TFPT scientific source excerpt Original repository path: experiments/theory-contracts/elliptic-moebius-source-bridge-20261026/source_recursion_20261028/EPSILON-WARD-SOURCE-AUDIT.txt Original file SHA-256: a8077ae13d433fc62aa57c5d649806d0d0dd0b02f92cdb6cd5b3cd1dd3bf9147 Inclusive original lines: 1-200 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 | SOURCE LEVEL EPSILON WARD AUDIT 000002 | =============================== 000003 | 000004 | 28 October 2026 000005 | 000006 | Question 000007 | -------- 000008 | 000009 | Does the original compiler already contain a multiplication law that forces 000010 | four carrier indices into Lambda^4(C^5), or was the epsilon tensor in the new 000011 | A4 x A4 current channel introduced only by a later covariance choice? 000012 | 000013 | Verdict 000014 | ------- 000015 | 000016 | There is a forgotten positive source relation. The old finite compiler 000017 | already contains the exact multiplication table 000018 | 000019 | Lambda^2 E x Lambda^2 E -> Lambda^4 E, 000020 | b_ij wedge b_kl = epsilon(i,j,k,l) f_m, 000021 | 000022 | where {m,i,j,k,l} is the five-slot carrier. It is nonzero exactly on the 000023 | 15 doily lines. The later Pfaffian audit fixes the only apparent sign 000024 | difference: the global character chi(m)=(-1)^m is precisely the conversion 000025 | from the four-index wedge orientation to the five-index volume orientation. 000026 | Consequently 000027 | 000028 | epsilon(m,i,j,k,l) = chi(m) epsilon(i,j,k,l). (1) 000029 | 000030 | After identifying Lambda^4 E with the dual fundamental by the already fixed 000031 | volume form, fixing m in (1) gives exactly the five carrier matrices C_m of 000032 | the verified E8 A4 x A4 bracket. Thus the carrier epsilon tensor is not a 000033 | new covariance selector. It is the old P2/compiler Hodge multiplication, 000034 | now realized as an actual first OPE coefficient in the stipulated E8_1 000035 | current algebra. 000036 | 000037 | This closes an algebraic provenance arrow: 000038 | 000039 | finite five-slot compiler multiplication 000040 | -> canonical volume-form signs 000041 | -> C_m on Lambda^2(C^5) 000042 | -> actual E8 current bracket coefficient. 000043 | 000044 | It does not by itself prove source saturation. The old source explicitly 000045 | marks as open the physical premise that the Calderon boundary grammar uses 000046 | the quadratic one-step exhaustion. The primitive P1 local-net definition 000047 | does not list the E8 current insertions, and neither seam reflection nor the 000048 | regular Z4 Q-system chooses the full 50-current inclusive contraction. The 000049 | remaining physical arrow is therefore smaller than before but still real: 000050 | the raw seam must realize the charged E8 extension fields and use the closed 000051 | current line, or an equivalent source operation, that contracts the whole 000052 | (5,10) insertion module with its compact adjoint. 000053 | 000054 | 1. Exact comparison with the A4 response 000055 | ---------------------------------------- 000056 | 000057 | Let e_0,...,e_4 be the ordered carrier basis and let 000058 | 000059 | |ij> = e_i wedge e_j, i = epsilon(m,i,j,k,l) |kl>, (2) 000064 | 000065 | when {k,l} is the complement of {m,i,j}; set the result to zero if m is in 000066 | {i,j}. The old doily table instead records the unordered factorization 000067 | 000068 | |ij> wedge |kl> = epsilon(i,j,k,l) f_m. (3) 000069 | 000070 | The canonical Pfaffian/volume character recorded in the old paper is 000071 | 000072 | chi(m) = sign(m,0,...,m-1,m+1,...,4)=(-1)^m. 000073 | 000074 | Equations (2) and (3) are therefore the same signed table by (1). Each m 000075 | has three unordered doily factorizations, or six directed matrix entries. 000076 | Over all m this is 15 unordered lines and 30 directed entries. Each pair 000077 | ij excludes exactly three possible m, so the same multiplication gives 000078 | 000079 | sum_m C_m^* C_m = 3 I_10. (4) 000080 | 000081 | This is the carrier part of the previously verified gain 12=3 times4. 000082 | The accompanying checker compares every directed sign and proves (4) over 000083 | the integers. It reuses the two published conventions and introduces no 000084 | new representation. 000085 | 000086 | The decisive implication is stronger than a dimension match. The old 000087 | compiler table and the new E8 bracket have identical support, orientation, 000088 | and normalization on the carrier. The common tensor is unique only after 000089 | these marked basis and volume data are retained; the exact sign comparison 000090 | shows that they are retained here. 000091 | 000092 | 2. Why two tempting older products are not this Ward identity 000093 | ------------------------------------------------------------- 000094 | 000095 | The September current-product theorem contains a different genuine 000096 | antisymmetric bracket, 000097 | 000098 | [C_a,X_ib] = Y_(i,a wedge b). (5) 000099 | 000100 | Its types are 000101 | 000102 | (1,4) x (bar5,4) -> (bar5,Lambda^2 4). 000103 | 000104 | The carrier bar5 index i is a spectator; antisymmetry acts on the four A3 000105 | family indices. Equation (5) therefore cannot be substituted for (2). 000106 | Its associated grade-two product is bar5 tensor Sym^2(4), again with the 000107 | four-index symmetry on the family factor. 000108 | 000109 | The local product of the five native quartics is different again. Those 000110 | fields commute in the same profile, so their ordered multiplication factors 000111 | through Sym^2 L. Its kernel is Lambda^2 L of dimension10. That ten is a 000112 | space of relations, not a physical Lambda^2(C^5) output. Generic fermionic 000113 | Pfaffian antisymmetry on the eight original CAR arcs does not change this 000114 | type. A map from arc indices to the five carrier slots would still be 000115 | needed. This is the smallest exact reason that the old native Bell10 or 000116 | Sym^2(C^4) product cannot be used as the carrier Hodge Ward merely because 000117 | the number ten occurs in both places. 000118 | 000119 | The older Gr(2,5) Pfaffian result is relevant only because it explicitly 000120 | identifies its 15 monomials and the global sign character with the same 000121 | five-slot Hodge table. The unrelated flavor Plucker readouts 55/117 and 000122 | the family Lambda^2(F) live on a different selector datum and do not provide 000123 | the local current multiplication used here. 000124 | 000125 | 3. What the E8 OPE now forces 000126 | ----------------------------- 000127 | 000128 | Inside the stipulated compact E8_1 current source, the three actual modules 000129 | are 000130 | 000131 | input (10,bar5), 000132 | insertion (5,10), 000133 | output (bar10,5). 000134 | 000135 | The first OPE pole, equivalently the zero-mode Lie bracket, is the unique 000136 | marked tensor product of the two exterior maps 000137 | 000138 | (5 x Lambda^2 5 -> Lambda^3 5) 000139 | tensor 000140 | (Lambda^2 5 x Lambda^4 5 -> 5). 000141 | 000142 | After the fixed dual-coordinate conversion its coefficient is literally 000143 | 000144 | epsilon(m,i,j,k,l) A_ab. (6) 000145 | 000146 | All 600 nonzero coefficients and both neutral A4 actions were already 000147 | checked in one Chevalley cocycle chart. Equation (1) now shows that the 000148 | carrier factor in (6) is the prior compiler multiplication, rather than a 000149 | tensor selected after observing the Petersen channel. 000150 | 000151 | Compact adjunction supplies the positive current two-point metric. Closing 000152 | an insertion line in the E8 vacuum therefore uses the canonical 000153 | coevaluation sum over an orthonormal (5,10) basis. Algebraically this gives 000154 | 000155 | sum_(m,ab) B_(m,ab)^* B_(m,ab) = 12 I_50. (7) 000156 | 000157 | Hence, if the source operation is this closed inclusive current line, its 000158 | normalization is forced to 1/12 and the reduced carrier map is the Petersen 000159 | channel. No independent covariance selector or fitted weight remains. 000160 | 000161 | The conditional phrase in the preceding paragraph is load bearing. A 000162 | local OPE tells us the amplitude once fields are inserted. It does not 000163 | declare that a physical process inserts every member of one irreducible 000164 | module, closes the line with the vacuum metric, and forgets the outgoing 000165 | family label. That is the remaining saturation statement. 000166 | 000167 | 4. Sewing, reflection, Q-system, and the first missing source arrow 000168 | ------------------------------------------------------------------ 000169 | 000170 | The primitive P1 definition gives a relative spectral action from supplied 000171 | D_rel, D_ref, profile and eta phase. Its local Euclidean source algebra is 000172 | declared later as the algebra generated by supported field insertions. No 000173 | specific E8 current field, no map from a variation of D_rel to a root 000174 | current, and no inclusive current trace is defined there. 000175 | 000176 | Seam reflection does fix the compact adjoint pairing once a current field is 000177 | present. It exchanges a root with its opposite and cancels the Chevalley 000178 | cocycle sign in the two-point norm. It does not generate the other 49 000179 | insertion directions from one root and does not select the (5,10) charge 000180 | sector. Reflection therefore fixes the metric used in (7), but not the 000181 | physical occurrence of (7). 000182 | 000183 | The regular Z4 Q-system is also insufficient at this exact point. Its 000184 | multiplication x is associative, Frobenius, and reconstructs an abstract 000185 | index-four extension. Its four simple-current summands are one-dimensional 000186 | charge sectors. This theorem does not supply the five-slot Hodge tensor or 000187 | the 50 root-current insertion map. The same paper explicitly leaves open 000188 | the concrete QWZ-collar identification with the E8_1 net and the construction 000189 | of the non-bilinear spin/twist fields. Thus no Q-system uniqueness theorem 000190 | may be used to claim source saturation before the concrete extension fields 000191 | exist. 000192 | 000193 | The first missing identification can now be stated without a broad field 000194 | dictionary slogan: 000195 | 000196 | construct one raw-seam localized charged field in the selected Ramond 000197 | sheet whose OPE with the sourced D8 currents generates the marked 000198 | (5,10) E8 insertion module and whose reflected closed two-point sewing 000199 | is the standard coevaluation of that module. 000200 |