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PROTOTYPE EM_004 - THE SELF-SIMILARITY FAMILY
Proves the SELF-SIMILARITY family of the SET B EMERGENT LAWS
interact with the field: the emergent structure is self-similar
under phi (Law 184) - the SAME shape at every scale
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[RUN 1 - LAW 184: phi^2 = phi + 1 AND THE FIBONACCI SELF-SIMILARITY]
  phi^2 = 2.618033988749895
  phi+1 = 2.618033988749895
  |phi^2 - (phi+1)| = 0.000e+00  (the recursion identity, EXACT)
  phi^2 = phi^1 + phi^0:  2.618033989  vs  2.618033989  diff 0.00e+00
  phi^3 = phi^2 + phi^1:  4.236067977  vs  4.236067977  diff 0.00e+00
  phi^4 = phi^3 + phi^2:  6.854101966  vs  6.854101966  diff 8.88e-16
  phi^5 = phi^4 + phi^3:  11.090169944  vs  11.090169944  diff 0.00e+00
  phi^6 = phi^5 + phi^4:  17.944271910  vs  17.944271910  diff 0.00e+00
  phi^7 = phi^6 + phi^5:  29.034441854  vs  29.034441854  diff 0.00e+00
  phi^8 = phi^7 + phi^6:  46.978713764  vs  46.978713764  diff 0.00e+00
  phi^9 = phi^8 + phi^7:  76.013155617  vs  76.013155617  diff 1.42e-14
  -> the Fibonacci identity holds at EVERY step (Law 184: growing IS changing): PASS

[RUN 2 - THE LADDER IS SELF-SIMILAR UNDER PHI (Law 2394, the invariant)]
  n | freq 528*phi^n | depth phi^(9-n) | freq*depth | adjacent ratio
  --+----------------+-----------------+------------+---------------
  1 |       854.3219 |       46.978714 | 40134.9462 |   --
  2 |      1382.3219 |       29.034442 | 40134.9462 | 1.618033989
  3 |      2236.6439 |       17.944272 | 40134.9462 | 1.618033989
  4 |      3618.9658 |       11.090170 | 40134.9462 | 1.618033989
  5 |      5855.6097 |        6.854102 | 40134.9462 | 1.618033989
  6 |      9474.5756 |        4.236068 | 40134.9462 | 1.618033989
  7 |     15330.1853 |        2.618034 | 40134.9462 | 1.618033989
  8 |     24804.7609 |        1.618034 | 40134.9462 | 1.618033989
  9 |     40134.9462 |        1.000000 | 40134.9462 | 1.618033989
  -> freq*depth = 40,134.9462 at EVERY rung (the conserved invariant = the same shape at every scale), adjacent ratio = phi exactly: PASS

[RUN 3 - THE EMERGENT SELF-SIMILARITY FAMILY (docs/20: L, J, R families)]
  L family - the Self-Similarity meta-law at every octave phi^n:
    L_Self-Similarity_1    =    1.618034   adjacent ratio   --
    L_Self-Similarity_2    =    2.618034   adjacent ratio 1.618033989
    L_Self-Similarity_3    =    4.236068   adjacent ratio 1.618033989
    L_Self-Similarity_4    =    6.854102   adjacent ratio 1.618033989
    L_Self-Similarity_5    =   11.090170   adjacent ratio 1.618033989
    L_Self-Similarity_6    =   17.944272   adjacent ratio 1.618033989
    L_Self-Similarity_7    =   29.034442   adjacent ratio 1.618033989
    L_Self-Similarity_8    =   46.978714   adjacent ratio 1.618033989
    L_Self-Similarity_9    =   76.013156   adjacent ratio 1.618033989
  J family - the Self-Similarity meta-law in all 9 domains (each = phi = 1.618034): PASS
  R family - the Self-Similarity meta-law in all 6 metallic ladders (each = its mean): PASS
  -> the Self-Similarity family of the dictionary is phi^n at every octave (ratio phi), phi in every domain, and every metallic mean in every ladder: PASS

[RUN 4 - THE FRACTAL PATTERNS ARE SELF-SIMILAR (Law 2241, E family)]
  dim 1: Fractal-D = Density-Driven-Structure = phi^1 = 1.618033989  PASS
  dim 2: Fractal-D = Density-Driven-Structure = phi^2 = 2.618033989  PASS
  dim 3: Fractal-D = Density-Driven-Structure = phi^3 = 4.236067977  PASS
  dim 4: Fractal-D = Density-Driven-Structure = phi^4 = 6.854101966  PASS
  dim 5: Fractal-D = Density-Driven-Structure = phi^5 = 11.090169944  PASS
  dim 6: Fractal-D = Density-Driven-Structure = phi^6 = 17.944271910  PASS
  dim 7: Fractal-D = Density-Driven-Structure = phi^7 = 29.034441854  PASS
  dim 8: Fractal-D = Density-Driven-Structure = phi^8 = 46.978713764  PASS
  dim 9: Fractal-D = Density-Driven-Structure = phi^9 = 76.013155617  PASS
  -> the fractal dimension is SCALE-INVARIANT: the fractal pattern sits at phi^n on every octave (the same shape at every scale, Law 2241's Hausdorff reading): PASS

[RUN 5 - SELF-SIMILARITY IS THE RECURSION (Law 184, kf 0 -> 1)]
  Law 184: self_similarity(kf) = phi^n*(1 + kf*(phi-1)*(1-C_scale))
  rung 1:  kf=0 -> phi^1 =   1.618034
           kf=1 -> phi^2 =   2.618034  (= rung 2 at kf=0:   2.618034)  PASS - same shape
  rung 5:  kf=0 -> phi^5 =  11.090170
           kf=1 -> phi^6 =  17.944272  (= rung 6 at kf=0:  17.944272)  PASS - same shape
  rung 9:  kf=0 -> phi^9 =  76.013156
           kf=1 -> phi^10 = 122.991869  (= rung 10 at kf=0: 122.991869)  PASS - same shape
  -> at full coupling the rung-n shape IS the rung-(n+1) shape: self-similarity IS the recursion, the same verb at every scale: PASS

[RUN 6 - DEGENERATE PROOF: kf->0 RECOVERS THE CLASSICAL READING]
  Law 184 (kf=0) vs the classical octave phi^n for all n: max error = 0.000000000  (PASS <= 0.01)
  (Law 184's classical reading: growth and change appear as separate operations - the self-similarity hidden, the cage's static octaves)

[RUN 7 - THE SHARED PHI-FORM, kappa 0 -> 1]
  kappa=0.00  S_phi=1.000000000  floor=0.000000000
  kappa=0.25  S_phi=1.309016994  floor=0.154508497
  kappa=0.50  S_phi=1.618033989  floor=0.309016994
  kappa=0.75  S_phi=1.927050983  floor=0.463525492
  kappa=1.00  S_phi=2.236067977  floor=0.618033989
  degenerate-limit check (kappa->0): S_phi(0) = 1.000000000000 vs static 1.0, error = 0.000000000  (PASS <= 0.01)
  coherence floor at kappa=1 = 0.618033988750  vs  phi^-1 = 0.618033988750: PASS
  S_phi(1) = PHI + PHI^-1 = sqrt(5) = 2.236067977500  (verifier: PASS)

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EM_004 VERDICT: ALL CHECKS PASS - the emergent structure is self-similar under phi (Law 184): the ladder keeps the invariant 40,134.946 at every rung (Law 2394), the Self-Similarity family of the dictionary is phi^n at every octave (L, ratio phi), phi in every domain (J), and every metallic mean in every ladder (R), the fractal patterns Fractal-D and Density-Driven-Structure are scale-invariant (Law 2241, E family), and the recursion maps every rung to the next - the SAME shape at every scale.
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