VP GEOMETRIC FRACTION — CLEAN DERIVATION
======================================================================
alpha = 0.007297, alpha^2 = 5.324638e-05
GWT predictions:
  1/2^(d/2) = 0.353553  (confined, mass ratio)
  (d^2-1)/d^2 = 8/9 = 0.888889  (free, alpha)
  (d^2-1)/d = 8/3 = 2.666667  (gluon, alpha_s)

THE BREATHER SELF-ENERGY OVERLAP
------------------------------------------------------------

VP_self = <phi^2 * delta_H> / <phi^2>
       = sum[phi(i)^2 * delta_H(i)] / sum[phi(i)^2]

This is the fraction of the on-site potential that's modified by
the cross-component coupling, weighted by where the breather lives.

  n      VP_self      VP_lead  phi_max  width   sum_phi2   sum_phi2dH
----------------------------------------------------------------------
  1    -0.758896    -1.129043   1.0000    171    35.7937   -27.163739
  2    -0.758896    -1.129043   1.0000     85    17.9348   -13.610691
  3    -0.758896    -1.129043   1.0000     57    11.9989    -9.105946
  4    -0.758896    -1.129043   1.0000     43     9.0439    -6.863420
  5    -0.758896    -1.129043   1.0000     35     7.2816    -5.525946
  6    -0.758896    -1.129043   1.0000     29     6.1157    -4.641177
  7    -0.758896    -1.129043   1.0000     25     5.2910    -4.015287
  8    -0.758896    -1.129043   1.0000     23     4.6797    -3.551400
  9    -0.758897    -1.129043   1.0000     21     4.2110    -3.195735
 10    -0.758901    -1.129042   1.0000     19     3.8425    -2.916065
 11    -0.758911    -1.129032   1.0000     17     3.5471    -2.691910
 12    -0.758918    -1.129005   1.0000     15     3.3068    -2.509621
 13    -0.758903    -1.128958   1.0000     15     3.1094    -2.359746
 14    -0.758845    -1.128901   1.0000     15     2.9460    -2.235560
 15    -0.758732    -1.128856   1.0000     13     2.8102    -2.132185
 16    -0.758565    -1.128849   1.0000     13     2.6973    -2.046041
 17    -0.758355    -1.128903   1.0000     13     2.6036    -1.974467
 18    -0.758123    -1.129024   1.0000     13     2.5266    -1.915467
 19    -0.757891    -1.129206   1.0000     11     2.4641    -1.867534
 20    -0.757679    -1.129426   1.0000     11     2.4146    -1.829522
 21    -0.757502    -1.129652   1.0000     11     2.3770    -1.800567
 22    -0.757368    -1.129849   1.0000     11     2.3503    -1.780037
 23    -0.757283    -1.129987   1.0000     11     2.3340    -1.767492
 24    -0.757250    -1.130044   1.0000     11     2.3277    -1.762669

ANALYSIS: WHAT DETERMINES VP_SELF?
------------------------------------------------------------

  n  <phi^4>/<phi^2>  phi_max^2      ratio   kurtosis    VP_self
--------------------------------------------------------------
  1         0.686376     1.0000     0.6864      78.54  -0.758896
  2         0.686376     1.0000     0.6864     156.76  -0.758896
  3         0.686376     1.0000     0.6864     234.30  -0.758896
  4         0.686376     1.0000     0.6864     310.86  -0.758896
  5         0.686376     1.0000     0.6864     386.10  -0.758896
  6         0.686376     1.0000     0.6864     459.70  -0.758896
  7         0.686376     1.0000     0.6864     531.36  -0.758896
  8         0.686376     1.0000     0.6864     600.76  -0.758896
  9         0.686376     1.0000     0.6864     667.63  -0.758897
 10         0.686375     1.0000     0.6864     731.66  -0.758901
 11         0.686369     1.0000     0.6864     792.59  -0.758911
 12         0.686353     1.0000     0.6864     850.15  -0.758918
 13         0.686324     1.0000     0.6863     904.09  -0.758903
 14         0.686289     1.0000     0.6863     954.19  -0.758845
 15         0.686262     1.0000     0.6863    1000.26  -0.758732
 16         0.686258     1.0000     0.6863    1042.14  -0.758565
 17         0.686291     1.0000     0.6863    1079.67  -0.758355
 18         0.686365     1.0000     0.6864    1112.70  -0.758123
 19         0.686475     1.0000     0.6865    1141.10  -0.757891
 20         0.686609     1.0000     0.6866    1164.71  -0.757679
 21         0.686746     1.0000     0.6867    1183.40  -0.757502
 22         0.686866     1.0000     0.6869    1197.04  -0.757368
 23         0.686950     1.0000     0.6869    1205.55  -0.757283
 24         0.686984     1.0000     0.6870    1208.86  -0.757250

KEY RATIO: VP_self / phi_max^2
------------------------------------------------------------

If VP_self = -(pi^2/6) * C * phi_max^2, then C = VP_self / (-(pi^2/6) * phi_max^2)
C encodes the SHAPE of the breather (kurtosis-like).

  n      C_shape      VP_self   -(pi^2/6)*phimax^2
--------------------------------------------------
  1     0.461354    -0.758896            -1.644934
  2     0.461354    -0.758896            -1.644934
  3     0.461354    -0.758896            -1.644934
  4     0.461354    -0.758896            -1.644934
  5     0.461354    -0.758896            -1.644934
  6     0.461354    -0.758896            -1.644934
  7     0.461354    -0.758896            -1.644934
  8     0.461354    -0.758896            -1.644934
  9     0.461354    -0.758897            -1.644934
 10     0.461357    -0.758901            -1.644934
 11     0.461363    -0.758911            -1.644934
 12     0.461367    -0.758918            -1.644934
 13     0.461357    -0.758903            -1.644934
 14     0.461322    -0.758845            -1.644934
 15     0.461254    -0.758732            -1.644934
 16     0.461152    -0.758565            -1.644934
 17     0.461024    -0.758355            -1.644934
 18     0.460883    -0.758123            -1.644934
 19     0.460743    -0.757891            -1.644934
 20     0.460614    -0.757679            -1.644934
 21     0.460506    -0.757502            -1.644934
 22     0.460424    -0.757368            -1.644934
 23     0.460373    -0.757283            -1.644934
 24     0.460353    -0.757250            -1.644934

PHYSICAL VP CORRECTION
------------------------------------------------------------

The VP correction to the breather frequency/mass:
  delta_m/m = alpha^2 * (d-1) * VP_self

(d-1) = 2 transverse components (y and z)
alpha^2 comes from the tunneling physics (not in the Hessian)
VP_self is the geometric overlap (from the Hessian)

  n      VP_self  (d-1)*VP_self     alpha^2*..     GWT pred
------------------------------------------------------------
  1    -0.758896      -1.517793  -8.081698e-05 1.882544e-05
  2    -0.758896      -1.517793  -8.081698e-05 1.882544e-05
  3    -0.758896      -1.517793  -8.081698e-05 1.882544e-05
  4    -0.758896      -1.517793  -8.081698e-05 1.882544e-05
  5    -0.758896      -1.517793  -8.081698e-05 1.882544e-05
  6    -0.758896      -1.517793  -8.081698e-05 1.882544e-05
  7    -0.758896      -1.517793  -8.081698e-05 1.882544e-05
  8    -0.758896      -1.517793  -8.081697e-05 1.882544e-05
  9    -0.758897      -1.517794  -8.081704e-05 1.882544e-05
 10    -0.758901      -1.517803  -8.081752e-05 1.882544e-05

COMPARISON:
  n=1: (d-1) * VP_self = -1.517793
  1/2^(d/2)           = 0.353553
  Ratio               = -4.2930
  So VP_self          = -0.758896
  1/[2*(d-1)*2^(d/2)] = 0.088388

VP_self(n=1) = -0.75889638

Searching for clean expression:
            -(16/pi^2)*(2/3)/(pi^2/6) =  -0.65702286, ratio = 1.155053

Continuum limit integrals:
  int(arctan(1/cosh(t))^2 dt) = 1.43612864
  int(arctan(1/cosh(t))^4 dt) = 0.60804392
  Ratio int(f^4)/int(f^2)     = 0.42339099
  (4/pi)^2 * ratio            = 0.68637561
  VP_self_continuum = -(pi^2/6) * (4/pi)^2 * ratio = -1.12904263

  (d-1) * VP_self_continuum = -2.25808526
  1/2^(d/2) = 0.35355339
  Ratio: -6.386830

