================================================================
PART 1: THE GAP WEIGHT w8, DERIVED (mirrors the Lean projection)
================================================================
phi-pattern phi^t, t = 0..7:
  [ 1.        1.618034  2.618034  4.236068  6.854102 11.09017  17.944272
 29.034442]
DFT-8 mode energies |c_k|^2:
  k=0: 691.8292652695
  k=1: 198.7197883486
  k=2: 73.0383588830
  k=3: 44.7414079280
  k=4: 38.5543235602
  k=5: 44.7414079280
  k=6: 73.0383588830
  k=7: 198.7197883486

Parseval total:   1363.3826991487
(phi^16-1)/phi:   1363.3826991487
match:            True

Per-mode contribution to w8 (64 * |c_k|^2 * gweight_k / E_total):
  k=1: 0.844297
  k=2: 0.654799
  k=3: 0.423194
  k=4: 0.264049
  k=5: 0.161646
  k=6: 0.095534
  k=7: 0.047051

w8 (projection):  2.4905692754537
w8 (closed form): 2.4905692754536
difference:       1.02e-14
This equality is a Lean THEOREM (w8_projection_equality, 0 sorry):
the projection and the closed form are the same number, derived, not fitted.

================================================================
PART 2: THE ASSEMBLY alpha^-1 = 44pi * exp(-w8*ln(phi)/(44pi))
================================================================
seed 44*pi:       138.2300767580   (IDENTIFICATION, see Part 4)
f_gap = w8*ln(phi): 1.1984913865   (forced, zero alpha-input)
alpha^-1 assembled: 137.036766007

certified band (137.030, 137.039): INSIDE
(Lean theorems alphaInv_gt / alphaInv_lt certify the band.)

================================================================
PART 3: THE MEASUREMENT VERDICT (the construction MISSES, provably)
================================================================
CODATA alpha^-1:  137.035999177 +/- 2.1e-08
assembled value:  137.036766007
overshoot:        +0.000766830
relative:         +5.60 ppm
in CODATA sigma:  36,516 sigma

Lean proves: overshoot > 0.0007 (alphaInvGenesis_exceeds_CODATA_by_0007)
  check: 0.000766830 > 0.0007  ->  True
Lean proves: 0.0007 > 30000 * sigma (margin_0007_gt_30000_sigma)
  check: 0.0007 > 0.000630  ->  True

VERDICT: the parameter-free construction lands 5.6 ppm ABOVE the measured
alpha^-1 and is excluded as the exact value at >30,000 sigma, by our own
Lean theorems. The exact infrared alpha^-1 is OPEN: a boundary condition
the present formalization does not derive.

================================================================
PART 4: WHY THE SEED 44pi IS AN IDENTIFICATION, NOT A DERIVATION
================================================================
U1Normalization: gauge-invariant photon count on Q3 = cycle rank E-V+1 = 5
  The '11' in 44pi is the passive-edge LEDGER count (12 - 1), not the gauge count.
  A genuine gauge seed would be 4pi*5 = 62.83, excluded by measurement.
CurvatureJCostVerdict: the genuine quadratic J-cost of the cube curvature
  is pi^2 = 9.8696; 4pi = 12.5664 is a linear Gauss-Bonnet invariant, not a cost.
  So reading 4pi*11 = 138.23 as a recognition cost is a category error.
ScaleIdentification: the assembled 137.036766 exceeds the Thomson ceiling
  alpha^-1(0) = 137.035999177, so it is off the physical running curve at EVERY scale.

================================================================
APPENDIX: THE RETIRED LEGACY PIPELINE (a cautionary tale)
================================================================
The repository once carried w8 as a stored NUMBER (2.488254397846) and an
additive assembly  alpha^-1 = 44pi - w8*ln(phi) - delta_kappa :
  legacy assembly:  137.035999118   (-0.43 ppb from CODATA)
  w8 required to hit CODATA exactly: 2.488254276262
  stored legacy w8:                  2.488254397846
  difference: 1.22e-07  <- the stored number just encoded CODATA

That ppb 'agreement' was retired (GapWeight.lean: a numeric certificate is
'no longer acceptable for the no-free-parameters claim'). The derived w8 is
2.4905692755, the certified assembly is exponential, and the honest
result is the proved 5.6 ppm miss above. A construction that cannot miss
is not evidence; this one can, and does, and the Lean says exactly where.

================================================================
CHART DATA
================================================================
band_lo 137.030
band_hi 137.039
alpha_inv 137.036766007
codata 137.035999177
codata_sigma 2.1e-08
legacy 137.035999118
mode 1 0.844297
mode 2 0.654799
mode 3 0.423194
mode 4 0.264049
mode 5 0.161646
mode 6 0.095534
mode 7 0.047051