BOND EIGENVALUE FLOW — RAW PHYSICS
======================================================================
Started: 2026-03-26 23:49:15

PART 1: SINGLE KINK-ANTIKINK WELL
-------------------------------------------------------
Eigenvalues below mass gap (omega^2 < 1): 3

 idx      omega^2      omega    binding       type
--------------------------------------------------
    0    -0.372375  -0.610225  +1.372375    TACHYON
    1    +0.125387  +0.354100  +0.874613      BOUND
    2    +0.938052  +0.968531  +0.061948      BOUND
    3    +1.000039  +1.000020  -0.000039  band edge
    4    +1.000152  +1.000076  -0.000152  band edge
    5    +1.000354  +1.000177  -0.000354  band edge
    6    +1.000610  +1.000305  -0.000610  band edge
    7    +1.000984  +1.000492  -0.000984  band edge
    8    +1.001372  +1.000686  -0.001372  band edge
    9    +1.001928  +1.000964  -0.001928  band edge
   10    +1.002438  +1.001218  -0.002438  band edge
   11    +1.003187  +1.001592  -0.003187  band edge

PART 2: TWO KINK-ANTIKINK WELLS — EIGENVALUE FLOW
-------------------------------------------------------
Every integer R from 4 to 60. All eigenvalues below omega^2 = 1.1

  R n<gap        ev_0        ev_1        ev_2        ev_3        ev_4        ev_5        ev_6        ev_7        ev_8        ev_9 
--------------------------------------------------------------------------------------------------------------------------------
  4     4   -0.240764   -0.128583   +0.187812   +0.870071   +1.000038   +1.000158   +1.000339   +1.000630   +1.000941   +1.001418 
  5     4   -0.240764   -0.128583   +0.187812   +0.870071   +1.000038   +1.000158   +1.000339   +1.000630   +1.000941   +1.001418 
  6     5   -0.492612   -0.330757   -0.060563   +0.198838   +0.828498   +1.000040   +1.000147   +1.000356   +1.000587   +1.000988 
  7     5   -0.492612   -0.330757   -0.060563   +0.198838   +0.828498   +1.000040   +1.000147   +1.000356   +1.000587   +1.000988 
  8     6   -0.380083   -0.369165   +0.092196   +0.133415   +0.877275   +0.990060   +1.000040   +1.000170   +1.000359   +1.000678 
  9     6   -0.380083   -0.369165   +0.092196   +0.133415   +0.877275   +0.990060   +1.000040   +1.000170   +1.000359   +1.000678 
 10     6   -0.373134   -0.372105   +0.120657   +0.126798   +0.913350   +0.970067   +1.000040   +1.000165   +1.000362   +1.000660 
 11     6   -0.373134   -0.372105   +0.120657   +0.126798   +0.913350   +0.970067   +1.000040   +1.000165   +1.000362   +1.000660 
 12     6   -0.372461   -0.372353   +0.124673   +0.125656   +0.924950   +0.956535   +1.000041   +1.000165   +1.000366   +1.000659 
 13     6   -0.372461   -0.372353   +0.124673   +0.125656   +0.924950   +0.956535   +1.000041   +1.000165   +1.000366   +1.000659 
 14     6   -0.372385   -0.372373   +0.125276   +0.125437   +0.930188   +0.948627   +1.000041   +1.000166   +1.000369   +1.000662 
 15     6   -0.372385   -0.372373   +0.125276   +0.125437   +0.930188   +0.948627   +1.000041   +1.000166   +1.000369   +1.000662 
 16     6   -0.372376   -0.372375   +0.125369   +0.125396   +0.933176   +0.944161   +1.000041   +1.000167   +1.000372   +1.000666 
 17     6   -0.372376   -0.372375   +0.125369   +0.125396   +0.933176   +0.944161   +1.000041   +1.000167   +1.000372   +1.000666 
 18     6   -0.372375   -0.372375   +0.125384   +0.125388   +0.935015   +0.941627   +1.000042   +1.000168   +1.000375   +1.000671 
 19     6   -0.372375   -0.372375   +0.125384   +0.125388   +0.935015   +0.941627   +1.000042   +1.000168   +1.000375   +1.000671 
 20     6   -0.372375   -0.372375   +0.125386   +0.125387   +0.936166   +0.940168   +1.000042   +1.000169   +1.000378   +1.000675 
 21     6   -0.372375   -0.372375   +0.125386   +0.125387   +0.936166   +0.940168   +1.000042   +1.000169   +1.000378   +1.000675 
 22     6   -0.372375   -0.372375   +0.125387   +0.125387   +0.936886   +0.939315   +1.000042   +1.000170   +1.000381   +1.000681 
 23     6   -0.372375   -0.372375   +0.125387   +0.125387   +0.936886   +0.939315   +1.000042   +1.000170   +1.000381   +1.000681 
 24     6   -0.372375   -0.372375   +0.125387   +0.125387   +0.937334   +0.938810   +1.000043   +1.000171   +1.000384   +1.000686 
 25     6   -0.372375   -0.372375   +0.125387   +0.125387   +0.937334   +0.938810   +1.000043   +1.000171   +1.000384   +1.000686 
 26     6   -0.372375   -0.372375   +0.125387   +0.125387   +0.937612   +0.938509   +1.000043   +1.000173   +1.000387   +1.000691 
 27     6   -0.372375   -0.372375   +0.125387   +0.125387   +0.937612   +0.938509   +1.000043   +1.000173   +1.000387   +1.000691 
 28     6   -0.372375   -0.372375   +0.125387   +0.125387   +0.937782   +0.938329   +1.000043   +1.000174   +1.000391   +1.000697 
 29     6   -0.372375   -0.372375   +0.125387   +0.125387   +0.937782   +0.938329   +1.000043   +1.000174   +1.000391   +1.000697 
 30     6   -0.372375   -0.372375   +0.125387   +0.125387   +0.937887   +0.938220   +1.000044   +1.000176   +1.000394   +1.000703 
 31     6   -0.372375   -0.372375   +0.125387   +0.125387   +0.937887   +0.938220   +1.000044   +1.000176   +1.000394   +1.000703 
 32     6   -0.372375   -0.372375   +0.125387   +0.125387   +0.937952   +0.938154   +1.000044   +1.000177   +1.000397   +1.000708 
 33     6   -0.372375   -0.372375   +0.125387   +0.125387   +0.937952   +0.938154   +1.000044   +1.000177   +1.000397   +1.000708 
 34     6   -0.372375   -0.372375   +0.125387   +0.125387   +0.937991   +0.938114   +1.000045   +1.000179   +1.000401   +1.000714 
 35     6   -0.372375   -0.372375   +0.125387   +0.125387   +0.937991   +0.938114   +1.000045   +1.000179   +1.000401   +1.000714 
 36     6   -0.372375   -0.372375   +0.125387   +0.125387   +0.938015   +0.938090   +1.000045   +1.000180   +1.000404   +1.000720 
 37     6   -0.372375   -0.372375   +0.125387   +0.125387   +0.938015   +0.938090   +1.000045   +1.000180   +1.000404   +1.000720 
 38     6   -0.372375   -0.372375   +0.125387   +0.125387   +0.938029   +0.938075   +1.000045   +1.000182   +1.000408   +1.000726 
 39     6   -0.372375   -0.372375   +0.125387   +0.125387   +0.938029   +0.938075   +1.000045   +1.000182   +1.000408   +1.000726 
 40     6   -0.372375   -0.372375   +0.125387   +0.125387   +0.938038   +0.938066   +1.000046   +1.000183   +1.000411   +1.000732 
 41     6   -0.372375   -0.372375   +0.125387   +0.125387   +0.938038   +0.938066   +1.000046   +1.000183   +1.000411   +1.000732 
 42     6   -0.372375   -0.372375   +0.125387   +0.125387   +0.938044   +0.938061   +1.000046   +1.000185   +1.000415   +1.000739 
 43     6   -0.372375   -0.372375   +0.125387   +0.125387   +0.938044   +0.938061   +1.000046   +1.000185   +1.000415   +1.000739 
 44     6   -0.372375   -0.372375   +0.125387   +0.125387   +0.938047   +0.938057   +1.000046   +1.000186   +1.000418   +1.000745 
 45     6   -0.372375   -0.372375   +0.125387   +0.125387   +0.938047   +0.938057   +1.000046   +1.000186   +1.000418   +1.000745 
 46     6   -0.372375   -0.372375   +0.125387   +0.125387   +0.938049   +0.938055   +1.000047   +1.000188   +1.000422   +1.000751 
 47     6   -0.372375   -0.372375   +0.125387   +0.125387   +0.938049   +0.938055   +1.000047   +1.000188   +1.000422   +1.000751 
 48     6   -0.372375   -0.372375   +0.125387   +0.125387   +0.938050   +0.938054   +1.000047   +1.000190   +1.000425   +1.000758 
 49     6   -0.372375   -0.372375   +0.125387   +0.125387   +0.938050   +0.938054   +1.000047   +1.000190   +1.000425   +1.000758 
 50     6   -0.372375   -0.372375   +0.125387   +0.125387   +0.938051   +0.938053   +1.000048   +1.000191   +1.000429   +1.000764 
 51     6   -0.372375   -0.372375   +0.125387   +0.125387   +0.938051   +0.938053   +1.000048   +1.000191   +1.000429   +1.000764 
 52     6   -0.372375   -0.372375   +0.125387   +0.125387   +0.938052   +0.938053   +1.000048   +1.000193   +1.000433   +1.000771 
 53     6   -0.372375   -0.372375   +0.125387   +0.125387   +0.938052   +0.938053   +1.000048   +1.000193   +1.000433   +1.000771 
 54     6   -0.372375   -0.372375   +0.125387   +0.125387   +0.938052   +0.938053   +1.000049   +1.000195   +1.000437   +1.000778 
 55     6   -0.372375   -0.372375   +0.125387   +0.125387   +0.938052   +0.938053   +1.000049   +1.000195   +1.000437   +1.000778 
 56     6   -0.372375   -0.372375   +0.125387   +0.125387   +0.938052   +0.938052   +1.000049   +1.000196   +1.000441   +1.000785 
 57     6   -0.372375   -0.372375   +0.125387   +0.125387   +0.938052   +0.938052   +1.000049   +1.000196   +1.000441   +1.000785 
 58     6   -0.372375   -0.372375   +0.125387   +0.125387   +0.938052   +0.938052   +1.000049   +1.000198   +1.000445   +1.000792 
 59     6   -0.372375   -0.372375   +0.125387   +0.125387   +0.938052   +0.938052   +1.000049   +1.000198   +1.000445   +1.000792 
 60     6   -0.372375   -0.372375   +0.125387   +0.125387   +0.938052   +0.938052   +1.000050   +1.000200   +1.000449   +1.000799 

PART 3: EIGENVALUE PAIR TRACKING
-------------------------------------------------------
At large R, each single-well eigenvalue appears twice (degenerate).
As R decreases, they split into bonding + antibonding.
Track the splitting for each pair.

At R=60 (isolated wells):
  ev[0] = -0.37237471
  ev[1] = -0.37237471
  ev[2] = +0.12538659
  ev[3] = +0.12538659
  ev[4] = +0.93805214
  ev[5] = +0.93805233
  ev[6] = +1.00004988
  ev[7] = +1.00019980

Identified 10 eigenvalue pairs at R=60:
  Pair 0: ev[0,1] avg=-0.372375 split=2.28e-15
  Pair 1: ev[2,3] avg=+0.125387 split=1.39e-15
  Pair 2: ev[4,5] avg=+0.938052 split=1.94e-07
  Pair 3: ev[6,7] avg=+1.000125 split=1.50e-04
  Pair 4: ev[8,9] avg=+1.000624 split=3.50e-04
  Pair 5: ev[10,11] avg=+1.001521 split=5.52e-04

SPLITTING vs R for each pair:
  R    pair0_avg       split0    pair1_avg       split1    pair2_avg       split2    pair3_avg       split3 
---------------------------------------------------------------------------------------------------
  4    -0.184673   0.11218150    +0.528941   0.68225928    +1.000098   0.00011993    +1.000485   0.00029145 
  5    -0.184673   0.11218150    +0.528941   0.68225928    +1.000098   0.00011993    +1.000485   0.00029145 
  6    -0.411685   0.16185468    +0.069137   0.25940059    +0.914269   0.17154197    +1.000251   0.00020884 
  7    -0.411685   0.16185468    +0.069137   0.25940059    +0.914269   0.17154197    +1.000251   0.00020884 
  8    -0.374624   0.01091770    +0.112805   0.04121911    +0.933668   0.11278535    +1.000105   0.00012981 
  9    -0.374624   0.01091770    +0.112805   0.04121911    +0.933668   0.11278535    +1.000105   0.00012981 
 10    -0.372620   0.00102854    +0.123728   0.00614063    +0.941709   0.05671625    +1.000103   0.00012477 
 11    -0.372620   0.00102854    +0.123728   0.00614063    +0.941709   0.05671625    +1.000103   0.00012477 
 12    -0.372407   0.00010811    +0.125164   0.00098356    +0.940742   0.03158496    +1.000103   0.00012426 
 13    -0.372407   0.00010811    +0.125164   0.00098356    +0.940742   0.03158496    +1.000103   0.00012426 
 14    -0.372379   0.00001161    +0.125357   0.00016053    +0.939407   0.01843903    +1.000103   0.00012459 
 15    -0.372379   0.00001161    +0.125357   0.00016053    +0.939407   0.01843903    +1.000103   0.00012459 
 16    -0.372375   0.00000125    +0.125383   0.00002630    +0.938668   0.01098524    +1.000104   0.00012523 
 17    -0.372375   0.00000125    +0.125383   0.00002630    +0.938668   0.01098524    +1.000104   0.00012523 
 18    -0.372375   0.00000014    +0.125386   0.00000431    +0.938321   0.00661167    +1.000105   0.00012602 
 19    -0.372375   0.00000014    +0.125386   0.00000431    +0.938321   0.00661167    +1.000105   0.00012602 
 20    -0.372375   0.00000001    +0.125387   0.00000071    +0.938167   0.00400113    +1.000105   0.00012689 
 21    -0.372375   0.00000001    +0.125387   0.00000071    +0.938167   0.00400113    +1.000105   0.00012689 
 22    -0.372375   0.00000000    +0.125387   0.00000012    +0.938100   0.00242834    +1.000106   0.00012782 
 23    -0.372375   0.00000000    +0.125387   0.00000012    +0.938100   0.00242834    +1.000106   0.00012782 
 24    -0.372375   0.00000000    +0.125387   0.00000002    +0.938072   0.00147597    +1.000107   0.00012880 
 25    -0.372375   0.00000000    +0.125387   0.00000002    +0.938072   0.00147597    +1.000107   0.00012880 
 26    -0.372375   0.00000000    +0.125387   0.00000000    +0.938060   0.00089777    +1.000108   0.00012980 
 27    -0.372375   0.00000000    +0.125387   0.00000000    +0.938060   0.00089777    +1.000108   0.00012980 
 28    -0.372375   0.00000000    +0.125387   0.00000000    +0.938056   0.00054627    +1.000109   0.00013083 
 29    -0.372375   0.00000000    +0.125387   0.00000000    +0.938056   0.00054627    +1.000109   0.00013083 
 30    -0.372375   0.00000000    +0.125387   0.00000000    +0.938054   0.00033244    +1.000110   0.00013188 
 31    -0.372375   0.00000000    +0.125387   0.00000000    +0.938054   0.00033244    +1.000110   0.00013188 
 32    -0.372375   0.00000000    +0.125387   0.00000000    +0.938053   0.00020233    +1.000111   0.00013296 
 33    -0.372375   0.00000000    +0.125387   0.00000000    +0.938053   0.00020233    +1.000111   0.00013296 
 34    -0.372375   0.00000000    +0.125387   0.00000000    +0.938052   0.00012314    +1.000112   0.00013405 
 35    -0.372375   0.00000000    +0.125387   0.00000000    +0.938052   0.00012314    +1.000112   0.00013405 
 36    -0.372375   0.00000000    +0.125387   0.00000000    +0.938052   0.00007495    +1.000112   0.00013517 
 37    -0.372375   0.00000000    +0.125387   0.00000000    +0.938052   0.00007495    +1.000112   0.00013517 
 38    -0.372375   0.00000000    +0.125387   0.00000000    +0.938052   0.00004562    +1.000113   0.00013630 
 39    -0.372375   0.00000000    +0.125387   0.00000000    +0.938052   0.00004562    +1.000113   0.00013630 
 40    -0.372375   0.00000000    +0.125387   0.00000000    +0.938052   0.00002777    +1.000114   0.00013745 
 41    -0.372375   0.00000000    +0.125387   0.00000000    +0.938052   0.00002777    +1.000114   0.00013745 
 42    -0.372375   0.00000000    +0.125387   0.00000000    +0.938052   0.00001690    +1.000115   0.00013862 
 43    -0.372375   0.00000000    +0.125387   0.00000000    +0.938052   0.00001690    +1.000115   0.00013862 
 44    -0.372375   0.00000000    +0.125387   0.00000000    +0.938052   0.00001029    +1.000116   0.00013980 
 45    -0.372375   0.00000000    +0.125387   0.00000000    +0.938052   0.00001029    +1.000116   0.00013980 
 46    -0.372375   0.00000000    +0.125387   0.00000000    +0.938052   0.00000626    +1.000117   0.00014100 
 47    -0.372375   0.00000000    +0.125387   0.00000000    +0.938052   0.00000626    +1.000117   0.00014100 
 48    -0.372375   0.00000000    +0.125387   0.00000000    +0.938052   0.00000381    +1.000118   0.00014223 
 49    -0.372375   0.00000000    +0.125387   0.00000000    +0.938052   0.00000381    +1.000118   0.00014223 
 50    -0.372375   0.00000000    +0.125387   0.00000000    +0.938052   0.00000232    +1.000119   0.00014346 
 51    -0.372375   0.00000000    +0.125387   0.00000000    +0.938052   0.00000232    +1.000119   0.00014346 
 52    -0.372375   0.00000000    +0.125387   0.00000000    +0.938052   0.00000141    +1.000120   0.00014472 
 53    -0.372375   0.00000000    +0.125387   0.00000000    +0.938052   0.00000141    +1.000120   0.00014472 
 54    -0.372375   0.00000000    +0.125387   0.00000000    +0.938052   0.00000086    +1.000122   0.00014599 
 55    -0.372375   0.00000000    +0.125387   0.00000000    +0.938052   0.00000086    +1.000122   0.00014599 
 56    -0.372375   0.00000000    +0.125387   0.00000000    +0.938052   0.00000052    +1.000123   0.00014729 
 57    -0.372375   0.00000000    +0.125387   0.00000000    +0.938052   0.00000052    +1.000123   0.00014729 
 58    -0.372375   0.00000000    +0.125387   0.00000000    +0.938052   0.00000032    +1.000124   0.00014860 
 59    -0.372375   0.00000000    +0.125387   0.00000000    +0.938052   0.00000032    +1.000124   0.00014860 
 60    -0.372375   0.00000000    +0.125387   0.00000000    +0.938052   0.00000019    +1.000125   0.00014992 

PART 4: DOES THE TACHYON STABILIZE?
-------------------------------------------------------
Single well: mode 0 has omega^2 = -0.372 (TACHYON).
In the double well, does the lowest eigenvalue remain negative?
If it becomes positive at some R, the tachyon 'heals' in pairs.

  R    ev_lowest     status
----------------------------
  4    -0.240764    TACHYON
  5    -0.240764    TACHYON
  6    -0.492612    TACHYON
  7    -0.492612    TACHYON
  8    -0.380083    TACHYON
  9    -0.380083    TACHYON
 10    -0.373134    TACHYON
 11    -0.373134    TACHYON
 12    -0.372461    TACHYON
 13    -0.372461    TACHYON
 14    -0.372385    TACHYON
 15    -0.372385    TACHYON
 16    -0.372376    TACHYON
 17    -0.372376    TACHYON
 18    -0.372375    TACHYON
 19    -0.372375    TACHYON
 20    -0.372375    TACHYON
 30    -0.372375    TACHYON
 40    -0.372375    TACHYON
 50    -0.372375    TACHYON
 60    -0.372375    TACHYON


PART 5: WHERE DO THE EIGENVECTORS LIVE?
-------------------------------------------------------
For each eigenvalue at R=10, compute the weight on well A vs well B.

R = 10, wells at 251 and 261
Well A region: [241, 261]
Well B region: [251, 271]

 idx    omega^2     wt_A     wt_B   wt_mid      A-B         type
------------------------------------------------------------
    0  -0.373134   0.8061   0.8061   0.0000  -0.0000      BONDING
    1  -0.372105   0.8046   0.8046   0.0000  +0.0000     ANTIBOND
    2  +0.120657   0.7505   0.7505   0.0000  +0.0000      BONDING
    3  +0.126798   0.7490   0.7490   0.0000  -0.0000     ANTIBOND
    4  +0.913350   0.8055   0.8055   0.0025  -0.0000      BONDING
    5  +0.970067   0.6552   0.6552   0.0338  +0.0000     ANTIBOND
    6  +1.000040   0.0000   0.0000   1.0000  -0.0000  delocalized
    7  +1.000165   0.0001   0.0001   0.9999  +0.0000  delocalized
    8  +1.000362   0.0002   0.0002   0.9996  -0.0000  delocalized
    9  +1.000660   0.0003   0.0003   0.9995  +0.0000  delocalized

PART 6: EIGENVECTOR LOCALIZATION AT R=6 (close range)
-------------------------------------------------------
R = 6, wells at 253 and 259
 idx    omega^2     wt_A     wt_B      A-B         type
-------------------------------------------------------
    0  -0.492612   1.0000   1.0000  +0.0000    symmetric
    1  -0.330757   0.9999   0.9999  -0.0000      antisym
    2  -0.060563   0.9995   0.9995  +0.0000    symmetric
    3  +0.198838   0.9989   0.9989  -0.0000      antisym
    4  +0.828498   0.9748   0.9748  +0.0000    symmetric
    5  +1.000040   0.0000   0.0000  -0.0000      antisym
    6  +1.000147   0.0010   0.0010  +0.0000    symmetric
    7  +1.000356   0.0002   0.0002  +0.0000      antisym
    8  +1.000587   0.0039   0.0039  -0.0000    symmetric
    9  +1.000988   0.0006   0.0006  +0.0000      antisym

SUMMARY — RAW OBSERVATIONS (NO INTERPRETATION)
======================================================================

Single well: 3 eigenvalues below mass gap
  Lowest: omega^2 = -0.372375 (TACHYON)

Double well lowest eigenvalue range: [-0.492612, -0.240764]
  TACHYON persists at ALL separations R=4..60

Sub-gap eigenvalue count vs R:
  R=  4: 4 eigenvalues below mass gap
  R=  6: 5 eigenvalues below mass gap
  R=  8: 6 eigenvalues below mass gap
  R= 10: 6 eigenvalues below mass gap
  R= 15: 6 eigenvalues below mass gap
  R= 20: 6 eigenvalues below mass gap
  R= 30: 6 eigenvalues below mass gap
  R= 60: 6 eigenvalues below mass gap

Completed: 2026-03-26 23:49:16
