute scale;}\\
&m=\hbar/(a c)\text{ is a **reading** after a scale-break, not a local mint.}
\end{aligned}
}
\]

| Layer | Does | Does not |
|-------|------|----------|
| Hopf / Clifford / \(T(p,q)\) / DN | Ratios, \(n=2\) class, form | SI \(m_e\) |
| Mach closure \(p=2\), \(k_1=3/8\), \(k_2=1\) | Why \(\alpha_G\sim 10^{-40}\) | Unique \((R,m)\) without an anchor |
| **M13** mixed-mass + α double-copy | Percent-class relation among \(\{G,H_0,\alpha,m_e,m_p\}\) | Free α; free local \(m_e\) |
| Soft \(\mathcal{P}_{\mathrm{EMC}}\) | Join \(m\leftrightarrow a\) | Hard join / hold closed |

**Effort rule:** do **not** spend WPs on reproducing \(m_e\). Allowed theory WPs attack **\(R\)**: static vs quantized vs present-epoch; what Hubble expansion is if \(R\) is not \(\dot R/R=H_0\); whether \(\sqrt{6}\) is a law for \(R\) or a brick.

**Parked as \(m_e\) factories:** GE linear kernel; Robin \(f\); F1 \(\alpha^3/2\); MCL Compton pixel; CODATA invert (NS-S1).

---

## 3. M13 as a three-gate hypothesis

**Statement (M13).** Packing length = electron Compton; Mach gravitational seat = proton; α double-copy **[M]**:
\[
G
=
\frac{3\alpha}{8\sqrt{6}}
\frac{\hbar^{2} H_0}{c\, m_e^{2}\, m_p}.
\]
Equivalent invert (M14 / SC4 dual). α is **INPUT + [M] placement**, not a derived theorem. \(\sqrt{6}\) is **[M]**. Hypothesis status: **[M/P]**, not HARD_SCALE_PASS.

Census invert (lab \(G,\alpha,m_e,m_p\)): \(H_0^{\mathrm{M13}}=68.97\,\mathrm{km\,s^{-1}\,Mpc^{-1}}\).

**Convention condition (R-XCHECK / R-XCHECK-D).** This invert is of the **published book** (Green `3/8` used in the raw \(GNm/R\) slot, plus mixed-mass and \(\sqrt{6}\)). Sol showed that `3/8` is \(K_{\mathrm{Green}}\) only at \(\Omega x^2=1\), and that the same-book raw coefficient \(3\pi/4\) moves the invert to \(\sim10.98\). The contact is therefore **convention-conditional [P]**, not a one-convention derivation. It is **not** retired: G-H0 / G-EXT stay **OPEN** on the published book until a single-functional rederivation exists. Do not patch `3/8` after seeing \(H_0\). Do not promote \(\Omega=1/6\) or \(\Omega=\alpha/(12\pi)\) as the repair.

### Gate G-H0 — Hubble invert **[P]**

| Field | Freeze |
|-------|--------|
| Prediction | \(H_0=69.0\,\mathrm{km\,s^{-1}\,Mpc^{-1}}\) (census \(68.97\)) |
| Lives in | Planck \(67.4\) – SH0ES \(73.0\) |
| **Falsify if** | A tension-**resolved** consensus \(H_0\) lies **outside** \([66,73]\), or excludes \(69.0\) at the hypothesis’s claimed percent class |
| Now | **OPEN** |
| Honesty | Using only \(H_0=69\) recycled from this invert is **circular** (B-circ). This gate waits on *external* Hubble resolution. The invert itself is of the published (spliced) coefficient book — convention-conditional, not withdrawn |

### Gate G-EXT — independent \(H_0\) contact

| Field | Freeze |
|-------|--------|
| Test | Insert **external** \(H_0\) (Planck or SH0ES), predict \(G\) or \(m_e\); do **not** recycle \(69\) |
| Census | Planck: \(\delta G/G\approx-2.3\%\); SH0ES: \(\delta G/G\approx+5.8\%\) (linear in \(H_0\)); \(m_e\) errors are the cube-root (SC4: \(\sim-0.8\%\) / \(\sim+1.9\%\)) |
| **Falsify if** | Planck-class independent \(H_0\) gives \(|\delta G/G|>5\%\) or \(|\delta m_e/m_e|>3\%\), **or** *both* Planck and SH0ES give \(|\delta G/G|>8\%\), with no new named brick |
| Now | **OPEN** · percent-class · SH0ES \(+5.8\%\) on \(G\) *is* the Hubble tension, not an extra fail |

### Gate G-R — ontology / dynamics of \(R\)  ★ theory residual

If \(R(t)=\sqrt{6}\,c/H(t)\) tracks an expanding 3-sphere and M13 is an *eternal* law with fixed masses,
\[
\frac{\dot G}{G}=-H_0
\sim 7\times 10^{-11}\,\mathrm{yr}^{-1}
\sim 700\times\text{ LLR}.
\]
Naive expanding \(R\): **already FIRE** (M12). Census \(\approx 716\times\).

M13-as-law **survives only if** one of:

| Ontology | Meaning |
|----------|---------|
| **R-static** | Cosmic \(S^3\) radius is not the Hubble flow; expansion is something else |
| **R-locked** | \(R\) is quantized / discrete and sits in a ground state (\(\dot R=0\)) |
| **R-epoch** | M13 is a present-epoch constraint, not a differential-in-time law |

**This is the licensed theory residual.** A proposed \(R\)-ontology is itself falsifiable once stated (e.g. “expanding \(R\) + constant \(G\)” is already dead).

**Corollary (not a fourth M13 algebra gate):** M16 — Mach sum is **baryonic** only; adding \(\Omega_m/\Omega_b\) destroys percent contact. Separate **[P]**.

---

## 4. What theoretical work is for

| Do | Do not |
|----|--------|
| Say what \(R\) *is* (static / locked / epoch) | Another local \(511\,\mathrm{keV}\) formula |
| Say what Hubble expansion is if \(\dot R\neq H_0 R\) | GE-6 compare; Robin; F1; MCL-1 pixel |
| Possibly derive \(\sqrt{6}\) as a law for \(R\) | Invert \(c_\varepsilon^{2}\to\alpha\) |
| Keep M13’s three gates honest | Recycle \(H_0=69\) as a cosmology theorem |
| GE-J only if a design *computes* \(\sigma\) | Treat GE-J as an \(m_e\) mint |

**R-ONT-0…2:** DONE · lock underived · \(\sqrt{6}\) **[M]**.  
**R-EFF:** **DONE** · ontology **unnamed** · `a_geom_r_eff_ontology_honesty_freeze.md`.  
**R-XCHECK / R-XCHECK-D:** **DONE** · splice held · directional chips restored.  
**R-MCR:** **DONE** · `PASS_R_MCR_NO_SOLUTION` · no one-convention PRIMARY Hubble hit.  
**R-MCR-2:** **DONE** · `PASS_R_MCR2_FAIL_DERIVE_K` · remainder \(k\) **ruled out** as a derivation · one-convention Sciama+\(\alpha\) is \(H_0\sim11\) · published book stays convention-conditional **[P]**.

---

## 5. Integrity gates

| Gate | Status |
|------|--------|
| PASS_THEORY_NAMED | **PASS** |
| PASS_THREE_GATES_NAMED | **PASS** |
| PASS_H0_IN_BAND | **PASS** (\(68.97\in[66,73]\)) |
| PASS_EXT_PLANCK_CLASS | **PASS** |
| PASS_NAIVE_R_FIRE | **PASS** (M12 held) |
| PASS_NO_ME_MINT | **PASS** |
| PASS_GE_LINEAR_STILL_PARK | **PASS** |
| PASS_NO_PROMOTE | **PASS** |
| PASS_INTEGRITY | **PASS** |

**Status:** `PASS_M13_THREE_GATE_FREEZE`

---

## 6. Kill board

| ID | Kill |
|----|------|
| **K-AST-1** | M13 sold as free local \(m_e\) theorem |
| **K-AST-2** | \(H_0=69\) recycled as the only “confirmation” |
| **K-AST-3** | Expanding \(R=\sqrt{6}\,c/H(t)\) + constant \(G\) |
| **K-AST-4** | New Compton-pixel / Green-self-energy \(m_e\) factory |
| **K-AST-5** | Free α from M13 double-copy slogan |
| **K-AST-6** | Close hold / TOE from this freeze |
| **K-SC4-*** / **K-GE-*** | Held |

---

## 7. Handoff

| Option | Note |
|--------|------|
| **R-ONT-0** | **DONE** · spin-2 coherence length |
| **R-ONT-1** | **DONE** · lock not derived from C3 · `a_geom_r_ont1_lock_results.md` |
| **R-ONT-2** | **DONE** · \(\sqrt{6}\) stays **[M]** · `a_geom_r_ont2_sqrt6_brick_results.md` |
| Observe G-H0 / G-EXT | Wait on Hubble resolution; do not retune bricks |
| R-MCR-2 remainder \(k\) | **DONE** · **FAIL_DERIVE** · coefficient line **closed** |
| GE-J | Still **gated** (compute \(\sigma\)); not an \(m_e\) programme |
| Stop \(m_e\) minting | **Default** |

---

## 8. One-line

> **§AST: scale from unnamed \(R_{\mathrm{eff}}\); M13 three-gate; remainder k ruled out; \(H_0\simeq69\) convention-conditional [P]; √6 brick; stop naming \(R\); stop minting \(m_e\); no promote.**