TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/seam-joint-source-origin-audit-20261016/MINIMALITY_SELFCONSISTENCY_PROOF.txt
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This bounded excerpt is source evidence; it is not a new proof/checker run.
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000130 | 5. New decisive control AT the minimum carrier rank
000131 | 
000132 | Use only the covariant exterior ingredients of S2:2400-2439. Let E=V+W be
000133 | Hermitian and carry a unit complex volume, with b=dim V>=1,s=dim W>=1.
000134 | Take two NONZERO positive even-degree exterior fermion legs and one bosonic
000135 | fundamental h in W. This is an invariant-type audit, NOT an identification
000136 | of the actual source fields or their coupling.
000137 | 
000138 | At g=b+s=5 the single canonical tensor
000139 | 
000140 |   T(alpha,beta,h)=vol_E(alpha wedge beta wedge h),
000141 |   alpha,beta in Lambda2 E, h in W,
000142 | 
000143 | has the following nonzero invariant restrictions:
000144 | 
000145 |   (b,s)   first fermion       second fermion     boson   dimensions
000146 |    1,4    V tensor W         Lambda2 W          W       4,6,4
000147 |    2,3    Lambda2 V          Lambda2 W          W       1,3,3
000148 |    3,2    V tensor W         Lambda2 V          W       6,3,2
000149 |    4,1    Lambda2 V          Lambda2 V          W       6,6,1
000150 | 
000151 | The determinant degrees are respectively(b,s) in EVERY row. Each is
000152 | invariant under the full connected block group G=S(U(V) x U(W)), not just
000153 | under an equal-dimension surrogate. All fermion legs lie in exterior degree2.
000154 | The one-dimensional determinant leg at(2,3) is charged under the remaining
000155 | U1, so it is not a trivial constant fermion. All tensors are restrictions
000156 | of the SAME vol_E, with no split-specific fitted coefficient or chosen Yukawa.
000157 | This tensor is available in the given exterior class; it is NOT proved to be
000158 | the actual primitive source vertex.
000159 | 
000160 | Primitivity scope is the holomorphic COVARIANT polynomial invariant ring,
000161 | without independent conjugate/dual field variables or their norm invariants.
000162 | The SU(b),SU(s) centers and the trace-balanced U1 require total fundamental
000163 | degrees(P,Q)=(bm,sm), m a positive integer, for every positive invariant.
000164 | Thus its carrier degree is at least b+s=5. Every displayed trilinear has
000165 | carrier degree2+2+1=5 and is primitive: a product of two positive invariant
000166 | factors would have degree at least10. This does not assert primitivity in
000167 | an enlarged *-algebra with quadratic field norms.
000168 | 
000169 | Rank5 is the FIRST possible carrier rank in this class. For even g the
000170 | total degree of two even legs plus one fundamental is odd, whereas gm is
000171 | even. For g=3 both nontrivial even legs have degree2, so total degree5 is
000172 | not divisible by3. Rank1 has no two-factor decomposition; rank2/4 are
000173 | excluded by parity. This analytic argument is backed by an exhaustive
000174 | finite degree-signature check below5, not an unbounded numerical scan.
000175 | 
000176 | Requiring a UNIQUE primitive carrier channel up to exchanging the fermion
000177 | legs does not rescue the selection. At rank5 both legs must have degree2;
000178 | the only component types of Lambda2 E are Lambda2 V, V tensor W, Lambda2 W.
000179 | Their unordered pairs satisfying total degrees(b,s-1) number1,2,1,1 for
000180 | (b,s)=(1,4),(2,3),(3,2),(4,1). The extra(2,3) channel pairs V tensor W with
000181 | itself. Each channel has determinant-contraction multiplicity one: in each
000182 | block, only the full wedge can contract these covariant columns to its
000183 | determinant. Thus uniqueness removes(2,3), but leaves(1,4),(3,2),(4,1).
000184 | Even the naming convention b>s still leaves both(3,2) and(4,1). These are
000185 | carrier channels, not counts of independent family/flavor couplings.
000186 | 
000187 | For(4,1) the wedge pairing Lambda2 V x Lambda2 V -> Lambda4 V is symmetric.
000188 | A scalar contraction of two Grassmann Weyl spinors is also symmetric in
000189 | the species labels; there is no automatic Fermi-exchange zero. An additional
000190 | source-derived flavor or sector constraint could distinguish it, but is not
000191 | silently supplied here.
000192 | 
000193 | All four rows still have the same16-dimensional even packet. For the
000194 | primitive internal generator X=(-s)I_V+(b)I_W, induced by sums on exterior
000195 | degrees0,2,4, every row has Tr X=Tr X^3=0. These particular mixed-gravity/U1
000196 | cubic trace tests therefore do not remove the competing splits either.
000197 | This is not a claim of all global anomalies, a chiral measure, or a3+1D QFT.
000198 | 
000199 | The exact checker verifies2652 Lie-generator tensor coefficients over the
000200 | integers. The nonzero coefficient counts for the four rows are12,3,6,6.
000201 | It also checks the four trace pairs and the conditional occupancy solutions.
000202 | The complete unordered degree-two component enumeration gives channel counts
000203 | 1,2,1,1; the one-dimensional determinant contractions justify the counts
000204 | analytically rather than treating a list of degree signatures as a full
000205 | representation-theoretic proof by itself.