The same A is assumed on both correctly typed legs; no source origin of A, no identification of the two-mode carrier with C5, and no universal TP exclusion. Other family weights or channels remain possible; interaction formula assumes the specified finite generator decomposition.
Read the exact result, assumptions and evidence
Result
For A on C5, D=Lambda²A and original bilinear C_m:H→H*, D^T C_m D=sum_n adj(A)_(n,m)C_n, including singular A. Invertible A preserves the whole span and the concrete five-cycle the given equally weighted channel. A common scalar normalizes the simple equally weighted channel to trace preservation iff adj(A)adj(A)*=rI, equivalently AA*=sI for invertible A. A=diag(1,1,1,1,1/2) gives effects 1/2 and 1/4. An exact Duhamel integral of [V,a†] gives the free creator-relation defect for supplied H=dGamma(h)+V; the existing two-mode counterexample gives 1/256.
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 massless geometric Cauchy graph and R066 quarter identity. No identification of the graph with the physical whole-state covariance, no general Higgs-coupled matrix identity and no derivation of a physical spacetime dimension. The coupled example is a fixed comparison field, not the dynamical physical source and not a test of the comoving clock.
Read the exact result, assumptions and evidence
Result
For the given free scalar two-face projection, all nonzero Fourier modes of the commutator quadratic response equal exactly the Dirichlet-to-Neumann matrix of a strip of width beta/2 by pointwise identities and telescoping. At beta=4pi and alpha=1/4 or 3/4 the constant relative eigenvalue is half as large: N_source=N_strip−Pi_relative/(2pi), and N_strip/2≤N_source≤N_strip as forms. The normal-ordered variance is Q/2. In the given coupled 2×2 background, the same scalar face operations change nonzero modes as well: at m=0.3, k=1 gives N11≈1.54580712 versus 2.00001395 at m=0 (two internal components), with explicit sum-tail bound 1.26e−7 and floating-point arithmetic.
Required assumptions
Actual whole two-face graph and local matrix-operation class
Given original G=[[1-C,X],[X,C]],C(p)=(1+exp(beta p))^-1,X=(2cosh(beta p/2))^-1,p=r+alpha,alpha1/4,3/4,beta4pi,existing wholeCAR source,unchanged M and periodic8fold B=8(-i∂x)+diag(k+alpha). Entire bounded matrix multiplication for reduction,entire smooth unitary multiplication for Fock implementation. No originalOS/Padm field intertwiner assumed.
Exact fixed source/color Cartan and charge-lattice interface with inherited chiral vertex monodromy. Algebraic center transport is not physical gauge/support/state identification;center neutrality is not SU3 singlet. Retain existing AIA parent,actual spin/defect/covers or nonlocal charged maps;no universal point-local1D dictionary required. No new chosen fermion sector,physical96 map,measure,time,3+1 or spin2. 58 own exact controls plus all-lattice proof,one actual analytic The reviewed review;no new Lean or ledger promotion.
Read the exact result, assumptions and evidence
Result
At unchanged original M,M^T w=(1,-1,0,-1,-1,-1,0,-1),so given Z3 color center extends to all integer NS charges. Exactly±(e1-e2) among six actual color A2 roots lie in M D8;other four have half-integral source charge in every coordinate and full monodromy−1 with each original odd vector generator. This stipulated root-current action is not an internal full SU3 action on the unchanged NS charge space. Common center-neutral even lattice D0=(M D8)∩ker(t mod3) has index6 in E8 and determinant36. E8_center and (M Z8)_center are distinct even/odd index2 neighbors over D0,both determinant9;the latter does not contain the former. R069 no odd overlattice of the ENTIRE E8_center remains unchanged.
Required assumptions
Original fixed color action and actual folded NS charge frame
Retain unchanged E8 root representation, actual rational M in fold-results.json, NS charge space M Z8 and even M D8, R069 color A2=e1-e2,e2-e3 and center w=e1+e2-2e3. Reuse existing four-module AIA parent with actual cocycle/spin/quarter transport. Assume no common physical96-matter map, local inclusion, new gauging, measure law or spacetime from this data identification.
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 marked color center on whole stipulated E8 lattice field source
Actual original rightup color triplet with fixed family vector and even color source
Given original96 Phi(H),gamma_L+1,gamma_R−1,fixedp1/4,ell>0. Samefixed eigenvector of GIVEN Yu*Yu,y²>0,inall3originalrightupcolors;notnewphysicalstateselection. M*M=u diag(Yu*Yu,Yd*Yd) givesactualt(u)=1/2+p tanh(ell sqrt(p²+y²u))/(2sqrt(p²+y²u)) on(a,b],a1/2,b(1+tanh(ell p))/2. Reader assumespositiveevenstateinvariantunderoriginalconstantSU3;finite modularlogrequiresfaithfuldensity. Forcednonlinearity and positivemixturetestONLYadditionallyunderR054positive normalizedwholeHmixturewithnonconstantradialu,includingwholedet/Pf,Ref,Eta,Z_F,Z0. Generalreaderdoesnotrequiremixture. No primitiveP1measure,physicaltimeorfull96modularrestrictionclaim.
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 quark Higgs consumer and actual positive conditional source mixture
Given whole96 Phi(H)=L(H)+DM with original q and fixed supplied Y/MR;existing constant common quarter circle p=1/4,ell>0,h=p gammaF−i gammaF Phi,second-face C_H. Existing charged left-quark CAR subalgebra is conditionally quasifree with A_H tensor I_color3 and zero anomalous pairs. Actual full normalized H weight must give a positive mixture including Z_F/reference/eta and all factors. Strictness requires nonconstant u=HdaggerH and a nonzero quarkY. No new physical96-fold integration,measure derivation,general variable H(x)/connection or3+1D assumption.
Observable algebra, OS reflection, state, collar operator, time evolution, carrier connection, family holonomy, and readout are distinct data until an original map relates them.
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.
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.
Existing C2 group with untwisted raw flat spin pullbacks
On the existing marked parity curve E:y²=x⁴−1,study h=(x,−y),Q=(ix,y),kappa=(bar x,bar y) and flat square-lattice spin characters. The explicit raw one-particle calculation uses the untwisted line alpha with chi(A)=chi(B)=−1. Complex-linear pullbacks U and anti-linear Real structures j are distinct;no fitted U1 phase. This is not the normal twist alpha tensor p*O1 with its two-dimensional kernel. An original P1 Spin/Pin or Calderon transport is not presumed.
Analytic theorem in supplied constant-quarter/H source consumer with positive normalized conditional mixture. Forty targeted controls including original96 builder and independent rational64-state Fock defect4/27;no physical weight selection or newLean. No general nongaussian claim for fixed u,complex weights,other connections or extra conditional fermion vertices. No local4D gauge observable,scattering or C_Sigma/P_F/mu origin derived. Primitive quasifree source BEFORE H integration remains possible. Reuse R026 cumulants and earlier conformal Wick averaging in their own scopes.
Read the exact result, assumptions and evidence
Result
O=3N²−sum_cd E_cd E_dc with E_cd=sum_i a_icdagger a_id is pure quartic and invariant under constant internal colorU3 and commonU6 frames. Any A_H tensor I3 gives <N²>=9m²+3tau,<Q>=3m²+9tau,<O>=24m². On original q(H) quark block,m(u)=3−p/2 sum_y tanh(ell sqrt(p²+y²u))/sqrt(p²+y²u) is strictly monotone for any supplied nonzero quarkY,independent of H direction. Thus full positive nonconstant-radial mixture class has exact defect24Var_mu(m(u))>0 from the Gaussian state with identical full two-point data. Covariance log remains an exact two-point reader,but not a whole H-mixture reconstruction.
Required assumptions
Original quark Higgs consumer and actual positive conditional source mixture
Given whole96 Phi(H)=L(H)+DM with original q and fixed supplied Y/MR;existing constant common quarter circle p=1/4,ell>0,h=p gammaF−i gammaF Phi,second-face C_H. Existing charged left-quark CAR subalgebra is conditionally quasifree with A_H tensor I_color3 and zero anomalous pairs. Actual full normalized H weight must give a positive mixture including Z_F/reference/eta and all factors. Strictness requires nonconstant u=HdaggerH and a nonzero quarkY. No new physical96-fold integration,measure derivation,general variable H(x)/connection or3+1D assumption.
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.
Extends existingSchur/memory results with same geometric spectralreader,rather than relabeling them new.36controls andgeneric algebra. No numericnormalblocks,physicalgap ornewmetric chosen. Finite logdetjets are notFPzeta0 orautomatica2/a4;no multiplicativity of separatezetadets. IndefiniteR gives no universalSchursign. Distinguish parallel/nonparallelP andmovingframes. OriginalB_Ea→Brel/Tr_rel mapnotderived;F1remainsopen.
Read the exact result, assumptions and evidence
Result
PK²P−(PKP)²=PKQKP=(QKP)*QKP≥0;for[P,K0]=0 andK0+tV defectt²PVQVP. Original96P+hasP+PhiP+=0 whileTrP+L(H)²=A|H|²/2. K−z=[[A,B],[C,E]],R=E^-1,S=A−BRC full inverse retains all4blocks. Unweightedfulltrace=TrR−Tr(S^-1 dzS),dzS=−I−BR²C;apply actual weightedtrace to allblocks. (K²+omega²)^-1 uses both K±iomega resolvents;retain same reference/eta and original continuation. Full finite source keepsdetEdetS,two shifted legs and directQterm. Complete affineHjet recursion andlogdetH²/H⁴ with nonzeroB0/C0 andnoncommutingblocks executed.
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 Clifford/form grades,parallel projection and complete relative block reader
Existing whole96 Higgs consumer L(H)+DM,unchanged Y/MR;given conditional spin4 product with fixed Clifford connection. Distinguish ordinary internal two-form from Clifford-valued full notation. Heat comparison assumes parallel spin-compatible orthogonalP and compact interior variation;P+ is actual internal test,not identified with normalP or printedI48/P2. Block formulas exact for same finite regulated operator;infinite domain/relative trace/continuation require original contract. FixedP for affine Hjets;moving frames retain actual induced terms. No choice of physical normal operator,measure,profile,gap or free parameters.
Given normal Higgs modes and distinct module types
Given3+2 carrier V=(3,1)-1/3+(1,2)1/2 with conjugation on End_C(V),given colorless H=(1,2)1/2,real-linear insertion;normal2D spin,weakSU2 and spacetime spin distinct. Reuse original normal B_Ea and Higgs/wedge results. Whole kernel/Gram formulas apply only when actual normal realization is given selfadjoint B or supplied doubling with finite constant kernel and actual scale fBf/domain,prescribed S and insertions on their domains. No new profile,normal metric,physical measure,or index-operator/Brel identification.
Complete linear realization for specified original single-beta L/J alphabet on neutral grade4:204 exact original-vector identities, positive metric,15 source controls and6 structure controls.68 is neither spacetime dimension nor whole E8 source. Values are response amplitudes, not event probabilities. No new Lean formalization, ledger promotion or full physical origin. Old rank10 was only a lower bound.
Read the exact result, assumptions and evidence
Result
Exact original Fock polynomials close under L,J,Jdagger in dimension68. Actual positive metric and adjoint closure prove reachability, observability and minimality for all word lengths in this alphabet. Code5 needs memory63 with J multiplicities18,14,17,14 at1,i,-1,-i. Positive original forms F_w give both old defect Delta=sum F_w and new rank5 clock defect sum(w-1)F_w=-2(F_-1+F_i), with F_i=F_-i and ranks4,5,5,5. Original unit f4 has L² response6144 and LJL response0; bare repeated code compression gives1024. Full nested Schur reconstruction through10 equals direct reduction to5 exactly.
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 neutral root-pair moment fields
Use existing E8_1 neutral_source_response reader:64 chirally selected roots, original omission of one SU4 family weight line, B48=12T_E8+R_neut, AR=4P_A2-8P_U1. Family omission is an explicit existing input, not derived here from P1; neither it nor the R port or physical coupling is newly selected.
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.
Original positive adjoint two-insertion response
Compute o(W_beta), W_beta=E_beta,0 R_neut, with unchanged original real transverse v_beta. Ordered return is positive Fock Gram o(W)*o(W) between code or original trimer states without intermediate projection. Allow W=sum a_beta W_beta for arbitrary complex input coefficients; distinct lattice charges force nonnegative weights |a_beta|². Nonadjoint returns, other complex transverse fields, longer histories or physical instruments are not thereby selected.
Specified ordered response family of original single pair
Examine mathematically specified moments M_n=Pi L_beta^n Pi,n>=0, L_beta=o(W_beta)*o(W_beta), for same original doubled root(0,0,0,0,0,0,-2,-2). Pi/Q only split full amplitudes; no projection is executed. This tests composition/memory of the already determined source, not P1 selection of physical history, coupling, preparation or time. Full coherent multiroot processes must additionally retain nonneutral final charge sectors.
Original single-root return with native Cartan J
Same complete original single-beta S and actual Fock adjoint as SOURCE-RETURN-MEMORY.txt, L=Sdagger S on neutral grade4 dimension726 with the original five quartics. Native linear J with Ju=i u,Jubar=-i ubar and its adjoint retained. Algebraic word responses of this supplied source only; no selected physical history or full E8 alphabet claimed.
Complete finite proof and rational counterexample. The bounded projector identity extends to infinite spaces but unbounded H retains domain requirements. This is a criterion for closed lossless reduction, not a universal physical PF condition.
Read the exact result, assumptions and evidence
Result
At one positive scale K−K²=(QwGW)*(QwGW); K is a projection exactly when VV* reduces H, in which case K=G(V*HV). Single-face equality at every scale is insufficient: symmetric ±a compressed by (1,1)/sqrt2 gives1/2. At a=log2 and ell1 the full compressed cross block is2/5 instead of1/2 with exact defect9/100 I.
Required assumptions
Whole Fermi two-face projection and common isometric compression
Finite self-adjoint H, positive ell and one isometry V on both faces. G has the full Fermi and cross blocks. This tests closed lossless reduction and does not impose reduction on an open physical subsystem.
Existing mathematical NS Hardy source contract, not a newly proved physical finite P1 selection. The graph-lifted shift differs from both-face diagonal shift and relative matching; the whole-annulus shift has index zero. Graph membership alone does not establish finite time stationarity. The evaluated response is neutral and grading even, retaining all D5/charge slots, not an odd Higgs/Dirac source. NS frequency1/2 and quarter1/4 are related only via the old double-frame map with ell_quarter=2ell_NS and both quarter sectors for the seam shift; no physical scale/time equality is inferred.
Read the exact result, assumptions and evidence
Result
Actual signed G columns prove F_v117=Ind_V4^A4 chi0=pi_*L0q. The old spin cable is the correct induced family bundle without extra copies. Its given NS Hardy unit-shift defect has family rank three and transports by V_h into the whole graph, giving M_rs=<JE_r,M_mu E_s>. The actual induced mu01 input at Phi=0 gives nonzero M02=(4X*/pi)I Pgamma+, X*=(2cosh(ell_NS/2))^-1, I=0.1258739334853158906973-0.4697679151157443151336i. Actual q2/p1 open arcs add by their Kummer signs while diagonal pairs cancel. A closed primitive and strict analytic lower bound prove nonvanishing. Independent root checks give zero exact derivative remainder and continuous-branch quadrature residual below3.2e-76.
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.
Recovered genus2 cover with retained coefficients
Retain exactly the older three-sheet cover C:w³=(z²-1)/(z²+1), eta=z(1-w³), eta²=1-w⁶, six Weierstrass points p_a=omega^a,q_a=-omega^a, given z base and clock markings. Seek actual flat coefficient lines for existing v117 monodromy, not mere permutation equivalence or an assumed field intertwiner.
Supplied split carrier with actual q sign lines on genus-two cover
Given existing split carrier E3⊕E2 with unitary split connection onX and actual unitary flat q linesL0,L1,L2 on old coverC:w³=(z²−1)/(z²+1),η²=1−w⁶,product1 andπ*Fv117=L0⊕L1⊕L2. Pull these data toX×C and combine with declared alignment to mA slots. This does not selectX,E,mA or alignment as original P1 quantum source.
Relative spin and reference Hardy cable on original three-cover
Compare holomorphic Hardy shifts M_z on S0 tensor L and the same degree-zero coefficient line L over the original pi:C→P1_z. Spin half-forms give boundary norm; Fredholm convention is det kernel tensor inverse det cokernel. Relative line is spin line tensor inverse reference line. This defines a mathematical sewing contract, not an established identification with physical D_rel/D_ref or state.
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 free scalar Gaussian defect line,fixed positive ell and constant α. Width4log2 is diagnostic,not physical scale. Projector connection is not asserted compulsory physical choice;no new time or exclusion of other originally specified connections. Original circle character remains correct. Whole coupled h time gives U_t=e^{ith}Ue^{-ith},not generally scalar2π closed clutch family. No native clock intertwiner or missing time in full operational Seed inferred.
Read the exact result, assumptions and evidence
Result
Existing scalar defect family Tφ=R−e^{iφ}U has periodic frame eφ,n=e^{iφn}y_n/||y||. Given circle action satisfies e^{itB}y(φ)=e^{itα}y(φ+t). Natural orthogonal-projector connection instead has〈eφ,∂φeφ〉=iμ and holonomy e^{-2πiμ},μ=〈n〉. Old allowed diagnostic control α=1/4,ell=4log2 gives μ=0.1855335532291087 and holonomy0.3940689714−0.9190808701i,not quarter phase. Exact rational partial sums and geometric tail bounds certify difference. Separately specified flat orbit transport still has circle phase+i;family equation does not select it as projector connection.
Required assumptions
Relative unit seam on explicitly assigned collar faces
In existing product collar,test original relative unit gluing v1=Uv0 on an explicitly assigned pair of labeled faces. Scalar block has h e_n=(n+α)e_n and U e_n=e_(n+1),α=1/4 or3/4. Actual original Γ±-to-face assignment,orientation and three-cabled spin trace map are not assumed derived.
Constant odd circle-source coefficient with existing sea and full boundary graph
In the existing Clifford chart on L2(S1,K) with finite K let h0=(-i∂theta+alpha)gamma,alpha=1/4 or3/4,gamma*=gamma,gamma²=I. Supply constant selfadjoint odd Phi,{gamma,Phi}=0,and V=-i gamma Phi,h=h0+V on common periodic H1. Compare given CAR sea projectors P_-(h0),P_-(h) and complete pure two-face projector G_ell(h),ell>0,including crossings. This is the existing source class with classical background, not physical selection from P1 or quantization of dynamic bosons.
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.
Entire supplied constant coefficient class with explicitly retained closed scalar clutch domain. No arbitrary single channel or physical three-copy choice. Invertible Φ removes both defects;index0 alone does not trivialize every determinant action. Controls are not derived Yukawa values. Variable Φ retains−i[A,Φ],normal-dependent V another derivative,and noncommuting U a boundary form. No exclusion of all P1 sources or variable interactions. Original face assignment,branched spin-domain map,physical source selection and time remain actual maps. Scalar unit defect and old same-spin once norm retain valid scope.
Read the exact result, assumptions and evidence
Result
On unchanged scalar unit-clutch domain,LΦ=∂r+Aγ_F−iγ_FΦ is closed Fredholm. Whole energy identity ||LΦψ||²=||L0ψ||²+||Φψ||² also holds for adjoint. Extra boundary term vanishes by U*VU=V;nonzero massless A/U boundary term remains in ||L0ψ||². Kernel/cokernel of existing whole boundary mismatch are precisely kerΦ∩Fminus and kerΦ∩Fplus. Whole index−Trγ_F=0 for48+48. Previously exactly checked v252 controls immediately give(0,0),(2,2),(1,1),without rerun. Constant original δΦ acts between surviving Gaussian defects as−i Pker,+ δΦ Pker,−. Reuse primitive NS clutch u_cov from solved quarter dictionary.
Required assumptions
Whole constant odd coefficient on same scalar unit-clutch domain
Test supplied whole h=Aγ_F−iγ_FΦ with constant Hermitian odd Φ,internally scalar A=−i∂θ+α and U=e^{i_oldθ}I on D_U={ψ∈H1:ψ(ell)=Uψ(0)}. Hence [A,Φ]=0 and U*ΦU=Φ. Original Γ±-to-face assignment remains an explicit premise. Cited v252 coefficients are previously checked constructor inputs,not selected physical masses. Variable Φ,noncommuting connections or charged noncentral clutches change these premises.
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.
Constant odd circle-source coefficient with existing sea and full boundary graph
In the existing Clifford chart on L2(S1,K) with finite K let h0=(-i∂theta+alpha)gamma,alpha=1/4 or3/4,gamma*=gamma,gamma²=I. Supply constant selfadjoint odd Phi,{gamma,Phi}=0,and V=-i gamma Phi,h=h0+V on common periodic H1. Compare given CAR sea projectors P_-(h0),P_-(h) and complete pure two-face projector G_ell(h),ell>0,including crossings. This is the existing source class with classical background, not physical selection from P1 or quantization of dynamic bosons.
Actual full v252 gauge lift and original P2 comparison
Given the full unchanged v252 constructor pi,J_F,gamma_F,D_F and its symbolic Yukawa/Majorana inputs. Compare real gauge lift U=pi(u)J_F pi(u)J_F^-1 on the connected unimodular branch lambda det(m)=1 with original oriented P2 carrier E3+E2,Y=(-1/3)^3+(1/2)^2. No full Spin10 source action is presumed. A cyclically fixed neutral family line is considered only with an expressly supplied sigma=(123) and eigenvalue1; its original physical selection is not assumed.