OMEGA geometry, numerical difference, and exact recovery verifier v1.0
Q = 339949771344778 ; S = 1000000000000000
PASS 1: symbolic six-mark gaps, seed radii, and Gram minor
        Coefficient identities hold in Q(phi)[s]; positivity uses the stated geometry assumptions.
PASS 2: exact Machin / integer-square-root certificate for ell < lambda < u
        sqrt5 certificate integer: 2236067977499789696409173668731276235440618359611525724270897
        ell = 339949771344778929576702809321/1000000000000000000000000000000
        u   = 169974885672389464788351404661/500000000000000000000000000000
        0 < q < lambda < 1/2; Q < S*lambda < Q+1
PASS 3: first-disagreement certificate; determinant 1 and denominator sum 42189065
        q floor at d* = 14342162; lambda floor at d* = 14342163
PASS 4: symbolic local and accumulated differences, bounds, and ideal-scale recovery
        55 seed/horizon cases; X is an indeterminate, never a floating approximation to lambda
PASS 5: fail-closed record validation; global identity alone is insufficient
        false record: terminal 0; residuals 179899542689556 169974885672389 ; last inverse is 1/2
PASS 6: fibers, residue period, ordered-history example, and canonical extinction bound
        first-residue inverse = 281419755786509 mod 500000000000000
        repeated quantization 26 -> 8 -> 2; floor(lambda^2 * 26) = 3
PASS 7: exhaustive canonical audit
        states: 32768
        horizon: 23
        transitions / prefix invariants: 753664
        complete ledger round trips: 32768
        first-residue seed inversions: 32768
        maximum extinction time: 10
PASS 8: verifier contains no assert statements
SOURCE SHA-256: e044133aceefb442bfa226bdf61359a58bca413e12196f090ff19eef3ef84ab5
ALL 8 GROUPS PASS