TFPT scientific source excerpt Original repository path: experiments/theory-contracts/elliptic-moebius-source-bridge-20261026/PUSHFORWARD-ANCHOR.txt Original file SHA-256: cbe5383c9df7b8f1b1095566f909ce3a357d10e9b3cd1bd15c89b8c72af870a2 Inclusive original lines: 1-190 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 | ELLIPTIC PUSHFORWARD AND THE P2 ANCHOR 000002 | ====================================== 000003 | 26 October 2026. Bounded original source audit and exact local monodromy test. 000004 | Atlas audit frame: revision 4a17a404b58c05f460121df8. 000005 | 000006 | Verdict 000007 | ------- 000008 | 000009 | There is an exact and useful common bundle statement. On the already present 000010 | double cover 000011 | 000012 | p:E -> P1, E: y^2=x^4-1, 000013 | 000014 | and for the already identified nontrivial Q-fixed two-torsion spin line alpha, 000015 | 000016 | p_*O_E = O + O(-2), p_*alpha = O(-1) + O(-1). 000017 | 000018 | Therefore 000019 | 000020 | V_ell := (p_*O_E/O) + p_*alpha = O(-2) + O(-1)^2. (1) 000021 | 000022 | This is exactly the holomorphic P2 anchor with splitting exponents {1,1,2}, 000023 | rank 3, degree -4 and h0=0. It is a real convergence of the existing elliptic 000024 | bosonic and spin data on the same marked base. It is stronger than the earlier 000025 | observation that several constructions merely produce the number 3. 000026 | 000027 | The natural connection does not agree with the original P2 flavor connection. 000028 | Its local conjugacy class at every branch mark has order 2, whereas P2 fixes an 000029 | order 3 cusp class. Thus (1) identifies the underlying holomorphic bundle only. 000030 | It supplies neither the P2 Deligne connection nor its irreducible flavor 000031 | representation. No change to AX.P2.01 or QGEO.SYM follows. 000032 | 000033 | 1. Why the bundle equality is exact 000034 | ----------------------------------- 000035 | 000036 | The prior curve calculation gives p_*O_E=O+O(-2) and, for a nontrivial 000037 | elliptic two-torsion alpha, p_*alpha=O(-1)+O(-1). See 000038 | 000039 | experiments/theory-contracts/original-admissible-source-closure-20261025/ 000040 | ORIGINAL-BASES-AND-ELLIPTIC-SPIN.txt:55-65 000041 | SHA256 bf6b2336e24a39aef26b9907c24f033a47a8a1318f8f19ae561e4c477680ce67. 000042 | 000043 | Removing the invariant scalar O from p_*O_E leaves the trace-free deck line 000044 | O(-2). Adding the full rank 2 spin pushforward yields (1). Consequently 000045 | 000046 | rank V_ell = 3, 000047 | deg V_ell = -2-1-1 = -4, 000048 | h0(V_ell) = 0, 000049 | a(V_ell) = (1,1,2). 000050 | 000051 | This matches the Birkhoff-Grothendieck object in P2.TYPING exactly. v491 proves 000052 | that three positive integral splitting exponents summing to four have the 000053 | unique multiset {1,1,2}; v499 types the P2 anchor as the Deligne extension 000054 | O(-2)+O(-1)^2 under its stated rank, cusp, irreducibility and unitarity 000055 | premises. The matching object is the bundle, not the parabolic weights. 000056 | 000057 | 2. Smallest decisive connection test 000058 | ------------------------------------- 000059 | 000060 | Let x=t^2 near one branch mark. The odd local function t spans the trace-free 000061 | part of p_*O_E and obeys 000062 | 000063 | dt = (1/2)(dx/x)t. 000064 | 000065 | Its canonical exponent is 1/2 and its local monodromy is -1. A flat 000066 | two-torsion line alpha is locally trivial near the ramification point because 000067 | the lifted small loop contracts on E. Its pushforward has an even and an odd 000068 | sheet combination. Their exponents are 0 and 1/2 and their monodromies are 000069 | +1 and -1. Hence the natural direct-sum pushforward has, at each of the four 000070 | marks, 000071 | 000072 | residues {0, 1/2, 1/2}, 000073 | monodromy spec {1,-1,-1}, 000074 | chi_ell(lam) = (lam-1)(lam+1)^2, 000075 | trace = -1, 000076 | order = 2. (2) 000077 | 000078 | The residue trace is one per mark. Deligne bookkeeping therefore gives 000079 | 000080 | deg V_ell = -4(0+1/2+1/2) = -4, 000081 | 000082 | which explains why the same underlying bundle type appears. 000083 | 000084 | The original P2 local class is different. v117 fixes an exact matrix M0 with 000085 | 000086 | chi_P2(lam)=lam^3-1, 000087 | spec M0={1,omega,omega^2}, 000088 | trace M0=0, 000089 | M0^3=I. 000090 | 000091 | Its canonical Deligne exponents are {0,1/3,2/3}, again with trace one per 000092 | mark and degree -4. The two local matrices cannot be conjugate because their 000093 | characteristic polynomials, traces and orders differ. The equality of degree 000094 | and splitting therefore cannot be promoted to equality of logarithmic 000095 | connections. 000096 | 000097 | There is a second independent obstruction. The connection naturally induced 000098 | on (1) preserves the explicit rank 1 summand p_*O_E/O, so it is reducible. 000099 | v117 proves that the intended representation generated by U and M0 is the 000100 | irreducible three-dimensional standard-times-sign representation of S4. P2 000101 | uses precisely this irreducibility to obtain stability. A connection 000102 | intertwiner cannot turn the reducible pushforward representation into that 000103 | irreducible representation. 000104 | 000105 | One can put another irreducible logarithmic connection with the P2 cusp class 000106 | on the same holomorphic bundle. That connection is additional data. It is not 000107 | the natural connection induced from O_E and alpha and therefore does not give 000108 | the requested source derivation. 000109 | 000110 | 3. A notation trap resolved 000111 | --------------------------- 000112 | 000113 | v117 calls (0,1/2,1/2) the harmonic diagonal and proves 000114 | 000115 | diag M0 = (0,-i/2,+i/2). 000116 | 000117 | Those are diagonal entries and the associated amplitude delta=1/2. They are 000118 | not the eigenvalues of the Deligne residue. The residue spectrum in v499 and 000119 | the active P2 setup is {0,1/3,2/3}. Thus the phrase harmonic diagonal does not 000120 | repair the mismatch in (2). 000121 | 000122 | 4. Was the boson-spin pushforward already the original construction? 000123 | -------------------------------------------------------------------- 000124 | 000125 | No such identification occurs in the bounded original set requested here or 000126 | in the targeted Atlas search for the combined pushforward. The original chain 000127 | does establish the same bundle type by another route: 000128 | 000129 | four marks + rank 3 + cusp residues + unitarity and irreducibility 000130 | -> degree -4 + h0=0 000131 | -> O(-2)+O(-1)^2. 000132 | 000133 | It does not form (p_*O_E/O)+p_*alpha. v499 explicitly separates the genus 1 000134 | Z2 pillowcase cover from the genus 2 Z3 Biswas weight cover. The denominator 000135 | of the flavor weights is 3, not the deck order 2. Its explicit decomposable 000136 | Z3 regular-representation model yields O+O(-2)^2 and is rejected as unstable. 000137 | The surviving {1,1,2} bundle is selected through stability of the original 000138 | cusp representation, not through the elliptic direct sum. 000139 | 000140 | There is a canonical mathematical reason to inspect (1): both terms already 000141 | exist on the same C2 cover, the first is the nontrivial scalar deck sector, 000142 | and the second is the unique Q-fixed nontrivial spin pushforward. Removing the 000143 | trivial scalar gives exactly one plus two dimensions with no fitted degree. 000144 | There is no original physical rule in the checked sources that declares this 000145 | boson-spin direct sum to be the generation carrier or couples its natural 000146 | connection to the flavor field. v499 records this precise module 000147 | identification as residual R1. Therefore selecting both terms as one physical 000148 | module would still be an added premise. 000149 | 000150 | 5. Executable finite certificate 000151 | -------------------------------- 000152 | 000153 | The comparison was replayed with exact SymPy arithmetic and no floats: 000154 | 000155 | M_ell = diag(-1,1,-1) 000156 | trace=-1, det=1, 000157 | charpoly=(lambda-1)(lambda+1)^2, 000158 | M_ell^2=I, M_ell^3!=I. 000159 | 000160 | M0 = the exact v117 matrix 000161 | trace=0, det=1, 000162 | charpoly=(lambda-1)(lambda^2+lambda+1), 000163 | M0^2!=I, M0^3=I. 000164 | 000165 | sum{0,1/2,1/2}=sum{0,1/3,2/3}=1, 000166 | -4 times either sum = -4. 000167 | 000168 | This is the smallest decisive test. It retains the positive bundle match and 000169 | excludes the stronger connection identification before any new model is built. 000170 | 000171 | 6. Original sources and status 000172 | ------------------------------ 000173 | 000174 | P2 exact cusp matrix and irreducible S4 representation: 000175 | 000176 | verification/v117_monodromy_weyl_a3.py:1-43,129-182 000177 | SHA256 55eca6a64144f6e49789f3f1bbe84e256764e7ce92b207044be693f8684d5df5. 000178 | 000179 | The matrix properties are exact [I]. Its identification with the analytic 000180 | A0* ODE monodromy remains numerical [N], as v117 itself states at lines 000181 | 184-200. The active P2 target independently prescribes the order 3 cusp 000182 | conjugacy class, so the local-class obstruction is exact for that target. 000183 | 000184 | P2 Deligne residues, degree and honest module residual: 000185 | 000186 | verification/v499_p2_weights_deligne_bg.py:1-89,184-229,408-484 000187 | SHA256 73098e62b3148d1cddb31e60c16a424159f985b874aa0dee1737260fb81017b9. 000188 | 000189 | documents/changelog/sections/007-source.tex:47-94 000190 | Atlas paper-source:d9dbc0d2c52e1775385a984c