TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/mixed-current-origin-20261014/REGISTER-GRADE-LIMITS-GAUSS-AND-DIRAC.txt
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000001 | UR.SEAM.MIXED_CURRENT_ORIGIN.01 -- continuation 22 October 2026
000002 | All-word register obstruction; exact conditional Gauss completion;
000003 | compatibility with the independently checked full Dirac cancellation
000004 | Verdict: PARTIAL. No microscopic P1/E8 origin or chiral measure derived.
000005 | 
000006 | 1. Decisive question, success and stopping criteria
000007 | 
000008 | The preceding report excluded the unchanged sharp half flux. The remaining
000009 | question is whether smoothing/renormalizing EXISTING half-flux operations
000010 | can evade this exclusion while keeping the original local register, both
000011 | edges, full positive state words and physical time. It cannot in the
000012 | register-charged class specified below. This stronger result does not assume
000013 | a bounded or unitary limit and is independent of the sharp HS divergence.
000014 | 
000015 | The next comparison asks exactly what a local Gauss rule would change. Its
000016 | success criterion is an explicit all-size isometry preserving the entire
000017 | EVEN physical Hilbert space, original Hamiltonian and all state words,
000018 | while making the given half flux a finite local generator. It is NOT an
000019 | E8 construction. The first new premise is the local Gauss/observable rule;
000020 | because it is not supplied by the checked original P1 definitions, no
000021 | profile optimization or continuum spinor claim is made downstream of it.
000022 | 
000023 | Original sources: v1033 symbolic_construction and register_hamiltonian;
000024 | MMST sections/005-subsection-the-neutral-net-and-its-index.tex:332-356;
000025 | the primitive physical-net definition in archive--tfpt-45 source01:302-343.
000026 | The fixed original source, fold, Hamming/E8/J, native4/5/20/carry, time and
000027 | unit-winding results remain intact. No physical gate is closed.
000028 | 
000029 | 2. Exact register grading of the FULL operation menu
000030 | 
000031 | In the actual enlarged v1033 model let Z=diag(1,i,-1,-i), ZX=iXZ, and
000032 |   D_A=X tensor exp(i pi N_A/2), F_A=D_A^2.
000033 | Matter operators commute with Z. The ORIGINAL register Hamiltonian is
000034 |   H=sum_r |r><r| tensor dGamma(h_r),
000035 | so [H,Z]=0. Normal-ordering constants diagonal in r do not change this.
000036 | The original real/imaginary time operations therefore have register grade0.
000037 | Functions of Z, all matter CAR operations, smooth matter current dressings
000038 | and their adjoints also have grade0. D_A has grade1, F_A grade2.
000039 | 
000040 | Define B_q={T: Z T Z*=i^q T}. Then products add grades modulo4 and
000041 | adjoints reverse them. A general grade2 operator has the exact form
000042 |   T=sum_(r=0..3) |r+2><r| tensor T_r
000043 | with arbitrary matter matrices T_r. They need not be Gaussian or unitary.
000044 | Any arbitrarily long, cutoff-dependent, coherently summed original history
000045 | of total grade2 satisfies Z T Z*=-T. Endpoint smearing and scalar field
000046 | renormalization preserve it. This is not just the one bare string.
000047 | 
000048 | For bounded T the quantitative obstruction is
000049 |   ||[Z,T]||=2||T||.
000050 | More generally if ||ZTZ*+T||<=eta, then
000051 |   ||[Z,T]||>=2||T||-eta.
000052 | Thus normalized approximate local operators cannot approach this charged
000053 | class in norm while commuting with a remote surviving Z.
000054 | 
000055 | 3. A FOUR-WORD obstruction including unbounded field limits
000056 | 
000057 | Let T_N(f) be ANY grade2 candidate, including smearing and N-dependent
000058 | renormalization with divergent operator norms. Suppose the requested FULL
000059 | word limit retains a bounded local unitary Z in a region disjoint from f,
000060 | and a positive nonzero field norm
000061 |   n=lim omega_N(T_N(f)* T_N(f))>0.
000062 | The original exact register relation gives
000063 |   omega_N(T_N* Z_N* T_N Z_N)=-omega_N(T_N* T_N).       (1)
000064 | Ordinary locality of a bosonic spinor CURRENT with the remote EVEN Z would
000065 | give +n for the limiting left side, whereas (1) gives -n. Contradiction.
000066 | Graded locality gives the same condition because Z is even.
000067 | 
000068 | No strong-operator convergence, bounded limit, sharp-profile assumption,
000069 | quasifree state or common Fock representation is required for (1). Only
000070 | the joint words, their adjunction/positivity, retained local Z and nonzero
000071 | field norm are used. For a field annihilating the vacuum, apply the same
000072 | test on a cyclic-domain vector A Omega, whose full words are part of the
000073 | contract. A nonzero generator cannot evade all such positive norms.
000074 | 
000075 | Equivalently exact polynomial relations descend to the GNS domain of a
000076 | positive all-word limit. The identities Z*Z=ZZ*=I and ZTZ*=-T survive.
000077 | A local charged limit disjoint from Z can therefore only be zero.
000078 | 
000079 | The original A/A*-admissible restriction does not repair this if both Z
000080 | and T preserve the physical subspace: restricting an operator identity
000081 | retains it. Projecting to one Z eigenspace kills a grade2 field rather
000082 | than generating it. Removing Z from that local observable algebra or
000083 | reassigning it a global support changes the physical local-net contract.
000084 | 
000085 | This excludes ALL regularized grade2 endpoint fields preserving this
000086 | original register locality/full-word contract. It does NOT identify every
000087 | possible P1 spinor with register grade2 or exclude a different derived net.
000088 | 
000089 | 4. The shared strip flux is another all-word constraint
000090 | 
000091 | In the SAME x orientation, both edge background Wilson labels in the
000092 | register model are Z_L=Z_R=Z. The actual r is common to the whole cylinder.
000093 | For opposite boundary orientation replace one by its adjoint consistently.
000094 | In particular (Z_L-Z_R)* (Z_L-Z_R)=0 is an exact operator identity.
000095 | For ANY candidate T_N, not just a finite string,
000096 |   omega_N(T_N* (Z_L-Z_R)* (Z_L-Z_R) T_N)=0.           (2)
000097 | 
000098 | On an initial definite common-flux vector, a proposed field flipping
000099 | the left holonomy and retaining the right gives instead 4||T Omega||^2.
000100 | It cannot be a faithful all-word limit retaining these label operators.
000101 | An additional right absorber, an altered physical label map or a changed
000102 | background family is outside (2). This is the exact shared-flux class,
000103 | not an exclusion of all chiral boundary theories or the known beta route.
000104 | 
000105 | 5. A concrete conditional Gauss completion -- no new physical energy
000106 | 
000107 | This shows constructively how a changed locality rule can evade (1).
000108 | It is NOT a derivation of that rule from P1 and not a same-local-net
000109 | equivalence. Use the existing circle of L columns and all its actual QWZ
000110 | matter modes; n_x denotes column particle number. Work in total EVEN
000111 | matter parity, which contains the mass1,width8 half-filled source states.