TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/marked-seam-monodromy-20261023/THEORY.txt
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000285 | 6. Native 4/5/6 diagram and local-coefficient invariants
000286 | 
000287 | Let V=C4 be the native Gaussian coordinate space, W=exterior2 V, and let
000288 | P be the EXISTING five-dimensional symmetric quartic code subspace of
000289 | Sym4 V. Its exact sparse basis is extracted from process._code_basis:
000290 | sum z_i^4; the three paired-square sums; and z0z1z2z3, with the existing
000291 | normalizations. The inclusion P->Sym4 V and antisymmetrizer W->V tensor V
000292 | are G0-equivariant. They define a common associated-bundle diagram on X.
000293 | They are not proposed isomorphisms between spaces of different dimensions.
000294 | 
000295 | The actual J actions are i I4 on V, -I6 on W, and I5 on P. For R the P
000296 | action is diag(1,1,1,1,-1); its W action has three eigenvalues-1 and three+1.
000297 | Consequently the two valid order4 monodromies J and JR yield
000298 |   representation       H0(J) H1(J)   H0(JR) H1(JR)
000299 |   V, rank4                 0     8        0      8
000300 |   exterior2 V, rank6        0    12        3     15
000301 |   quartic P, rank5          5    15        4     14.
000302 | Thus equal clock order and even the same V cohomology do not identify
000303 | the marked quartic/exterior transport systems. These invariants separate
000304 | the examples without changing any native representation.
000305 | 
000306 | PROOF OF THE COHOMOLOGY FORMULAS. X retracts onto a bouquet of3circles.
000307 | For a rank r local system F, its cellular complex is
000308 |   C0=F -> C1=F^3,
000309 |   delta(v)=((M0-1)v,(M1-1)v,(M2-1)v).
000310 | Over a field H0 is the common fixed space and dimH1=2r+dimH0;
000311 | all higher groups vanish. The displayed eigenvalues give the table.
000312 | For the integral E8 local system with all meridians J,
000313 |   H0=0, H1=Z^16 direct_sum (Z/2)^4, Hk=0(k>=2).         (6.1)
000314 | Indeed a unimodular change of the three cochain components takes delta
000315 | to((J-1)v,0,0); its cokernel is L^2 direct_sum L/(J-1)L. The Gaussian
000316 | free-module description gives L/(1-i)L=F2^4. Independently, the exact
000317 | integral matrix has Smith entries1,1,1,1,2,2,2,2.
000318 | 
000319 | All complex vector bundles on this graph homotopy type are topologically
000320 | trivial; ordinary positive-degree Chern classes cannot detect these flat
000321 | monodromies. Their local-system cohomology and the lift classes can.
000322 | Likewise H2(X,A_rho)=0 does not force a group class in H2(V,Z) to vanish:
000323 | these are different cohomology groups. The peripheral-class obstruction
000324 | in(4.1) is a relative constraint, despite existence of unrestricted lifts.
000325 |