TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/marked-seam-monodromy-20261023/THEORY.txt
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000015 | 1. Category of marked seam geometries
000016 | 
000017 | Work with compact finite-CW pairs (Y,Sigma), and U=Y minus Sigma. For the
000018 | surface subcategory Sigma is a finite ordered set of punctures. A reflected
000019 | pair includes an involution tau of Y preserving Sigma. Additional clock
000020 | actions, such as a C4 action, are optional explicitly marked automorphisms.
000021 | 
000022 | A lattice transport datum is a functor H:Pi_1(U)->Lat_q. The target has
000023 | free finite-rank integral lattices with an even integral symmetric form;
000024 | arrows are lattice isometries. Its finite quadratic datum is
000025 |   q_L(x mod2L)=(x,x)/2 mod2.
000026 | It is well defined since replacing x by x+2y changes this by
000027 | 2(x,y)+2(y,y). Transport preserves q_L. An independently chosen finite
000028 | quadratic module may instead be an additional marking, stated separately.
000029 | 
000030 | The local system L and, after choosing a basepoint and fiber chart, the
000031 | representation rho are descriptions of H, not independent arbitrary
000032 | inputs. Changing the fiber chart conjugates rho. For a reflected object
000033 | include an isometric natural transformation T:H -> tau*H with
000034 | T_(tau x) T_x=1. This reflection marking is real/integral; in a complex
000035 | chart it may be semilinear and must not be called a complex reflection.
000036 | 
000037 | A morphism consists of a map of marked pairs f, avoiding Sigma' on U,
000038 | and a natural fiberwise isometry H -> f*H'. It respects reflections and
000039 | any clock/peripheral markings that were declared. Composition uses pullback
000040 | and composition of natural transformations. Isomorphisms have f a marked
000041 | homeomorphism and invertible fiber maps. Allowed homotopies are homotopies
000042 | of such maps through the same pair/reflection constraints, together with
000043 | the induced compatible transport identifications.
000044 | 
000045 | The direct-sum product uses
000046 |   (Y,Sigma) times (Z,Lambda)
000047 |    =(Y times Z, (Sigma times Z) union (Y times Lambda)),
000048 |   H_product=pr_Y*H direct_sum pr_Z*K, q_product=q_H orthogonal_sum q_K.
000049 | Its complement is U times V. The unit is a point with rank0 lattice.
000050 | Ranks add, so the rank8 subcategory alone is not product-closed. Tensor
000051 | products are a different optional operation and are not substituted here.
000052 | 
000053 | For seam gluing of oriented punctured surfaces replace punctures by boundary
000054 | circles with collars. Supply an orientation-reversing boundary gluing and
000055 | a fiber isometry F. If M,N are the corresponding positive peripheral
000056 | holonomies, the descent condition is F M F^-1=N^-1. Under this condition
000057 | the functors glue; necessity is the common loop relation and sufficiency
000058 | follows by the fundamental-groupoid pushout/van Kampen presentation.
000059 | Reflection markings must agree on the gluing collars. The gluing map and
000060 | fiber identification are data, not a presumed unique composition.
000061 | 
000062 | Thus transport is a groupoid. Conjugacy classes alone do not have a
000063 | single-valued multiplication: for example in S3 the product of two
000064 | transpositions can be identity or a3cycle. Defect composition on classes
000065 | is instead the correspondence {(a,b):a in C,b in D,ab in E}/G. This does
000066 | not by itself define a semisimple fusion category or a higher category.
000067 | 
000068 | A concrete reflected base object uses tau(z)=conjugate(z) on CP1 and
000069 | T0(x0,x1,...)=(x0,-x1,x2,-x3,...). T0 preserves E8 and its quadratic
000070 | datum; T0 J T0=J^-1 matches the reversal/permutation of positive meridians.
000071 | Thus the all-J base local system below admits this reflection marking.
000072 | An extra reflection lift on its SIGN extension is additional mathematical
000073 | marking and is not imposed in the finite classification of Section4.
000074 | T0 is distinct from the complex-linear reflection R of Section5.
000075 | 
000076 | 2. Existing lattice and the actual factor set
000077 | 
000078 | Let L be the standard even unimodular E8 lattice in R8. Retain the actual
000079 | native quarter-clock J(x0,x1,...)=(-x1,x0,-x3,x2,...). It preserves L,
000080 | J^2=-1 and J^4=1. Let epsilon be the existing ordered lattice cocycle,
000081 | retained literally by the source extraction in checker.py. Define
000082 |   E_x E_y=epsilon(x,y) E_(x+y),
000083 |   [E_x,E_y]=(-1)^((x,y)).
000084 | This central extension of L is denoted Lhat. All sign automorphisms below
000085 | fix its central generator. The sign-lift kernel is
000086 |   A=Hom(L,{+1,-1})=Hom(L,F2).
000087 | 
000088 | For any isometry g fix an integral ordered basis b_i and define a lift
000089 | S_g by E_(b_i) -> E_(g b_i). Inner-product invariance preserves all
000090 | generator commutators; freeness of L gives an invertible automorphism.
000091 | For x=sum n_i b_i let
000092 |   product_i E_(b_i)^n_i=c_B(n) E_x,
000093 |   product_i E_(g b_i)^n_i=c_(gB)(n) E_(g x),
000094 |   eta_g(x)=c_(gB)(n)/c_B(n).
000095 | Negative powers use E_b^-1=epsilon(b,-b) E_-b. This is an all-integral-
000096 | vector definition, not just an evaluation on roots.
000097 | 
000098 | The COMPLETE factor formula is
000099 |   d(g,h)(x)=eta_h(x) eta_g(hx)/eta_(gh)(x),
000100 |   c(g,h)(y)=d(g,h)((gh)^-1 y),
000101 |   S_g S_h=D_(c(g,h)) S_(gh),                            (2.1)
000102 | where D_a(E_y)=a(y) E_y. The product projects to gh; the ratio projects
000103 | to identity and hence is a character. Equivalently divide the two
000104 | cocycle identities for eta to prove d(x+y)=d(x)d(y).
000105 | 
000106 | With left action (g.a)(y)=a(g^-1 y), associativity gives
000107 |   c(g,h)+c(gh,k)=g.c(h,k)+c(g,hk)                        (2.2)
000108 | in additive F2 notation. Formula(2.1) computes any requested pair in
000109 | any specified lattice-isometry subgroup. This is a MODULE-valued group
000110 | 2cocycle; the extension need not be central. Indeed S_g D_a S_g^-1=D_(g.a).
000111 | The complete preimage of a subgroup is central over A only if its action
000112 | on A is trivial. A character-valued factor is not a scalar multiplier.
000113 | These statements are standard lattice-extension algebra, recalled with
000114 | the actual cocycle conventions; see Okada§3 and AppendixA, arXiv2412.19430.
000115 | The new classifications below are not attributed to that paper.
000116 |