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This is a raw source document. Integrity tags [E] [M] [O] [F] are coloured for readability. Content is as written in the research file.

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.
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