PRIMARY PYTHON CERTIFICATE
canonical placements: 40
possible cancelling tied-minimum strata: 20
pair-minimum cancellation strata: 18
quotient modulus and periods: 33215 36 12
empty quotient families: [3, 4, 7, 8, 9, 16, 17]
total quotient states: 23
mod 27 survivor (family,state,low/high): (12, (3, 1, 4, 8), LH)
terminal factor pairs: [(1, 175), (5, 35), (7, 25)]
unique solution (x,m,g,K): (3, 1, 4, 8)
presentation: (4+3)(1+8*3)+3^4 = 2^8
cost and excess: 29 13
certificate: PASS

INDEPENDENT C++20 AUDIT
canonical placements: 40
possible cancelling tied-minimum strata: 20
pair-minimum cancellation strata: 18
quotient modulus and periods: 33215 36 12
empty quotient families: 3 4 7 8 9 16 17
total quotient states: 23
mod 27 survivor: family 12, state (3,1,4,8), LH
terminal factor pairs: (1,175) (5,35) (7,25)
unique solution (x,m,g,K): (3,1,4,8)
cost and excess: 29 13
independent audit: PASS

GENERAL-REMAINDER DIAGNOSTIC (NOT A COMPLETENESS THEOREM)
canonical f=0 placements: 40
states after modulus 33215: 149
Q roots surviving modulo 729: 44
raw lifted modulo-729 patterns: 2032
patterns after the new universal valuation law: 1953
placements still represented: 20
conclusion: the next 3-adic digit branches rather than closes the general remainder
diagnostic: PASS
