BARE HESSIAN MULTI-MODE BOND MODEL
======================================================================
Started: 2026-03-25 21:30:47
d = 3, gamma = 0.0650898425

PART 1: SINGLE KINK WELL — ALL BREATHER MODES
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Bound states (below mass gap omega^2=1): 3

mode      omega^2      omega  GWT omega     err%   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

PART 2: ALL MODE SPLITTINGS VS SEPARATION R
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Each single-well eigenvalue splits into bonding + antibonding.
Splitting = tunnel coupling strength for that mode.

   R        spl_0        spl_1        spl_2
-------------------------------------------
   4   0.11218150   0.68225928   0.00011993
   6   0.16185468   0.25940059   0.17154197
   8   0.01091770   0.04121911   0.11278535
  10   0.00102854   0.00614063   0.05671625
  12   0.00010811   0.00098356   0.03158496
  14   0.00001161   0.00016053   0.01843903
  16   0.00000125   0.00002630   0.01098524
  18   0.00000014   0.00000431   0.00661167
  20   0.00000001   0.00000071   0.00400113
  22   0.00000000   0.00000012   0.00242834
  24   0.00000000   0.00000002   0.00147597
  26   0.00000000   0.00000000   0.00089777
  28   0.00000000   0.00000000   0.00054627
  30   0.00000000   0.00000000   0.00033244
  32   0.00000000   0.00000000   0.00020233
  34   0.00000000   0.00000000   0.00012314
  36   0.00000000   0.00000000   0.00007495
  38   0.00000000   0.00000000   0.00004562
  40   0.00000000   0.00000000   0.00002777
  42   0.00000000   0.00000000   0.00001690
  44   0.00000000   0.00000000   0.00001029
  46   0.00000000   0.00000000   0.00000626
  48   0.00000000   0.00000000   0.00000381
  50   0.00000000   0.00000000   0.00000232

PART 3: BONDING SHIFT V_n(R) = E_bond_n(R) - E_single_n
-------------------------------------------------------
Negative = attractive (bonding). This is the energy gain per mode.

   R          V_0          V_1          V_2
-------------------------------------------
   4  +0.13161056  +0.06242517  +0.06198541
   6  -0.12023717  -0.18594951  -0.10955470
   8  -0.00770789  -0.03319068  -0.06077708
  10  -0.00075924  -0.00472930  -0.02470181
  12  -0.00008609  -0.00071408  -0.01310226
  14  -0.00001012  -0.00011029  -0.00786426
  16  -0.00000121  -0.00001721  -0.00487643
  18  -0.00000015  -0.00000271  -0.00303693
  20  -0.00000002  -0.00000043  -0.00188583
  22  -0.00000000  -0.00000007  -0.00116598
  24  -0.00000000  -0.00000001  -0.00071801
  26  -0.00000000  -0.00000000  -0.00044069
  28  -0.00000000  -0.00000000  -0.00026980
  30  -0.00000000  -0.00000000  -0.00016488
  32  -0.00000000  -0.00000000  -0.00010063
  34  -0.00000000  -0.00000000  -0.00006136
  36  -0.00000000  -0.00000000  -0.00003739
  38  +0.00000000  -0.00000000  -0.00002278
  40  +0.00000000  -0.00000000  -0.00001387
  42  +0.00000000  -0.00000000  -0.00000844
  44  +0.00000000  -0.00000000  -0.00000514
  46  +0.00000000  -0.00000000  -0.00000313
  48  +0.00000000  -0.00000000  -0.00000190
  50  -0.00000000  -0.00000000  -0.00000116

PART 4: TOTAL BOND ENERGY BY OCCUPATION
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V_total(R) = sum of V_n(R) over occupied modes.

Occupations (like electron configurations):
  1 mode:  H-like (1 electron)
  2 modes: He-like (2 electrons)
  3 modes: Li/Be-like
  4 modes: C-like (4 valence)
  All:     all bound modes

   R        1 mode       2 modes           all       3 modes
------------------------------------------------------------
   4   +0.13161056   +0.19403573   +0.25602114   +0.25602114
   6   -0.12023717   -0.30618667   -0.41574137   -0.41574137
   8   -0.00770789   -0.04089858   -0.10167566   -0.10167566
  10   -0.00075924   -0.00548854   -0.03019035   -0.03019035
  12   -0.00008609   -0.00080017   -0.01390243   -0.01390243
  14   -0.00001012   -0.00012041   -0.00798467   -0.00798467
  16   -0.00000121   -0.00001842   -0.00489485   -0.00489485
  18   -0.00000015   -0.00000285   -0.00303978   -0.00303978
  20   -0.00000002   -0.00000045   -0.00188627   -0.00188627
  22   -0.00000000   -0.00000007   -0.00116605   -0.00116605
  24   -0.00000000   -0.00000001   -0.00071802   -0.00071802
  26   -0.00000000   -0.00000000   -0.00044069   -0.00044069
  28   -0.00000000   -0.00000000   -0.00026980   -0.00026980
  30   -0.00000000   -0.00000000   -0.00016488   -0.00016488
  32   -0.00000000   -0.00000000   -0.00010063   -0.00010063
  34   -0.00000000   -0.00000000   -0.00006136   -0.00006136
  36   -0.00000000   -0.00000000   -0.00003739   -0.00003739
  38   +0.00000000   -0.00000000   -0.00002278   -0.00002278
  40   +0.00000000   -0.00000000   -0.00001387   -0.00001387
  42   +0.00000000   -0.00000000   -0.00000844   -0.00000844
  44   +0.00000000   -0.00000000   -0.00000514   -0.00000514
  46   +0.00000000   +0.00000000   -0.00000313   -0.00000313
  48   +0.00000000   -0.00000000   -0.00000190   -0.00000190
  50   -0.00000000   -0.00000000   -0.00000116   -0.00000116

PART 5: MORSE WELL ANALYSIS
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        1 mode: R_eq=    6, D_e=0.12023717, Morse_a=1.0659
       2 modes: R_eq=    6, D_e=0.30618667, Morse_a=0.9357
           all: R_eq=    6, D_e=0.41574137, Morse_a=0.2538
       3 modes: R_eq=    6, D_e=0.41574137, Morse_a=0.2538

PART 6: SPLITTING RATIOS — σ/π FROM EIGENVALUE STRUCTURE
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The ratio split_1/split_0 is the REAL pi/sigma coupling ratio.
This should relate to W_PI = cos(pi/d) = 0.5

   R      split_0      split_1      split_2    s1/s0    s2/s0
----------------------------------------------------------
   4   0.11218150   0.68225928   0.00011993   6.0817   0.0011
   6   0.16185468   0.25940059   0.17154197   1.6027   1.0599
   8   0.01091770   0.04121911   0.11278535   3.7754  10.3305
  10   0.00102854   0.00614063   0.05671625   5.9702  55.1423
  12   0.00010811   0.00098356   0.03158496   9.0978 292.1560
  14   0.00001161   0.00016053   0.01843903  13.8225 1587.6565
  16   0.00000125   0.00002630   0.01098524  20.9962 8770.2888
  18   0.00000014   0.00000431   0.00661167  31.8928 48911.9767
  20   0.00000001   0.00000071   0.00400113  48.4449 274247.0158
  22   0.00000000   0.00000012   0.00242834  73.5868 1542097.7196
  24   0.00000000   0.00000002   0.00147597 111.7777 8684178.8537

SPLITTING DECAY RATES:
mode  decay_rate  ratio_to_m0      eps_n    width
--------------------------------------------------
     0    1.091814       1.0000   0.065044    15.37
     1    0.899257       0.8236   0.129812     7.70
     2    0.214842       0.1968   0.194031     5.15

KEY RATIO:
  Mode 0 (σ) decay rate: 1.091814
  Mode 1 (π) decay rate: 0.899257
  Ratio mode1/mode0: 0.823636
  V8 W_PI = cos(π/d) = 0.500000

  At R_eq=6: split_1/split_0 = 1.602676

SUMMARY
======================================================================

Single kink well: 3 bound states

        1 mode: D_e = 0.120237 at R = 6
       2 modes: D_e = 0.306187 at R = 6
           all: D_e = 0.415741 at R = 6
       3 modes: D_e = 0.415741 at R = 6

The eigenvalue splittings ARE the bond — no perturbation needed.
Each mode's splitting = its contribution to the bond energy.
The σ/π ratio comes from the DECAY RATE ratio of the splittings.

Completed: 2026-03-25 21:30:49
