Follow the published construction into its registered claims, then explore reviewed research questions, attempts and the connections that remain open.
Quantum and boundary structure
The published foundation in this area
32 authored claim references lead into the original ledger and paper sections. This source record remains available even where no separate reviewed research packet is attached.
Finite E8, Pauli/Clifford, code, state, covariance, and instrument constructions supply internal quantum models and scoped guarantees. A code or successful channel is not yet a selected, repeatable physical source with the same parent dynamics.
Starting structure in the Atlas orientation
Concrete finite code and operator structures exist, with local results within explicitly specified source and instrument menus.
Which connection is sought
Open items include common physical preparation, source, interaction, repetition, record preservation, and connection to the TFPT parent.
How it connects
Codes and clocks specify internal information and symmetry structures; gravity additionally requires local geometry and dynamics.
Constructive all-mode theorem in the already given original neutral vacuum source, using primary current-net theorems and independent analytic/executable review. Not an unverified continuum extrapolation of the finite v438 ladder. SAME gapped H remains excluded as Borchers P; P is a different derived generator. The elliptic Gibbs state is not invariant under this P and must not be replaced. No full charged-sector theorem, equality with original P1 Theta/time, automatic eight-arc word transport or3+1D closure. Unbounded-generator sum equalities are on the common core with the corresponding closure.
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Result
Existing currents give L_n^0=Lcal_n^alpha-bJ_n-b²delta_n0/2 with the same Omega and L0^0=H_alpha onQ0. P=L0^0-(L1^0+L-1^0)/2 is dGamma of an operator exactly diagonalized to x/2 by the complete Laguerre basis: P>=0, spectrum[0,infinity), kernelC Omega. With D=(L1^0-L-1^0)/(2i), explicit weighted dilation supplies the full affine group law. The actual local vacuum current net has a standard half-line inclusion with Delta_M^(it)=exp(i2pi tD) and the full Borchers relation. The original t_v quarter onQ0 is precisely R=exp(-i pi L0^0/2); its four positive gapless parabolic generators have opposite-pair averageH_alpha. Local algebras and modular groups are transported together.
Required assumptions
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.
Actual neutral vacuum current space of the quarter source
Retain the supplied one-copy quarter CAR source alpha1/4 or3/4, negative sea Omega, actual level1 currents J_n with adjoints, H_alpha=N+Q²/2+(alpha-1/2)Q, common finite-energy core, existing local current net and original transported stress/winding comparison. Work precisely on the neutral cyclic Q0 space; elliptic Gibbs/HS state and charged sectors are distinct data. No physical P1 source map is assumed.
Original quarter CAR source with fixed fold and current dictionary
The already supplied single-channel CAR source with actual quarter domains alpha=1/4,3/4, negative-D sea, positive original excitation time, unitary eightfold F and actual channel diagonalization is fixed. M,O,U+ and the conformal NS point-splitting reference are the same original dictionary data; the single delta counterterm follows from the kernel difference. Even CAR current composites retain full arc correlations, charges, domains, all modes, original time and the existing subtraction prescription including finite derivative counterterms. No E8 root/spinor extension is assumed for this neutral current subalgebra. Identity with the original P1 net, Theta, collar and reference sector is not supplied. Real profiles and pulse strengths are intervention data.
Necessary theorem about the same joint state and specified algebras. Does not exclude every interaction or future entanglement. Full projection and commutation are not inferred from a thermal face or incomplete neutral data. In a finite full Slater block, commutation with its entire even part already suffices via rank1 support. Positive conditional mixtures need actual positive disintegration; complex eta weights are not one. The source identity presupposes an actual bilinear action/integration and is not a derivation from P1. Physical joint response remains open.
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Result
The existing full CAR projection-covariance theorem gives a pure restriction omega_P. For b>=0 from a commuting algebra, phi_b(a)=omega(ab) is positive and dominated by ||b||omega_P; purity forces omega(ab)=omega_P(a)omega(b). All equal-time connected mixed responses of those algebras vanish. A positive mixture of conditional CAR states with full mean projector P consists almost everywhere of the same state. An actual original coupled source with nonzero mixed response therefore cannot have that full free reference projector; it needs its own joint response or a proved independent source sector. The correct conditional complex source identity retains K(phi), or its J derivatives.
Required assumptions
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.
Full projector covariance in a positive joint source
Conditional theorem: a positive normalized joint state has full projection covariance P on the actually fully realized CAR algebra. The further unital C* algebra commutes with this CAR algebra. These assumptions are a source contract to test, not already derived P1 data; one face or incomplete neutral measurements do not replace them.
Only the direct selfdual finite Pfaffian ansatz and universal internal J repair at fixed D. The Kramers countercase belongs to the existing constructor class and does not assert selected physical masses. Independent complex barpsi/psi, other correctly derived local sources and the4D product action remain possible. KO4 modulo8 does not derive metric dimension4, Lorentz signature or P1 quantization. Exact18 control checks and independent analytic review; no full suite or ledger promotion.
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Result
For antiunitary J and selfadjoint D, B(x,y)=<Jx,Dy> has transpose sign epsilon epsilon_prime. The actual v252 KO6 block therefore gives a symmetric form; its standalone Grassmann quadratic term vanishes. Gamma_F J_F changes both signs and does not repair it. An actual Majorana constructor output moreover has four simple eigenvalues in the full96 space; Kramers excludes EVERY square-minus antiunitary commuting with that same D there. The Majorana-off control permits an explicit internal alternative. The ordinary local KO4 times KO6 product instead supplies the correct KO2 Pfaffian type if its original spinor source and common domain are actually derived.
Required assumptions
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.
Actual finite KO6 block and tested selfdual ansatz
Given the existing v252 constructor with selfadjoint D_F, antiunitary J_F, J_F²=+I, J_F D_F=D_F J_F and J_F Gamma_F=-Gamma_F J_F. Test the additional direct Pfaffian identification through B_F=<J_F.,D_F.> and a universal internal rechoice of J at unchanged D_F. Control parameters of the published constructor class are not selected physical TFPT values. A total spacetime domain or Grassmann source law is not presumed derived from P1.
Only identification of this same source energy with the Borchers translation generator. Other gapless generators, correctly constructed local standard pairs and HSMI remain possible. Lean formalizes the elementary spectral-set statement, not Stone, spectral theory or the actual P1 source. Other KMS/spectral-profile identifications still require their actual operator dictionaries.
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Result
Using this same gapped quarter energy as a modularly scaled translation fails already at the full spectral gap: the group law and Stone uniqueness imply unitary equivalence of H and H/2. The least positive spectral point would have to be both1/4 and1/8. Lean grind proves the elementary spectral-set lemma with only the three standard axioms. Independent review confirms the analytic connection and scope. The source remains a conditionally supplied model; neither original-P1 source derivation nor its universal refutation follows.
Required assumptions
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.
The same given quarter energy as Borchers translation
Use exactly the already given full chiral quarter source H_alpha, alpha1/4 or3/4, with spectrum containing0 and least positive spectral point1/4. The proposed standard-pair bridge identifies U(s)=exp(is H_alpha) as its strongly continuous translation group and requires unitary modular conjugation V_t U(s) V_t*=U(exp(-2pi t)s) for all real s,t. This is an additional proposed identification to be tested, not an original P1 consequence.
Counterproof only to the generic positive-weight preservation inference in cited original passages,not the entire geometric/BRST measure theorem under all additional conditions. Finite test is no full P1/P2 countermodel,no TFPT prediction,and no newly chosen physical source. Source Bernstein cone means completely monotone profile,not a general Bernstein function. No actual original field map or completeT1-T8 goal closed. Preserve existing given-source paths,positive transfer constructionv330,Schur/projector/Weyl results. Do not introduce full ambient geometric measure construction as a prerequisite for every neutral local response;direct local OS embedding remains possible. P1 already assumesRP;only a proposed derivation shortcut is excluded.
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Result
The generic RP preservation inference in the old constructive-measure proof is false. A strictly positive RP base with eigenvalues19,1 becomes non-RP under w=exp(-Tr exp(-D²)),despite positive weights,reflection covariance and gap>=1. Exact Delta=e^-1(1-e^-4)²>75/256>1/9>log(10/9);normalized reflected sign response changes from1/19 to-0.12394264612509166625. General nonconstant positive Laplace profiles haveDelta>0;generic failure uses r in(exp(-Delta),1),while fixed9/10 base requires the stated threshold. Finite W>=0 is the universal Schur-multiplier preservation criterion;for a fixed base K0 elementwise-times W>=0 is exact. A suitable reflected factorization is sufficient for repair. Source-independent positive reference denominator cancels in normalized responses and repairs neither false RP nor missing cross coherence.
Required assumptions
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.
Finite test of the generic spectral RP preservation inference
Test only the generic inference used in old measure proofs that a pointwise positive reflection-symmetric spectral weight exp(-Tr f(D²)) preserves existing RP. Test class: classical reflected finite halves,K0=[[10,9],[9,10]],D(s,t)=[[2s,1],[1,2t]],s,t=plus/minus1,f(u)=exp(-u). No full geometric/BRST/branch or P1/P2 realization of these data assumed. General-profile formula uses finite positive Laplace measure;the9/10 threshold is not asserted for arbitrarily small profiles.
Source-reviewed inventory and reuse audit with explicit coverage: automated inventory is not manual full proof/archive review,not a full current suite run,and not a universal absence claim. Existing model arguments are not upgraded to formal or complete physical proofs. Original sources and ledger unchanged. Earlier local-source criticism is recovered knowledge,not a new scientific gap. Full-CAR ground-state reduction requires the same gapped Bogoliubov action,H>=0,H Omega=0 in the same representation; neutral route directly transports full functional. No unnecessary full E8 observable extension for neutral response.
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Result
Inventoried current registry with all707 present modules,788 ledger rows and3108 project Python/Lean/Wolfram sources; closely inspected24 strong Python closure candidates and11 formal interfaces for actual source premises. Reuse the existing QWZ to CAR field/time map,v746/v779 net blocks,v745 analytic sector bound and neutral octic composition. v1015 is a finite QWZ times deck response block; its ZERO wording assumes external source identification. Inspected Lean compositions assume realization implications as axioms,while Calderon/Yukawa interfaces explicitly retain upstream provenance. Two targeted current runs reproduce v622 at10/10 and v1030 at27/27; the latter retains microscopic FEGEN/ALG2 as open. Original master TFPT.TOE.COMPLETE.01 requires AND(T1..T8),all eight remain open in current ledger. Joint two-face source is a current provenance bridge,not sufficient total closure.
Required assumptions
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.
P1/P2 remain declared starting axioms
This program checks which additional source, state, event, and observable structures follow from accepted TFPT principles; it does not revoke or replace P1/P2.
Original P1, boundary, seed and local restriction definitions
Reviewed originals: P1 declares oriented seam, reflection-positive kernel, unit winding and c3; stronger T_boundary includes A_+ as data; operational seed takes A_loc, time, reflection, state, seam class and collar operator as inputs. A_adm(O) is A/A*-invariant restriction of supplied local insertions. Compare without P2/E8/target spinor assumptions and without presuming an original gauge/microscopic localization intertwiner.
Analytic all-mode theorem in the explicitly existing normal source representation, supplemented by exact finite path controls, not an all-mode inference from finite samples. W(f) must already be bounded in the source von Neumann algebra; no norm CAR membership, bounded imaginary-time conjugates or arbitrary odd OS words asserted. The GNS partner is not identified with a physical P1 second boundary. Older papers32 and45 assume the positive field kernel or its reflection positivity. Actual local physical P1 source map, original measure/coupling, physical time, EM and complete TFPT remain open. The old spatial type exclusion retains its stated scope.
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Result
The old Schottky reflection rho(t,s)=(-t,s) does not reverse spatial profiles; its neutral Weyl profile map is -I and is not excluded by the previous -r_s result. In the existing HS source space, W_R(f)=L_Wf and W_L(f)=R_Wf* have opposite Weyl phases directly from right multiplication. J_HS W_R J_HS=W_L is conjugate-linear multiplicative; centered finite bilinears obey j_L(A)=-R_j(A) and J n_R J=I-n_L. The joint functional Tr(sqrt(rho)Wi*sqrt(rho)Wj) is the positive HS Gram of rho^1/4 Wi rho^1/4 without a Gaussian premise. For the given thermal source, fully ordered heat products K_F handle bounded even insertions: ||K_F||2<=sqrt(Z_beta)product||Aj|| and S(ThetaF,G)=Tr K_F* K_G/Z_beta. Strong approximation extends the existing polynomial free functional to existing bounded neutral Weyls. The old Green midpoint X and all four graph blocks are reused, not chosen anew. Exact NS modes pi,3pi test the same original implementation path, including null relations, particle-hole contribution and reflected noncommuting histories.
Required assumptions
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.
Normal even sea CAR source with canonical graded HS doubling
For the algebraic source test use the already given sea Fock/CAR carrier,with faithful even normalized traceclass density rho and its standard HS space,Omega=sqrt(rho),L_A X=AX,R_A X=XA,Gamma X=Pi X Pi. Gamma is the canonical GNS implementer of source parity fixing Omega. Partner fields are represented in the fixed mode-index chart by b_j=−Gamma R_aj;this assumes no physical second P1 boundary. The finite saturation theorem covers arbitrary faithful even finite densities;the separate all-mode route requires the actual normal HS carrier,a complete covariance eigenbasis and saturation in every mode. For the free elliptic connection,rho is the already derived original thermal density,not a newly chosen geometry or temperature. TomitaJ,OSTheta,classical doubling and physical time remain distinct until actually mapped.
Free chiral Cauchy quantization of the canonical elliptic annulus
Treat the dividing half of the given square curve in its canonical flat conformal representative as a product collar. Compute the Cauchy graph of Dplus=partial_t+B_Sigma,B_Sigma=−i partial_s with NS boundary,then interpret it using the expressly free chiral CAR construction as a quasifree two-boundary state. The labelled second-face compression is the covariance;the other is complementary. This Gaussian interpretation does not derive the original relative spectral-action measure,P1 time or EM/Quillen response.
Existing normal thermal source with bounded neutral insertions
Use the already given thermal sea representation of the free elliptic NS source with H>=0, finite Z_beta=Tr exp(-beta H), W=beta/2, rho=exp(-beta H)/Z and its HS standard space. Neutral even insertions including existing current Weyls are bounded in the source von Neumann algebra. Their imaginary-time conjugates are not assumed everywhere bounded; ordered heat products define the functional. The general HS Gram formula needs only an existing normal density, not Gaussianity. This comparison premise asserts neither physical P1 source, second geometric field family nor physical time identification.
Exact necessary algebra identity and complete declared finite menu, not a classification of all profile maps or sources. The excluded candidate was expressly conditional, not an established OS bridge. No exclusion of OS Grassmann symbol products or existing positive Green/null relations. The opposite-face pair is an algebraic target candidate; actual P1 field map, state/RP, physical time/locality/EM and complete TFPT remain open.
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Result
For Theta(W(f))=W(Lf), scalar sign c and product-order sign o, the ordinary Weyl relation forces sigma(Lf,Lg)=c o sigma(f,g). A conjugate-linear order-reversing map requires symplectic L. The previously conditional Theta(Wf)=W(-r f) is antisymplectic on the same single chiral current and is excluded under these type premises. A compactly supported local witness g=f-prime has integral f g-prime=-integral(f-prime)^2 nonzero. A common field minus or Weyl characters do not fix the central phase. Conjugate-linear multiplicative and linear order-reversing reconstructed types pass the single-current test. Oppositely oriented faces with swap and r pass the anti/anti algebraic test conditionally.64 declared variants are checked exactly by matrix forms and directly multiplied rationally scaled Weyl words, with independent normal/-OO review.
Required assumptions
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.
Given eightfold even CAR quotient with marked charge and source time
Use existing fold eta_r(x)=psi((x+2pi r)/8)/sqrt8, local even rank8 CAR algebra on punctured quotient circle, fixed D8 Cartan chart and q=(1)^8, original current normal ordering/Weyl rule, and declared K_alpha source time. DHR sector labels/local von Neumann statement use standard locally normal even-CAR reference/Haag-dual vacuum-net representation. Assume neither normal local P1 single-interval identification,primitiveOS time,spin-structure sum nor bounded sharp spinor field.
Ordinary Weyl products and explicitly specified reflection type
Compare the given chiral current Weyl relations under a real linear profile map L, retaining scalar linearity/conjugation and product preservation/reversal as separate explicit types. Transfer of the OS Grassmann word type to ordinary Weyl products is the tested identification hypothesis, not a proved P1 property. An oppositely oriented face pair remains a conditional target type.
Exact comparison within given actual current representation,not new physics or P1-selected weights. ProvesPhi alone cannot identify source andspinor Kraus operators are not necessary for everyPhi realization. NS14 does not embed allH40 states inrawNS vacuum.4+6 is internal grading of ten pair labels,not new physical3+2 or spacetime dimension. No exclusion of neutral DHR route or allP1 sources.
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Result
Same FoldM splits Ward input40/output40 asNS20+spinor20 andcreation30 asNS14+spinor16. All360 signed columns are168 sector-preserving and192 sector-changing. UniformNS14 has effects3,4,5 and no scalar TP normalization. Positive weights1/6 on8 actualNS roots and1/3 on6 give exactTP onH40 and family gainI4/3 for everyC_m,hence identical fullPhi marginal even for entanglement. This different process preserves source sector. Physical color e3-e2 takesNS e2+e6 tospinor e3+e6;NS14 support lacks full color covariance. Source gradingw=Me1=(e1+e2+e6+e7)/2 factorsPC tensorPF withcarrier dimensions4+6;pureNS20 inputs have only52 of100 carrier matrix directions.
Required assumptions
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.
Given eightfold even CAR quotient with marked charge and source time
Use existing fold eta_r(x)=psi((x+2pi r)/8)/sqrt8, local even rank8 CAR algebra on punctured quotient circle, fixed D8 Cartan chart and q=(1)^8, original current normal ordering/Weyl rule, and declared K_alpha source time. DHR sector labels/local von Neumann statement use standard locally normal even-CAR reference/Haag-dual vacuum-net representation. Assume neither normal local P1 single-interval identification,primitiveOS time,spin-structure sum nor bounded sharp spinor field.
Given E8 fold with retained source charge and P2 grading
Use the same explicit Weyl matrix M from v16-operator-pullback-20261014/fold-results.json, mu_alpha=(j+alpha q)/8 with q=(1)^8 and alpha=1/4 or3/4, Qsource=(Mq).H, and original P2 Cartan charge X=(-2,-2,-2,3,3,0,0,0). Given D5+A3 target frame and compact E8 current representation. Do not assume raw local P1 spinor-field map, executed spectral filter, new physical time or changed source phase.
Conditional normalized-source criterion,not proof P1 suppliesL_gamma or executes a chosen intervention. Priority is original local iota_I preserving support,state,reflection,true time and marked Ward response. For claimed correlation kernel do not require added measurement instrument. Full field/spacetime claims retain stronger conditions.
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Result
Pure observable response can be written T(a)=sum L_beta†aL_beta+sum L_gamma†rho_s(a)L_gamma for given localized star automorphism and bounded local observable coefficients. Unitality fixes normalization; each term is CP. Individual charged spinor need not belong toA. Full local E8/Q-system extension is stronger realization for charged individual fields and0<->s field coherences. Spinor endomorphism and smooth transport now constructed on given quotient net are not another missing choice there.
Required assumptions
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.
Given eightfold even CAR quotient with marked charge and source time
Use existing fold eta_r(x)=psi((x+2pi r)/8)/sqrt8, local even rank8 CAR algebra on punctured quotient circle, fixed D8 Cartan chart and q=(1)^8, original current normal ordering/Weyl rule, and declared K_alpha source time. DHR sector labels/local von Neumann statement use standard locally normal even-CAR reference/Haag-dual vacuum-net representation. Assume neither normal local P1 single-interval identification,primitiveOS time,spin-structure sum nor bounded sharp spinor field.
Given E8 fold with retained source charge and P2 grading
Use the same explicit Weyl matrix M from v16-operator-pullback-20261014/fold-results.json, mu_alpha=(j+alpha q)/8 with q=(1)^8 and alpha=1/4 or3/4, Qsource=(Mq).H, and original P2 Cartan charge X=(-2,-2,-2,3,3,0,0,0). Given D5+A3 target frame and compact E8 current representation. Do not assume raw local P1 spinor-field map, executed spectral filter, new physical time or changed source phase.
Closes algebraic DHR existence/smooth neutral line on given quotient net. Proven bounded implementers,localized Ad action,and local vN transporters in Haag-dual vacuum-net representation; do not infer norm-C* membership from kernel. Sharp step violatesH1/2; exponent2 and renormalized matrix elements are determined,general closability/UV operator normalization require energy bounds/domains. Quotient interval has eight cover arcs. No raw P1 single-interval,OS-time,field-channel or executed-CP proof.
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Result
On given even quotient CAR net g_I=exp(i a_I/2),with exterior values a_I=0,2pi,defines localized automorphism. Branch signs are trivial parity on even words. Fixed q chart labels s=q/2,Q=4,h=1; odd pseudolocality alone does not distinguish s/c. For two localizations b_IJ=(a_I-a_J)/2 is smooth periodic and existing W(-b_IJ oF) implements transport. Exact fold gives J_tot,n=J_cover,8n,level8 and8(1/2)^2=2=s². Original CCR,adjoint and Hall/Wick functionals determine smooth transport histories. K_alpha time gives exp(-itn); unscaled cover time exp(-i8tn). Retain Q zero modes and charged-field action.
Required assumptions
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.
Given eightfold even CAR quotient with marked charge and source time
Use existing fold eta_r(x)=psi((x+2pi r)/8)/sqrt8, local even rank8 CAR algebra on punctured quotient circle, fixed D8 Cartan chart and q=(1)^8, original current normal ordering/Weyl rule, and declared K_alpha source time. DHR sector labels/local von Neumann statement use standard locally normal even-CAR reference/Haag-dual vacuum-net representation. Assume neither normal local P1 single-interval identification,primitiveOS time,spin-structure sum nor bounded sharp spinor field.
Given E8 fold with retained source charge and P2 grading
Use the same explicit Weyl matrix M from v16-operator-pullback-20261014/fold-results.json, mu_alpha=(j+alpha q)/8 with q=(1)^8 and alpha=1/4 or3/4, Qsource=(Mq).H, and original P2 Cartan charge X=(-2,-2,-2,3,3,0,0,0). Given D5+A3 target frame and compact E8 current representation. Do not assume raw local P1 spinor-field map, executed spectral filter, new physical time or changed source phase.
Sector-resolved response needs actual designated branch with scalar effect. Exclusive C5 support of unread entire process is required only for unconditional purePhi total response. Orthogonal output grades already define mathematical instrument components; added hardware record is unnecessary. Full process retains actual2/5 Lambda1 carry. No physical event-time law,reset or raw P1 use follows. Process and state remain distinct.
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Result
Previously verified full60 current process has canonical CP outputs E+ from carrier degree2 to3 and E- to1. Creationgain3 times familygain3 and annihilationgain2 times3 with totalgain15 give E+*(I)=3I40/5 and E-*(I)=2I40/5. Thus Q+=(5/3)E+ is CPTP and equals direct30 normalization1/9. Tr_F Q+=Phi Tr_F holds for entangled inputs. More generally L_r=sum_m C_m tensorF_rm with effect pI and independence of25 C_m†C_n forces operator-valued sum_r F_rm†F_rn=(p/3)delta_mn I, hence Tr_F E=pPhi Tr_F without uniform weights.
Required assumptions
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.
Given local E8_1 vacuum representation and explicit cocycle
Positive level-one E8 lattice vacuum representation in orthonormal D5+D3 coordinates, declared simple-root cocycle, level-one Heisenberg Ward rules and internal quarter character v=(0^5,1/2^3). Insertions are separated. No raw-P1 pullback or physical time law is assumed or derived from this representation.
Given E8 fold with retained source charge and P2 grading
Use the same explicit Weyl matrix M from v16-operator-pullback-20261014/fold-results.json, mu_alpha=(j+alpha q)/8 with q=(1)^8 and alpha=1/4 or3/4, Qsource=(Mq).H, and original P2 Cartan charge X=(-2,-2,-2,3,3,0,0,0). Given D5+A3 target frame and compact E8 current representation. Do not assume raw local P1 spinor-field map, executed spectral filter, new physical time or changed source phase.