# BF-1/2/3 — Surface Freeze, Energy Contact, Hierarchy Honesty

### L1b \(\Sigma\) frozen · \(E_\Sigma=(\alpha/2)mc^2\) at Compton · hard join still OPEN · \(\mathcal{P}_{\mathrm{EMC}}\) stands

**CPT Framework — BF-2D programme**  
**Date:** 2026-08-10  
**Status:** **RESULTS FREEZE** · **`PASS_BF1_2_3_SURFACE_FREEZE_CONTACT_HIERARCHY`**  
**Design:** `a_geom_bulk_face_mach_surface_energy_transition_design.md`  
**Script:** `scripts/investigations/s3_geometry/bf2d_bf1_surface_freeze_contact.py`  
**JSON:** `scripts/investigations/s3_geometry/data/bf2d_bf1_surface_freeze_contact.json`  
**Priors:** L1b · \(\mathcal{P}_{\mathrm{EMC}}\) · Probe B/B2  

Tags: **[E process]** · **[E honesty]** · **[E algebra]** · **[M packaging]** · **[O residual]**

---

## 0. One-line

> **BF-1 freezes primary surface \(\Sigma=\) L1b match sphere with EM shell energy \(E_\Sigma=\alpha\hbar c/(2R_m)\) and separate metric wall budget — no free \(r_{\mathrm{min}}\); BF-2 shows at Compton \(R_m=a\), \(E_\Sigma=(\alpha/2)mc^2\ll mc^2\), while forcing \(E_\Sigma=mc^2\) reintroduces \(R=r_e/2\) (Probe B1 class **KILL**); BF-3: surface book is local scale \(a\) (avoids B3 volume-tube kill) but does not hard-join to \(m_{\mathrm{Mach}}\); soft \(\mathcal{P}_{\mathrm{EMC}}\) stands; hard join OPEN.**

---

## 1. BF-1 — Surface \(\Sigma\) and density freeze

### 1.1 Primary surface (frozen)

| Item | Content |
|------|---------|
| \(\Sigma\) | **L1b thin match sphere** (SC1 charge packaging) |
| Default radius | \(R_m=a\) under soft packaging, or geometric \(R_m=a\sin(\chi_m/a)\) from L1 |
| Free volume cutoff \(r_{\mathrm{min}}\) | **Forbidden** |

### 1.2 Energy density books on \(\Sigma\) (not collapsed)

| ID | Content | Tag |
|----|---------|-----|
| **u_EM_shell** | Classical electrostatic shell ⇒ \(E_\Sigma=\alpha\hbar c/(2R_m)\) | **[E under L1b packaging]** |
| **u_metric_wall** | Thin-shell metric stress from \(\Delta k\) (L1) | **[E/M]** separate budget |

**Do not double-count** EM self-energy and metric wall as one rest-mass theorem (L1b honesty).

### 1.3 Killed surface options

| Option | Verdict |
|--------|---------|
| Choose \(R_m=r_e/2\) so \(E_\Sigma=mc^2\) | **KILL** — free classical-radius cutoff (B1 class) |
| Smear shell energy on Clifford area | **[M]** redistribution only; still \(O(\alpha)mc^2\) |
| Ad hoc \(u=E_{\mathrm{path}}/a^2\) on sphere | **KILL** — smuggles factor \(4\pi\) |

---

## 2. BF-2 — Contact board

| Book | \(E/(mc^2)\) | vs hard join |
|------|-------------:|--------------|
| **S0** soft \(E_{\mathrm{path}}=\hbar c/a\) with \(a=\hbar/(mc)\) | **1** by construction | Soft glue **[M]** · not hard theorem |
| **S1** shell at Compton \(R_m=a\) | \(\alpha/2\approx 0.00365\) | **FAIL** hard join (exterior \(O(\alpha)\) only) |
| **S1b** shell forced to \(E=mc^2\) | 1 only if \(R=r_e/2\) | **KILL B1 class** |

**Path vs surface:**

\[
\frac{E_{\mathrm{path}}}{E_\Sigma}
=
\frac{2}{\alpha}
\sim 274
\quad(R_m=a).
\]

**Finding:** 2D EM surface budget at Compton scale is the **EP★ exterior** piece, not the full rest energy. Hard join is not achieved by equating shell self-energy to \(m_{\mathrm{Mach}}c^2\).

---

## 3. BF-3 — Hierarchy honesty (vs Probe B3)

| Item | Content |
|------|---------|
| Scale \(a/R_U\) | \(\sim 10^{-39}\) (order) |
| B3 kill | Tubular **volume** energy on \(S^3(R_U)\) suppressed by \(\sim a/R_U\) |
| Surface book | Lives at **local** seat/match scale \(a\) |
| Avoids B3? | **Yes** as a definition of local \(E_\Sigma\) (not a cosmos tube integral) |
| Hard-joins \(m_{\mathrm{Mach}}\)? | **No** — still needs a principle linking cosmos Mach mass to local surface energy without soft glue |

**Reading:** BF-3 does **not** revive tubular join; it clarifies that local 2D surface energy is allowed to exist at scale \(a\), while hard bulk↔surface theorem remains open.

---

## 4. Hard join status after BF-1…3

| Item | Status |
|------|--------|
| Soft \(\mathcal{P}_{\mathrm{EMC}}\) | **STANDS** (operational) |
| Surface \(\Sigma\) + \(u_{\mathrm{EM}}\) frozen | **PASS** (no free \(r_{\mathrm{min}}\)) |
| Contact \(E_\Sigma\sim mc^2\) at geometric Compton \(R\) | **FAIL** (\(O(\alpha)\)) |
| Force full rest via \(R=r_e/2\) | **KILL** |
| Hard join theorem | **STILL OPEN** |
| Public promote | **NOT APPROVED** |

**Next (BF-4):** transition principle linking \(m_{\mathrm{Mach}}\) to surface density scale **or** honest FAIL that strengthens “soft package mandatory” (B2-class).

---

## 5. Kill board

| ID | Result |
|----|:------:|
| BF-1 \(\Sigma\) frozen without free \(r_{\mathrm{min}}\) | **PASS** |
| Force \(R=r_e/2\) | **KILL** |
| Shell \(O(\alpha)\) not hard join | **PASS** |
| Soft glue control | **PASS** |
| Surface local ≠ B3 redo | **PASS** |
| Hard join still OPEN | **PASS honesty** |
| Public promote | **OPEN / NO** |

**Verdict:** **`PASS_BF1_2_3_SURFACE_FREEZE_CONTACT_HIERARCHY`**

---

## 6. Allowed / forbidden language

| Allowed | Forbidden |
|---------|-----------|
| “L1b shell self-energy \(\alpha/(2R_m)\) at Compton is \(O(\alpha)mc^2\)” | “Surface integral derives hard \(E=mc^2\) via \(R=r_e/2\)” |
| “Soft \(\mathcal{P}_{\mathrm{EMC}}\) equates path energy to \(m_{\mathrm{Mach}}c^2\) by glue” | “Soft glue is hard bulk–face theorem” |
| “2D surface book avoids B3 volume-tube kill” | “Local surface energy harvests Mach bulk without principle” |
| “Hard join remains OPEN” | Public hard-join chip flip |

---

## 7. One-line banked

> **BF-1…3: Σ=L1b sphere frozen; E_Σ=(α/2)mc² at Compton; force R=r_e/2 killed; hard join OPEN; P_EMC stands.**