TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/seam-joint-ward-state-20261014/PROOF.txt
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Inclusive original lines: 742-850
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.

000742 | 
000743 | This section works forward from the retained chiral CAR source, rather than
000744 | identifying fields by their dimensions. It does not select that chiral free
000745 | source, its eightfold localization, or its Hamiltonian from raw P1/P2. The
000746 | primitive P1 spectral-flow class alone is not an explicit microscopic collar
000747 | operator. Unit winding below means the chosen SOURCE circle theta, not an
000748 | identified P1 seam-core circle. Winding four is its fourth power. Under an
000749 | additionally specified double-cover map it may instead represent core
000750 | winding two; neither identification of the P1 circle is derived here.
000751 | 
000752 | Take source Fourier modes exp(i(m+alpha)theta), m in Z, alpha=1/4 or3/4,
000753 | and the fixed sea m<0. Multiplication by exp(i theta) is unit winding.
000754 | Writing m=8 ell+k, its fourth power sends
000755 | 
000756 |   (ell,k) -> (ell,k+4)          k<4,
000757 |              (ell+1,k-4)       k>=4.
000758 | 
000759 | This is an actual CAR one-particle action. Its off-diagonal polarization
000760 | block has rank four, so the standard charged chiral Fock representation
000761 | implements it, up to one central phase, by W4. This is a restricted-Fock
000762 | implementer, not the number-conserving second quantization on the complete
000763 | finite QWZ cylinder. The latter contains a compensating opposite edge.
000764 | For positive-energy Fock implementers and charge shifts see Carey-Langmann,
000765 | arXiv:math-ph/9805010, sections3.2-3.3 and4.1. The implementer is global;
000766 | it is not already a localized vertex field with its oscillator factors.
000767 | 
000768 | The sheet fold has theta=(x+2pi j)/8. Consequently exp(4i theta) becomes
000769 | exp(ix/2)(-1)^j. The sheet Fourier transform sends (-1)^j to the channel
000770 | permutation P:k<->k+4. Thus the actual complex-eight source-bundle action is
000771 | 
000772 |   B4(x)=exp(ix/2)P.
000773 | 
000774 | Its realification on the sixteen real SOURCE fermion components V16_source is
000775 | 
000776 |   O4(x)=[[cos(x/2)P,-sin(x/2)P],
000777 |          [sin(x/2)P, cos(x/2)P]].
000778 | 
000779 | This is not yet the TFPT target V16=(10,1)+(1,6): the E8 Weyl map M8
000780 | does not by itself transport the elementary Clifford generators or the
000781 | source charge lattice to that target. That further operator map is not
000782 | derived here. This source orthogonal map preserves the CAR form. It is
000783 | antiperiodic as a matrix,
000784 | but is a legitimate endomorphism of the already twisted bundle: for
000785 | D_alpha=diag(exp(2pi i(k+alpha)/8)), P D_alpha=-D_alpha P, and hence
000786 | B4(x+2pi)D_alpha=D_alpha B4(x). In particular the real quarter phase has
000787 | not been discarded. With this one-particle convention,
000788 | B4^dagger(-i partial_x)B4=(-i partial_x)+1/2. Reversing the field adjoint
000789 | convention reverses the winding and the associated charge together.
000790 | 
000791 | On channel charges the unit-winding map is
000792 | 
000793 |   T1(p)=(p7+1,p0,p1,...,p6).
000794 | 
000795 | Let q=(1)^8, s=q/2, r0=(1/2)^4(-1/2)^4, and a=s+r0=(1^4,0^4).
000796 | Then the fourth power is T(p)=Pp+a, while T1^8(p)=T^2(p)=p+q.
000797 | The shift therefore retains actual integer charge carry; it is not a
000798 | four-state cyclic register. Nonzero current oscillators are permuted by
000799 | P, preserving their total oscillator level. For all charges and finite
000800 | current descendants in the common chiral finite-energy core,
000801 | 
000802 |   [Q,W4]=4 W4,
000803 |   [K_alpha,W4]=W4(Q/2+3/4+alpha/2).
000804 | 
000805 | More generally W_n has charge n and time increment
000806 | n Q/8+n(n-1)/16+n alpha/8. The charge-parity law is
000807 | Pi W_n Pi=(-1)^n W_n; W1 is odd, W4 is even. These are CAR implementers,
000808 | not independent canonical CCR creation operators. The normal-ordering
000809 | constant follows from the occupied modes0,1,2,3: their original energy is
000810 | 6+4alpha=7 or9, and their fold energy is7/8 or9/8. No physical scale,
000811 | native mediator gap Delta or interaction constant has been identified.
000812 | 
000813 | 17. EXACT MARKING/MEMORY RELATION AND THE DECISIVE OPERATOR DISTINCTION
000814 | 
000815 | For the given K, T has exactly the same energy increment as the E8 charge
000816 | translation tau_s:p->p+s. This explains a previously easy-to-misread match:
000817 | K(a)=K(s)=3/4+alpha/2, even though a is an actual integer CAR charge with
000818 | conformal weight2, while s is a half-integral E8 root with conformal weight1.
000819 | 
000820 | There is also a positive relation to the existing neutral memory label.
000821 | On the stipulated Ls extension define the AFFINE LABEL map
000822 | 
000823 |   rho(p)=Pp+r0.
000824 | 
000825 | Exactly, rho^2=id, Q(rho p)=Q(p), K(rho p)=K(p), and T=tau_s composed rho.
000826 | The map exchanges0<->r0 and D8<->D8+s. For every integer n,
000827 | T^n(0)=n s+(n mod2)r0, with the odd parity convention also for negative n.
000828 | Thus the true source winding combines charge-four marking with an
000829 | alternating neutral offset in precisely the earlier memory direction.
000830 | Neither tau_s nor rho separately preserves the raw D8 charge lattice;
000831 | their product T does. This is a derived joint affine factorization, not
000832 | a physical derivation of a separate local memory bit or of an E8 extension.
000833 | 
000834 | Crucially the coarse agreement does not give an operator identification.
000835 | For fine g_j cocharacter nu_k=(k+j)/8,
000836 | 
000837 |   nu.(T(p)-p-s)=r0.p-1.
000838 | 
000839 | For integer CAR charges the residual character is(-1)^Q. Hence g agrees
000840 | with the s-translation only on the even charge sector, and C=g^2 agrees
000841 | on all integer charges. Testing only C loses this statistical distinction.
000842 | The unmodified geometric segment transition is t_alpha=exp(2pi i alpha Q/8)
000843 | g_0, not g_0 itself. Its W4 character is exp(2pi i(3/4+alpha/2))(-1)^Q,
000844 | so it retains the original quarter phase. Calling g_0 the bare geometric
000845 | translation would silently repeat the earlier U1-compensation error.
000846 | For the neutral Cartan probe beta=e0-e4, the stronger failure is explicit:
000847 | 
000848 |   beta.T(p)=1-beta.p,       beta.(p+s)=beta.p.
000849 | 
000850 | The source winding reflects this neutral observable; the E8 translation