TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/mixed-current-origin-20261014/REGISTER-GRADE-LIMITS-GAUSS-AND-DIRAC.txt
Original file SHA-256: 94dc3fe2f4b612b2a6f081db8339165f6b7beadfb764f4e3ae143b922d9aa8dd
Inclusive original lines: 112-235
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.

000112 | 
000113 | Write the old register label r=b+2w, b,w in {0,1}. Introduce redundant
000114 | Z2 link qubits with Pauli tau_x^z,tau_x^x and retain b. Define
000115 |   G_x=(-1)^n_x tau_(x-1)^x tau_x^x.
000116 | They commute, square to I, and product_x G_x=(-1)^Ntotal.
000117 | The physical rule G_x=1 therefore includes the even constraint. It is
000118 | the NEW LOCAL GAUSS PREMISE, not the old global GSO Gram projection.
000119 | 
000120 | The Hamiltonian uses the SAME original hopping coefficients. Replace the
000121 | forward horizontal link coefficient by tau_x^z, with an additional i^b
000122 | on the old seam link L-1; retain every vertical/onsite term. Include no
000123 | electric energy term or new coupling. Gauge invariance follows because
000124 | G_x reverses the two adjacent link signs and the column matter sign.
000125 | 
000126 | 6. Explicit all-size isometry and its full-state/time proof
000127 | 
000128 | Let |r,n> be an old register/occupation basis with even n. Start with link
000129 | bits z_x=w delta_(x,L-1). Set t_0=0, and for all t_1,...,t_(L-1) in Z2,
000130 |   z_x(t)=w delta_(x,L-1)+t_x+t_(x+1) mod2, t_L=t_0,
000131 |   E|r,n>=2^(-(L-1)/2) sum_t
000132 |          (-1)^(sum_(x=1..L-1) t_x n_x) |b,z(t),n>.    (3)
000133 | Every link configuration of Wilson parity w occurs exactly once. The
000134 | gauge orbits are disjoint, so E*E=I. Summing the orbit gives G_x E=E for
000135 | x!=0; its product and even parity imply G_0 E=E too. The orbit count shows
000136 | E is onto the full Gauss-invariant space, not just an embedding of a few
000137 | low-energy states. Dimensions match exactly: 4 dim(F_even).
000138 | 
000139 | The new H commutes with all G_x. On the gauge-fixed seed configuration it
000140 | is exactly dGamma(h_(b+2w)). Hence
000141 |   H_new E=E H_old.                                    (4)
000142 | This proves exact equality of real/imaginary time and every transported
000143 | ordered state word at every energy. For any original density rho use
000144 | rho_new=E rho E*. Positivity, adjunction, pure states and the actual mixed
000145 | fixed-filling Ramond state are preserved. It does NOT make the original
000146 | pure NS state equal the prescribed mixed R state.
000147 | 
000148 | Here H is the ACTUAL microscopic register Hamiltonian, with one common
000149 | energy reference if desired. Independent sector-by-sector vacuum
000150 | subtractions f(Z) must also be transported to f(W); they can become
000151 | nonlocal Wilson-loop terms. No unproved local-time statement for such a
000152 | different reference is made. Nor is any particular quarter vacuum thereby
000153 | proved to be the full register Hamiltonian's selected physical vacuum.
000154 | 
000155 | Define the GLOBAL Wilson operator
000156 |   W=i^b product_x tau_x^z.
000157 | Then W E=E Z. The old register observable survives; it is no longer assigned
000158 | to one remote local cell. No local gauge identifies different holonomies.
000159 | 
000160 | For A={0,...,a}, multiplying its Gauss laws yields on physical states
000161 |   tau_a^x E=E [X_register^2 (-1)^N_A]=E F_A.            (5)
000162 | Thus the old half string is a LOCAL electric LINK operator in this new
000163 | observable net. Equations(1) and(5) are compatible: W is a global loop,
000164 | not a remote local observable. Merely deleting Z in the old net would
000165 | not have proved (3)-(5).
000166 | 
000167 | Only the half operation is localized by this Z2 completion. No ordinary
000168 | local quarter-twist field, h1 spinor, or E8 algebra is inferred. Bare
000169 | cross-window CAR bilinears require gauge Wilson paths in the new net;
000170 | their old v746 support cannot be retained unchanged. Exact finite local
000171 | generators still require controlled renormalized continuum field limits.
000172 | 
000173 | 7. What this has and has not derived from original P1
000174 | 
000175 | The complete algebraic construction (3)-(5) uses no new physical Hilbert
000176 | dimension relative to the specified EVEN register source, and no altered
000177 | physical Hamiltonian/state. The new premise is the GAUSS RULE AND LOCAL
000178 | OBSERVABLE ASSIGNMENT. It changes the physical net and so is not just a
000179 | passive coordinate change of the original register-local model.
000180 | 
000181 | The checked original primitive source01:319-343 defines the physical net
000182 | through already specified local insertions and A/A*-invariant restriction.
000183 | It supplies neither these G_x nor an identification of its Padm with this
000184 | Gauss sector. Its later carrier/determinant selectors cannot be substituted
000185 | by name. MMST:332-356 explicitly retains the added-register origin gap.
000186 | Therefore the construction remains conditional; P1 origin is not claimed.
000187 | It nevertheless removes an unnecessary stronger assumption: correcting
000188 | the register locality need not require an extra PHYSICAL Hilbert sector
000189 | or a new time law. The required local constraint can now be stated exactly.
000190 | 
000191 | For literature comparison, not a new TFPT premise: Radicevic,
000192 | Spin Structures and Exact Dualities in Low Dimensions, arXiv:1809.07757v3,
000193 | section2, treats parity Gauss constraints, global sectors and exact lattice
000194 | dualities. The local/global distinction is essential. The present equations
000195 | are derived directly for the existing register and actual hopping data.
000196 | 
000197 | [OMITTED: non-scientific conversation or execution metadata]
000198 | 
000199 | The original check_dirac_lift.py was replayed with ALL ORIGINAL ASSERTIONS
000200 | ACTIVE; only its result-file write was suppressed. Both quarters pass.
000201 | Its analytic normal-complement lemma also checks directly:
000202 |   N C N*=1-C, [N,X]=[N,Y]=0,
000203 |   c_(1-C)(X,Y)=-c_C(X,Y) by trace-class cyclicity,
000204 |   c_(1-C)(X,Y)= c_C(X,Y) by unitary covariance.
000205 | Thus the complete vector cocycle is zero, even on one boundary face.
000206 | The full Dirac graded components are NOT the two physical strip edges and
000207 | NOT the two collar faces; all three distinctions must remain explicit.
000208 | 
000209 | Any isometric gauge completion transporting the COMPLETE source state and
000210 | the SAME vector current operations preserves every ordered amplitude,
000211 | including a trivial implementer-commutator phase. A changed local support
000212 | assignment by itself cannot choose a different chiral determinant measure.
000213 | This is a general word-preservation statement, NOT an unproved identification
000214 | of the QWZ register source with the full Dirac Cauchy quantization.
000215 | 
000216 | [OMITTED: non-scientific conversation or execution metadata]
000217 | [OMITTED: non-scientific conversation or execution metadata]
000218 | [OMITTED: non-scientific conversation or execution metadata]
000219 | [OMITTED: non-scientific conversation or execution metadata]
000220 | [OMITTED: non-scientific conversation or execution metadata]
000221 | 
000222 | 9. Exact controls and scope
000223 | 
000224 | check_gauge_locality_completion.py proves the finite (3)-(5) identities
000225 | exactly on the entire 128-dimensional physical even carrier of a three-
000226 | column,one-transverse-site sign/CAR control, using actual v1033 coefficients.
000227 | This small control is not substituted for the mass1,width8 source. In
000228 | addition ALL 48 sign/background/local-gauge cases for the ACTUAL width8
000229 | three-column one-body QWZ operator obey the same covariance exactly.
000230 | The all-size,all-state result is the written gauge-orbit proof above.
000231 | 
000232 | The all-word exclusion includes smooth dressings and unbounded renormalized
000233 | field candidates only when their original register grade, local Z support
000234 | and full joint word contract are retained. The explicit Gauss completion
000235 | exits exactly that LOCAL-NET premise; it keeps both edges/shared flux and