TFPT scientific source excerpt Original repository path: experiments/theory-contracts/elliptic-moebius-source-bridge-20261026/charged_source_alternatives_20261027/SECTOR-TRANSPORT-CORRECTION.txt Original file SHA-256: 65110e2226bb31beee107314e0244131d084196b74d5d07be4438b0c36bf9268 Inclusive original lines: 1-160 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 | [OMITTED: non-scientific conversation or execution metadata] 000002 | [OMITTED: non-scientific conversation or execution metadata] 000003 | 000004 | VERDICT 000005 | 000006 | The original winding does not create a Ramond spinor from an NS state. 000007 | This follows from its exact original action, not from a numerical analogy. 000008 | The proposed square-root shortcut and membership in the unchanged NS CAR 000009 | algebra fail. Local spinor intertwiners in an appropriate extension remain 000010 | a viable conditional route. No conclusion against all Ramond constructions 000011 | or against categorical reconstruction is drawn. 000012 | 000013 | 1. THE ACTUAL FOUR SECTORS 000014 | 000015 | Use the retained D8 modules in the order (0,v,s,c): 000016 | 000017 | H_NS=H_0+H_v, H_R=H_s+H_c, 000018 | H_E8=H_0+H_s, H_E8,twisted=H_v+H_c. 000019 | 000020 | These are two DIFFERENT decompositions of the same supplied parent space. 000021 | The exact original operator W_ext=E_e0 Rhat has sector action 000022 | 000023 | W_ext:(0 v)(s c). 000024 | 000025 | It maps the E8 module to its g-twisted module while preserving NS and R 000026 | separately. If P_R projects onto H_s+H_c, then [P_R,W_ext]=0. Since W_ext 000027 | is unitary and P_R reduces it, spectral calculus gives 000028 | 000029 | [P_R,f(W_ext)]=0 000030 | 000031 | for every bounded Borel function f, including every chosen spectral branch 000032 | of a square root. Thus P_R f(W_ext) Omega_NS=0. A non-functional square root 000033 | would be an additional operator whose source origin and locality are not 000034 | provided by W_ext. 000035 | 000036 | The global charge shift X=E_s has a different sector action (0 s)(v c). 000037 | It is already available on the four-sector parent; its global existence 000038 | is not the missing local source theorem. The full affine W_ext includes 000039 | charge translations and cocycles, so its four-sector permutation must not 000040 | be substituted for its entire infinite-dimensional operator. 000041 | 000042 | 2. CORRECTION OF OUR OWN WORDING 000043 | 000044 | The previous WEAKEST-DIRECT-E8-SOURCE-ROUTE.txt contained the phrase 000045 | 'its half integral powers' after discussing W_ext. That phrasing invited an 000046 | incorrect interpretation. The proved identity is 000047 | 000048 | W_ext Y(e^beta,z) W_ext^-1 000049 | = lambda(beta) C(e0,R beta) z^(-(R beta).e0) Y(e^(R beta),z). 000050 | 000051 | It is the exponent of the FIELD COORDINATE z that is half integral for 000052 | spinor roots. W_ext itself sends 0+s to v+c. There is no statement here 000053 | about W_ext^(1/2) generating s from 0. The original SAME-SOURCE proof 000054 | states both facts correctly; the later summary is repaired in place and 000055 | its old version is retained with this audit. 000056 | 000057 | 3. THE CORRECT LOCAL OBJECT 000058 | 000059 | Take v=e1 and s=(1/2,...,1/2). Then exp(2 pi i v.s)=-1, whereas s has 000060 | integral pairing with all D8 roots. Thus the s-charge field is local 000061 | relative to the D8 current theory and gives the bosonic E8 extension 000062 | V_D8+H_s. It is not mutually local with the vector fermions of the NS CAR 000063 | theory. Combining v and s in one ordinary local or graded-local algebra 000064 | is not justified by this construction. 000065 | 000066 | In the unchanged NS representation the Spin(10) kernel fixes the vector 000067 | fermions and vacuum, hence its whole generated algebra. It cannot produce 000068 | an operator transforming in the spinor M_h on that same representation. 000069 | Bounded affiliation with the same local von Neumann algebra is membership 000070 | and does not evade this obstruction. Unbounded affiliated operators also 000071 | cannot manufacture a disjoint target sector from that unchanged algebra. 000072 | 000073 | A well-typed target is a localized charged intertwining field relative to 000074 | the even D8 current net on an appropriate sector Hilbert space, or a field 000075 | of the local E8 extension. Smeared vertex fields generally are closable and 000076 | energy bounded rather than bounded. Energy bounds, domains, strong locality 000077 | and the original source identification must be stated separately. A bounded 000078 | global defect implementer is a different object from a smeared primary field. 000079 | 000080 | Once a full marked s-sector is realized, its ground space already contains 000081 | (16,4)+(bar16,bar4); no extra arbitrary choice of a single vector is needed 000082 | for the extension theorem. Likewise the existing common Ramond lift sign is 000083 | intertwiner-basis gauge, not a reopened physical parameter. Extra original 000084 | markings matter only when actually fixed by the source contract. 000085 | 000086 | 4. FURTHER CORRECTIONS TO THE FIRST EXTERNAL REPORT 000087 | 000088 | For {psi0^a,psi0^b}=delta_ab the chirality involution uses gamma_a=sqrt(2)psi0^a: 000089 | 000090 | Gamma=i^8 gamma_1...gamma_16=2^8 psi0^1...psi0^16. 000091 | 000092 | Its square is one; the unscaled product has square 2^-16. The full complex 000093 | zero-mode module has dimension256, each half-spin module128, and each 000094 | marked ground component64. A fusion-intertwiner space of dimension1 is a 000095 | space of operator families, not any of those ground spaces. 000096 | 000097 | A single chiral theory of16 real fermions still has all120 D8 currents. 000098 | Spacetime chirality and the internal D8 Lie algebra are different labels. 000099 | Complete Dirac vector Ward charges must be compared by an explicit map, 000100 | not equated with the internal chiral currents by name. 000101 | 000102 | Unit spectral flow phi:0->2pi returns to the same spin boundary condition 000103 | and changes occupation. Half flow changes the NS/R boundary condition. 000104 | The free orientable-double deck of a Moebius strip has no fixed point; 000105 | a Ramond puncture is not automatically an existing deck fixed point. 000106 | 000107 | [OMITTED: non-scientific conversation or execution metadata] 000108 | [OMITTED: non-scientific conversation or execution metadata] 000109 | [OMITTED: non-scientific conversation or execution metadata] 000110 | [OMITTED: non-scientific conversation or execution metadata] 000111 | [OMITTED: non-scientific conversation or execution metadata] 000112 | 000113 | 5. THE FIRST TEST MUST TEST THE RIGHT TRANSITION 000114 | 000115 | For a claimed spinor field X_s the minimal sector test is 000116 | 000117 | P_s X_s P_0 != 0, 000118 | 000119 | with its D8 charge determined on the retained current algebra. In contrast, 000120 | 000121 | P_s W_ext P_0 = 0. 000122 | 000123 | Equation32 of NEUTRAL-NET-AND-DEFECT localizes the affine winding action; 000124 | it is not, by itself, a test that a proposed operator creates the s-sector. 000125 | The two localization questions are distinct. The existing global E_s passes 000126 | the sector test but still needs the genuine local field/source construction. 000127 | This prevents spending a new localization proof on the wrong charged object. 000128 | 000129 | [OMITTED: non-scientific conversation or execution metadata] 000130 | [OMITTED: non-scientific conversation or execution metadata] 000131 | [OMITTED: non-scientific conversation or execution metadata] 000132 | [OMITTED: non-scientific conversation or execution metadata] 000133 | [OMITTED: non-scientific conversation or execution metadata] 000134 | [OMITTED: non-scientific conversation or execution metadata] 000135 | [OMITTED: non-scientific conversation or execution metadata] 000136 | 000137 | 6. WHAT IS REUSED AND WHAT THE SOURCE STILL MUST SUPPLY 000138 | 000139 | The same-conformal extension theorem, four-sector AIA, complete affine 000140 | winding, lift-sign gauge and theta-to-lattice-to-VOA pair are already 000141 | available with their stated assumptions. None is reclassified as missing. 000142 | The original physical P1 source must supply either a charged local sector 000143 | map in that class, or an equivalent independently justified global 000144 | extension-selection property. Selection is not proved merely by giving a 000145 | standard extension model or by localizing an operator in the wrong algebra. 000146 | The alternative chiral-sewing test is developed in the adjacent report. 000147 | 000148 | SOURCES 000149 | 000150 | [OMITTED: non-scientific conversation or execution metadata] 000151 | [OMITTED: non-scientific conversation or execution metadata] 000152 | [OMITTED: non-scientific conversation or execution metadata] 000153 | [OMITTED: non-scientific conversation or execution metadata] 000154 | [OMITTED: non-scientific conversation or execution metadata] 000155 | [OMITTED: non-scientific conversation or execution metadata] 000156 | [OMITTED: non-scientific conversation or execution metadata] 000157 | [OMITTED: non-scientific conversation or execution metadata] 000158 | [OMITTED: non-scientific conversation or execution metadata] 000159 | [OMITTED: non-scientific conversation or execution metadata] 000160 | [OMITTED: non-scientific conversation or execution metadata]