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Original repository path: experiments/theory-contracts/joint-source-clock-field-closure-20261102/CANONICAL-SECTOR-RECONSTRUCTION-AND-ACTUAL-FIXEDPOINT.txt
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000001 | CANONICAL MARKED SECTOR RECONSTRUCTION AND THE ACTUAL SOURCE FIXEDPOINT
000002 | 2 November 2026. Read-only canonical sources; all new files only in work/.
000003 | 
000004 | CENTRAL RESULT
000005 | 
000006 | There is a stronger constructive route than assuming that the primitive
000007 | observable source itself is E8 or already charge complete. GIVEN the existing
000008 | even8-CAR quotient source A, its fixed charge chart and its constructed marked
000009 | bosonic DHR automorphism rho_s, reconstruct its unique index2 local simple
000010 | current extension B up to equivalence. Its lattice is D8 plus the marked
000011 | spinor, hence E8 and selfdual. No full target D5 x A3 inclusion is needed
000012 | before this reconstruction. The old whole-class obstruction only excludes
000013 | placing the full compiler plus order4 generator INSIDE unchanged A.
000014 | 
000015 | The extension is reconstructive field/sector data. It does not require every
000016 | charged current to be a physical observable. Every gauge-even output word
000017 | can be reduced back to A with rho_s and its intertwiner multiplication.
000018 | No independent mismatch of physical neutral outputs against that DHR
000019 | representation has been proved. A pointwise projection of each current to
000020 | A does lose actual compiler operations, as the exact source test below shows.
000021 | 
000022 | This removes the independent selfduality postulate inside this GIVEN source
000023 | class. It does not derive the eightfold quotient, CAR source, its level/count
000024 | metric or physical P1 dictionary from the primitive kernel.
000025 | 
000026 | 1. THE EXACT GIVEN INPUT, NOT A NEW SOURCE
000027 | 
000028 | Use the prior original A8(I)=CAR_even(F^-1 I), one quotient interval being
000029 | eight cover arcs. Retain its source state, original time, charge chart,
000030 | adjoint and localization. The original smooth half-kink construction supplies
000031 | the localized transportable rho_s with s=q/2, q=(1,...,1), d(rho_s)=1,
000032 | rho_s^2 equivalent id, s^2=2 and h_s=1. The branch choice selects the already
000033 | marked s rather than the other D8 spinor c. No W4 square root or new spin sum
000034 | is used. Smooth differences have the actual source Weyl implementers.
000035 | 
000036 | Originals under experiments/theory-contracts/
000037 | elliptic-moebius-source-bridge-20261026/local_source_origin_20261028/neutral/:
000038 |  HALF-WINDING-DHR-SOURCE-ADDENDUM.txt:30-112;
000039 |  CLOSED-HALF-KINK-SOURCE-WEYL-LINE.txt:24-55,77-116,117-177;
000040 |  NEUTRAL-DHR-BILOCAL-SOURCE-TEST.txt:18-95,103-175,225-239.
000041 | Atlas handles research-source:6544d002215be4919b8c4793 and
000042 | research-source:55ba404e7350afa1c83ea289 were hash-validated on read.
000043 | 
000044 | 2. DIRECT D8 PLUS MARKED SPINOR COMPLETION
000045 | 
000046 | In the source's fixed standard current metric,
000047 |  D8={n in Z8 : sum n even}, s=(1/2,...,1/2),
000048 |  L=D8 union (D8+s).
000049 | Since 2s=(1,...,1) lies D8, [L:D8]=2. For n in D8, n.s=sum n/2 is integral
000050 | and s^2=2. Thus L is integral and even. detL=detD8/2^2=4/4=1.
000051 | This is the explicit standard E8 lattice, not a numerical identification
000052 | from equal dimensions. The marked branch fixes which spinor extension is
000053 | used relative to the named source charges. The original fixed M transports
000054 | the whole lattice, clock, metric and grading to the target chart.
000055 | 
000056 | This proves source-class selfduality without first supposing that target
000057 | U=D5 x A3 is contained in the unextended A. The CAR level/count metric is
000058 | already part of the given source and is not derived here from raw P1.
000059 | 
000060 | 3. EXPECTATION, STATE AND NONZERO ENDPOINT NORM
000061 | 
000062 | Use a representative rho of the marked sector. Because it is an order2
000063 | bosonic dimension1 sector, its local Z2 simple current extension exists.
000064 | Write rho^2=Ad u with unitary u in A and rho(u)=u after coherent choice.
000065 | The extension is generated by A and a charged unitary F with
000066 |  F a=rho(a) F, F^2=u, F*F=FF*=1.
000067 | Every element has the unique form a+bF. The multiplication is
000068 |  (a+bF)(c+dF)=(ac+b rho(d)u)+(ad+b rho(c))F.
000069 | The bosonic braiding makes the corresponding local net extension local.
000070 | For the fixed order2 boson, Hom(rho^2,id) is one-dimensional. A scalar
000071 | change of its multiplication is absorbed by rephasing F. Thus there is no
000072 | additional inequivalent Q-system choice in this particular Z2 extension
000073 | class. This is not a general uniqueness claim for arbitrary sector systems.
000074 | 
000075 | Let gamma(a+bF)=a-bF. The canonical expectation is explicitly
000076 |  E=(id+gamma)/2, E(a+bF)=a.
000077 | For ANY already given state omega_A, omega_B=omega_A o E is positive and
000078 | normalized. It is the UNIQUE gamma-invariant state restricting to omega_A:
000079 | gamma invariance forces the expectation of bF to vanish. For its GNS vector,
000080 |  ||F Omega_B||^2=omega_B(F*F)=1.
000081 | The charged sector is therefore nonzero in the reconstructed Hilbert space.
000082 | This is a Hilbert norm, not a computed primitive P1 reflected endpoint norm.
000083 | One still needs the actual P1 domain and Theta dictionary to equate those.
000084 | 
000085 | No independent state is needed for gauge-even readouts: for any x in B with
000086 | gamma(x)=x, x lies A and omega_B(x)=omega_A(x). States on B with the same
000087 | restriction to A cannot disagree on those observables. The canonical
000088 | gamma-invariant extension is a representation choice with a precise rule,
000089 | not a new physical measure. Coherent charged-field readouts are a stronger
000090 | target than these gauge-even outputs and need their actual physical meaning.
000091 | 
000092 | 4. SOURCE TIME LIFT, ITS EXISTENCE CRITERION AND UNIQUENESS
000093 | 
000094 | Let alpha_t be the ALREADY GIVEN strongly continuous source time on A.
000095 | The original half-kink Weyl transport proves covariance of rho's sector
000096 | under that time. Choose continuous intertwiners z_t with z_0=1 such that
000097 |  Ad z_t o rho = alpha_t o rho o alpha_-t.
000098 | A lift is alpha_t^B(a)=alpha_t(a), alpha_t^B(F)=z_t F. It preserves all
000099 | relations exactly when
000100 |  z_t rho(z_t) u = alpha_t(u),
000101 |  z_(t+s) = alpha_t(z_s) z_t.
000102 | The first formula is the explicit square/Q-system compatibility condition;
000103 | it must not be omitted merely because a transporter intertwines rho.
000104 | 
000105 | In the one-dimensional intertwiner spaces the square discrepancy is scalar.
000106 | Choose its continuous square-root phase starting at1. Once square
000107 | compatibility holds, the composition discrepancy of two proposed time lifts
000108 | is in Aut(B/A)={1,gamma}. It is continuous in (t,s), hence constantly1 on
000109 | R2. This gives the group law. Similarly two continuous lifts of the SAME
000110 | alpha_t starting at identity differ in this discrete group, hence coincide.
000111 | Thus time extension is canonical in this fixed bosonic local extension
000112 | class, provided the actual sector covariance/continuity hypotheses hold.
000113 | The source Weyl construction supplies that covariance on the stipulated
000114 | quotient. It does not identify source time with primitive P1 OS time.
000115 | 
000116 | E intertwines the lifted time. If omega_A is stationary, omega_A o E is
000117 | stationary and gives a canonical unitary time implementation in its GNS
000118 | space. No arbitrary charged chemical-potential shift remains: a phase
000119 | exp(i ct) on F would violate F^2=u unless c=0 with A time fixed.