TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/seam-positive-kernel-search-20261015/PRIMITIVE-INTEGRAL-COMPLETION-20261102.txt
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000019 | 2. THEOREM WITHOUT EVENNESS, UNIMODULARITY OR MINIMALITY INPUT
000020 | Let n>=3, L0=D_(n+1) orthogonal A_(n-1), with its standard positive root
000021 | norm2. Let L be ANY integral lattice of the same full rank containing L0.
000022 | Suppose v=(d,a) in L has primitive A discriminant projection: [a]=b[lambda1]
000023 | in A*/A=Z_n, gcd(b,n)=1. Then n=4 and L is one of the two marked index4
000024 | E8 completions of D5+A3. In particular L is automatically even/unimodular.
000025 | 
000026 | Integral means all pairings in L are integers, not just a determinant test.
000027 | The positive standard D/A pairing is still an explicit candidate premise.
000028 | The theorem does not assume an E8 source, rank8, cyclic whole glue, maximal
000029 | locality, diagonal glue, a selected chirality or the smallest candidate.
000030 | 
000031 | Proof of n4. Integrality gives L subset L0*. Every D_m dual vector has
000032 | integer or half-integer coordinates, and 4 d.d is integer for all four
000033 | classes, whether D_m*/D_m is cyclic or Z2 times Z2. In A_(n-1),
000034 | 
000035 |  a.a = b^2 (n-1)/n  modulo Z.
000036 | 
000037 | This follows from a=b lambda1+r, r in A, lambda1.r integer and r.r even.
000038 | Because v.v=d.d+a.a is integer, 4a.a is integer, hence n divides 4b^2.
000039 | Primitivity gives gcd(b^2,n)=1. Euclid's lemma implies n divides4; with
000040 | n>=3, n4 is the only possibility. This is an all-n proof, not a scan.
000041 | 
000042 | Proof of the whole completion. At n4 the primitive A3 classes are b=1,3,
000043 | with quadratic norm3/4 modulo2Z. Integrality of v.v forces a D5 spinor
000044 | class s or c, of norm5/4 modulo2Z; trivial and vector classes cannot work.
000045 | Thus v.v=0 modulo2Z. The joint class has order4. The generated lattice
000046 | L1=L0+Zv is even integral, index4 and determinant16/4^2=1. Since L is
000047 | integral and contains L1, it lies in L1*=L1. Hence L=L1: a larger integral
000048 | extension is impossible. Four spinor/family representatives form two
000049 | inverse-paired marked glue subgroups. The two orientations are retained.
000050 | No selected physical chirality is inferred from their abstract isometry.
000051 | 
000052 | The standard A3 to D3 map from the existing puncture-metric theorem sends
000053 | lambda1 to(1/2)^3. Together with s=(1/2)^5, the generator becomes(1/2)^8.
000054 | Twice that generator adds the odd-odd integer coset to D5+D3, giving D8;
000055 | then L=D8 union(D8+(1/2)^8), the explicit usual E8 lattice.
000056 | 
000057 | 3. WHOLE PREMISE CLASS AND NECESSARY COUNTERCONTROLS
000058 | The independent finite replay computes all four D dual norm classes and
000059 | ALL primitive b at every n3..96, using Fraction arithmetic. Only n4
000060 | survives. This replay checks arithmetic and implementation; the general
000061 | argument above supplies the infinite-class conclusion.
000062 | 
000063 | Drop primitive projection: n16, s=(1/2)^17, a=2 lambda1_A15. Then
000064 | s.s=17/4, a.a=15/4, v.v=8 and v pairs integrally with all roots. This is
000065 | R101's valid higher completion input. In particular 2lambda1 is NOT
000066 | lambda2: lambda2.lambda2=7/4, differing by2 in this example.
000067 | 
000068 | Drop integrality: n5, s=(1/2)^6 and a=lambda1_A4 have norm23/10. They are
000069 | perfectly valid rational weights, but do not define an integral extension.
000070 | 
000071 | Change the standard pairing: multiply the whole n5 form by20. Rootpairings
000072 | remain integral, roots have norm40, v.v=46 and all generator pairings are
000073 | integral, so an even integral extension with the original primitive A4
000074 | weight label exists. This label refers to the ORIGINAL A4 weight quotient;
000075 | the discriminant group of the rescaled physical lattice is a different
000076 | object. The example shows why source normalization cannot be dropped.
000077 | 
000078 | Keep only FULL DOUBLED locality: on L0 plus L0 with bilinear form B_L-B_R,
000079 | the diagonal generator (s,a;s,a) has norm0 for ANY n, and integral pairings
000080 | with each root copy. At n5 its positive sum norm is23/5, not0. Thus a
000081 | nonchiral doubled field can be mutually local while each positive chiral
000082 | half has fractional norm23/10. This is a counterexample to inferring the
000083 | one-sided integral hypothesis from whole doubled locality. It is not an
000084 | identification of this free doubled class with the actual TFPT source.
000085 | A braiding form, positive Hilbert norm, energy and geometric Double Cover
000086 | are distinct data until the original source identifies them.
000087 | 
000088 | 4. FUNCTIONS, DIFFERENTIALS AND THE ALREADY EXISTING METRIC THEOREM
000089 | For a reduced fourpoint divisor D on P1, V=H0(O(D)) has dimension5. Its
000090 | principal-parts sequence is
000091 | 
000092 |  0 -> C -> H0(O(D)) -> H0(O(D)|_D) -> 0,
000093 | 
000094 | whose last space has dimension4. Summation-zero residues instead describe
000095 | meromorphic DIFFERENTIALS H0(K(D)), dimension3; on P1 their residues have
000096 | one sum constraint. Both derive from the same divisor, but they are not
000097 | the false short exact sequence0->C->V5->U3->0.
000098 | The actual example f=1/(z-1) has f dz residue1 at1 and residue-1 at infinity.
000099 | Thus its residues on four finite marked points alone need not sum to0.
000100 | Multiplication by an unchosen differential cannot silently identify these
000101 | spaces or the physical P2 three-plus-two split.
000102 | 
000103 | Clifford(V plus V*) has a canonical nondegenerate evaluation pairing and
000104 | spin module, without choosing a positive metric on V. Over real coordinates
000105 | this evaluation is neutral. The scaling(v,phi)->(tv,t^-1 phi) preserves it
000106 | for every t>0, so it alone does not choose a positive physical norm. P1
000107 | already supplies operative Hilbert/kernel data; these are not declared absent.
000108 | 
000109 | The existing puncture-metric-joint-completion-20261102 theorem ALSO remains:
000110 | under its fixed original additive spinor-meridian extension, W(D5) timesC4,
000111 | and integral UNIMODULAR full pairing, the entire invariant positive metric
000112 | has x=a=b=1. That theorem and R102 remove different assumptions. Combining
000113 | them cannot prove either premise from its own consequence: using metric
000114 | rigidity to prove unimodularity and then using unimodularity for metric
000115 | rigidity would be circular. Source levels and actual insertion remain typed.
000116 | 
000117 | 5. MODEL COUNTEREXAMPLES, NOT NEW SOURCE LAWS
000118 | Gemini's proposed unmodified Dirichlet-to-Neumann form has q(1,1)=0 since