Appendix E. Geometric origin of the electron mass and direct measurement

Appendix E. Geometric origin of the electron: This appendix assumes the following inputs are locked. In the universal regime where δ=1/π² applies. The goal is to fix (i) the electron radius rₑ, (ii) the electron resistance integral S, and (iii) the electron mass mₑ in closed form via define→substitute→eliminate.

This appendix assumes the following inputs are locked. In the universal regime where δ=1/π² applies.

Corrected in place (v0.5.0). Earlier versions of this appendix ran the mass resistance integral to the event-rate radius rₑ=(D_anch/2)δ, which inserts a spurious factor π² into mₑ (yielding ≈ 5.04 MeV). Per the §13.5.4 canon the mass-bearing length is the Compton length λ_(C,e)=D_anch/2; rₑ=r₀δ governs only the event rate νₑ=1 (§9.3). The integral below is corrected accordingly; the historical error is recorded here, not hidden.

E.0 Locked inputs and goals

This appendix assumes the following inputs are locked.

  1. Action-unit constant: h.
  2. Operational anchor speed constant: c_ref.
  3. Realized length (volume-particle diameter): a.
  4. Anchor length: D_anch.
  5. Rectification coefficient: δ (in the universal regime, δ=1/π²).
  6. Lattice unit energy (single source of truth):
    \begin{equation} U_{\mathrm{lat}}:=\frac{h\,c_{\mathrm{ref}}}{a}. \end{equation}

The goal is to fix (i) the electron radius rₑ, (ii) the electron resistance integral S, and (iii) the electron mass mₑ in closed form via define→substitute→eliminate.

E.1 Definition of the electron radius rₑ (canonical)

Define the electron radius as follows.

\begin{equation} \boxed{ r_e:=\frac{D_{\mathrm{anch}}}{2}\,\delta } \end{equation}

In the universal regime where δ=1/π² applies,

\begin{equation} \boxed{ r_e=\frac{D_{\mathrm{anch}}}{2\pi^2} } \end{equation}
Role restriction (§13.5.4). rₑ is the event-rate radius: it sets νₑ=1 via sₑ=r₀/rₑ=1/δ and nothing else. It is not the mass-bearing length; the mass integral of E.2 runs to the Compton length λ_(C,e)=r₀=D_anch/2.

E.2 Operational definition of the electron resistance integral S (layer integral)

Define the radial coordinate R on 0≤ R≤ λ_(C,e) (the mass-bearing Compton length, §13.5.4). If the thickness of one layer is fixed as the VP diameter a, then the number of layers (dimensionless) in an infinitesimal interval dR is

\begin{equation} dN(R):=\frac{dR}{a}. \end{equation}

Define the electron resistance integral S by the following integral.

\begin{equation} \boxed{ S:=\int_{0}^{\lambda_{C,e}}\frac{dR}{a} } \end{equation}

Since a is locked as a constant, expanding the integral gives

\begin{align} S &=\int_{0}^{\lambda_{C,e}}\frac{dR}{a} =\frac{1}{a}\int_{0}^{\lambda_{C,e}} dR =\frac{1}{a}\Bigl[R\Bigr]_{0}^{\lambda_{C,e}} =\frac{\lambda_{C,e}}{a}. \end{align}

Therefore

\begin{equation} \boxed{ S=\frac{\lambda_{C,e}}{a} } \end{equation}

Since λ_(C,e)=D_anch/2,

\begin{equation} \boxed{ S=\frac{D_{\mathrm{anch}}}{2a} } \end{equation}

(No δ enters: the spurious universal-regime form S=D_anch/(2aπ²) of earlier drafts is withdrawn; see the correction banner above and §13.5.4.)

E.3 Definition of the electron mass mₑ and closed form

Applying the mass=resistance axiom (operational definition) to the electron,

\begin{equation} \boxed{ m_e:=\frac{U_{\mathrm{lat}}}{S} } \end{equation}

we define it as above. Substituting (AppE_S_final) gives

\begin{align} m_e &=\frac{U_{\mathrm{lat}}}{\lambda_{C,e}/a} =U_{\mathrm{lat}}\frac{a}{\lambda_{C,e}}. \end{align}

Also, substituting (AppE_Ulat) into (AppE_me_expand1) gives

\begin{align} m_e &=\left(\frac{h\,c_{\mathrm{ref}}}{a}\right)\frac{a}{\lambda_{C,e}} =\frac{h\,c_{\mathrm{ref}}}{\lambda_{C,e}} =\frac{2h\,c_{\mathrm{ref}}}{D_{\mathrm{anch}}}. \end{align}

Therefore, under the same lock version,

\begin{equation} \boxed{ m_e c^{2}=\frac{h\,c}{\lambda_{C,e}}=\frac{2hc}{D_{\mathrm{anch}}}=0.511\ \mathrm{MeV} } \end{equation}

it is fixed in this closed form — coinciding with the §13.5.4 calibration node (an anchor, not a prediction).

E.4 Sensitivity (dependence on changes in locked values)

From (AppE_me_final), when h,c_ref are locked,

\frac{dm_e}{m_e}=-\frac{d\lambda_{C,e}}{\lambda_{C,e}}=-\frac{dD_{\mathrm{anch}}}{D_{\mathrm{anch}}}.

(δ no longer enters the mass; it governs only the event rate, §9.3.)

E.5 Deterministic verification script (electron mass evaluation)

# verify_appendix_E.py
# Purpose: compute r_e, S, m_e using only LOCK values from registry, and
#         leave a log (identifiers/hashes) of the computation process as a deterministic script.

import json
import hashlib
from math import pi

def sha256_file(path: str) -> str:
    h = hashlib.sha256()
    with open(path, "rb") as f:
        while True:
            b = f.read(1024 * 1024)
            if not b:
                break
            h.update(b)
    return h.hexdigest()

def read_json(path: str) -> dict:
    with open(path, "r", encoding="utf-8") as f:
        return json.load(f)

def main():
    canon = read_json("registry/canon_lock.json")
    realz = read_json("registry/realization_lock.json")
    prot  = read_json("registry/protocol_lock.json")
    snap  = read_json("registry/registry_snapshot.json")

    # [LOCK] constants
    h_Js = canon["constants"].get("h_Js", 6.62607015e-34)
    c_ref = realz["inputs"]["c_ref_m_s"]
    a_m   = realz["inputs"]["a_m"]
    D_anch = canon["inputs"]["D_anch_m"]
    assert abs(D_anch/4.852620477e-12 - 1) < 1e-6, "canon D_anch not reconciled to v0.4.1 (2*lambda_Ce=4.8526 pm)"

    # delta: either explicit numeric or derived from pi
    delta_item = canon["constants"].get("delta_rect", "1/pi^2")
    if isinstance(delta_item, (int, float)):
        delta = float(delta_item)
    else:
        delta = 1.0 / (pi**2)

    GeV_to_J = prot["unit_conversions"]["GeV_to_J"]

    # [DERIVED] U_lat
    U_lat_J = (h_Js * c_ref) / a_m
    U_lat_GeV = U_lat_J / GeV_to_J

    # [DERIVED] lambda_Ce (mass-bearing length, S13.5.4), event-rate radius r_e, S, m_e
    lam_Ce = D_anch / 2.0            # Compton length = mass-bearing length
    r_e = lam_Ce * delta             # event-rate radius (nu_e = 1 only; NOT used for mass)
    S = lam_Ce / a_m
    m_e_GeV = U_lat_GeV / S
    assert abs(m_e_GeV / 5.10999e-4 - 1) < 2e-3, "m_e must reproduce 0.511 MeV (S13.5.4 canon)" 

    out = {
        "registry_snapshot_id": snap["registry_snapshot_id"],
        "locks": snap["locks"],
        "hashes": {
            "canon_lock_sha256": sha256_file("registry/canon_lock.json"),
            "realization_lock_sha256": sha256_file("registry/realization_lock.json"),
            "protocol_lock_sha256": sha256_file("registry/protocol_lock.json"),
            "registry_snapshot_sha256": sha256_file("registry/registry_snapshot.json"),
        },
        "derived": {
            "U_lat_J": U_lat_J,
            "U_lat_GeV": U_lat_GeV,
            "r_e_m": r_e,
            "S_dimless": S,
            "m_e_GeV": m_e_GeV,
        }
    }

    print(json.dumps(out, indent=2, ensure_ascii=False))

if __name__ == "__main__":
    main()