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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000067 | 3. Unit winding/index is weaker than the original constant-increment Ward
000068 | 
000069 | Keep the actual original seam operator Z=M_exp(i theta) fixed. For ANY
000070 | smooth scalar multiplier u, direct differentiation gives
000071 |  [B,M_u]=-i a M_(u'), hence [B,Z]=a(theta) Z.
000072 | Consequently preserving the SAME constant Ward [B,Z]=kappa Z is equivalent
000073 | to a(theta)=kappa pointwise. For the actual normalized original source
000074 | kappa=1, so a=1. This follows without minimizing a guessed metric cost.
000075 | For a given degree1 symbol u=exp(i phi) the corresponding condition is
000076 | a phi'=kappa. Integration of phi' gives omega=kappa. The degree alone
000077 | does not give the pointwise equation or select the symbol.
000078 | 
000079 | POSITIVE INTRINSIC CONSTRUCTION: the SAME B determines its uniform winding
000080 |  u_B(theta)=exp(i x(theta)), [B,M_(u_B)]=omega M_(u_B).
000081 | Conversely the preceding ODE proves every smooth unimodular wind1 symbol
000082 | with a positive constant increment equals u_B up to global phase, and
000083 | the increment must be omega. The basepoint in x is only that common
000084 | phase. Thus spectral spacing, proper travel time and uniform source
000085 | coordinate are outputs of ONE operator, without a fitted profile/clock.
000086 | Its time-quarter operator is also fixed as
000087 |  Q_B=exp(i T B/4)=U[exp(i pi beta/2)Q4_x]U*,
000088 |  Q_B^4=exp(2pi i beta)I.
000089 | This is a mathematical intrinsic source construction. Identifying it
000090 | with a separately frozen geometric mark/lift or a physical P1 event is
000091 | still an operator dictionary question, not a consequence of equal order.
000092 | 
000093 | In contrast, Z* B Z=B+a. The entire selfadjoint path
000094 |  B_s=B+s a, 0<=s<=1,
000095 | has beta_s=beta+s, omega_s=omega and EXACT eigenvalues
000096 |  E_n(s)=omega(n+beta+s).
000097 | For 0<beta<1, precisely n=-1 crosses zero upwards at s=1-beta. There are
000098 | no endpoint zero modes and the spectral flow is+1 for EVERY positive a.
000099 | Thus unit spectral flow does NOT imply constant unit Ward. This is an
000100 | exact theorem for the whole scalar class, not a dimension-count argument.
000101 | 
000102 | The Ward is an existing identity of the stipulated source B_alpha,Z.
000103 | It is NOT derived here from raw P1's nontrivial/unit spectral-flow class.
000104 | Nor is its one-particle constant increment equated with the charged-Fock
000105 | Ward: the latter retains charge-dependent sea terms and original factor8.
000106 | No physical second, autonomous clock or altered normalization is obtained.
000107 | 
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 |