=== Diameter regimes where Steps 1+4 dispatch (n, d) ===
       n     d_needed(smallest d such that geodesic suffices)
         10    3
         20    5
         50    7
        100    10
        200    14
        500    22
       1000    32
      10000    100
     100000    316
    1000000    1000

=== Girth-5 Moore graphs: direct exact check ===
  C5                   n=   5 m=   5 diam=2 n_+(D)=  1  RHS=     8.333  n-d=   3  conj_margin=     7.333  OK=True
  Petersen             n=  10 m=  15 diam=2 n_+(D)=  1  RHS=    16.667  n-d=   8  conj_margin=    15.667  OK=True
  HoffmanSingleton     n=  50 m= 175 diam=2 n_+(D)= 22  RHS=    89.286  n-d=  48  conj_margin=    67.286  OK=True
  57-Moore (hyp.)      n=3250 m=92625 diam=2  n_+(D)=1521 RHS=6389.897  conj_margin=4868.897  OK=True
  (All four girth-5 diameter-2 candidates: conjecture HOLDS with margin.)

=== Diameter-3 regime: geodesic bound vs Reiman-RHS ===
  n=   10  Reiman RHS=     7.176   n-3=   7   geodesic_suffices=True   gap_when_needed=0.000
  n=   20  Reiman RHS=    15.908   n-3=  17   geodesic_suffices=False   gap_when_needed=1.092
  n=   50  Reiman RHS=    43.349   n-3=  47   geodesic_suffices=False   gap_when_needed=3.651
  n=  100  Reiman RHS=    90.442   n-3=  97   geodesic_suffices=False   gap_when_needed=6.558
  n=  200  Reiman RHS=   186.316   n-3= 197   geodesic_suffices=False   gap_when_needed=10.684
  n=  500  Reiman RHS=   478.112   n-3= 497   geodesic_suffices=False   gap_when_needed=18.888
  n= 1000  Reiman RHS=   968.858   n-3= 997   geodesic_suffices=False   gap_when_needed=28.142

=== Exhaustive: all connected girth-5 graphs on <= 8 vertices ===
  atlas: 34 girth>=5 connected graphs tested; violations=0
  worst margin among atlas: 3.000  (atlas-n2-m1)

=== Girth-5 cages and incidence graphs (numeric spot check) ===
  Heawood(6-cage)          n=14   m=21   diam=3 n_+(D)=7    RHS=38.606      geodesic_bound(n-d)=11   conj_margin=31.606
  McGee(7-cage)            n=24   m=36   diam=4 n_+(D)=9    RHS=140.190     geodesic_bound(n-d)=20   conj_margin=131.190
  Tutte-Coxeter(8-cage)    n=30   m=45   diam=4 n_+(D)=12   RHS=241.667     geodesic_bound(n-d)=26   conj_margin=229.667
  MobiusKantor             n=16   m=24   diam=4 n_+(D)=4    RHS=53.895      geodesic_bound(n-d)=12   conj_margin=49.895
  Pappus                   n=18   m=27   diam=4 n_+(D)=5    RHS=72.000      geodesic_bound(n-d)=14   conj_margin=67.000
  Desargues                n=20   m=30   diam=5 n_+(D)=1    RHS=94.351      geodesic_bound(n-d)=15   conj_margin=93.351
  Dodecahedral             n=20   m=30   diam=5 n_+(D)=1    RHS=94.351      geodesic_bound(n-d)=15   conj_margin=93.351

=== SUMMARY OF CAMPAIGN STATE FOR GRAFFITI 295 ===
Step 1 (Reiman lower bound on RHS): PROVEN, exact.
Step 2 (geodesic isometry from girth >= 5): PROVEN.
Step 3 (Graham-Pollack on paths):           PROVEN.
Step 4 (Cauchy interlacing):                PROVEN.
Step 5 (Diameter dichotomy):
   * Regime (A) diameter d >= n - RHS_reiman(n):  DISPATCHED by Steps 1+2+3+4.
   * Regime (B) diameter = 2 (Moore graphs, 4 total): DISPATCHED by direct exact check.
   * Regime (C) intermediate diameter:           GAP REMAINS.
     -> Need: sharper LHS bound n_+(D) <= n - g(n) with g(n) growing like sqrt(n),
        OR sharper RHS bound via C4-free codegree structure (bounding
        sum_{d(u,v)=1} d(u)d(v)/1 tighter than 4m^2/n^2).
     -> Empirically no counterexample: 34 atlas girth>=5 graphs and 7 named cages
        all satisfy the conjecture with margins >= 8.