TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/seam-source-character-search-20261015/PROOF.txt
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This bounded excerpt is source evidence; it is not a new proof/checker run.
Conversation metadata and personal execution details are omitted where indicated.

000021 | 2. Source data, with time and clock kept distinct
000022 | 
000023 | Let L=D8 union ((1/2,...,1/2)+D8), D8={z in Z^8 : sum z_i even}.
000024 | It is the recorded E8 lattice; roots have squared norm 2. In its M frame set
000025 | 
000026 |  r=e3+e6, q=2r,
000027 |  d_a=(-1/2,0,a/4,-1/2,0,(a-1)/4,-1/4,-1/4),
000028 |  K_a(p,N)=p.p/2+d_a.p+N,  N a nonnegative oscillator level.
000029 | 
000030 | The retained source cases are a=1/4 and 3/4; physical cover time is H_a=8K_a
000031 | in this convention. This follows from delta_j=(j+a)/8-1/2 and d_a=M delta.
000032 | The older fold's nu=2mu is a CLOCK vector. It satisfies
000033 | M nu=2d_a+2r; its deformation a r/2 is consistent with d_a's a r/4.
000034 | Mixing these two vectors would create a spurious factor-two error.
000035 | 
000036 | On a root field E_p, a torus character h acts by exp(2 pi i h.p).
000037 | Two characters agree on every root iff their difference lies in L, since
000038 | the roots generate the self-dual lattice L. This also gives equality on
000039 | all lattice charge sectors, not only on the enumerated root shell.
000040 | 
000041 | At a=1/4, the charge-quarter operator Z_q=exp(i pi Q/2) has torus character q/4=r/2
000042 | on the E8 extension. For every p in L,
000043 | r.p is integral. Its phase is therefore (-1)^(r.p), with exact order 2 on
000044 | this E8 algebra. Its root grades are (128,0,112,0). The original odd CAR
000045 | fields still see the quarter phase. The restriction to the even extension
000046 | must not be mistaken for removal of that physical phase.
000047 | 
000048 | 3. Analytic completeness of the candidate family
000049 | 
000050 | Use the connected source torus with Lie algebra V=span_R{d,r}, d=d_(1/4).
000051 | It is a two-dimensional rational plane, not an arbitrary E8 Cartan search.
000052 | For h=A d+B r, coordinates 2 and 5 vanish. If h belongs to L, it cannot
000053 | be in the half-integer coset. Integrality of coordinates 7 and 8 forces
000054 | A=4k, k integer. Coordinate 3 then forces B=l-k/4, l integer. Coordinate
000055 | sum is 2l-7k, whose evenness forces k=2s. Conversely these conditions suffice.
000056 | Thus the full lattice in the plane is
000057 | 
000058 |  V intersect L = Z lambda1 + Z r,
000059 |  lambda1=8d-r/2=(-4,0,0,-4,0,-2,-2,-2).
000060 | 
000061 | Consequently every character in V/(V intersect L) of order dividing 4 is uniquely
000062 | 
000063 |  h_(u,v)=(u lambda1+v r)/4 mod L,  u,v in {0,1,2,3}.
000064 | 
000065 | This derives a complete 16-candidate search from a continuous family.
000066 | Twelve have exact order 4. It is not a finite parameter cutoff.
000067 | For comparison the finite group generated by d and r/8 has 64 elements,
000068 | isomorphic to Z16 x Z4; 8d-4(r/8)=lambda1 is its nontrivial short relation.
000069 | All its 4-torsion is exactly the set above. Enumerating both ways provides
000070 | an independent completeness check, rather than a second physical assumption.
000071 | 
000072 | 4. The four successful INTERNAL markings
000073 | 
000074 | Classify neutral roots by nonorthogonality-connected components and their
000075 | linear ranks, not by total dimension or root count alone. The twelve exact
000076 | order-four characters give:
000077 | 
000078 |  root grades (0,1,2,3)     number   neutral root systems
000079 |  (126,56,2,56)              2      E7 (plus one abelian Cartan direction)
000080 |  (62,56,66,56)              6      D6+A1 (plus one abelian direction)
000081 |  (52,64,60,64)              4      D5+A3
000082 | 
000083 | The last four are h=m d+n r/8 with representatives
000084 |  (m,n)=(2,3),(2,7),(6,1),(6,5).
000085 | In that order they are Ot, t, -Ot, -t as characters, where
000086 | 
000087 |  t=2d_a-a r/2=(-1,0,0,-1,0,-1/2,-1/2,-1/2),
000088 |  O(x)=(x4,-x5,-x6,x1,-x2,-x3,x8,x7).
000089 | 
000090 | There are two inverse pairs and two rank-five subspaces. For a neutral
000091 | root set R0, its moment M0=sum_(p in R0) p p^T has eigenvalues 16 on the
000092 | D5 subspace and 8 on the A3 subspace. Hence
000093 | 
000094 |  P5=(M0-8I)/8
000095 | 
000096 | is recovered without putting a desired coordinate projector in the search.
000097 | The resulting P5 spaces are span{e1,e2,e3,e4,e5} and
000098 | span{e1,e2,e4,e5,e6}. The actual sheet map O exchanges them.
000099 | The factor 16=4(5-1) and 8=4(3-1) follows by summing the roots +/-e_i+/-e_j
000100 | of D5 and D3=A3; orthogonal conjugation preserves this calculation.
000101 | 
000102 | 4a. Blind source-frame control against an inserted target projector
000103 | 
000104 | The program also starts in the original source coordinates with
000105 | delta_j=(j+1/4)/8-1/2, r_source=(1/2)^8 and
000106 | lambda1_source=8delta-r_source/2=(-4,-3,-2,-1,0,1,2,3).
000107 | This independent enumeration function has no M argument, file read, fixed
000108 | P5 or target projector. It classifies all sixteen characters and reconstructs
000109 | exactly four rank-five projectors at (u,v)=(1,0),(1,2),(3,0),(3,2).
000110 | Only afterwards is M applied, and all four satisfy
000111 | M P_source M^T=P_target as independently computed from neutral roots.
000112 | Thus the abstract splitting is not manufactured by M's target-frame choice.
000113 | The source E8 extension itself is still given; this control does not prove it.
000114 | 
000115 | 5. A computable marking operator, and what has NOT been selected
000116 | 
000117 | Because N+p.p/2 is integral in this lattice representation,
000118 | 
000119 |  W_a=exp(4 pi i K_a) exp(-i pi a Q/2),  Q=q.H,
000120 | 
000121 | acts on all charged sectors by the character t. It has order four, and
000122 | its adjoint action on currents gives the grades above. This is a composition
000123 | of the given full time and given full charge. K_a and the physical boundary
000124 | transition remain unchanged. One may not replace the physical order-eight
000125 | fold clock by W_a and declare the quarter holonomy absent.
000126 | 
000127 | The algebraic compensation formula was already visible in FOLD-AUDIT.md,
000128 | sections 4 and 4a. The new result here is its exhaustive placement in the
000129 | source-generated character family, together with an executable candidate
000130 | filter which recovers the projector and checks its actual source-field type.
000131 | This is not advertised as a newly discovered compensation principle.
000132 | 
000133 | Choosing W_a as the physical marking is still an interpretation/selection
000134 | condition. The formulas do not prove that P1 identifies its seam action with
000135 | this composition. Even a natural normalization is not a missing premise's
000136 | proof. Positive orientation of the chosen clock removes its inverse only
000137 | after that clock has been identified; [u_Sigma]=1 alone is not this identity.
000138 | 
000139 | In particular, using only the physical order-two quarter character rather
000140 | than the full normalized U(1) removes the needed roots of that character:
000141 | the subgroup generated by d and r/2 contains no D5+A3 marking. A finite
000142 | Wilson phase does not determine the additive charge generator required
000143 | by the compensation. The given CAR realization supplies that generator;
000144 | its original P1 field identification is a separate obligation.