TFPT scientific source excerpt Original repository path: experiments/theory-contracts/mixed-current-origin-20261014/P1-GAUSS-AXIOM-DECISION.txt Original file SHA-256: 92b2e8a3b8f08fe651ddb09c32fa4aeae410d0e2baf3ae7f894f134f953d3d75 Inclusive original lines: 107-229 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. 000107 | The MMST original explicitly says that even the four-level register was 000108 | added to the seam model: documents/articles--2026-08-30-- 000109 | mmst_charged_scaling_limit_en/sections/ 000110 | 005-subsection-the-neutral-net-and-its-index.tex:331-349. 000111 | Thus a claim about the primitive model cannot start by treating this 000112 | enlarged register algebra as an already proved P1 source. 000113 | 000114 | 4. Concrete physical difference of the two local nets 000115 | 000116 | Old net: Z is an even LOCAL register observable; F_[0,a] is a string. 000117 | New net: W=i^b product_x tau^z_x is a GLOBAL Wilson loop; tau^x_a is local. 000118 | The relation F Z=-Z F is preserved as tau^x W=-W tau^x. Products and 000119 | adjoints are not the issue; support is changed. 000120 | 000121 | More explicitly, assume the register cell is disjoint from a candidate 000122 | endpoint interval O. If a support-preserving faithful comparison retained 000123 | both Z and a nonzero local F in those disjoint regions, ordinary even 000124 | locality would require [Z,F]=0. The actual identity gives [Z,F]=2ZF, 000125 | which is nonzero for unitary F. Hence the existing E cannot be a local-net 000126 | equivalence with this old assignment fixed. This requires no E8 premise. 000127 | 000128 | The same change affects matter insertions. An old local even hopping 000129 | c_x* c_(x+1) changes sign under G_x and G_(x+1). In the existing gauge 000130 | description its local observable representative includes tau^z_x; 000131 | cross-region bilinears require the corresponding Wilson path. Calling 000132 | the undressed old hopping the same gauge-invariant local generator would 000133 | change the operator rule. This is a consequence of (3.1), not a new model. 000134 | 000135 | Conditional consequence: WITH THE OLD FULL LOCAL REGISTER NET FIXED, 000136 | identifying this E as a support-preserving re-expression is incompatible. 000137 | This does not establish incompatibility with EVERY primitive P1 source, 000138 | since that primitive-to-old-register local-net identification is missing. 000139 | 000140 | 5. Why two complete P1 alternatives have not been demonstrated 000141 | 000142 | The old and new models are globally isometric on the stated even carrier; 000143 | E transports H, state and all supplied operator words. That proves their 000144 | common algebraic/state/time identities under global transport. 000145 | It does not show that BOTH realize the same complete primitive P1 datum: 000146 | - the primitive local insertion map has not been identified with either; 000147 | - the primitive P_adm has not been identified with P_G; 000148 | - the primitive geometric reflection/positive-time source regions and 000149 | collar/reference data have not been intertwined with the NEW support; 000150 | - the primitive winding-one operator has not been mapped to this model. 000151 | These are specific missing comparisons, not fresh hypothetical objections. 000152 | They prevent promoting the global isometry to two full P1 countermodels. 000153 | 000154 | Accordingly the verdict MIT P1 VERTRAEGLICH, ABER NICHT AUSGEWAEHLT would 000155 | currently overclaim. A net supplied as part of a fully specified seed is 000156 | not free to be changed while calling that the same seed. Conversely no 000157 | primitive condition has been shown to forbid every gauge-net seed. 000158 | 000159 | 6. Compatibility of the two existing additional constructions 000160 | 000161 | (a) Neutral pair. For an interval I=[a,b], multiplying the ALREADY GIVEN 000162 | Gauss constraints cancels all internal electric links and gives 000163 | product_(x in I) G_x=(-1)^{N_I} tau^x_(a-1) tau^x_b. 000164 | On the Gauss space, therefore, 000165 | E (-1)^{N_I} E*=tau^x_(a-1) tau^x_b |Ran(P_G). (6.1) 000166 | The same identity holds componentwise for a finite disjoint interval union, 000167 | including the eight-preimage set of the existing neutral pair. 000168 | E is onto the specified physical space, so products, adjoints and state 000169 | words are preserved. Both representatives have support in the interval 000170 | closure with its boundary links in the specified lattice convention. 000171 | This restricted pair compatibility does not imply a support-preserving 000172 | equivalence of the WHOLE nets, as the Z/F example already disproves. 000173 | Nor does it prove that these intervals are the original P1 regions. 000174 | 000175 | (b) Global spinor Phi. Its global H/Q/vacuum identities remain valid. 000176 | Its pullback of a target local even Weyl operator is already known to be 000177 | Pi_even W+Pi_odd W*, which fails the original local CAR support test. 000178 | Therefore it does not meet the requested unchanged full-local-assignment 000179 | condition. E preserves products/adjoints globally but does not supply a 000180 | new local Phi dictionary. In particular the existing E is defined on a 000181 | total-even register/matter space, whereas Phi is specified on the full 000182 | single-chiral source, including odd intermediates; composing their names 000183 | without a domain and locality map is not a construction. 000184 | Restricting the observable menu to remove an old locality witness changes 000185 | the comparison. It must not be reported as compatibility with the old 000186 | full local assignment. No new post-Gauss spinor map is attempted here. 000187 | 000188 | 7. The exact missing definition, and the four-way decision 000189 | 000190 | The missing object is an ORIGINAL SOURCE/LOCALIZATION IDENTIFICATION: 000191 | which primitive supported quantum field insertions represent this 000192 | register/matter model and, if the gauge net is intended, which represented 000193 | local redundancy makes exactly G_x=1 the physical rule? 000194 | 000195 | A concrete completion would need a primitive-to-microscopic map iota that 000196 | fixes region support and adjoints and establishes the relevant identities, 000197 | not merely matching dimensions. For the proposed Gauss identification the 000198 | unproved statements include 000199 | iota H_adm = Ran(P_G), 000200 | iota A_adm(O) iota* = A_G(O) for the specified regions, (7.1) 000201 | together with the original state/time/reflection/winding comparison. 000202 | This states what is missing; it is NOT introduced as a new axiom here. 000203 | 000204 | On the operational-seed formulation, A_loc must already be specified. 000205 | On the boundary formulation, A_+ is part of the input and the microscopic 000206 | field/region realization of (2.2) is not supplied for this comparison. 000207 | On scalar P1 alone, the relevant gauge generators are not defined at all. 000208 | Thus the undecided step is a precise identification/definition gap, not 000209 | an unexplained request for more computation or another extension. 000210 | 000211 | Decision table: 000212 | AUS P1 ERZWUNGEN -- not established; no derivation of (3.1)/(7.1). 000213 | MIT P1 VERTRAEGLICH, ABER NICHT AUSGEWAEHLT -- not established as a 000214 | two-complete-P1-model statement; global word equivalence is insufficient. 000215 | MIT P1 UNVEREINBAR -- proved ONLY if the unchanged old full local 000216 | register assignment is included in the comparison, not for bare P1. 000217 | AUFGRUND KONKRET BENANNTER DEFINITIONSLUECKE UNENTSCHEIDBAR -- the 000218 | overall primitive-P1 classification supported by the original sources. 000219 | 000220 | Relative to the checked explicit inputs, the local Gauss rule is an 000221 | ADDED physical locality premise. This does not mean its independence 000222 | from every possible full P1 formulation has been proved. Zero formerly 000223 | additional physical premises have been derived by this decision test. 000224 | The requested positive completion criterion is not met. 000225 | 000226 | Stop point: no new sector, dressing, gauging, target E8 argument or 000227 | downstream TFPT motivation is offered to fill (7.1). The primitive P1 000228 | description is not yet a complete specified physical source FOR THIS 000229 | LOCAL-NET QUESTION. This does not refute the conditional TFPT mathematics.