BREATHER MODE SHIFT — SELF-CONSISTENT BOND CORRECTION
======================================================================
Started: 2026-03-25 18:19:51
d = 3, gamma = 0.0650898425

PART 1: ISOLATED BREATHER MODE FREQUENCIES
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Lattice: 512 sites, kink_width = 3
mode      omega^2      omega  GWT_omega   error%   status
     0  -0.37237471   0.610225   0.997882    38.85%    BOUND
     1   0.12538659   0.354100   0.991539    64.29%    BOUND
     2   0.93805223   0.968531   0.980995     1.27%    BOUND
     3   1.00003937   1.000020   0.966298     3.49%     band
     4   1.00015242   1.000076   0.947507     5.55%     band
     5   1.00035431   1.000177   0.924704     8.16%     band
     6   1.00060964   1.000305   0.897984    11.39%     band
     7   1.00098405   1.000492   0.867462    15.34%     band
     8   1.00137160   1.000686   0.833265    20.09%     band
     9   1.00192826   1.000964   0.795540    25.82%     band
    10   1.00243815   1.001218   0.754445    32.71%     band
    11   1.00318652   1.001592   0.710155    41.04%     band
    12   1.00380913   1.001903   0.662857    51.15%     band
    13   1.00475825   1.002376   0.612752    63.59%     band
    14   1.00548430   1.002738   0.560052    79.04%     band

Bound states below mass gap: 3

PART 2: MODE FREQUENCY SHIFT VS SEPARATION R
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Two protons at separation R. Track lowest eigenvalue shifts.

   R           E0           E1     shift_E0     shift_E1        split
---------------------------------------------------------------------------
   6  -0.49261187  -0.33075720  -0.12023717  +0.04161751   0.16185468
   8  -0.38008260  -0.36916490  -0.00770789  +0.00320981   0.01091770
  10  -0.37313394  -0.37210540  -0.00075924  +0.00026931   0.00102854
  12  -0.37246080  -0.37235269  -0.00008609  +0.00002202   0.00010811
  14  -0.37238483  -0.37237321  -0.00001012  +0.00000149   0.00001161
  16  -0.37237592  -0.37237466  -0.00000121  +0.00000004   0.00000125
  18  -0.37237485  -0.37237472  -0.00000015  -0.00000001   0.00000014
  20  -0.37237472  -0.37237471  -0.00000002  -0.00000000   0.00000001
  22  -0.37237471  -0.37237471  -0.00000000  -0.00000000   0.00000000
  24  -0.37237471  -0.37237471  -0.00000000  -0.00000000   0.00000000
  26  -0.37237471  -0.37237471  -0.00000000  -0.00000000   0.00000000
  28  -0.37237471  -0.37237471  -0.00000000  -0.00000000   0.00000000
  30  -0.37237471  -0.37237471  -0.00000000  -0.00000000   0.00000000
  32  -0.37237471  -0.37237471  -0.00000000  -0.00000000   0.00000000
  34  -0.37237471  -0.37237471  -0.00000000  -0.00000000   0.00000000
  36  -0.37237471  -0.37237471  -0.00000000  -0.00000000   0.00000000
  38  -0.37237471  -0.37237471  -0.00000000  -0.00000000   0.00000000
  40  -0.37237471  -0.37237471  -0.00000000  +0.00000000   0.00000000
  42  -0.37237471  -0.37237471  -0.00000000  +0.00000000   0.00000000
  44  -0.37237471  -0.37237471  -0.00000000  -0.00000000   0.00000000
  46  -0.37237471  -0.37237471  -0.00000000  -0.00000000   0.00000000
  48  -0.37237471  -0.37237471  +0.00000000  +0.00000000   0.00000000
  50  -0.37237471  -0.37237471  -0.00000000  -0.00000000   0.00000000

PART 3: SHIFT DECAY RATE — DOES IT MATCH eps_n?
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GWT prediction: eps_1 = sin(gamma) = 0.06504389
Expected decay rate of shift: exp(-0.0650 * R)

Fitted decay rate: 1.087175
eps_1 = 0.065044
Ratio (decay/eps_1): 16.7145
2*eps_1 = 0.130088 (kink-kink repulsion rate)

  n=1: eps_n = 0.065044, ratio = 16.7145
  n=2: eps_n = 0.129812, ratio = 8.3750
  n=3: eps_n = 0.194031, ratio = 5.6031
  n=4: eps_n = 0.257428, ratio = 4.2232
  n=7: eps_n = 0.440027, ratio = 2.4707

PART 4: CROSS-MODE SHIFTS
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How does a mode-n breather shift when near a mode-m breather?
(Asymmetric: wide mode feels narrow mode differently than reverse)

Cross-mode overlap integrals:
  n   m    eps_n    eps_m  width_n  width_m overlap_at_R10 overlap_at_R20
    1   1   0.0650   0.0650    15.37    15.37   2.170777e+01   1.509603e+01
    1   4   0.0650   0.2574    15.37     3.88   6.801644e+00   3.356864e+00
    1   7   0.0650   0.4400    15.37     2.27   4.024434e+00   1.916359e+00
    4   1   0.2574   0.0650     3.88    15.37   6.801644e+00   3.356864e+00
    4   4   0.2574   0.2574     3.88     3.88   1.038358e+00   1.939265e-02
    4   7   0.2574   0.4400     3.88     2.27   3.392792e-01   2.459023e-03
    7   1   0.4400   0.0650     2.27    15.37   4.024434e+00   1.916359e+00
    7   4   0.4400   0.2574     2.27     3.88   3.392792e-01   2.459023e-03
    7   7   0.4400   0.4400     2.27     2.27   3.999706e-02   1.554854e-05

PART 5: SELF-CONSISTENT MODE SHIFT
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Iteration: shift changes width, width changes overlap, repeat.

   R iter      omega        eps    width     shift%
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   8    0   0.385443   1.071712     0.93   -61.3739%
   8    1   0.613522   1.173204     0.85   -38.5176%
   8    2   0.612401   1.172619     0.85   -38.6299%
   8    3   0.612403   1.172620     0.85   -38.6298%
   8    4   0.612403   1.172620     0.85   -38.6298%
   8    5   0.612403   1.172620     0.85   -38.6298%
  --> Converged in 6 iterations

  10    0   0.412100   1.081585     0.92   -58.7026%
  10    1   0.610436   1.171594     0.85   -38.8269%
  10    2   0.610429   1.171590     0.85   -38.8276%
  10    3   0.610429   1.171590     0.85   -38.8276%
  --> Converged in 4 iterations

  12    0   0.416873   1.083413     0.92   -58.2242%
  12    1   0.610251   1.171498     0.85   -38.8454%
  12    2   0.610251   1.171498     0.85   -38.8454%
  12    3   0.610251   1.171498     0.85   -38.8454%
  --> Converged in 4 iterations

  16    0   0.399834   1.076971     0.93   -59.9318%
  16    1   0.610226   1.171484     0.85   -38.8479%
  16    2   0.610226   1.171484     0.85   -38.8479%
  --> Converged in 3 iterations

  20    0   0.399812   1.076963     0.93   -59.9340%
  20    1   0.610225   1.171484     0.85   -38.8480%
  20    2   0.610225   1.171484     0.85   -38.8480%
  --> Converged in 3 iterations

  30    0   0.399852   1.076978     0.93   -59.9300%
  30    1   0.610225   1.171484     0.85   -38.8480%
  30    2   0.610225   1.171484     0.85   -38.8480%
  --> Converged in 3 iterations

  40    0   0.399852   1.076978     0.93   -59.9300%
  40    1   0.610225   1.171484     0.85   -38.8480%
  40    2   0.610225   1.171484     0.85   -38.8480%
  --> Converged in 3 iterations

PART 6: BOND CURVE — FIXED MODE vs SELF-CONSISTENT
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Compare bond energy with and without self-consistent mode shifting.

   R      V_fixed    V_selfcon   correction     corr_%
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   6  -0.12023717  -0.08734091  +0.03289626   +27.3595%
   8  -0.00770789  -0.00266055  +0.00504734   +65.4828%
  10  -0.00075924  -0.00024880  +0.00051044   +67.2299%
  12  -0.00008609  -0.00003208  +0.00005401   +62.7373%
  14  -0.00001012  -0.00000431  +0.00000581   +57.3730%
  16  -0.00000121  -0.00000058  +0.00000063   +51.7738%
  18  -0.00000015  -0.00000008  +0.00000007   +46.1262%
  20  -0.00000002  -0.00000001  +0.00000001   +40.5763%
  22  -0.00000000  -0.00000000  +0.00000000   +35.2572%
  24  -0.00000000  -0.00000000  +0.00000000   +30.2793%
  26  -0.00000000  -0.00000000  +0.00000000   +25.7284%
  28  -0.00000000  -0.00000000  +0.00000000   +21.6339%
  30  -0.00000000  -0.00000000  +0.00000000    +0.0000%
  32  -0.00000000  -0.00000000  +0.00000000    +0.0000%
  34  -0.00000000  -0.00000000  +0.00000000    +0.0000%
  36  -0.00000000  -0.00000000  -0.00000000    +0.0000%
  38  -0.00000000  -0.00000000  +0.00000000    +0.0000%
  40  -0.00000000  -0.00000000  +0.00000000    +0.0000%
  42  -0.00000000  +0.00000000  +0.00000000    +0.0000%
  44  -0.00000000  -0.00000000  -0.00000000    +0.0000%

WELL DEPTH COMPARISON:
  Fixed:         R_eq =   6, D_e = 0.12023717
  Self-consistent: R_eq =   6, D_e = 0.08734091
  Depth change: -27.36%

SUMMARY
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Key findings:
  1. Isolated breather: omega_1 = 0.997882, eps_1 = 0.065044
  2. Shift decay rate vs eps_1:
     Fitted: 1.087175, eps_1: 0.065044, ratio: 16.7145
  3. Self-consistent convergence: typically 2-3 iterations
     R=  8: shift = -38.6298%, converged in 6 iterations
     R= 10: shift = -38.8276%, converged in 4 iterations
     R= 12: shift = -38.8454%, converged in 4 iterations
     R= 16: shift = -38.8479%, converged in 3 iterations
     R= 20: shift = -38.8480%, converged in 3 iterations
     R= 30: shift = -38.8480%, converged in 3 iterations
     R= 40: shift = -38.8480%, converged in 3 iterations
  4. Bond depth correction from self-consistency:
     Fixed D_e = 0.12023717
     Self-consistent D_e = 0.08734091
     Correction = -27.36%

Completed: 2026-03-25 18:19:52
