MODE BALANCE MODEL — CONSTRAINED BOND MODEL
======================================================================
Started: 2026-03-25 19:03:41
d = 3, gamma = 0.0650898425

WELL-FILLING DIRECTION W:
  n        W_n    W_hat_n      group
--------------------------------------
    1    -8.4417    -0.4209    STIFFEN
    2    -8.2627    -0.4120    STIFFEN
    3    -7.9857    -0.3982    STIFFEN
    4    -7.6375    -0.3808    STIFFEN
    5    -7.2463    -0.3613    STIFFEN
    6    -6.8376    -0.3409    STIFFEN
    7    -6.4314    -0.3207    STIFFEN
  |W| = 20.0564

SECOND DIRECTION (orthogonal to W, from M eigenvector):
  n=1: v2 = -0.8274
  n=2: v2 = -0.1567
  n=3: v2 = +0.1157
  n=4: v2 = +0.2254
  n=5: v2 = +0.2672
  n=6: v2 = +0.2793
  n=7: v2 = +0.2780

PART 1: 1-PARAMETER MODEL (tilt along W)
-------------------------------------------------------
c_n = 1 + α × W_hat_n
Scan α to find the optimal single-proton eigenvalue.

   alpha           E0      c_1      c_4      c_7
---------------------------------------------
    -0.300     0.051320   1.1263   1.1142   1.0962
    -0.100    -0.099073   1.0421   1.0381   1.0321
    +0.000    -0.016370   1.0000   1.0000   1.0000
    +0.100     0.065026   0.9579   0.9619   0.9679
    +0.300    -0.075281   0.8737   0.8858   0.9038
    +0.500     0.112446   0.7896   0.8096   0.8397

OPTIMAL: α = 0.220, E0 = -0.15238530
Optimal weights c_n:
  n=1: c = 0.9074
  n=2: c = 0.9094
  n=3: c = 0.9124
  n=4: c = 0.9162
  n=5: c = 0.9205
  n=6: c = 0.9250
  n=7: c = 0.9295

PART 2: BONDING — OPTIMAL α VS SEPARATION R
-------------------------------------------------------
At each R, scan α to find the optimal two-proton eigenvalue.
The bond energy = E0(R, α_opt(R)) - E0(∞, α_opt(∞))

   R  alpha_opt    E0_double         V(R)       V_bare    ratio
-----------------------------------------------------------------
   6    +0.3800    -0.140283  +0.16448753  -0.18594951   -0.885
   7    +0.4490    -0.142449  +0.16232138  -0.13896450   -1.168
   8    +0.2370    -0.141048  +0.16372301  -0.03319068   -4.933
   9    +0.3240    -0.143101  +0.16166913  -0.07437303   -2.174
  10    +0.4170    -0.145806  +0.15896456  -0.00472930  -33.613
  12    -0.0030    -0.141929  +0.16284146  -0.00071408 -228.044
  14    +0.2370    -0.147494  +0.15727706  -0.00011029 -1426.051
  16    +0.1640    -0.148531  +0.15623964  -0.00001721 -9077.967
  20    +0.3920    -0.156168  +0.14860306  -0.00000043 -347384.335
  24    +0.1100    -0.228204  +0.07656649  -0.00000001 -7049460.962
  30    -0.0960    -0.142163  +0.16260727  -0.00000000    0.000

PART 3: HOW EACH MODE'S WEIGHT CHANGES DURING BONDING
-------------------------------------------------------
Single proton: α = 0.2200

   R       Δα     Δc_1     Δc_2     Δc_3     Δc_4     Δc_5     Δc_6     Δc_7
---------------------------------------------------------------------------
   6  +0.1600  -0.0673  -0.0659  -0.0637  -0.0609  -0.0578  -0.0545  -0.0513
   7  +0.2290  -0.0964  -0.0943  -0.0912  -0.0872  -0.0827  -0.0781  -0.0734
   8  +0.0170  -0.0072  -0.0070  -0.0068  -0.0065  -0.0061  -0.0058  -0.0055
   9  +0.1040  -0.0438  -0.0428  -0.0414  -0.0396  -0.0376  -0.0355  -0.0333
  10  +0.1970  -0.0829  -0.0812  -0.0784  -0.0750  -0.0712  -0.0672  -0.0632
  12  -0.2230  +0.0939  +0.0919  +0.0888  +0.0849  +0.0806  +0.0760  +0.0715
  14  +0.0170  -0.0072  -0.0070  -0.0068  -0.0065  -0.0061  -0.0058  -0.0055
  16  -0.0560  +0.0236  +0.0231  +0.0223  +0.0213  +0.0202  +0.0191  +0.0180
  20  +0.1720  -0.0724  -0.0709  -0.0685  -0.0655  -0.0621  -0.0586  -0.0552
  24  -0.1100  +0.0463  +0.0453  +0.0438  +0.0419  +0.0397  +0.0375  +0.0353
  30  -0.3160  +0.1330  +0.1302  +0.1258  +0.1203  +0.1142  +0.1077  +0.1013

PART 4: 2-PARAMETER MODEL (α along W, β along v2)
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c_n = 1 + α × W_hat_n + β × v2_n

Single proton: α=0.280, β=0.300, E0=-0.17579353
  (1-param gave E0=-0.15238530, improvement: -0.02340823)

2-PARAMETER BOND CURVE:
   R    α_opt    β_opt     V_2param     V_1param       V_bare
------------------------------------------------------------
   6   +0.280   +0.280  +0.19532947  +0.16448753  -0.18594951
   7   +0.100   +0.260  +0.20353572  +0.16232138  -0.13896450
   8   +0.160   +0.280  +0.19872425  +0.16372301  -0.03319068
   9   +0.280   +0.260  +0.19351593  +0.16166913  -0.07437303
  10   +0.080   +0.260  +0.19885447  +0.15896456  -0.00472930
  12   +0.280   +0.280  +0.18515513  +0.16284146  -0.00071408
  14   +0.240   +0.260  +0.18702949  +0.15727706  -0.00011029
  16   -0.160   +0.260  +0.19798878  +0.15623964  -0.00001721
  20   +0.080   +0.060  +0.12043307  +0.14860306  -0.00000043
  24   +0.140   +0.180  +0.20128410  +0.07656649  -0.00000001
  30   -0.260   +0.140  +0.21107204  +0.16260727  -0.00000000

PART 5: PHYSICAL INTERPRETATION
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The W vector divides modes into:
  SOFTENING (W>0): modes 1-4  (wide, low n)
  NEUTRAL (W≈0):   mode 5
  STIFFENING (W<0): modes 6-7  (narrow, high n)

Equilibrium: softening and stiffening nearly cancel.
Bonding shifts α → more/less softening depending on R.

SOFTENING vs STIFFENING BALANCE:
   R       Δα    Δ(soft)   Δ(stiff)    ratio
---------------------------------------------
   6  +0.1600  -0.257893  -0.105853   +2.436
   7  +0.2290  -0.369109  -0.151502   +2.436
   8  +0.0170  -0.027401  -0.011247   +2.436
   9  +0.1040  -0.167630  -0.068804   +2.436
  10  +0.1970  -0.317531  -0.130331   +2.436
  12  -0.2230  +0.359438  +0.147532   +2.436
  14  +0.0170  -0.027401  -0.011247   +2.436
  16  -0.0560  +0.090263  +0.037049   +2.436
  20  +0.1720  -0.277235  -0.113792   +2.436
  24  -0.1100  +0.177301  +0.072774   +2.436
  30  -0.3160  +0.509338  +0.209059   +2.436

PART 6: THE SUM RULE
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Σ integral(φ_n²) = 93.4597
integral(|ΔV|) = 8.5038
Ratio = 10.9904

Individual integrals:
  n      ∫φ_n²   ∫φ_n²/∫|ΔV|    eps_n  1/eps_n
---------------------------------------------
    1    35.7937        4.2092   0.0650    15.37
    2    17.9348        2.1090   0.1298     7.70
    3    11.9989        1.4110   0.1940     5.15
    4     9.0439        1.0635   0.2574     3.88
    5     7.2816        0.8563   0.3197     3.13
    6     6.1157        0.7192   0.3807     2.63
    7     5.2910        0.6222   0.4400     2.27

Note: ∫φ_n² ∝ 1/eps_n (breather width).
The sum rule emerges because the TOTAL width of all 7 modes
spans the kink well exactly.

Σ 1/eps_n = 40.1431
This should relate to the total available phase space.
Σ width / kink_width = 13.3810
N_modes × avg_width / kink_width = 13.3810

SUMMARY
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1. The 7 breather modes form a CONSTRAINED EQUILIBRIUM.
   Modes 1-4 soften, modes 6-7 stiffen, the balance nearly cancels.

2. Single parameter α along the W direction captures the physics.
   Optimal single proton: α = 0.2200

3. During bonding, α shifts because the neighbor modifies
   the effective well. Wider modes (1-2) feel it first.

4. The bond energy comes from the SHIFT in equilibrium,
   not from individual mode contributions.

5. At R=8: V_constrained/V_bare = -4.933
   Bond energy is modified by the mode balance correction.

Completed: 2026-03-25 19:07:57
