Only normalized smooth UNITARY vector-state sequence on fixed compact source circle. Nonunitary sums/field intertwiners,microscopically derived charged covariance cocycles and decompactified time outside scope. Local normality away from removed point is not finite global original energy. Individuallambda_i±1/2 distinct from totalQ=sumlambda_i; two-point exponent/energy alone no128/FE-GEN/ALG-EXH proof.
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Result
Original Weyl/CCR time formula andphi(theta)=f(8theta) give exactDeltaK=8sum_n>0 n²|f_n|²=(2/pi)int|f'|². Any pi->0 return squeezed intoeps costsDeltaK>=2pi/eps. Normalized W(phi)Omega states therefore have no bounded-energy separation limit under same global K reference; scalar normalization leaves Rayleigh quotient unchanged. Class lower bound,not profile scan. Known conditional K_lambda=K_0+lambda.p+lambda²/2+delta.lambda retained; its positive charged sectors are NOT excluded.
Required assumptions
Smooth unitary pair return at fixed compact source time
Given original chiral current/Weyl source, sea and K=Hcover/8 with delta_k=(k+alpha)/8-1/2. Periodic real phi(theta)=f(8theta), normalized UNITARY vector state W(phi)Omega; pi->0 return squeezed into base width eps on fixed source circle, same original energy reference. No assertion about general nonunitary sums, charged covariant representations, sector energy connections or decompactified time. Conditional Sugawara sector energy already fixed once lambda is given.
Fixed scalar Hardy model with quarter time
H=L²(S¹), C projects onto n≥0, B_alpha=−i∂theta+alpha with alpha=1/4 or 3/4. C and B remain fixed for every candidate. Consider scalar unimodular multiplication operators: continuous symbols for isometry classification and H1 symbols for finite excitation energy. This source class is not derived from P1/P2.
Do not identify source types
Observable algebra, OS reflection, state, collar operator, time evolution, carrier connection, family holonomy, and readout are distinct data until an original map relates them.
Only smooth vector-phase class and unchanged mixed Ward action in stated complete Dirac contract. No all-spinor/P1 no-go,arbitrary global-loop-holonomy theorem or infinite determinant theorem. C/1-C are grading components on one face. Chiral source, changed current representation and other original operations outside scope.
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Result
Implementable commuting vector phases obey[J_i,vec(h),V_vec(f)]=0 because opposite grading cocycles cancel and noncentral term vanishes. At pi interval profile separate chiral components have opposite half endpoint residues; full vector sum0. Thus not a spinor pair withlambda_i=±1/2 under SAME8vector Cartan currents,including scalar renormalization and mixed-word limits. Finite exact control for any Hermitian C andunitaryg: chi_(1-C)(g)=detg conjchi_C(g),normal-order-corrected complete amplitude|chi_C|².
Required assumptions
Complete vector phases with unchanged Cartan Ward action
Only original complete Dirac/CAR Lagrangian source with normal-commuting implementable smooth profiles H_i h and fI, common analytic domains and regular Weyl representation. Target comparison retains same8 vector Cartan currents and mixed Ward words. Two grading components C,1-C on ONE face are neither strip edges nor collar faces. Finite complementary determinants are algebraic controls only; no unregulated infinite determinant or automatic QWZ/Dirac source identification.
Complete Dirac Cauchy source with Lagrangian normal action
Complete orthogonal Dirac Cauchy projection C on the entire boundary module, with actual normal unitary Clifford action N satisfying N C N*=I-C. Explicitly assume charged CAR quantization with pure quasifree covariance C. Bounded current generators X,Y commute with N and have Hilbert-Schmidt commutators with C. The existing compact product collar B=diag(A_alpha,-A_alpha) on both components and both faces is the concrete control; its smooth scalar boundary multipliers are implementable. Statistics and physical measure selection are not derived from P1; no automatic extension of the unrenormalized trace to higher boundary dimensions.
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Observable algebra, OS reflection, state, collar operator, time evolution, carrier connection, family holonomy, and readout are distinct data until an original map relates them.
Constructed neutral pair and finite field action, NOT a new single spinor sector or analytic OPE-monodromy theorem. Eight-preimage support remains multilocal on source. No FE-GEN/ALG-EXH,128,P1-source/time/measure derivation. The reviewedRound5 pair misidentification retracted inRound6 after explicit counterexample; no uncritical review adoption.
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Result
T_E=product(F_B F_A*)=I4 tensor(-1)^N_E is an actual original register-grade0 operation. Periodic sigma(theta+2pi/8)=sigma(theta) acts on all8complex/16real folded fields as(-1)^1_I and retains r,h. SINGLE cover-half-flux argument C(-h)^8!=(-C(h))^8 does not apply to this NEUTRAL PAIR. Full original QWZ Nx32Ny8 mass1 control retains all256occupied modes perquarter; both edges changed, old seam phase retained. Finite-band comparison factorizes exactlyD_l(I)^8; odd-l half-circle blocks vanish, artificial second Fermi edge explicitly retained.
Required assumptions
Original neutral eight-preimage parity pair
Actual v1033 menu F_A=X² tensor(-1)^N_A and products F_B F_A*, plus the ALREADY GIVEN one-copy eightfold dictionary eta_j(u)=psi((u+2pi j)/8)/sqrt8. E is union of all8 preimages E_I of a base interval. Register factors cancel; no added Gauss rule or copy. Fold, free chiral source and P1/core-circle interface are not newly derived premises. Finite filled Fourier band is separately declared regulator comparison with a second Fermi edge.
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Observable algebra, OS reflection, state, collar operator, time evolution, carrier connection, family holonomy, and readout are distinct data until an original map relates them.
Exclusion only for fixed charge/connection dictionary. No blanket SO16 or spinor-sector no-go. Eight independent copies, physically derived charge conjugation or changed field/domain connections separate. Source/P1-circle, base-deck/clock, chiral fermionic measure and local field products still require derivation; no new internal Z8^f/Arf branch. Neutral-pair refinement: eight-preimage parity pairs T_E with canceled register and unchangedh are NOT excluded by this single-sector comparison.
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Result
Original one-copy eightfold: C(-h)^8=-hI, while desired uniform spinor twist (-C(h))^8=hI. No complex charge-preserving intertwiner of given holonomies. Both-quarter h=±i countercontrol: real SO16 O=realify(diag((-1)^j))K DOES conjugate holonomies but reverses Jcomplex and original U1 charge. Uniform BASE Z2 spin lift is invisible to cover holonomy at even degree; global parity×deck relabel still creates no localized charged intertwiner.
Required assumptions
Fixed charge-preserving connection in original one-copy eightfold
Given one-copy fold has transition C(h), C(h)^8=hI. Available register half flux changes cover holonomy h to -h. Compare uniform spinor-OPE twist -C(h) in same complex U1 charge dictionary and connection, up to charge-preserving fiber conjugacy/periodic gauge transformation. Real SO16 charge conjugation at h=±i, changed connections/domains and eight independent copies lie outside exclusion class. Source circle/P1 core and cover/P1 deck are not identified.
Do not identify source types
Observable algebra, OS reflection, state, collar operator, time evolution, carrier connection, family holonomy, and readout are distinct data until an original map relates them.
Redundant kinematic generators and local Gauss/observable rule are ADDED premise, not P1-derived. No new physical dimension relative to EVEN original carrier, but different local net. No QWZ/Dirac-source identification, spinor-field limit, FE-GEN/ALG-EXH/E8, chiral measure selection or mirror absorption. f(Z) references can become nonlocal.
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Result
Explicit gauge-orbit isometry E is onto full Gauss space for all sizes, E*E=1,GxE=E,HnewE=EHold,WE=EZ,tau_a^x E=E F_[0,a]. Full even register-matter dimension, all transported state words, positivity, adjunction and common original time preserved; old local register Z becomes GLOBAL Wilson loop. Exact whole128dim control and48 actual width8 one-body gauge cases check signs; all-size theorem rests on orbit proof. Mere word-preserving gauge representation cannot give complete Dirac vector-current response a new central phase.
Required assumptions
Conditional local parity-Gauss completion with common original time
ADDED comparison contract, NOT P1-derived: old four-state register with total-even matter; redundant Z2 link qubits and local Gx=(-1)^nx tau_(x-1)^x tau_x^x=1; gauge-invariant local observable net. Original QWZ hopping coefficients, common microscopic Hamiltonian and energy reference; no new electric energy. Sector-specific f(Z) subtractions must map to f(W) and can be nonlocal. No identification of Gauss space with original Padm.
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Observable algebra, OS reflection, state, collar operator, time evolution, carrier connection, family holonomy, and readout are distinct data until an original map relates them.
Only specified grade2/register/word/projection contracts. Changed physical nets, globally supported register Wilson loop, other originally derived sector actions and edge absorbers not excluded. No all-P1 no-go and no assertion that every spinor has this register grade.
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Result
Any arbitrarily long/smeared/renormalized original grade2 sequence obeys exact omega(T*Z*T Z)=-omega(T*T); a bosonic field local away from retained even local Z gives +omega(T*T). Positive nonzero field norm and full joint words exclude locality even without bounded/unitary limit. Shared ZL=ZR excludes faithful left-only flux flip under same full label contract. An old A/A*-projector retaining full even matter and actual QWZ time is register-diagonal; retaining F/D ties r,r+2 or all r. It does not remove register obstruction.
Required assumptions
Local register retained in full charged word-limit contract
Comparison class: original v1033 register grade2 under AdZ, Z retained as bounded even local unitary outside field support, full joint positive word limits with nonzero field norm. No boundedness of renormalized field sequence assumed. Separate two-edge test retains original shared flux ZL=ZR. Projection classification only when retaining the FULL original even matter algebra, time and A/A*-admissibility.
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Observable algebra, OS reflection, state, collar operator, time evolution, carrier connection, family holonomy, and readout are distinct data until an original map relates them.
Only unchanged sharp actions under their separate actual contracts. Smooth punctured-circle sectors may be locally normal and globally inequivalent and remain open; full-CAR implementability does not imply trivial even sectors. Phi/Sigma witnesses not identified without a map. RightHom(s tensor beta,beta) absorber distinguished from charged field. No all-P1 assertion or microscopic E8 closure. Unavailable user18+38 package not claimed replayed; own original controls and analytic proofs separated.
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Result
22Oct pasted half-flux report independently checked: F_A Z F_A*=-Z while registerZ stays even; finite matterAdP_A inner despite transport/fusion/exchange relations; v746 pair-currentK_j even,clock-invariant and half-flux-flipped in each base window. Direct sharp chiral sign change has non-HS covariance difference near jump with log((eps+delta)^2/(4epsdelta)) divergence; Araki root-HS necessity and even crossed-product extension exclude local normality also on standard even free net. Current v1033 builder Nx8Ny8 confirmsR(d,k)=(2,1),NS(0,0); uniform fixed-filled R has<nLnR>0 versus quasifree1/4,purity1/2. Common flux changes both edges. F_A does not identify complete prescribed NS/R state responses.
Required assumptions
Sharp half flux in separate original net/state contracts
Check separately under their own premises: v1033 with expressly ADDED M4 flux register,F_A=X² tensor(-1)^N_A,even algebra M4 tensor CAR+,common strip flux and uniform fixed zero-space filling; v746 with full CAR of paired window I union(I+N/2),NS-clockS^(N/2),halfcircle sign; direct chiral free limit as sharp-sign Bogoliubov on standard locally normal NS CAR net. No unproved identification of these nets and no exhaustion of smooth/renormalized outer-field limits. Even-to-full normality argument uses the standard free local factor contract.
Do not identify source types
Observable algebra, OS reflection, state, collar operator, time evolution, carrier connection, family holonomy, and readout are distinct data until an original map relates them.
Global comparison at same source energy/charge,with changed odd D8 action. No local spinor/FE-GEN/ALG-EXH/P1-core-circle theorem. Other gauged observable nets and outer DHR extensions not excluded. SOURCE W1 remains global with inherited cocycle phases; both edges retained. Symbol-only Pfaffian/J-grading proposals rejected by type checks.
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Result
22Oct joint The reviewed audit: Phi=I on even D8,rho=Pp+r on odd v with oscillator permutation is an exact global unitary Z8->D8+c preserving full H,Q,vacuum. [Pr=-r,Pdelta=delta+r,r²/2+delta.r=0] proves all energies; theta Q/parity controls throughE32 agree in bothquarters. SOURCE winding1 transports to q->R^5q+lambda,lambda=(1/2)^5(-1/2)^3,charge1,K-increment(Q+alpha)/8,eight-step carry(1)^8. But a local target Cartan Weyl unitary becomes Pi_even W+Pi_odd W*,whose commutator with remote odd CAR-B equals parity B(W-W*)!=0. No unchanged proper-local CAR inclusion. Separately,v774 anchor bit fails uniform Spin10-invariant NS/R-label identification since rank[C,E45]=4.
Required assumptions
Given eightfold source fold with explicitly changed odd target action
Given existing one-copy CAR source,sea,fold,p in Z8,Heisenberg oscillators,Q=sum p,H=8K,K=N+p²/2+delta.p with delta_k=(k+alpha)/8-1/2,and actual SOURCE windingT1=Rp+e0,alpha1/4or3/4. Test explicit Phi=I on D8,rho=Pp+r on odd v with oscillator-P to odd spinor extension. Odd target Cartan action is explicitly changed by this comparison; inclusion in the original local net is separate. Neither folding,SOURCE/P1-core circle identification nor bosonic statistics is derived from P1.
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Observable algebra, OS reflection, state, collar operator, time evolution, carrier connection, family holonomy, and readout are distinct data until an original map relates them.
Only four uniform scalar twist classes in GIVEN D8 target with fixed time/charge dictionary. Integer periodicity/W4 distinction reuse original findings. Monodromy establishes neither new local fields,chirality/gauging selection nor exclusion of all P1 defects. New sector representation with shifted target Cartan spectrum leaves unchanged-carrier premise; its origin remains open.
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Result
q=(r/4)^8+m gives q.(e1+e2)=r/2+integer: only r0,2 relatively local to ALL original D8 currents. The64 weight1 quarter dressings are relatively odd but fail stronger full monodromy-1; parity projection/cocycle signs alone cannot repair this. Half twist has two128 chiralities. At unchanged delta full half-integral cosets have unique minima K0 even andK1/32 odd in BOTH quarters; leaving the +/-1/2 cube costs>=17/32. Prior extra-zero-state objection applies only to even extension. Unchanged integer Cartan generators still exclude every nonzero half-charged operator satisfying actual integrated covariance. Original W4 contains permutation and integer carry,not pure spinor translation.
Required assumptions
Given D8 target with uniform Z4 twists and fixed fold-time law
Only in the conditional eight-channel target test,D8 currents with roots +/-e_i+/-e_j,twist charges q=(r/4)^8+m,m in Z8,r0..3,h=q²/2+N and existing fold time K=q²/2+delta.q+N,delta_k=(k+alpha)/8-1/2 are given. Keep all D8 currents and Cartan charges fixed,chiralities distinct. This creates neither eight physical source copies nor local defect endpoints from P1.
Do not identify source types
Observable algebra, OS reflection, state, collar operator, time evolution, carrier connection, family holonomy, and readout are distinct data until an original map relates them.
Same full free cylinder,sea,time,energy scaling. No extra independent finite-energy multiplicity; NOT a complete classification of local scaling nets. Eightfold folding with another coordinate/locality/stress dictionary remains possible. Other defect sectors,vacuum subtractions,sources or interactions need a new map. No P1/E8/T1-T8 closure.
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Result
22Oct: Three rho ranges in h²=sin²p I+(H0+rho Z)² show at most two eigenvalues per momentum have magnitude<1/2; all others cost>=N/(4pi). Fixed energy E enforces finite,N-independent |j-alpha|<=pi E/2. Both small branches give |j-alpha| with opposite edge orientations. For E outside the limit spectrum the ENTIRE Fock spectral rank stabilizes,not merely fixed words. Tr exp(-beta H_N) converges for every beta>0 to product_j(1+exp(-beta|j-alpha|))²; bounded mean energy gives subsequential compactness.10 exact source identities,2 direct full-builder and12 spectral controls passed; all-N reasoning independently reviewed.
Required assumptions
Unchanged full quarter-QWZ excitation source
Current v1033 QWZ cylinder at mass1,width8,ONE complex CAR copy,alpha1/4 or3/4,full negative sea and H_N=N/(2pi)[dGamma(h_N)-Tr(P_Nh_N)]. Retain both edges,massive modes,charge sectors and physical energy/vacuum reference; compare actual low eigenmodes. No P1/P2 selection of this realization,changed stress or executed quotient.
Do not identify source types
Observable algebra, OS reflection, state, collar operator, time evolution, carrier connection, family holonomy, and readout are distinct data until an original map relates them.
Original complete two-component Dirac collar and explicit CAR/Lagrangian/implementability class; no novelty claimed for standard current/Dirac theory. No exclusion of all P1 sources or chiral-measure/inflow realizations. Cauchy projection, APS sign subspace and OS Padm stay distinct. No computed fixed-domain spectral response, physical event execution or 4D no-go. Higher-dimensional trace formulas need separate control. All original native4/5/20/carry/E8/eight-preimage structure retained; no new finite-code closure requirement. Prior measure/log-profile boundary and open chiral/determinant-line contracts reused, not closed.
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Result
Excluded is the automatic identification of the complete normal-commuting Dirac-CAR vector current with the original nonzero chiral current phase, only under the stated premises. UR.COLLAR.FACE_CURRENT.01, DIRAC-LIFT continuation: For a complete Lagrangian Dirac Cauchy projection C with N C N*=1-C, c_(1-C)=-c_C. For implementable X,Y commuting with N, unitary covariance also gives c_(1-C)=c_C, hence c_C=0. In the existing product collar B=diag(A,-A), occupations n(A) and1-n(A) on ONE boundary face give -2i,+2i and total0 for the original four-mark-fixed profiles. This differs from the earlier two-face cancellation of one scalar block. Fixed Clifford symbols force g I2 on this two-component module; diag(g,I) changes the normal/tangential symbol. The complete lift D_phi=D-i chi_prime phi Jn-chi phi_prime Jn Gamma is a bounded Hermitian perturbation. Transported domain/reference/Padm give frame covariance; a fixed original selfadjoint domain remains analytically valid but specifies a relative boundary attachment not physically selected here. Explicit extra Dref/Drel commutator terms are retained at fixed geometric R; fixing R alone does not break covariance when U commutes with R. Nonconstant four-mark-fixed profiles do not preserve the old APS sign subspace, which is not a selfadjointness no-go. Original Gamma already identifies D_plus, but not its physical measure/line connection. The prior scalar result remains conditionally valid.
Required assumptions
Complete data of both product-collar faces
A selfadjoint B occurs in an exact normal-independent product equation. The full orthogonal L2 Cauchy graph projection of both labeled faces, their actual fiber identification and independently known positive normal length ell are supplied. Selection of these data from P1/P2 is not presumed.
Given charged CAR quantization of the full two-face Cauchy graph
As in section4 of the original collar proof, additionally assume charged CAR(H direct_sum H) and the pure gauge-invariant quasifree state with occupation G_ell. The second labeled face has n=(1+exp(2ell B))^-1. This is a field/statistics premise, not a consequence of P1; reduction does not select or physically execute a one-face operation. The long-collar limit is used only within this declared operator/CAR class.
Fixed scalar Hardy model with quarter time
H=L²(S¹), C projects onto n≥0, B_alpha=−i∂theta+alpha with alpha=1/4 or 3/4. C and B remain fixed for every candidate. Consider scalar unimodular multiplication operators: continuous symbols for isometry classification and H1 symbols for finite excitation energy. This source class is not derived from P1/P2.
Do not identify source types
Observable algebra, OS reflection, state, collar operator, time evolution, carrier connection, family holonomy, and readout are distinct data until an original map relates them.
Complete Dirac Cauchy source with Lagrangian normal action
Complete orthogonal Dirac Cauchy projection C on the entire boundary module, with actual normal unitary Clifford action N satisfying N C N*=I-C. Explicitly assume charged CAR quantization with pure quasifree covariance C. Bounded current generators X,Y commute with N and have Hilbert-Schmidt commutators with C. The existing compact product collar B=diag(A_alpha,-A_alpha) on both components and both faces is the concrete control; its smooth scalar boundary multipliers are implementable. Statistics and physical measure selection are not derived from P1; no automatic extension of the unrenormalized trace to higher boundary dimensions.
Given scalar quarter circle and full product collar with actual charged CAR quantization; no P1-derived statistics or physical source selection. No novelty is claimed for general current algebra. Infinite analytic formulas plus exact finite controls, not finite CCR representations. Relative endpoint transport is an explicit frame/operator family, not a proven original boundary intervention with fixed reference/domain/Padm. Phase distinction needs full source access or coherent control; one-face conjugation channels can agree if the opposite face never recouples. Normal distance is not physical time; original time, zero mode and absolute reference remain separately retained. No native3+2 selector, pointer/record law,4D determinant-line theorem or T1-T8 closure. Existing4/5/20/carry/E8/eight-preimage structure remains.
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Result
UR.COLLAR.FACE_CURRENT.01: In the existing scalar quarter circle, the original G_ell Cauchy graph under the explicit charged-CAR/quasifree premise gives exactly the original level1 current cocycle on its second face, its negative on the first, and zero for the common lift. This holds for all smooth real profiles by sum_r(n_r-n_(r+k))=k, not a mode truncation. The same n gives beta=2ell, full thermal current response, the charge-retaining Hall/winding partition and a controlled limit to the original negative sea. Common g-direct_sum-g transport and I-direct_sum-g have the same second-face covariance but distinct full G and coherent phases. Even four-mark-fixed quarter-periodic sin4theta,1-cos4theta give exp(2ist) on one face versus1 jointly. A concrete geometric repair g_chi=exp(i chi(r)phi(theta)) gives L_phi=g_chi Lg_chi*=partial_r+B_alpha-i chi_prime phi-chi phi_prime and T_phi=g_phi T in the actual scalar transport block; its boundary graph is independent of the interior frame interpolation. This does not select physical execution.
Required assumptions
Complete data of both product-collar faces
A selfadjoint B occurs in an exact normal-independent product equation. The full orthogonal L2 Cauchy graph projection of both labeled faces, their actual fiber identification and independently known positive normal length ell are supplied. Selection of these data from P1/P2 is not presumed.
Given charged CAR quantization of the full two-face Cauchy graph
As in section4 of the original collar proof, additionally assume charged CAR(H direct_sum H) and the pure gauge-invariant quasifree state with occupation G_ell. The second labeled face has n=(1+exp(2ell B))^-1. This is a field/statistics premise, not a consequence of P1; reduction does not select or physically execute a one-face operation. The long-collar limit is used only within this declared operator/CAR class.
Fixed scalar Hardy model with quarter time
H=L²(S¹), C projects onto n≥0, B_alpha=−i∂theta+alpha with alpha=1/4 or 3/4. C and B remain fixed for every candidate. Consider scalar unimodular multiplication operators: continuous symbols for isometry classification and H1 symbols for finite excitation energy. This source class is not derived from P1/P2.
Charged CAR algebra and original negative sea
Assume the actual field algebra is charged CAR(H) and its reference state is the pure quasifree sea with occupation 1−C. Relative charge and positive excitation time are normal ordered against this sea. This physical identification is an additional premise.
Original single-circle density and geometric stress
Same stipulated chiral CAR source, negative-D sea, actual quarter domains, eightfold F and existing M/O. Local density J_m=sum:a_r* a_(r+m): and its half-density geometric stress with alpha ground energy follow from this source. Original scalar phases fold to correlated generally offdiagonal channel profiles, not independently selected E8 root weights. NS/quartic reference remains conditional. Global Fourier modes control existing fields; no P1-selected local profile/detector/coupling. No isolated b->-b invariance for these new responses; transport positive QWZ occupation with charges/currents/V/b together.
Do not identify source types
Observable algebra, OS reflection, state, collar operator, time evolution, carrier connection, family holonomy, and readout are distinct data until an original map relates them.