TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/joint-native-algebra-selection-20261016/SCALAR-FIRST-ORDER-SOURCE-FIBER-AND-SEAM-WARD.txt
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Inclusive original lines: 108-198
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000108 | 4. Four marks, quarter covariance and exact discriminators
000109 | 
000110 | Let theta_j=j pi/2 be the four retained marks. If these are held pointwise
000111 | fixed, the ordered arc travel times
000112 |  ell_j=integral_(theta_j)^(theta_(j+1)) dtheta/a
000113 | must be retained under marked coordinate equivalence. They sum to T but
000114 | need not be equal. Open arc gauge phases also matter if independently
000115 | fixed endpoint frames are part of the datum; those are NOT assumed here.
000116 | These are spatial propagation data, not three internal colors or a3+2 split.
000117 | 
000118 | An exact diagnostic family (|epsilon|<1) is
000119 |  a_(epsilon,k)=1/[1+epsilon cos(k theta)], c=alpha,
000120 |  x=theta+(epsilon/k)sin(k theta), eta=0, k=1 or4.
000121 | For the WHOLE parameter interval, T=2pi, beta=alpha, gap1/4 in both
000122 | quarters and spectral flow+1, but [B,Z]=a_(epsilon,k)Z differs fromZ
000123 | whenever epsilon!=0. These functions are diagnostic controls, not fitted
000124 | physical source candidates or full P1 countermodels.
000125 | 
000126 | For k1 the four arc times are
000127 |  (pi/2+epsilon, pi/2-epsilon, pi/2-epsilon, pi/2+epsilon).
000128 | Its coordinate map does not fix the four original marks. Dropping that
000129 | datum would spuriously erase a distinction in the marked source.
000130 | For k4, x fixes ALL four marks, all ell_j=pi/2, and it commutes with the
000131 | original geometric quarter rotation. The coefficients are quarter periodic.
000132 | Nevertheless the original constant Z-Ward still fails at epsilon!=0.
000133 | Thus even these given finite mark/quarter data plus gap/holonomy/unitSF
000134 | are weaker than the specified source time/winding commutator.
000135 | 
000136 | When every datum, including the winding multiplier, is transported withU,
000137 | the flat symbol becomes exp(i x(theta)); then a x'=omega and its Ward is
000138 | constant again. That is a transported presentation of the SAME source.
000139 | It must not be described as a new physical dynamics while freezing the
000140 | old winding or fields on only one side of the comparison.
000141 | 
000142 | In this PERIODIC chart define Q4 f(theta)=f(theta+pi/2). The actual charged
000143 | geometric quarter transition is Q_alpha=exp(i pi alpha/2)Q4, so
000144 | Q_alpha^4=exp(2pi i alpha)I. Its holonomy factor must not be dropped or
000145 | identified with an internal order4/nativeJ action.
000146 | 
000147 | Conversely, inside the fixed original Ward slice a=1, let this quarter
000148 | rotation also be fixed and commute withB. Then c has periodpi/2.
000149 | On the CONTINUOUS original holonomy branch average c=alpha, phi'=alpha-c, phi(0)=0
000150 | defines a quarter-periodic phase with phi(theta_j)=0. Thus g=exp(i phi)
000151 | commutes withZ,Q4, all position multipliers, and equals1 at all four marks.
000152 | The exact pair identity B=g B_alpha g* classifies this continuous marked
000153 | B/Z/Q_alpha fiber up to that gauge transport. No new principal direction is left.
000154 | Infinitesimally, delta a=0 and delta c=-phi' are gauge-commutator tangents.
000155 | This is a statement about the named complex one-particle data, not an
000156 | unproved classification with a physically fixed P1 real structure/state.
000157 | 
000158 | GLOBAL INTEGER CARRY: holonomy alone permits average c=alpha+N, N inZ.
000159 | Quarter-periodic small gauges reduce this to B_(alpha+N). For
000160 | g_N=exp(-iN theta), B_(alpha+N)=g_N B_alpha g_N*, and conversely
000161 | B_alpha=g_N* B_(alpha+N) g_N. This gauge fixes the quarter transition and
000162 | all four endpoint frames only for N in4Z. The exact joint invariant is
000163 |  exp(i pi B/2) Q_alpha^* = i^N I.
000164 | Thus with literal Q_alpha fixed there are four joint-phase residue classes,
000165 | unless the actual original time/quarter equality exp(i pi B/2)=Q_alpha
000166 | is also retained; then N=0 mod4. The remaining4Z large-gauge carry still
000167 | changes the named Fourier/sea labels and must be transported explicitly.
000168 | The originally fixed reference sea and B_alpha correspond to N=0. These
000169 | residues are NOT new particle species, an autonomous clock or a physical
000170 | 3+2 split. No global marked uniqueness is claimed by silently droppingN.
000171 | 
000172 | 5. State, local fields, normal order and original action are not forgotten
000173 | 
000174 | Functional calculus gives C_B=U 1_(B_beta<0) U* for the source spectral
000175 | covariance, since omega>0. On the original continuous quarter branch
000176 | beta=alpha this uses C_alpha; integer shifts require the retained carry
000177 | transport. It does NOT give C_B=C_alpha on a physically fixed reference.
000178 | The CAR algebra has the canonical unitary-induced automorphism, but a
000179 | same-representation Fock implementer needs its restricted-unitary/domain
000180 | analysis; it is not assumed from one-particle unitarity. The already
000181 | derived native quartics and all their local products must be transported
000182 | with their actual fields, reference and point-split subtraction if used.
000183 | 
000184 | In particular the quantum stress transformation has the central/Schwarz
000185 | term. An unchanged reference sea or stress cannot be inferred from the
000186 | one-particle normal form. Primary comparison: Fewster--Hollands,
000187 | arXiv:math-ph/0412028, sections3.1.3--3.2, equations3.18,3.27--3.32.
000188 | https://arxiv.org/pdf/math-ph/0412028
000189 | This is a boundary on the inference, not a new energy-minimization rule or
000190 | an assertion that arbitrary exponentiated geometric products have already
000191 | been controlled on the native states. No QEI weighting is imported.
000192 | 
000193 | The full P1 relative action is invariant under a covariant unitary change
000194 | ONLY if reflection, reference, bulk operator, domain and eta data are
000195 | transported compatibly. A circle normal form alone proves no such full
000196 | P1 invariance with some of those structures physically frozen. The
000197 | previous full gap-dependent action differential remains in force.
000198 |