TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/puncture-metric-joint-completion-20261102/ORIGIN-TOPOLOGY-RESULT.txt
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Inclusive original lines: 57-117
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This bounded excerpt is source evidence; it is not a new proof/checker run.
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000057 | EXACT CONNECTION TO THE ACTUAL GLUE, WITH THE ACTION TEST
000058 | 
000059 | There is a direct lattice bridge stronger than rankH1=3. Write the four oriented
000060 | meridians as gamma_i with sum gamma_i=0. Then
000061 |   H1(P1-D,Z)=Z^4/Z(1,1,1,1),
000062 |   H^1(P1-D,Z)=ker(sum:Z^4->Z).
000063 | The second is the A3 root lattice. With the specified mark counting metric,
000064 | the first is its dual/weight lattice. Projecting the standard mark vectors onto
000065 | the sum-zero subspace gives m_i=e_i-(1/4)1. The first three m_i have Gram
000066 | I3-(1/4)J3,det1/4. Roots e1-e2,e2-e3,e3-e4 have Cartan Gram
000067 |   [[2,-1,0],[-1,2,-1],[0,-1,2]], det4.
000068 | Their change-of-basis determinant in the m basis is4. Thus the puncture
000069 | relation itself yields A3*/A3=Z4. Each m_i has norm3/4, exactly the family
000070 | discriminant norm complementary to the D5 spinor norm5/4 in the E8 glue.
000071 | 
000072 | This identifies the mathematical origin of that discriminant from the four
000073 | punctures once the mark metric/root-length normalization is supplied. It does
000074 | not identify the physical seam with this integral metric, nor derive D5.
000075 | 
000076 | The crucial action comparison is also exact:
000077 |   m_i-m_j=e_i-e_j is in A3.
000078 | Hence ALL FOUR mark meridians belong to the SAME discriminant generator class.
000079 | Every permutation of the marks, including the actual four-cycle, acts trivially
000080 | on A3*/A3. The four marked weights are NOT the four glue sectors. They are four
000081 | weights within the same fundamental SU4 charge sector. The other glue classes
000082 | correspond to degree/center charge (e.g. exterior powers), not mark labels.
000083 | 
000084 | Consequently a direct identification of geometric cyclic mark permutation with
000085 | fusion translation through the four discriminant sectors fails at the actual
000086 | group action. Equivalently the mark-cycle matrix on C4 has trace0; the central
000087 | SU4 quarter iI4 on that same fundamental representation has trace4i. They are
000088 | not conjugate. This is a scoped obstruction to that particular identification,
000089 | not an obstruction to one source carrying both different operators or an
000090 | explicit sector-changing field construction.
000091 | 
000092 | This root/weight/action calculation was not found in the queried compact Atlas
000093 | records or the located RR contracts. Its ingredients are standard and the
000094 | D5+A3 lattice closure is already established. It is best treated as a newly
000095 | written bridge and guard against a false dictionary, not as a new E8 theorem.
000096 | 
000097 | LITERAL MAP TO THE ORIGINAL E8 GLUE GENERATOR
000098 | 
000099 | The actual source verification/v983_simple_current_generator.py:11-33 specifies
000100 |   omega_f=(3/4,-1/4,-1/4,-1/4), omega_s=(1/2,1/2,1/2,1/2,1/2),
000101 |   lambda=(omega_s,omega_f).
000102 | Thus omega_f is LITERALLY the first puncture meridian m0 in the same coordinates,
000103 | not merely another vector of the same norm. The probe imports the current v983
000104 | definitions and checks this equality directly. It then checks all16 even-minus
000105 | D5 spinor weights s and all4 m_i: the64 distinct vectors (s,m_i) have norm2 and
000106 | each differs from the original lambda by an element of the original D5+A3.
000107 | All64 therefore are actual first-coset roots from the given E8 realization.
000108 | This adds a direct topological reading of the existing family/glue data.
000109 | 
000110 | The same source ALREADY proves the complete root census52,64,60,64 and exact
000111 | order4 of lambda. Its order and norm are checked here; its full root enumeration
000112 | is deliberately reused, not rerun. After the given D5 spinor chirality and
000113 | diagonal gluing are supplied, one paired meridian root generates the entire
000114 | glued lattice together with D5+A3. The orientation/conjugate gluing alternatives
000115 | are not silently removed: v92_glue_uniqueness.py:104-111 exchanges the two
000116 | diagonal choices under a single-factor conjugation. This is not a derivation
000117 | of D5 or of physical source permission for the corresponding field.