
==============================================================================
SOURCE CONVENTIONS
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[source conventions] PASS  A,C,G,T order; C+G=T; K2P a_C=a_T
[source conventions] PASS  five explicit Lemma 4.1 coordinates and favorable-order Q factorization

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STRICT JSON PARSING AND CLOSED-SCHEMA MUTATION TESTS
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[strict JSON] PASS  all five canonical certificates have unique keys and closed schemas
[JSON mutation rejection] PASS  compact K2P duplicate top-level key: duplicate JSON object key 'schema_version'
[JSON mutation rejection] PASS  compact K2P unknown top-level field: closed JSON schema mismatch
[JSON mutation rejection] PASS  compact K2P unknown nested vertex field: closed JSON schema mismatch
[JSON mutation rejection] PASS  compact K2P non-standard NaN constant: non-standard JSON numeric constant 'NaN'
[JSON mutation rejection] PASS  continuous-time K2P duplicate top-level key: duplicate JSON object key 'schema_version'
[JSON mutation rejection] PASS  continuous-time K2P unknown top-level field: closed JSON schema mismatch
[JSON mutation rejection] PASS  K3P duplicate top-level key: duplicate JSON object key 'schema_version'
[JSON mutation rejection] PASS  K3P duplicate nested vertex key: duplicate JSON object key 'id'
[JSON mutation rejection] PASS  K3P unknown rooted-network field: closed JSON schema mismatch
[JSON mutation rejection] PASS  K3P unknown operative arc field: closed JSON schema mismatch
[JSON mutation rejection] PASS  K3P same-length row-shape multiplicity shift: closed JSON schema mismatch
[JSON mutation rejection] PASS  K3P coordinated embedded-and-sidecar extra fields: closed JSON schema mismatch
[JSON mutation rejection] PASS  K3P Jacobian sidecar duplicate key: duplicate JSON object key 'ambient_space'
[JSON mutation rejection] PASS  K3P continuous-time sidecar duplicate key: duplicate JSON object key 'method'

ALL STRICT JSON AND CLOSED-SCHEMA MUTATION CHECKS PASSED

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SIMPLE K2P COLLISION
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[field] PASS  sqrt(71) is irrational and rigorously isolated
[topology] PASS  rooted binary DAG suppresses to a strict level-two theta 3-blob
[parameters] PASS  all network, effective, and tree edges lie in Theta_0^circ
[parameters] PASS  exact global minimum transition entry is 1/120
[root splitting] PASS  all three comparison-tree edges admit strict stochastic half-time roots
[root splitting] PASS  the compatible theta root uses the certified K odot K factorization
[collision] PASS  all 16 factors, 64 Fourier coordinates, and 64 positive patterns; Q=0
[collision] PASS  exact global minimum = 1188799/79626240, attained by 8 patterns

ALL SIMPLE K2P CHECKS PASSED

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SIMPLE K2P DISPLAYED-TREE RECONSTRUCTION AND DIRECT PRUNING
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[field] PASS  sqrt(71) is irrational and rigorously isolated
[source convention] PASS  descendant-label rule reproduces all five explicit 3-sunlet coordinates of Lemma 4.1
[switch p,p]
  retained edges = rho->1, rho->u, u->p, u->q, p->r2, p->r3, r2->2, r3->3
  u->p descendants = {2,3}
  u->q descendants = {}
  core monomial = S_y*S_z*U_(y+z)
[switch p,q]
  retained edges = rho->1, rho->u, u->p, u->q, p->r2, q->r3, r2->2, r3->3
  u->p descendants = {2}
  u->q descendants = {3}
  core monomial = S_y*T_z*U_y*V_z
[switch q,p]
  retained edges = rho->1, rho->u, u->p, u->q, q->r2, p->r3, r2->2, r3->3
  u->p descendants = {3}
  u->q descendants = {2}
  core monomial = T_y*S_z*U_z*V_y
[switch q,q]
  retained edges = rho->1, rho->u, u->p, u->q, q->r2, q->r3, r2->2, r3->3
  u->p descendants = {}
  u->q descendants = {2,3}
  core monomial = T_y*T_z*V_(y+z)
[common K factor] PASS  K_x*K_(y+z)*K_y*K_z = K_x^2*K_y*K_z
[displayed trees] PASS  four monomials reconstructed from retained edges
[transition matrices] PASS  all exact 4x4 K2P matrices reconstructed from the stored Fourier vectors
[core matrix] PASS  graph mixture gives all 16 factors; M_AC=151/1440 and M_CC=71/1600
[Fourier] PASS  all 64 graph-derived network/tree coordinates agree exactly
[direct pruning] PASS  all 64 network/tree probabilities agree exactly
[direct pruning] PASS  certificate value is the exact global minimum at 8 patterns

ALL DISPLAYED-TREE CHECKS PASSED

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EDGEWISE STRICTLY CONTINUOUS-TIME K2P AND PROOF AUDIT
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[field] PASS  irreducible cubic root and sqrt(1423) isolated rigorously
[field] PASS  the six displayed monomials form a valid number-field basis
[topology] PASS  rooted binary DAG suppresses to a strict level-two theta 3-blob
[parameters] PASS  all rooted/effective/tree edges lie in Theta_0^circ and are edgewise strictly continuous-time K2P
[parameters] PASS  exact global minimum eigenvalue 1/16 and rate margin 11/900
[root splitting] PASS  all three comparison-tree edges admit strict stochastic half-time roots
[root splitting] PASS  the compatible theta root uses the certified K odot K factorization
[construction] PASS  cubic symmetric-ansatz equations verified exactly
[factorization] PASS  all 16 core identities M_yz=P_(y+z)R_yR_z
[Fourier] PASS  all 64 Fourier coordinates agree exactly
[patterns] PASS  all 64 probabilities agree, are positive, and sum to 1
[patterns] PASS  rigorous global minimum at 8 patterns lies in [0.0149867914232259, 0.0149867914232259]
[direct probabilities] PASS  ordinary Markov pruning independently matches all 64 Fourier-inverted probabilities
[induction] PASS  parent Q=0 exactly; one level-one child has fixed-order Q<0
[induction] INFO  fixed-order Q interval [-1.9199710723828273e-09, -1.9199710723828273e-09]
[induction] INFO  favorable-order Q interval [3.4284883265259252e-09, 3.4284883265259252e-09]
[induction] PASS  relabeling that child makes Q>0, confirming the order conflict
[sign test] PASS  an independent exact rational theta point has Q<0 in all six leaf orders
[sign test] PASS  exact global minimum pattern probability is 2920987217429243/200000000000000000 at 8 patterns

ALL EXTENDED K2P CHECKS PASSED

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EXACT FOUR-LEAF K2P CHERRY-GRAFT REGRESSION
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[field] PASS  irreducible cubic root and sqrt(1423) isolated rigorously
[field] PASS  the six displayed monomials form a valid number-field basis
[topology] PASS  rooted binary DAG suppresses to a strict level-two theta 3-blob
[parameters] PASS  all rooted/effective/tree edges lie in Theta_0^circ and are edgewise strictly continuous-time K2P
[parameters] PASS  exact global minimum eigenvalue 1/16 and rate margin 11/900
[root splitting] PASS  all three comparison-tree edges admit strict stochastic half-time roots
[root splitting] PASS  the compatible theta root uses the certified K odot K factorization
[construction] PASS  cubic symmetric-ansatz equations verified exactly
[factorization] PASS  all 16 core identities M_yz=P_(y+z)R_yR_z
[Fourier] PASS  all 64 Fourier coordinates agree exactly
[patterns] PASS  all 64 probabilities agree, are positive, and sum to 1
[patterns] PASS  rigorous global minimum at 8 patterns lies in [0.0149867914232259, 0.0149867914232259]
[four-leaf topology] PASS  exact rooted binary graft has 11 vertices, 12 arcs, and four displayed trees
[four-leaf topology] PASS  root suppression retains one maximal strict level-2 theta blob; the cherry is on a cut-edge side
[four-leaf topology] PASS  comparison tree is the binary quartet with split 14|23
[four-leaf edges] PASS  each new K edge has row (5/8,1/8,1/8,1/8) and CT margin 1/4
[four-leaf edges] PASS  suppressed K odot K stem has row (7/16,3/16,3/16,3/16) and CT margin 3/16
[four-leaf edges] PASS  the full graft retains the certified global CT-margin minimum 11/900
[four-leaf Fourier] PASS  literal graphs and graft formula agree on all 256 coordinates (64 consistent, 192 zero)
[four-leaf pruning] PASS  all 256 patterns agree by displayed-tree pruning, quartet pruning, Fourier inversion, and Markov extension
[four-leaf patterns] PASS  all probabilities are positive, sum to 1, and the certified exact minimum has 16 minimizers
[four-leaf scope] PASS  this is an exact n=4 regression of the shared cherry-graft identity
[four-leaf scope] INFO  iteration to arbitrary n requires the separate theorem-level graft lemma

ALL FOUR-LEAF GRAFT CHECKS PASSED

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K2P RANK AND COLLISION FAMILIES
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[field] PASS  Q(sqrt(71)) and its positive embedding are certified
[field] PASS  irreducible cubic root and sqrt(1423) isolated rigorously
[field] PASS  the six displayed monomials form a valid number-field basis
[K2P dimensions] PASS  16 consistent coordinates form 10 C<->T orbits; normalization gives ambient dimension 9; the certified tree minor gives dimension 6
[K2P rank] PASS  simple witness rank 9 with det = -4126104359487341/9539621664406901296012984320000000000
[K2P tree rank] PASS  exact 6x6 minor is positive; det = (0) + (21911761/305764761600000) sqrt(71)
[K2P rank] PASS  edgewise-strict-continuous-time witness rank 9; det in [-4.129731e-22,-4.129731e-22]
[K2P fiber] PASS  local fixed-output theta fiber has dimension 11 (20-9) at both rank-9 witnesses
[K2P geometry] PASS  derived collision-locus dimension 17 = 20-9+6 (codimension 3)
[K2P family] PASS  exact two-equation core has rank 2; derived local positive family dimension 6

ALL K2P RANK/FAMILY CHECKS PASSED

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K2P CERTIFICATE-FIELD MUTATION REJECTION TEST
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[mutation rejection] PASS  stored K_odot_K transition row: K_odot_K stored transition row

ALL K2P SEMANTIC MUTATION CHECKS PASSED

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K3P COLLISION, RANK, DIRECT PRUNING, AND EDGEWISE CONTINUOUS-TIME ANALYTIC-IFT DATA
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[sidecars] PASS  transport copies equal their embedded certificate sections
[field] PASS  irreducible Q(h), h^4=1/5, basis 1,h,h^2,h^3, with 2/3 < h < 7/10
[topology] PASS  rooted binary DAG and correct semi-directed root suppression
[topology] PASS  maximal theta 3-blob with three leaf sides and two reticulations
[topology] PASS  literal three-clause audit finds exactly six single-edge sets; each contracts to four incidences
[topology] PASS  edge-incidence/degree-two audit finds no proper suppressible substructure
[construction] PASS  C/G-symmetric ansatz reduction verified exactly
[construction] PASS  tau=1/3 forces kappa=3h^2 and b=h on the positive quartic branch
[parameters] PASS  every network/effective/tree edge lies in Theta_0^circ
[parameters] PASS  every transition probability is strictly positive
[parameters] PASS  both inheritance parameters equal 1/2
[root splitting] PASS  all three comparison-tree edges admit strict stochastic half-time roots
[root splitting] PASS  only the compatible u-side theta rooting is certified by K odot K
[collision] PASS  all four displayed-tree terms reconstructed exactly
[collision] PASS  all 16 core identities M[y,z]=P[y+z]B[y]B[z]
[collision] PASS  all 64 Fourier coordinates agree (16 consistent, 48 zero)
[collision] PASS  all 64 positive leaf-pattern probabilities agree and sum to 1
[direct probabilities] PASS  exact K3P Markov pruning on all four retained graphs matches the comparison tree and Fourier inversion for all 64 patterns
[K3P/K2P scope] PASS  U_C,U_G,U_T are pairwise distinct, excluding every globally character-relabelled K2P edge-parameter submodel
[K3P/K2P scope] PASS  the common output/tree lies in exactly the C=G globally character-relabelled K2P specialization
[Jacobian] PASS  specified 15 x 15 minor reconstructed by exact differentiation
[Jacobian] PASS  det J = h(10 h^2+1)/(2^61 3^4 5^14) > 0
[Jacobian] PASS  fixed semi-directed theta map has rank 15; the nonzero-minor locus is Zariski open
[Jacobian] PASS  local fixed-output theta fiber has dimension 14 (29-15)
[Jacobian] PASS  tree model has rank 9; local network preimage of the tree model has dimension 23 (codimension 6)
[edgewise continuous time] PASS  U and V are the only saturated positive-rate margins
[edgewise continuous time] PASS  exact tangent identity for the fixed-output IFT branch verified
[edgewise continuous time] PASS  saturated-margin derivatives are positive
[edgewise continuous time] PASS  algebraic hypotheses for the analytic IFT corollary verified

ALL K3P CHECKS PASSED

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K3P SEMANTIC MUTATION REJECTION TESTS
==============================================================================
[mutation rejection] PASS  coordinated Jacobian descriptor/column/pivot cycle: canonical Jacobian descriptor order
[mutation rejection] PASS  free-direction semantic relabeling: canonical continuous-time free-direction descriptors
[mutation rejection] PASS  reticulation-order reversal: canonical ordered reticulation descriptors
[mutation rejection] PASS  root-suppression source reassignment: singleton root-suppression source binding for e_u_p
[mutation rejection] PASS  coordinated actual p/q endpoint swap with stale descriptors: canonical rooted arc ID/endpoint/vector map
[mutation rejection] PASS  root-adjacent arc-ID swap: canonical rooted arc ID/endpoint/vector map
[mutation rejection] PASS  extra shadowed duplicate vertex row: closed JSON schema mismatch
[mutation rejection] PASS  same-length duplicate vertex identifier: duplicate vertex identifier
[mutation rejection] PASS  reticulation parent contradicts referenced arc: reticulation descriptor parent for e_p_r2
[mutation rejection] PASS  unknown top-level certificate field: closed JSON schema mismatch

ALL K3P SEMANTIC MUTATION CHECKS PASSED

ALL EXACT CHECKS PASSED
