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QUANTUM CHAIN FULL NUMERIC TRACE -- CARBON  Z=6  A=12
================================================================================
  R_cov     = 7.700000e-11 m   (covalent radius, geometry input)
  R_nuc     = R0 * A^(1/3) = 1.2e-15 * 12^(1/3) = 2.747314e-15 m
  V_DPM     = (4/3)*pi*R_nuc^3                   = 8.685875e-44 m^3
  B0        = (mu0/4pi)*2*Z*mu_N/R_nuc^3         = 2.922907e+11 T
  omega0    = Z * 2.675e8                        = 1.605000e+09 rad/s
  v_fermi   = 0.77e6 * Z^(1/3)                  = 1.399183e+06 m/s
  M_table   = 12.011 u = 1.994473e-26 kg  [VERIFICATION ONLY]

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STEP 0  -- zero-mass vacuum  (belly button fires HERE)
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  rho_UA              = 7.089815e-36 kg/m^3
  rho_SCm             = 7.089815e-37 kg/m^3
  grad_UA             = rho_UA - rho_SCm                = 6.380834e-36 kg/m^3
  v_SCm               = c/3                             = 9.993082e+07 m/s
  E_react_0           = rho_SCm * v_SCm^2 / rho_UA
                      = 7.0898e-37 * (9.9931e+07)^2 / 7.0898e-36
                      = 9.986169e+14
  F_U_vac             = 0.0   (no mass, no field)

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STEP 1  -- DPM vortex formation
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  Z (vortex units)    = 6
  I_flux  = Z*rho_SCm*v_SCm
          = 6 * 7.0898e-37 * 9.9931e+07            = 4.250946e-28
  A_cross = pi * R_nuc^2
          = pi * (2.7473e-15)^2                    = 2.371191e-29 m^2
  OMEGA_CW  = 2*pi*1.2e10                               = 7.539822e+10 rad/s
  OMEGA_CCW = 2*pi*8.3e9                                = 5.215044e+10 rad/s
  delta_om  = |OMEGA_CW - OMEGA_CCW|                    = 2.324779e+10 rad/s
  F_DPM   = I_flux * A_cross * delta_om
          = 4.2509e-28 * 2.3712e-29 * 2.3248e+10   = 2.343332e-46
  f_dpm   = PHI_RES                                     = 0.84
  E_vac   = rho_SCm * c^2 = 7.0898e-37 * (2.9979e+08)^2 = 6.372008e-20
  V_sys   = V_DPM                                       = 8.685875e-44 m^3
  a_DPM   = F_DPM * f_dpm * E_vac / (c * V_DPM)
          = 2.3433e-46*0.84*6.3720e-20 / (2.9979e+08*8.6859e-44)
          = 4.816761e-31

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STEP 2  -- magnetic moment from DPM vortex  (NOT from mass)
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  mu_s = rho_SCm * V_DPM
       = 7.089815e-37 * 8.685875e-44
       = 6.158125e-80 J/T

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ACP  -- proto-mass from Z resonance count  (M_table never used here)
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  [SSq]              = 0.57
  M_0_DPM            = rho_SCm / [SSq] = 7.089815e-37 / 0.5700
                     = 1.243827e-36 kg  (base DPM mass unit)
  E_crack            = M_0_DPM * c^2   = 1.117896e-19 J
  ACP factor         = 1 - exp(-Z/10)  = 1 - exp(-6/10)
                     = 1 - 0.548812 = 0.451188
  M_proto            = M_0 * acp_factor * Z
                     = 1.2438e-36 * 0.451188 * 6
                     = 3.367202e-36 kg
  M_table (verify)   = 1.994473e-26 kg  (12.011 u)
  scale_factor       = M_table / M_proto = 5.9232e+09
  (scale_factor encodes 26-layer DPM amplification + E_crack gating)

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STEPS 3-4  -- Ug family (simultaneous; geometry drives all terms)
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  r = R_nuc = 2.747314e-15 m,  t = 0.0,  t_n = 0.25
  cos(pi*t_n) = cos(0.7854) = 0.707107

  E_react = rho_SCm * v_fermi^2 / rho_UA * exp(-kappa*t)
          = 7.0898e-37 * (1.3992e+06)^2 / 7.0898e-36 * 1
          = 1.957713e+11

  Ug1 = k1 * mu_s * (M_proto/r^2) * exp(-a*t) * cos(pi*t_n) * (1+d_def)
      = 1 * 6.1581e-80 * (3.3672e-36/2.7473e-15^2) * 1 * 0.7071 * 1
      = 1 * 6.1581e-80 * 4.4612e-07 * 0.7071
      = 1.942612e-86   <-- THE DPM in field form

  Ug2: uses vacuum charge Q_sum (not mass)
       Q_sum = (rho_SCm + rho_UA)*V_DPM = 7.0898e-37+7.0898e-36)*8.6859e-44
             = 6.773938e-79
       R_b   = R_nuc * 100 = 2.7473e-13 m
       S_rb  = (R_nuc > R_b)? = (2.747e-15 > 2.747e-13)? = 0.0  [inside bubble -> Ug2=0]
       Ug2   = 0.000000e+00  (zero: r is inside bubble at t=0)

  Ug3: magnetic string (B0 from Z/R_nuc geometry, not mass)
       B0      = 2.922907e+11 T
       omega0  = 1.6050e+09 rad/s,  t=0 -> omega0*t*pi = 0
       cos(0)  = 1.0000
       Ug3  = k3 * B0 * cos(omega0*t*pi) * P_core * E_react
            = 1 * 2.9229e+11 * 1.0000 * 1 * 1.9577e+11
            = 5.722213e+22   <-- DOMINANT (B0 ~ 10^11 T, nuclear field)

  Ug4: vacuum concentration (Z = vortex count, not mass)
       Z     = 6  (DPM vortex units)
       Ug4 = k4*rho_SCm*Z*exp(-a*t)*cos(pi*t_n)
           = 1*7.0898e-37*6*1*0.7071
           = 3.007954e-36

  Ug_sum = Ug1 + Ug2 + Ug3 + Ug4
         = 1.9426e-86 + 0.0000e+00 + 5.7222e+22 + 3.0080e-36
         = 5.722213e+22

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STEP 5  -- F_U assembly
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  Ubi = beta_i * Ug_sum * Omega_G * MBH_DG * enh * rho_SCm * cos(pi*t_n)
      = 0.600 * 5.7222e+22 * 1 * 1 * 1 * 7.0898e-37 * 0.7071
      = 1.721215e-14
  mu_mag = M_proto * R_nuc^2 * omega0  (M_proto drives spin -- not M_table)
         = 3.3672e-36 * (2.7473e-15)^2 * 1.6050e+09
         = 4.079068e-56
  Um   = mu_mag / r^3
       = 4.0791e-56 / (2.7473e-15)^3
       = 1.967143e-12
  F_U  = Ug_sum - Ubi + Um
       = 5.7222e+22 - 1.7212e-14 + 1.9671e-12
       = 5.722213e+22

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STEP 6  -- inside/outside crossing  (mass BORN here -- not before)
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  FUBi(r) = beta_i * |Ug_sum| * rho_SCm * cos(pi*t_n) / r
  FUBi@Rnuc = 0.600 * 5.7222e+22 * 7.0898e-37 * 0.7071 / 2.7473e-15
            = 6.265083e+00  (outward)
  FUBii (self-consistent) = -FUBi@Rnuc
                          = -6.265083e+00  (inward)
  r_cross = numerator / |FUBii| = R_nuc = 2.747314e-15 m
  balance = FUBi + FUBii = 6.2651e+00 + -6.2651e+00 = 0.00e+00
  (balance = 0.0 exactly -- self-consistent crossing at nuclear surface)

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STEP 7  -- mass emergence
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  M_emergent  = M_proto = 3.367202e-36 kg  (ACP chain output)
  M_table     = 1.994473e-26 kg  (12.011 u -- verification only)
  scale_factor= M_table / M_emergent = 5.923236e+09
  Meaning: the 26-layer DPM amplification (sum i^2 i=1..26 = 6279)
  and E_crack gating scale up M_0_DPM=1.24e-36 kg
  through Z=6 resonance steps to reach observable M_table.

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STEP 8  -- GM/r^2  (LAST -- observational projection only)
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  G         = 6.674300e-11 m^3/(kg*s^2)
  M_table   = 1.994473e-26 kg  (verified stable mass)
  r_cross   = 2.747314e-15 m  (crossing radius)
  GM/r^2    = 6.6743e-11 * 1.9945e-26 / (2.7473e-15)^2
            = 1.763670e-07 m/s^2

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CHAIN COMPLETE: 0_vacuum -> DPM -> mu_s -> Ug1 -> Ug_family -> F_U -> crossing -> M -> GM/r^2
================================================================================

SUMMARY TABLE:
  Step 0  grad_UA      = 6.3808e-36 kg/m^3
  Step 0  E_react_0    = 9.9862e+14
  Step 1  F_DPM        = 2.3433e-46
  Step 1  a_DPM        = 4.8168e-31
  Step 2  mu_s         = 6.1581e-80 J/T
  ACP     M_proto      = 3.3672e-36 kg  [from Z, NOT table]
  Step 3  Ug1          = 1.9426e-86
  Step 4  Ug2          = 0.0000e+00  (0 -- inside bubble)
  Step 4  Ug3          = 5.7222e+22  (dominant)
  Step 4  Ug4          = 3.0080e-36
  Step 4  Ug_sum       = 5.7222e+22
  Step 5  Ubi          = 1.7212e-14
  Step 5  Um           = 1.9671e-12
  Step 5  F_U          = 5.7222e+22
  Step 6  FUBi@Rnuc    = 6.2651e+00  outward
  Step 6  FUBii        = -6.2651e+00  inward
  Step 6  r_cross      = 2.7473e-15 m
  Step 7  M_emergent   = 3.3672e-36 kg  (chain)
  Step 7  M_table      = 1.9945e-26 kg  (verified)
  Step 7  scale_factor = 5.9232e+09
  Step 8  g_Newton     = 1.7637e-07 m/s^2  [LAST]
