Equivariant flat coefficient lines of unchanged v117 source with actual clock direction. No selection of CAR field quantization or state,no identification of modularT with geometricQ,no new base inversion symmetry.
Read the exact result, assumptions and evidence
Result
Original geometric lift Q(z,w,eta)=(iz,w^-1,-i eta/w³) preserves q orbit and satisfies Q²=(−z,w,−eta). Kummer roots obey Q*h_a=epsilon_a omega/h_(-a-1),product epsilon_a=1. Actual local marked inertia t proportional s⁴ at q_0 forces epsilon_0/epsilon_2=−1 and epsilon_1=−1. The two inverse lifts are V4 frame-conjugate. Determinant-one coefficient representation fixes the remaining line with+1,square has two minus signs,spectrum1,i,−i. Explicit simultaneous basis change identifies U; positive quotient monodromy is contravariantly U^-1. C4 equivariance of existing coefficient lift is closed.
Required assumptions
Original P2 order-three cusp class and stable bundle anchor
Target the original P2 Deligne class with cusp weights0,1/3,2/3 and holomorphic anchor O(-2)+O(-1)^2. Exact v117 M0_EXACT has polynomiallambda³−1,trace0,order3;its joint representation with U is irreducible. Keep these exact matrix properties distinct from the numerical ODE identification there. Test whether the natural scalar/spin pushforward of existing C2 realizes this actual connection;physical selection of the direct sum as flavor module is not presumed.
Unchanged A0* Fuchsian equation and actual C4 clock
Fix exactly v115 A0*, U=diag(1,i,-i), residues U^k A0* U^-k at i^k and regular base point z=0. Retain base rotation, positive meridian on t=z⁴ and simultaneous fiber basis change. No new Riemann Hilbert solution or D4 reflection lift is assumed.
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.
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.
Exact flat coefficient lift with actual marked finite monodromy. No intertwiner of primitive CAR fields,state,positive time or support; no implicit selection of joint C2 NS field linearization. Older source already supplied curve and point intertwiner; line identification is separately proved here.
Read the exact result, assumptions and evidence
Result
On C:eta²=1-w⁶ the correct lines are L_a=O(q_a-q_(a+1)),q_a=-omega^a. Kummer roots h_a²=(w+omega^a)/(w+omega^(a+1)) multiply to1. V4 cover Zq of w sphere ramifies only at three q points and has genus0. Actual quotient coordinate t=((1+w³)/(1-w³))² gives degree24 with uniform ramification orders4,3,2 over0,1,infinity. Thus Zq is regular S4 triangle cover of same marked v117 class. Fiber product with z⁴=t has ordinary A4 image; intersection field is C(z²), since S4 has no C4 quotient. Over C this is exactly unramified genus5 V4 kernel cover. Hence pi*E_v117 decomposes into three expressly written unitary two-torsion lines. Equality follows from actual Kummer cover and target marks, not fitted target matrices.
Required assumptions
Original P2 order-three cusp class and stable bundle anchor
Target the original P2 Deligne class with cusp weights0,1/3,2/3 and holomorphic anchor O(-2)+O(-1)^2. Exact v117 M0_EXACT has polynomiallambda³−1,trace0,order3;its joint representation with U is irreducible. Keep these exact matrix properties distinct from the numerical ODE identification there. Test whether the natural scalar/spin pushforward of existing C2 realizes this actual connection;physical selection of the direct sum as flavor module is not presumed.
Unchanged A0* Fuchsian equation and actual C4 clock
Fix exactly v115 A0*, U=diag(1,i,-i), residues U^k A0* U^-k at i^k and regular base point z=0. Retain base rotation, positive meridian on t=z⁴ and simultaneous fiber basis change. No new Riemann Hilbert solution or D4 reflection lift is assumed.
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.
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.
Reused older analytic proof independently checked now. Closes ODE group identification incorrectly treated again as numerical-only. No new origin derivation selecting A0*, no complete field/state/time isomorphism. Physical gates remain open.
Read the exact result, assumptions and evidence
Result
The28Sept synthesis already contains analytic identification of unchanged v117 ODE: t=z⁴ and diag(1,z,z³) give three Fuchsian points of orders4,3,2; sole infinity resonance coefficient is exactly zero. Alternative frame diag(1,z,z^-1) gives semisimple exponents -1/2,0,-1/2 without resonance. Image is faithful S4 rotation representation; positive t0 meridian is U^-1. Marked pair is exactly v117 up to simultaneous basis change and remaining diagonal basis-sign convention. Diagonal0,-i/2,+i/2 follows analytically. Original four meridians have A4 image; U extends to S4. No new ODE fit needed.
Required assumptions
Original P2 order-three cusp class and stable bundle anchor
Target the original P2 Deligne class with cusp weights0,1/3,2/3 and holomorphic anchor O(-2)+O(-1)^2. Exact v117 M0_EXACT has polynomiallambda³−1,trace0,order3;its joint representation with U is irreducible. Keep these exact matrix properties distinct from the numerical ODE identification there. Test whether the natural scalar/spin pushforward of existing C2 realizes this actual connection;physical selection of the direct sum as flavor module is not presumed.
Unchanged A0* Fuchsian equation and actual C4 clock
Fix exactly v115 A0*, U=diag(1,i,-i), residues U^k A0* U^-k at i^k and regular base point z=0. Retain base rotation, positive meridian on t=z⁴ and simultaneous fiber basis change. No new Riemann Hilbert solution or D4 reflection lift is assumed.
Do not identify source types
Observable algebra, OS reflection, state, collar operator, time evolution, carrier connection, family holonomy, and readout are distinct data until an original map relates them.
Complete theorem in exactly specified genuine D4 class. Same abstract matrix class does not identify different marked base paths. Do not infer full D4 connection symmetry from C4 rotation. No physical P1 origin.
Read the exact result, assumptions and evidence
Result
J_D4=1/2-(x⁴+x^-4)/4 generates C(x)^D4. For genuine unitary SU3 descent on(infinity,2,4), c=diag(1,i,-i), regular cubic M=a², four half-meridian branches have traces0,2,2omega,2omega². Trace2 is impossible; complex branches are reducible. Unique survivor is a³=1,p=(1/2,1/4,1/4), the v117 matrix orbit. For genuinely linearized stable parabolic object canonical MS descent follows from uniqueness of adapted metric. Treat concrete A0* connection separately.
Required assumptions
Original P2 order-three cusp class and stable bundle anchor
Target the original P2 Deligne class with cusp weights0,1/3,2/3 and holomorphic anchor O(-2)+O(-1)^2. Exact v117 M0_EXACT has polynomiallambda³−1,trace0,order3;its joint representation with U is irreducible. Keep these exact matrix properties distinct from the numerical ODE identification there. Test whether the natural scalar/spin pushforward of existing C2 realizes this actual connection;physical selection of the direct sum as flavor module is not presumed.
Genuine stable unitary D4 orbifold descent
The expressly specified stable determinant-trivial parabolic rank3 branch has genuine mark-preserving D4 linearization with Coxeter fiber c=diag(1,i,-i) and regular cubic meridian M=a². Apply Mehta Seshadri to this object. Do NOT identify the concrete A0* ODE with it merely from C4 rotation.
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 full equatorial trace for supplied metric. No equality with old quarter state, positive annulus KMS state or physical time. Preserve smooth annulus results and original W8 Hardy comparison.
Read the exact result, assumptions and evidence
Result
Actual measure is |dz|=|dx|/(|c||y|). Closed half-form trace is L2 unitary to |dx|. The two bounded Cauchy idempotents gP_Hg^-1 and g^-1P_Hg, |g|²=|tan(theta)|^1/2, have exactly closed range H² by outer cyclic factors. Their orthogonal range projector is P_H⊕P_H. Integrable branch singularities do not remove H1 vertex coupling. Intrinsic boundary generator has variable speed 4K(-1)sqrt(|sin2theta|), failing constant base-winding Ward relation.
Required assumptions
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.
Original full four-mark inversion and distinct equatorial Real slots
The supplied marked base has x->ix and the inversion x->1/x already present in v512. Compare their actual lift to E:y²=x⁴−1,retaining the original four-marked base circle Gamma=|x|1. v510/v519 distinguish pointwise equatorial x->1/bar x from the OS family x->mu bar x acting along Gamma. These operations must not be identified in a source comparison. The old free carrier is named;no general physical P1 equality is presumed.
Ramified holomorphic half-form trace of the given C2 spin line
Retain actual E:y²=x⁴−1, its already supplied Q fixed nontrivial NS NS spin-line class alpha and normalized Szego kernel. Holomorphic trace includes sqrt(dz/dx), while ordinary flat coefficient p_*alpha remains separate. No extra Q linearization phase, positive equal-time CAR covariance or smooth circle domain is assumed.
Actual L2 half-form trace on full marked equator
Retain supplied square elliptic curve, flat metric and NS spin line. On the full ramified equator sqrt(dz/dx) transports actual L2 norm to |dx|. Keep holomorphic kernel, orthogonal analytic range, positive CAR covariance and H1 vertex domain distinct. No new channel metric or free vertex condition.
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.
Exact conditional free Cauchy/CAR family calculation and finite70-digit checks;not an admitted original compiler branch,derived P1 law or variance,common physical time or spacetime interaction. ModularS differs fromT;complex structure differs from flat U1 Quillen moduli;holomorphic spin kernel is not directly positive boundary covariance.
Read the exact result, assumptions and evidence
Result
Under the expressly changed modulus-family premise,both real structures and free Moebius quotient persist,but Q exchanges rho and minus rho. Actual primitive periods give I(rho)=2K((2−rho)/4),r=I(−rho)/I(rho),W=r/2,tau=ir and r_prime(0)=2Gamma(3/4)²/Gamma(1/4)². Free boundary covariance is c_n,alpha=[1+exp(4pi r(n+alpha))]^-1,alpha1/4,3/4. Its first variation is fixed and trace class;the original global holomorphic spin kernel independently confirms a nonzero variation. For actual lowest opposite NS modes,any positive shared geometry law gives exactly kappa4=−Var(c). Diagnostic averaging at rho plus/minus.1 yields−8.133849897e−6 with positive normalized two-mode density. In the retained original fixed-Q compiler rho0 disables this mechanism.
Required assumptions
Explicitly changed marked elliptic modulus family with transported spin
Consider E_rho:y²=x⁴+i rho x²−1 for real |rho|<2 with the continued original nontrivial spin partition,both original dividing real structures and free Moebius glide. Expressly continue free chiral Cauchy quantization on the two-face annulus normalized by its primitive A cycle. Q and the original inversion act between rho and minus rho fibers;their original fixed-fiber action is expressly relaxed. The mixing formula conditionally assumes an arbitrary positive normalized source-independent law on this new family. Neither admission in the fixed C2 compiler nor P1 measure support,variance,full marked graph domain,physical time or an interacting source is derived.
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.
Sufficient identification test for the existing chiral scalar two-face source;no physical purification freely selected from C_A,no complete Dirac vector source or new time. Nonprojection marginals still require higher-word comparison. Quantitative error only inside the tested reducing block;no global control from finite samples. Original P1 measure,joint closed real graph trace/domain and physical cross equation remain open. Prior four-field requirement is sharpened only for the full projection case.
Read the exact result, assumptions and evidence
Result
The original closed annulus graph has P_A=[[I-C_A,X_A],[X_A,C_A]],C_A=(I+exp(2WB_Sigma))^-1,X_A=(2cosh(WB_Sigma))^-1 and X_A²=C_A(I-C_A). Its full kernel is a projection,a single face is not. An actual positive P1 CAR source with this joint kernel would therefore inherit all previous free current/Weyl/native-composite correlations under the old domain/subtraction dictionary without an extra Gaussian premise. The missing physical cross equation is omegaP1(aL*(f)aR(g))=<f,X_Ag>. Three positive source controls share the same face marginals;dephased and product states even share the full diagonal covariance but have four-point values0 andp(1-p). On an actual finite P reducing block,delta=Tr[(I-P)C]+Tr[P(I-C)] gives ||rho-rhoP||1<=2sqrt(delta) for arbitrary positive states. All256 ordered four-field words are independently checked;normal/-OO byte-identical.
Required assumptions
Test a positive joint CAR source against the full geometric projection
Consider an arbitrary normalized positive state on the specified joint charged two-face CAR algebra,without assuming quasifreeness or U1 invariance. The geometric target is the existing closed chiral annulus graph P_A with both marked faces,W=1/2 and original NS B_Sigma. Actual identity C_source=P_A is an explicit condition of the uniqueness theorem and an unresolved P1 source equation,not a fact inferred from geometry or RP. A finite error test applies only on a genuinely P_A reducing CAR block. Complete Dirac vector source,OS Padm and physical time are not identified with this chiral scalar target. No freely chosen purification supplies missing physical cross correlations.
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.
Exact infinite CAR-state theorem under actual full projection covariance,not a new general CAR theorem and requiring no extra Gaussian premise. GNS uniqueness gives no implementability in an arbitrarily prechosen old Fock representation. A geometric Cauchy projection is not automatically presented as a physical covariance;P1 and its real joint field carrier remain to be identified.
Read the exact result, assumptions and evidence
Result
For an arbitrary positive CAR state with full covariance C=P=P*=P²,GNS zero norms force a(f)Omega=0 onkerP and a*(f)Omega=0 onranP. CAR normal ordering uniquely determines all finite words and hence the full state as the pure Fock/Slater state omegaP. Quasifreeness,gauge invariance and zero anomalous pairs follow instead of being assumed. This is the charged application of Araki1970 Lemma4.3. A positive ensemble mean with projection covariance P forces that same covariance and state almost surely;complex weights are outside this ensemble argument.
Required assumptions
Test a positive joint CAR source against the full geometric projection
Consider an arbitrary normalized positive state on the specified joint charged two-face CAR algebra,without assuming quasifreeness or U1 invariance. The geometric target is the existing closed chiral annulus graph P_A with both marked faces,W=1/2 and original NS B_Sigma. Actual identity C_source=P_A is an explicit condition of the uniqueness theorem and an unresolved P1 source equation,not a fact inferred from geometry or RP. A finite error test applies only on a genuinely P_A reducing CAR block. Complete Dirac vector source,OS Padm and physical time are not identified with this chiral scalar target. No freely chosen purification supplies missing physical cross correlations.
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.
Exact transported pair identity on the full given source;pulse values from a finite129-mode run with chosen duration/amplitude. No cyclic cutoff shift as an index proof. Full P1 measure/reference Ward identification remains open;the relative CAR response is not substituted into delta logR without proof.
Read the exact result, assumptions and evidence
Result
For the two actual commonly evolved states P_t=U C0 U*,Q_t=P_t+Pi_t and exactly(I-zQ_t)(I-zP_t)^-1=I-zPi_t,determinant1-z,without requiringC_t-C0 inL1. After50 pulse steps and50 steps with exactly restored D0,the t1 relative readouts differ:no pulse-.4739486483,positive+.01-.4701253569,negative-.01-.4780397086. Recursion matches direct evolution to4.8e-14. Causal kernel predicts pulse derivative.3956885828,finite symmetric difference.3957175880. Complete historical sector weight remainsQ_full=e^-2alpha R(g,alpha);its geometry response at fixed alpha comes fromR and is not calculated by the index.
Required assumptions
Retained zero-branch defect on a common Pin geometry history
The original normal-zero branch k=pi/2,b1,c+i carries the retained positive quarter defect Pi. Compare the given pure D0 sea C0 and C0+Pi,both evolved by the same D_s=exp(-s h/2)D0 exp(-s h/2). The profile and tested step/pulse histories are named diagnostic inputs. Relative geometry response is Tr(U Pi U* partial_sD_s). It is not defined into the primitive P1 action response or a thin local seam map.
Given flat Moebius Dirac boundary family and its full sea
The comparison fixes D=−i(sigma1 partial_x+sigma2 partial_w),Psi(x+1,−w)=c sigma1 Psi(x,w),c=±i,sigma1 Psi|boundary=b Psi|boundary,b=±1,and the canonical full two-dimensional negative D spectral polarization. This specified primary-literature family is not identified with all possible original P1 collars or covariances.
Controlled conformal normal geometry of the given Pin Dirac source
For the authorized source study,the same given Dirac comparison D0 on its fixed(c,b)Pin/boundary domain is varied by smooth real even profiles h:g_e=exp(2e h)g0,D_e=exp(-e h/2)D0 exp(-e h/2)after unitary density transport. Profiles/amplitude are explicitly chosen diagnostic interventions,not a P1-selected history. Original pure D0 sea remains initial state;thermal Green quantization stays separate.
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.
Historical SF sector action and explicit relative normalization
fullv27/v28 set J1=SF,J=d SF,d2,SF1 in S_def(alpha)=S_def0+alpha J. Review concerns the intended defect-only relative reweighting and its extension to complete measure-compatible sectors Z_s=e^(-alpha d s)Z_s^0(alpha),Z0=Z0^0(alpha). Constancy of R=Z_s^0/Z0^0 is an explicit condition, not assumed from SF.
Exact relative response form of the given source;no assertion of separately defined infinite vacuum quadratic operators or geometry-change Fock implementers. Long curves finitely tested. No identification with primitive action variation,selected history,native quartic words or physical time.
Read the exact result, assumptions and evidence
Result
On the same source R_h(s,t)=Tr(DeltaC_s(t)partial_sD_s) and exactly partial_sR_h|0=k<h²>+integral_0^t chi_hh. Full mode sum weights are(E±k)^3/(16E),chi(0)0,chi_prime(0)=-k<h_prime g_prime>;it converges for smooth profiles. At t1 the retained collar-preserving probe gives contact+.3861727422,causal part-.4054682615,total-.0192955193. Direct commutator error<3.6e-13,independent full-family derivative after extrapolation<3.3e-8. Omitting contact gives error.386. Actual400-step machine computes the same relative response;129/257-mode difference<5.4e-7.
Required assumptions
Retained zero-branch defect on a common Pin geometry history
The original normal-zero branch k=pi/2,b1,c+i carries the retained positive quarter defect Pi. Compare the given pure D0 sea C0 and C0+Pi,both evolved by the same D_s=exp(-s h/2)D0 exp(-s h/2). The profile and tested step/pulse histories are named diagnostic inputs. Relative geometry response is Tr(U Pi U* partial_sD_s). It is not defined into the primitive P1 action response or a thin local seam map.
Given flat Moebius Dirac boundary family and its full sea
The comparison fixes D=−i(sigma1 partial_x+sigma2 partial_w),Psi(x+1,−w)=c sigma1 Psi(x,w),c=±i,sigma1 Psi|boundary=b Psi|boundary,b=±1,and the canonical full two-dimensional negative D spectral polarization. This specified primary-literature family is not identified with all possible original P1 collars or covariances.
Controlled conformal normal geometry of the given Pin Dirac source
For the authorized source study,the same given Dirac comparison D0 on its fixed(c,b)Pin/boundary domain is varied by smooth real even profiles h:g_e=exp(2e h)g0,D_e=exp(-e h/2)D0 exp(-e h/2)after unitary density transport. Profiles/amplitude are explicitly chosen diagnostic interventions,not a P1-selected history. Original pure D0 sea remains initial state;thermal Green quantization stays separate.
Do not identify source types
Observable algebra, OS reflection, state, collar operator, time evolution, carrier connection, family holonomy, and readout are distinct data until an original map relates them.
Only the given conditional free quantization and correctly transported CAR observables; no Gaussian theorem for original S_prim/P1 measure and no assertion that connected current correlations vanish. Z_ref[0] cancels in Z_rel[J]/Z_rel[0], not automatically from geometric forces. The next decisive four-field test requires an original covariance-changing variation or higher fermion vertex.
Read the exact result, assumptions and evidence
Result
After the derived boundary transport, each given conditional free CAR word on the same marked curve is independent of sigma. Its identical normalized source functional therefore factors out of any source-independent integrable total geometric weight of nonzero normalization, including determinants, reference, Jacobian and anomalies. The connected fermion four-point defect stays exactly zero, as do all higher Wick rules. No particular geometric measure is selected or discarded. Pure conformal geometry variance is therefore not evidence of an interacting boundary source. With other varying data, integrating out this factor may change their effective weight; their covariance requires a separate test.
Required assumptions
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.
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.
Conformal metric family of the same marked massless annulus
Study the existing smooth midpoint-annulus source on the same complex square curve with fixed spin, cut and face markings and massless geometric chiral Dirac operator. g_sigma=exp(2sigma)g for arbitrary smooth real sigma, with canonical spin lift and bulk/boundary L2 transport derived from volume densities. On the fixed complex curve this includes all Hermitian metrics. Connection, mass and complex structure do not vary. An unchanged original reflection/clock requires invariant profiles; otherwise transport the symmetry as well. No identification with the unresolved full marked equator graph, primitive P1 measure or physical time. The averaging lemma expressly retains the given free conditional CAR quantization, source-independent integrable total weights and nonzero normalization.
Analytic smooth massless conformal Cauchy-data theorem, not a regulator-limit claim. The free state consequence uses the existing CAR quantization premise. Newly metric-defined stress, spectra, resolvents, APS projections, full marked graph traces, P1 origin and physical time are not identified; previous nonzero Schur/sea responses remain intact.
Read the exact result, assumptions and evidence
Result
The conformal Dirac law and boundary density give V_sigma=e^((d-1)sigma/2), which exactly cancels the harmonic-spinor factor:V_sigma Tr psi_sigma=Tr psi0. Thus V_sigma P_sigma V_sigma*=P0 for the orthogonal closed Cauchy-data projection. Both marked faces and regional supports are preserved. In common bulk L2 the required trace is T_sigma=e^-sigma_boundary/2 Tr for Dtilde=e^-sigma/2 D e^-sigma/2. Quarter/fold are transported afterwards. The existing labelled marginal and full free current source are identical for all compatible metrics on the same complex curve. An unweighted raw trace produces an exactly detected spurious covariance change.
Required assumptions
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.
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.
Conformal metric family of the same marked massless annulus
Study the existing smooth midpoint-annulus source on the same complex square curve with fixed spin, cut and face markings and massless geometric chiral Dirac operator. g_sigma=exp(2sigma)g for arbitrary smooth real sigma, with canonical spin lift and bulk/boundary L2 transport derived from volume densities. On the fixed complex curve this includes all Hermitian metrics. Connection, mass and complex structure do not vary. An unchanged original reflection/clock requires invariant profiles; otherwise transport the symmetry as well. No identification with the unresolved full marked equator graph, primitive P1 measure or physical time. The averaging lemma expressly retains the given free conditional CAR quantization, source-independent integrable total weights and nonzero normalization.