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.
Read the exact result, assumptions and evidence
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.
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.
Read the exact result, assumptions and evidence
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.
Conditional on the same full CAR source, support-preserving source time and local vacuum separation, not merely finite compression4I. Exact preparation from same sea by ONE bounded proper-local operator, including products fitting in a larger proper interval. No approximation/probability/cost bound, no ban on unbounded field insertions, other initial states, global protocols or all P1 physics. NS/quarter reference transport explicit; physical P1 net identification remains open. No classification of abstract AQFT-local CP maps without such local Kraus implementers.
Read the exact result, assumptions and evidence
Result
In the given full positive CAR vacuum net, local separation of Omega and A Omega=psi,Hpsi=Epsi,E>0 for bounded proper-local A imply alpha_t(A)=exp(itE)A. Then A*Omega would have energy-E; positivity and separation force A0. Actual original constructor confirms both quarters: five full source states neutral in every residue, only H_alpha32, Gram rank5, supports694/710/711/713/688. Thus the lemma covers all nonzero actual R_b4 target directions without a closed-code assumption. Positivity extends the exclusion to pure or mixed exact E32 outputs of normal instruments with finite/countable bounded Kraus operators in the same proper interval algebra: every Kraus vector would lie in that positive eigenspace and vanishes.
Required assumptions
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.
Actual native G31 event representation
Use the actual original Hamming/E8/J matrices and60 reflections, their four-dimensional action, characteristic P64=O2(G31), and existing S6 quotient. These are accepted mathematical core data, not a physical source derived from P1 in this round.
Symmetric quartic and actual erasure intertwiner
Use Sym^4 of the native four-representation, the actual five codeword sums, and the existing one-register erasure intertwiner. The symmetric tensor rule and its physical selection are not silently inferred from the full tensor product.
Actual positive Cartan quartic Fock embedding
Use original Construction-A roots with physical metric dot/2 and u_j=(h_2j-i h_2j+1)/sqrt2 as J=+i basis. Canonically embed the actual five Sym^4 code support sums as neutral grade4 Heisenberg polynomials in the given E8_1 lattice vacuum. No physical preparation follows.
Original displayed J-equivariant root isometry
Use existing S=_SIMPLE and T=_TARGET_SIMPLE from cartan_source._glue_deck_character_certificate: O=T S^-1/sqrt2 maps actual Construction-A roots r/sqrt2 to original Chevalley roots r/2, preserving metric and J. This is the existing concrete map, not a newly fitted embedding; physical P1 selection stays separate.
Full existing local E8 field algebra with observer five-code
Accepted field-algebra scope: In accepted original E8_1 lattice vacuum use all existing Cartan/root currents, normal products, derivatives and local smooth smearings; native five-code is input/observer slice, not assumed dynamically closed. Retain all exterior charges/modes and both actual twenties. P,Q only decompose analytic endpoint/memory formulas, never execute intermediate projection. Global mode products of local fields are not localized instruments. PhysicalP1 pullback, intervention history, state/time stay separate. Intrinsic tensor/real-form/simplex reconstruction is not a postulated physical potential/state rule.
Fixed conformal current reference in original fold-state transport
Continuation17Oct: same givenchiralCARquarterfold,sea,F Cα F*,existingM/literalO; no newcopies/clock/coupling/selectionrule. NativehidentifiedherewithFIXEDconformalNSpointsplitcurrents;countertermagainstsea-normalorderedHfollowskerneldifference. ThisreferenceidentificationisCONDITIONALdictionarypremise,notphysicalP1originfromM/O. OriginalQWZpositivej≥1andnegativeDfoldorientationwithcharge/current/particleholetransportseparate;bothquartersretained. E8DeltatwistedextensionandoriginalglobalW4fromexistingevenCAR/E8cocyclefactorizationremaingivenextension,nooddCAR/E8localityproof. Noreselectedgroundstate;sourcechargesinitiallyZ8. Fullproducts/oppositepolarization/exteriormodes/charges,noperformedcodeprojection;originali0boundaryvalues,nofittedcontactterms.
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.
Same full positive CAR source with locally separating sea
Given negative-D quarter-CAR representation with full H_alpha=sum(n+alpha):a_n* a_n:>=0,H Omega=0 and support-preserving time covariance. Omega separates each larger proper-interval algebra. Test class: bounded proper-local A with A Omega an exact positive-energy eigenvector. NS/quarter chart retains support, reference and time; no physical P1 net identification follows.
ONLY implication from named weakercircleconditions to fixedoperatorWard excluded; premise list deliberately does NOT include fixedWard. No completeP1/E8countermodel or physical3+2/eventno-go. Do not identify rawP1 seamindex symbol with givensourcecircle without actuallocalization/operator dictionary.
Read the exact result, assumptions and evidence
Result
For arbitrarya>0,Z*BZ=B+a,pathB_s=B+s a has eigenvaluesomega(n+beta+s):0<beta<1 gives exactlyone upwardcrossing independentofa. UnitSF is not constantunitWard. Exact controls a=(1+epsilon cos(ktheta))^-1,c=alpha,abs(epsilon)<1,k1/4 have samegap1/4,holonomy,SF1;k4 additionally preservesall4marks,equalarctimes/geometricquartercovariance. Yet[B,Z]=aZ differsfromZ forepsilon!=0. Controls not claimed completeP1models; under fullcovariantdatatransport they represent sameunmarkedsource.
Required assumptions
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.
Whole scalar positive first-order comparison class
Compare all smooth scalar first-order differential operators on already given oriented source circle, fixedL2(dtheta),H1per,positive nowhere-zero principal symbol. Periodic chart contains quarterphase in c/holonomy; do not add domain twist again. Original Balpha is reference point; comparison operators are not all claimed physically admissibleP1 data. Point defects,matrix fibers,nonlocal and general higher-dimensional sources are outside.
Ward retained as already given-source object, NOT derived fromP1 spectral flow. Separate continuous marked gauge fiber and globalinteger branches. No autonomous clock/new species/unchangedFockstate/physical intervention inferred. No3+2 derivation or closedC5 requirement.
Read the exact result, assumptions and evidence
Result
Whole class satisfies[B,Z]=aZ. Retaining same original constantWard[B,Z]=Z forcesa1 without metric minimization. Given wind1symbolu=exp(iphi) requiresa phi_prime=kappa,henceomegakappa. Additional fixed quarter rotation makesc quarterperiodic; continuous originalbranch mean c=alpha leaves quarterperiodic gauge transport fixingZ/all4marks. Globally retainmean c=alpha+N: actualQalpha=exp(i pi alpha/2)Q4,Qalpha^4=exp(2pi i alpha)I,joint invariant exp(i pi B/2)Qalpha*=i^N. Literalclock/endpointframes retainNmod4, exactoriginaltimequarter equality forcesN0mod4;4Zlargegauge/sea carry not deleted. Reference sea fixes originalN0branch.
Required assumptions
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.
Whole scalar positive first-order comparison class
Compare all smooth scalar first-order differential operators on already given oriented source circle, fixedL2(dtheta),H1per,positive nowhere-zero principal symbol. Periodic chart contains quarterphase in c/holonomy; do not add domain twist again. Original Balpha is reference point; comparison operators are not all claimed physically admissibleP1 data. Point defects,matrix fibers,nonlocal and general higher-dimensional sources are outside.
Actual fixed one-particle time-seam Ward of given source
Already given source Balpha=-i dtheta+alpha,Z=multiplication exp(itheta) satisfies[Balpha,Z]=Z, equivalently exp(-itBalpha)Z exp(itBalpha)=exp(-it)Z. Retaining THIS concrete operator relation is stronger than unit spectral flow. It is given-source datum, not a rule derived here from bareP1. Do not identify one-particle Ward with charge-dependent CAR sea term or physical second. Charged quarterlift Qalpha=exp(i pi alpha/2)Q4,reference sea and integer carry retained separately.
Only smooth scalar positive elliptic local differential class on given source. Not all minimalP1 fiber or matrix/defect/nonlocal/E8 classification. One-particle unitarity permits no reset of CAR reference sea,normal order,Schwarz/central term,J or relative action. Retain native sixrays/60flags/globalpair/full field response.
Read the exact result, assumptions and evidence
Result
Every operator in declared class has B=-i(a d+a_prime/2)+c,a>0,c real. T=integral1/a,omega=2pi/T,beta=integral c/a/(2pi),x_prime=omega/a,eta_prime=(omega beta-c)/a yield unitaryUf=sqrt(omega/a)exp(ieta)f(x),BU=Uomega(-i dx+beta) on fullH1 domains. Whole spectrum omega(Z+beta), no mode cutoff. SameB determines unique-up-to-phase uniform wind1 representative u_B=exp(ix),[B,u_B]=omega u_B andQ_B=exp(iTB/4),Q_B^4=exp(2pi i beta)I: spacing,proper-timecoordinate/chargedquarterreturn fromoneoperator,not newclock/interventionselection. Original fixedquarter holonomy andgap1/4 giveomega1; unmarked one-particle source sameBalpha up to coordinate/phase transport. Four fixed marks require all arc travel times/possible endpoint phases, not dimension/species identification.
Required assumptions
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.
Whole scalar positive first-order comparison class
Compare all smooth scalar first-order differential operators on already given oriented source circle, fixedL2(dtheta),H1per,positive nowhere-zero principal symbol. Periodic chart contains quarterphase in c/holonomy; do not add domain twist again. Original Balpha is reference point; comparison operators are not all claimed physically admissibleP1 data. Point defects,matrix fibers,nonlocal and general higher-dimensional sources are outside.
ONLY literal identification of this canonical gap-CAR component with sharp native3+2 mark excluded. Qgap itself NOT claimed P1-selected physical event; no measurement, executed projection, closedC5 requirement, no exclusion of all unsharp/nonlinear events or allP1. Boundary eigenmode/internal direction/physical species distinct; original sixrays/60flags/globalpair/E8 retained.
Read the exact result, assumptions and evidence
Result
Actual original gap definition selects mode0 foralpha1/4 and mode-1 foralpha3/4, gap1/4 each. Within same stipulated CAR, Qgap=sgn(E0):a_r*a_r: is full particle/hole projector without rank fitting. On complete actual R_b4 inputs, B=I*I,H=I*QgapI obey EXACTLY H>0 and B-H>0, both rank5. Correct metric response strictly between0 andI, nonscalar/nonidempotent; exterior H-HB^-1H strictly positive rank5, bothquarters. This direct spectral bridge from original scale function is not a sharp3+2 input mark. Qgap is global spectral mode, not insertion localized in proper interval.
Required assumptions
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.
Actual native G31 event representation
Use the actual original Hamming/E8/J matrices and60 reflections, their four-dimensional action, characteristic P64=O2(G31), and existing S6 quotient. These are accepted mathematical core data, not a physical source derived from P1 in this round.
Symmetric quartic and actual erasure intertwiner
Use Sym^4 of the native four-representation, the actual five codeword sums, and the existing one-register erasure intertwiner. The symmetric tensor rule and its physical selection are not silently inferred from the full tensor product.
Actual positive Cartan quartic Fock embedding
Use original Construction-A roots with physical metric dot/2 and u_j=(h_2j-i h_2j+1)/sqrt2 as J=+i basis. Canonically embed the actual five Sym^4 code support sums as neutral grade4 Heisenberg polynomials in the given E8_1 lattice vacuum. No physical preparation follows.
Original displayed J-equivariant root isometry
Use existing S=_SIMPLE and T=_TARGET_SIMPLE from cartan_source._glue_deck_character_certificate: O=T S^-1/sqrt2 maps actual Construction-A roots r/sqrt2 to original Chevalley roots r/2, preserving metric and J. This is the existing concrete map, not a newly fitted embedding; physical P1 selection stays separate.
Full existing local E8 field algebra with observer five-code
Accepted field-algebra scope: In accepted original E8_1 lattice vacuum use all existing Cartan/root currents, normal products, derivatives and local smooth smearings; native five-code is input/observer slice, not assumed dynamically closed. Retain all exterior charges/modes and both actual twenties. P,Q only decompose analytic endpoint/memory formulas, never execute intermediate projection. Global mode products of local fields are not localized instruments. PhysicalP1 pullback, intervention history, state/time stay separate. Intrinsic tensor/real-form/simplex reconstruction is not a postulated physical potential/state rule.
Fixed conformal current reference in original fold-state transport
Continuation17Oct: same givenchiralCARquarterfold,sea,F Cα F*,existingM/literalO; no newcopies/clock/coupling/selectionrule. NativehidentifiedherewithFIXEDconformalNSpointsplitcurrents;countertermagainstsea-normalorderedHfollowskerneldifference. ThisreferenceidentificationisCONDITIONALdictionarypremise,notphysicalP1originfromM/O. OriginalQWZpositivej≥1andnegativeDfoldorientationwithcharge/current/particleholetransportseparate;bothquartersretained. E8DeltatwistedextensionandoriginalglobalW4fromexistingevenCAR/E8cocyclefactorizationremaingivenextension,nooddCAR/E8localityproof. Noreselectedgroundstate;sourcechargesinitiallyZ8. Fullproducts/oppositepolarization/exteriormodes/charges,noperformedcodeprojection;originali0boundaryvalues,nofittedcontactterms.
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.
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.
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.
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 gap-normalized relative P1 action
Original source defines chi_seed=1/lambda_1^+(|B_Sigma|), S_prim=Tr f(D_rel/chi)-Tr f(D_ref/chi)+i pi Deltaeta/2 for declared spectral test profile. D_rel/ref are geometric odd/even parts; varying reflection, boundary domains, reference, state and eta must be retained. This is the given source function, not a new coupling or physical event rule. Scalar quarter CAR realization and NS/native field dictionary remain explicitly conditional.
Regulated trace or controlled trace-class source curve with legitimate domains. Circle lemma does not imply entire relative-action isospectrality with fixedR; same phi symbol is no physical insertion dictionary. Given scalarCAR/reference conditional relative toP1. Earlier P1-INSERTION-TANGENT/master/minimality boundaries reused. Physical deltaD/deltaB/embedding/event relation remains open.
Read the exact result, assumptions and evidence
Result
chi=1/lambda forces delta(D/chi)=lambda deltaD+deltaLambda D. For isolated simple signed gap branch deltaLambda=sgn(E0)<u0,deltaB u0>; entire corresponding term in relative action differential derived without commutativity. Clusters, constant nullity, moving R/domain/reference/eta correctly scoped. In actual scalar quarter circle, EVERY smooth real v gives Balpha+v unitarily equivalent to Balpha+mean(v). On fixed holonomy branch mean(v)=0, gap/chi unchanged exactly while Pgap moves. This supplies a source covector, not a selected physical source variation/event functional; scalar gauge orbit is not inequivalent-minimal-P1 model theorem.
Required assumptions
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.
Actual native G31 event representation
Use the actual original Hamming/E8/J matrices and60 reflections, their four-dimensional action, characteristic P64=O2(G31), and existing S6 quotient. These are accepted mathematical core data, not a physical source derived from P1 in this round.
Symmetric quartic and actual erasure intertwiner
Use Sym^4 of the native four-representation, the actual five codeword sums, and the existing one-register erasure intertwiner. The symmetric tensor rule and its physical selection are not silently inferred from the full tensor product.
Actual positive Cartan quartic Fock embedding
Use original Construction-A roots with physical metric dot/2 and u_j=(h_2j-i h_2j+1)/sqrt2 as J=+i basis. Canonically embed the actual five Sym^4 code support sums as neutral grade4 Heisenberg polynomials in the given E8_1 lattice vacuum. No physical preparation follows.
Original displayed J-equivariant root isometry
Use existing S=_SIMPLE and T=_TARGET_SIMPLE from cartan_source._glue_deck_character_certificate: O=T S^-1/sqrt2 maps actual Construction-A roots r/sqrt2 to original Chevalley roots r/2, preserving metric and J. This is the existing concrete map, not a newly fitted embedding; physical P1 selection stays separate.
Full existing local E8 field algebra with observer five-code
Accepted field-algebra scope: In accepted original E8_1 lattice vacuum use all existing Cartan/root currents, normal products, derivatives and local smooth smearings; native five-code is input/observer slice, not assumed dynamically closed. Retain all exterior charges/modes and both actual twenties. P,Q only decompose analytic endpoint/memory formulas, never execute intermediate projection. Global mode products of local fields are not localized instruments. PhysicalP1 pullback, intervention history, state/time stay separate. Intrinsic tensor/real-form/simplex reconstruction is not a postulated physical potential/state rule.
Fixed conformal current reference in original fold-state transport
Continuation17Oct: same givenchiralCARquarterfold,sea,F Cα F*,existingM/literalO; no newcopies/clock/coupling/selectionrule. NativehidentifiedherewithFIXEDconformalNSpointsplitcurrents;countertermagainstsea-normalorderedHfollowskerneldifference. ThisreferenceidentificationisCONDITIONALdictionarypremise,notphysicalP1originfromM/O. OriginalQWZpositivej≥1andnegativeDfoldorientationwithcharge/current/particleholetransportseparate;bothquartersretained. E8DeltatwistedextensionandoriginalglobalW4fromexistingevenCAR/E8cocyclefactorizationremaingivenextension,nooddCAR/E8localityproof. Noreselectedgroundstate;sourcechargesinitiallyZ8. Fullproducts/oppositepolarization/exteriormodes/charges,noperformedcodeprojection;originali0boundaryvalues,nofittedcontactterms.
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.
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.
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.
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 gap-normalized relative P1 action
Original source defines chi_seed=1/lambda_1^+(|B_Sigma|), S_prim=Tr f(D_rel/chi)-Tr f(D_ref/chi)+i pi Deltaeta/2 for declared spectral test profile. D_rel/ref are geometric odd/even parts; varying reflection, boundary domains, reference, state and eta must be retained. This is the given source function, not a new coupling or physical event rule. Scalar quarter CAR realization and NS/native field dictionary remain explicitly conditional.
Internal order4 character remains distinct from actual quarter holonomy. Retain odd CAR parity; no physical holonomy compensation or clock selection. Energy response forms are not an executed measurement or entire local E8 algebra.
Read the exact result, assumptions and evidence
Result
Existing full internal seam character v=(0^5,1/2^3) gives original CAR phases[-1,i,1,-i]^2 via same M. Exactly tv Jm tv^-1=i^mJm and likewise original Lcal_m^alpha; full products carry total Fourier grading including central OPE terms. Known global tv=exp(-4pi iK)exp(i pi(1+alpha/2)Q)=(-1)^Q Walpha^-1 reused on all charges, not claimed new. Entire32-mode energy response splits without profile fitting into four original character classes Grho summing to32B4, both quarters.
Required assumptions
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.
Actual native G31 event representation
Use the actual original Hamming/E8/J matrices and60 reflections, their four-dimensional action, characteristic P64=O2(G31), and existing S6 quotient. These are accepted mathematical core data, not a physical source derived from P1 in this round.
Symmetric quartic and actual erasure intertwiner
Use Sym^4 of the native four-representation, the actual five codeword sums, and the existing one-register erasure intertwiner. The symmetric tensor rule and its physical selection are not silently inferred from the full tensor product.
Actual positive Cartan quartic Fock embedding
Use original Construction-A roots with physical metric dot/2 and u_j=(h_2j-i h_2j+1)/sqrt2 as J=+i basis. Canonically embed the actual five Sym^4 code support sums as neutral grade4 Heisenberg polynomials in the given E8_1 lattice vacuum. No physical preparation follows.
Original displayed J-equivariant root isometry
Use existing S=_SIMPLE and T=_TARGET_SIMPLE from cartan_source._glue_deck_character_certificate: O=T S^-1/sqrt2 maps actual Construction-A roots r/sqrt2 to original Chevalley roots r/2, preserving metric and J. This is the existing concrete map, not a newly fitted embedding; physical P1 selection stays separate.
Full existing local E8 field algebra with observer five-code
Accepted field-algebra scope: In accepted original E8_1 lattice vacuum use all existing Cartan/root currents, normal products, derivatives and local smooth smearings; native five-code is input/observer slice, not assumed dynamically closed. Retain all exterior charges/modes and both actual twenties. P,Q only decompose analytic endpoint/memory formulas, never execute intermediate projection. Global mode products of local fields are not localized instruments. PhysicalP1 pullback, intervention history, state/time stay separate. Intrinsic tensor/real-form/simplex reconstruction is not a postulated physical potential/state rule.
Fixed conformal current reference in original fold-state transport
Continuation17Oct: same givenchiralCARquarterfold,sea,F Cα F*,existingM/literalO; no newcopies/clock/coupling/selectionrule. NativehidentifiedherewithFIXEDconformalNSpointsplitcurrents;countertermagainstsea-normalorderedHfollowskerneldifference. ThisreferenceidentificationisCONDITIONALdictionarypremise,notphysicalP1originfromM/O. OriginalQWZpositivej≥1andnegativeDfoldorientationwithcharge/current/particleholetransportseparate;bothquartersretained. E8DeltatwistedextensionandoriginalglobalW4fromexistingevenCAR/E8cocyclefactorizationremaingivenextension,nooddCAR/E8localityproof. Noreselectedgroundstate;sourcechargesinitiallyZ8. Fullproducts/oppositepolarization/exteriormodes/charges,noperformedcodeprojection;originali0boundaryvalues,nofittedcontactterms.
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.
Given c1 CAR source and explicitly conditional NS/native dictionary, not all E8/P1. Hall is a STATE/current intertwiner, NOT a point-local cover identification with weight32 field. Finite stress-mode words yes; arbitrary exponentiated geometry or new unsmeared coincidences require analytic control. Mixed amplitudes retaining implementer are not bare vacuum Pfaffians. No physical profile/reference/readout selection.
Read the exact result, assumptions and evidence
Result
Actual five neutral sourceE32 states map exactly by original Maya occupations to Schur states. All32 lowering-current VECTORS independently match Young ribbons, stress1/2 matches fixed Sugawara and full CCR, both quarters. All state coefficients saved. Kij(u,v)=<Pi,exp(sum um pm)Pj(pm+mvm)>Hall gives all mixed current correlations; Weyl response additionally retains Gaussian sum over ALL profile modes, not only1..32. Every finite smooth density/source-time history follows with original BCH phase. Actual folded quartic fields remain on eight original preimages: A_a=Ualpha diag(Vka)Ualpha*, scalar source operation gives Gphi(x+)dagger A_a Gphi(x-). Entire existing point splitting/reference subtraction conjugated, no second b shift. Separated Heisenberg field products follow from finite Wick Pfaffians using same ordered Calpha/1-Calpha.
Required assumptions
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.
Actual native G31 event representation
Use the actual original Hamming/E8/J matrices and60 reflections, their four-dimensional action, characteristic P64=O2(G31), and existing S6 quotient. These are accepted mathematical core data, not a physical source derived from P1 in this round.
Symmetric quartic and actual erasure intertwiner
Use Sym^4 of the native four-representation, the actual five codeword sums, and the existing one-register erasure intertwiner. The symmetric tensor rule and its physical selection are not silently inferred from the full tensor product.
Actual positive Cartan quartic Fock embedding
Use original Construction-A roots with physical metric dot/2 and u_j=(h_2j-i h_2j+1)/sqrt2 as J=+i basis. Canonically embed the actual five Sym^4 code support sums as neutral grade4 Heisenberg polynomials in the given E8_1 lattice vacuum. No physical preparation follows.
Original displayed J-equivariant root isometry
Use existing S=_SIMPLE and T=_TARGET_SIMPLE from cartan_source._glue_deck_character_certificate: O=T S^-1/sqrt2 maps actual Construction-A roots r/sqrt2 to original Chevalley roots r/2, preserving metric and J. This is the existing concrete map, not a newly fitted embedding; physical P1 selection stays separate.
Full existing local E8 field algebra with observer five-code
Accepted field-algebra scope: In accepted original E8_1 lattice vacuum use all existing Cartan/root currents, normal products, derivatives and local smooth smearings; native five-code is input/observer slice, not assumed dynamically closed. Retain all exterior charges/modes and both actual twenties. P,Q only decompose analytic endpoint/memory formulas, never execute intermediate projection. Global mode products of local fields are not localized instruments. PhysicalP1 pullback, intervention history, state/time stay separate. Intrinsic tensor/real-form/simplex reconstruction is not a postulated physical potential/state rule.
Fixed conformal current reference in original fold-state transport
Continuation17Oct: same givenchiralCARquarterfold,sea,F Cα F*,existingM/literalO; no newcopies/clock/coupling/selectionrule. NativehidentifiedherewithFIXEDconformalNSpointsplitcurrents;countertermagainstsea-normalorderedHfollowskerneldifference. ThisreferenceidentificationisCONDITIONALdictionarypremise,notphysicalP1originfromM/O. OriginalQWZpositivej≥1andnegativeDfoldorientationwithcharge/current/particleholetransportseparate;bothquartersretained. E8DeltatwistedextensionandoriginalglobalW4fromexistingevenCAR/E8cocyclefactorizationremaingivenextension,nooddCAR/E8localityproof. Noreselectedgroundstate;sourcechargesinitiallyZ8. Fullproducts/oppositepolarization/exteriormodes/charges,noperformedcodeprojection;originali0boundaryvalues,nofittedcontactterms.
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.
Twisted and unshifted stress/input pairs must not be identified. Prior scoped exclusions retained. Exact CAR window follows maximum source energy33; no physical time calibration or external coupling.
Read the exact result, assumptions and evidence
Result
Original c1 Sugawara identity 8K=Q(Q-1)/2+alpha Q+sum_m>0 J-m Jm is checked on full R_b,4 inputs as all32 response Grams summing to32B4; higher lowering modes vanish by positive source energy, not chosen truncation. Original Lcal_m^alpha has fixed r+m/2+alpha coefficients and ground weight(alpha-1/2)^2/2. Full vector Wards [Lcal1,J-1]=Q+alpha-1/2 and [Lcal1,Lcal-1]=2Lcal0 hold. Folded c8 geometry is twisted Yd(T), Lhat_n=Lcal_8n/8+21/64delta. Original Lcal+-1 changes fold energy by1/8 with cyclic root charge outputs, outside prior unshifted c8 stress/primary-vacuum theorem.
Required assumptions
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.
Actual native G31 event representation
Use the actual original Hamming/E8/J matrices and60 reflections, their four-dimensional action, characteristic P64=O2(G31), and existing S6 quotient. These are accepted mathematical core data, not a physical source derived from P1 in this round.
Symmetric quartic and actual erasure intertwiner
Use Sym^4 of the native four-representation, the actual five codeword sums, and the existing one-register erasure intertwiner. The symmetric tensor rule and its physical selection are not silently inferred from the full tensor product.
Actual positive Cartan quartic Fock embedding
Use original Construction-A roots with physical metric dot/2 and u_j=(h_2j-i h_2j+1)/sqrt2 as J=+i basis. Canonically embed the actual five Sym^4 code support sums as neutral grade4 Heisenberg polynomials in the given E8_1 lattice vacuum. No physical preparation follows.
Original displayed J-equivariant root isometry
Use existing S=_SIMPLE and T=_TARGET_SIMPLE from cartan_source._glue_deck_character_certificate: O=T S^-1/sqrt2 maps actual Construction-A roots r/sqrt2 to original Chevalley roots r/2, preserving metric and J. This is the existing concrete map, not a newly fitted embedding; physical P1 selection stays separate.
Full existing local E8 field algebra with observer five-code
Accepted field-algebra scope: In accepted original E8_1 lattice vacuum use all existing Cartan/root currents, normal products, derivatives and local smooth smearings; native five-code is input/observer slice, not assumed dynamically closed. Retain all exterior charges/modes and both actual twenties. P,Q only decompose analytic endpoint/memory formulas, never execute intermediate projection. Global mode products of local fields are not localized instruments. PhysicalP1 pullback, intervention history, state/time stay separate. Intrinsic tensor/real-form/simplex reconstruction is not a postulated physical potential/state rule.
Fixed conformal current reference in original fold-state transport
Continuation17Oct: same givenchiralCARquarterfold,sea,F Cα F*,existingM/literalO; no newcopies/clock/coupling/selectionrule. NativehidentifiedherewithFIXEDconformalNSpointsplitcurrents;countertermagainstsea-normalorderedHfollowskerneldifference. ThisreferenceidentificationisCONDITIONALdictionarypremise,notphysicalP1originfromM/O. OriginalQWZpositivej≥1andnegativeDfoldorientationwithcharge/current/particleholetransportseparate;bothquartersretained. E8DeltatwistedextensionandoriginalglobalW4fromexistingevenCAR/E8cocyclefactorizationremaingivenextension,nooddCAR/E8localityproof. Noreselectedgroundstate;sourcechargesinitiallyZ8. Fullproducts/oppositepolarization/exteriormodes/charges,noperformedcodeprojection;originali0boundaryvalues,nofittedcontactterms.
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.
Only given global free source time/W_n/Q phases, not all local fields. No closed-C5 axiom or performed projection. Total time acts on grade superpositions. E8 half-sector needs its lift; global W2 is not local root vertex.
Read the exact result, assumptions and evidence
Result
Given Ward increments are c_n(q)=Falpha(q+n)-Falpha(q), Falpha(q)=q(q-1)/16+alpha q/8. Thus K-Falpha(Q) is winding invariant and alpha independent. Actual CAR space factors via W_n as ell2(Z) tensor H_Q0, not an assumed rotor. Every finite ordered time/winding word has the stated exact normal form; all n=8z+r checked symbolically. On a fixed neutral R_b,k field mode the input multiplicity is unchanged. Charge-return acts scalarly in actual B_k metric, retaining all intermediate charges/modes. W1 lies in original odd vector sector; W2 is smallest even winding and ends at an actual E8 root.
Required assumptions
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.
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.