TFPT scientific source excerpt Original repository path: experiments/theory-contracts/puncture-metric-joint-completion-20261102/PROOF.txt Original file SHA-256: 86701d996dc9b5ed2bc228359c45699ac6f458eddfbc7daf152087324f250aa7 Inclusive original lines: 1-159 Line numbers below refer to the original source. This bounded excerpt is source evidence; it is not a new proof/checker run. Conversation metadata and personal execution details are omitted where indicated. 000001 | PUNCTURES, JOINT CHARGE COMPLETION AND METRIC SELECTION 000002 | 2 November 2026. Research contract, PARTIAL for fundamental TFPT. 000003 | Exact conditional metric theorem. No ledger or physical gate promotion. 000004 | 000005 | QUESTION AND OUTCOME 000006 | 000007 | Can the positive metric used in the four-puncture A3 family construction 000008 | be a consequence of joint charge completion rather than an independently 000009 | chosen residue metric? Yes, for the original fixed additive charge extension 000010 | and its actual W(D5) times C4 action. Integral unimodular completion determines 000011 | the entire invariant real metric uniquely. Evenness then follows. 000012 | 000013 | The original source, four square marks, D5 carrier and spinor-meridian glue 000014 | are not selected by this theorem. The physical reason for integral and 000015 | unimodular joint charge completion remains a necessary origin question. 000016 | 000017 | 1. THE INTEGRAL PUNCTURE MAP 000018 | 000019 | Let D have four oriented punctures on P1, with meridians gamma_i. Then 000020 | 000021 | H1 = Z^4/Z(1,1,1,1), H^1 = {n in Z^4 : sum n_i=0}. 000022 | 000023 | The canonical counting identification of the free marked generators gives 000024 | an injection H^1 -> H1. Its cokernel is Z/4: the sum of coordinates modulo4 000025 | is well defined on H1 and its kernel is the image of H^1. Each individual 000026 | meridian maps to1. This integral statement precedes a positive metric. 000027 | 000028 | Put V={u in R4:sum u_i=0}, P=I-J/4. Under the counting realization, 000029 | gamma_i maps to m_i=P e_i. Differences m_i-m_j=e_i-e_j are the A3 root 000030 | lattice R. The m_i generate its weight lattice R*. Their Gram determinants 000031 | are4 for a root basis and1/4 for a meridian basis. Every m_i has norm3/4. 000032 | 000033 | All four meridians have the SAME discriminant class: [m_i]=[m_j] in R*/R. 000034 | The cyclic permutation of the marks therefore acts as identity on Z/4. 000035 | It is not translation by a generator of that group. Translation changes 000036 | the zero sector and is an affine fusion operation, not a lattice isometry. 000037 | 000038 | This is a direct connection to the original generator: m_0 is literally 000039 | OMEGA_F in verification/v983_simple_current_generator.py. With the supplied 000040 | D5 spinor s=(1/2)^5, lambda=(s,m_0) is that source's actual LAM. 000041 | All16 same-chirality D5 spinor weights times the four meridians produce 000042 | 64 original norm-two roots in its first extension sector. This root family 000043 | and the52/64/60/64 full census are existing v983 results, not new E8 closure. 000044 | 000045 | 2. A CONCRETE PROBLEM IN THE ORIGINAL METRIC ARGUMENT 000046 | 000047 | documents/tfpt_1_architecture_e8/sections/011-source.tex:32-45 asserts a 000048 | four-by-four delta residue pairing on puncture loops whose sum is zero, 000049 | then infers a canonical positive I4 metric. That inference is invalid. 000050 | 000051 | If a functional ell_i descended to H1 with ell_i(gamma_j)=delta_ij, then 000052 | ell_i(sum_j gamma_j)=1, whereas sum_j gamma_j=0. Individual dz/(z-p_i) 000053 | also has an extra pole at infinity; differences or zero-sum residue vectors 000054 | are required on the sphere punctured only at D. The corrected projected 000055 | pairing is I-J/4. Duality alone supplies no positive physical inner product. 000056 | 000057 | Square C4, or even square D4, symmetry still permits TWO positive metric 000058 | coefficients on V. Thus the metric assertion cannot be repaired merely by 000059 | invoking the same square symmetry. The conditional joint-completion theorem 000060 | below supplies a different, explicit route. The original paper and ledger 000061 | have not been silently edited or promoted by this research contract. 000062 | 000063 | 3. THE WHOLE METRIC CLASS, NOT A PARAMETER SAMPLE 000064 | 000065 | Keep the ORIGINAL additive extension 000066 | 000067 | L0 = D5 plus A3, L = L0 + Z(s,m_0), [L:L0]=4. 000068 | 000069 | Here D5={z in Z5:sum z_i even}; A3={z in Z4:sum z_i=0}. 000070 | No inner product is used to define these additive modules. Their supplied 000071 | real actions are W(D5) on the five carrier coordinates and the square 000072 | four-cycle C4 on the meridians. Both preserve L: W(D5) fixes the spinor 000073 | class modulo D5, and C4 fixes the meridian class modulo A3. 000074 | 000075 | Consider ALL positive real symmetric forms invariant under this PRODUCT 000076 | action. W(D5) has no fixed vector on R5 and its symmetric commutant consists 000077 | only of scalars. Thus invariance forces carrier form x I5 and kills every 000078 | mixed carrier-family block. Orthogonality and its scale are not premises. 000079 | 000080 | On V, set v=(1,-1,1,-1)/2. The real C4 module splits into an irreducible 000081 | rotation plane and one sign line. Its entire positive symmetric cone is 000082 | 000083 | Q = a(P-vv^T)+b vv^T, x>0,a>0,b>0. 000084 | 000085 | Reflection symmetry is unnecessary. The actual checker solves the whole 000086 | linear invariant-form space, not just a chosen ansatz. 000087 | 000088 | In the simple puncture-difference basis, the root Gram is 000089 | 000090 | [[a+b, -b, b-a], [-b, a+b, -b], [b-a, -b, a+b]], 000091 | 000092 | with determinant4a^2 b. A D5 simple-root Gram has determinant4x^5. 000093 | The fixed index-four extension therefore has 000094 | 000095 | det(L) = (4x^5)(4a^2 b)/4^2 = x^5 a^2 b. 000096 | 000097 | If the FULL charge pairing is integral, its restriction to the root 000098 | sublattice is integral. Adjacent D5 roots have pairing -x, hence x is a 000099 | positive integer. The displayed A3 Gram gives b an integer and a+b an 000100 | integer, hence a is a positive integer too. If L is also unimodular, 000101 | 000102 | x^5 a^2 b = 1, x,a,b positive integers, 000103 | 000104 | so x=a=b=1. This proves uniqueness on the complete positive invariant cone. 000105 | Evenness, root lengths and a fixed D5 scale were not used to select it. 000106 | 000107 | 4. SUFFICIENCY AND EXPLICIT E8 IDENTIFICATION 000108 | 000109 | The unique candidate must still make the full extension integral and even. 000110 | At x=a=b=1, the root lattices are even; lambda has norm5/4+3/4=2 and 000111 | integral pairing with every root. Therefore every r+k lambda has even norm. 000112 | The index is4 and determinant1, as required. 000113 | 000114 | One can identify E8 explicitly without relying on a uniqueness citation. 000115 | The map 000116 | 000117 | O(u)=(u0+u1-u2-u3, u0-u1+u2-u3, u0-u1-u2+u3)/2 000118 | 000119 | is an isometry V->R3, sends A3 onto D3 and m0 to(1/2)^3. Twice lambda 000120 | joins the odd-odd integer coset to D5+D3, giving D8. Adding lambda gives 000121 | 000122 | D8 union (D8+(1/2)^8), 000123 | 000124 | [OMITTED: non-scientific conversation or execution metadata] 000125 | [OMITTED: non-scientific conversation or execution metadata] 000126 | [OMITTED: non-scientific conversation or execution metadata] 000127 | 000128 | 5. EXACT SCOPE AND THE FUNDAMENTAL NEXT QUESTION 000129 | 000130 | This removes a freely chosen positive metric, separate orthogonality and 000131 | evenness INSIDE the stated common charge completion. It does not derive 000132 | the additive extension, product symmetries, four square marks, chirality, 000133 | physical integral-unimodular completeness, source state, physical time, 000134 | locality or 3+1 dimensional interacting physics. It is a conditional rigidity 000135 | result, not a circular origin proof in which E8 selects its own assumptions. 000136 | 000137 | The next origin test is whether the SAME original P1 boundary source supplies 000138 | this charge lattice and integral-unimodular completeness independently of 000139 | inserting E8. A statement about residue duality, a passing lattice checker, 000140 | or a holomorphic E8 theory already assumed as input does not answer it. 000141 | 000142 | The common-source reading has a concrete mixed consequence: for fixed 000143 | spinor s and different marks i,j, v_i=(s,m_i) has norm2 and v_i.v_j=1; 000144 | v_i-v_j=(0,e_i-e_j) is the actual family root. In the accepted lattice 000145 | current realization E_vi(z)E_-vj(w) has a simple-pole family-root term; 000146 | i=j instead has the normalized double pole. This is a required joint 000147 | field response, not an independent numerical fit or physical measurement. 000148 | Any claimed original source dictionary must preserve the actual cocycle, 000149 | adjoint, state and clock in this test. 000150 | 000151 | 6. EXECUTION AND REVIEW 000152 | 000153 | check_metric_completion.py:30 exact structural checks, normal and -OO PASS. 000154 | The global uniqueness proof is the positive-integer product argument above, 000155 | not a finite parameter search. Two independent agents reviewed the complete 000156 | whole-space invariant-form calculation and the proof. The stronger 9x9 000157 | calculation independently confirms dimension3 and vanishing mixed blocks. 000158 | No Lean formalization, full verification-suite run or physical gate closure 000159 | is claimed. No changes to the original proof source or status ledger.