Follow the published construction into its registered claims, then explore reviewed research questions, attempts and the connections that remain open.
Quantum and boundary structure
The published foundation in this area
32 authored claim references lead into the original ledger and paper sections. This source record remains available even where no separate reviewed research packet is attached.
Finite E8, Pauli/Clifford, code, state, covariance, and instrument constructions supply internal quantum models and scoped guarantees. A code or successful channel is not yet a selected, repeatable physical source with the same parent dynamics.
Starting structure in the Atlas orientation
Concrete finite code and operator structures exist, with local results within explicitly specified source and instrument menus.
Which connection is sought
Open items include common physical preparation, source, interaction, repetition, record preservation, and connection to the TFPT parent.
How it connects
Codes and clocks specify internal information and symmetry structures; gravity additionally requires local geometry and dynamics.
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.
Excludes only unchanged geometricQ mark in actual theta-square representation. No refutation of an originally derived family/sewing transport explicitly retaining distinct paths and central phases. No general exclusion of elliptic flavor sources.
Read the exact result, assumptions and evidence
Result
On actual tau=i torus,Q:z->iz induces modularS on theta squares. It has projective order2 with eigenvalue ratio1:1:−1. U_v117 and its exactly conjugate normalizedT have projective order4 and three distinct eigenvalues. Thus no invertible linear or projective intertwiner sendsQ toU_v117. Positive joint dictionaryT->U andST->M remains valid.
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.
Theta-square modular action with distinct geometric Q
Test standard S/T action on three theta squares as a representation of a marked elliptic family, retaining weight and central phases. Geometric Q:z->iz at fixed tau=i is the S homology change, not an already supplied T moduli path. Do not assume this family action has the same origin as spatial monodromy of fixed four-punctured 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.
Concrete mathematical erratum of archived TFPT45/42 formula. Preserve originals. Exact v117 representation remains valid;analytic ODE match and later anchor/resonance premises retain their own status. No blanket P2 exclusion or silent paper edit.
Read the exact result, assumptions and evidence
Result
Literal TFPT45 archive section004 assigns D,PDP^-1,P²DP^-2 and their inverse product as fourth monodromy. Printed matrices have triple productI,so fourth monodromyI instead of cusp spectrum(1,omega,omega²). P³I does not implement a quarter turn. Unchanged v117 gives exact existence repair M_k=U^kM0_EXACTU^-k,k0..3 with productM_k=I and every required local class. This does not prove uniqueness from local spectra,det and abstractD4 alone. Active TFPT2 already states bare Z4 nonrigidity and global parabolic additional data. Two independent original-source reviews separate archive defect from current theory.
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.
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 joint representation equivalence after expressly named phases. No original P1/P2 origin proof or identification of spatial with moduli monodromy. New moduli path is an additional changed premise and specifically does not repair the fixed Q identification.
Read the exact result, assumptions and evidence
Result
Standard theta squares give raw group96,central mu4 and projective PSL2(Z4)=S4. Spin labels alone see onlyS3. For UTheta=−i A_T,MTheta=i A_S A_T, X=[[-1−i,0,0],[0,i,−i],[0,1,1]]/sqrt2 satisfies exactly XUThetaX*=U_v117 and XMThetaX*=M0_EXACT. Same X intertwines both generators;normalized group has24elements. Geometric Q:z->iz at tau=i is modularS with projective order2 and eigenmultiplicity1+2,whereas U_v117=T has order4 and three distinct eigenvalues. Isotrivial v493 supplies no elliptic T path. Origin needs an actual level4 family/sewing transport with distinct spatial mark and derived phase section.
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.
Theta-square modular action with distinct geometric Q
Test standard S/T action on three theta squares as a representation of a marked elliptic family, retaining weight and central phases. Geometric Q:z->iz at fixed tau=i is the S homology change, not an already supplied T moduli path. Do not assume this family action has the same origin as spatial monodromy of fixed four-punctured 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.
Excludes only explicitly tested fixed carriers and marks. No exclusion of all native E8 modules,other original transports or existing exact C12 theorem. P_nat/Q comparison is a named ansatz,not an assumed physical module identity.
Read the exact result, assumptions and evidence
Result
Fixed P_nat and v117 M are unmarked regular C3 modules,but ZP_nat=P_*Z together with ZQ=QZ or ZG=GZ has rank9 and only Z0. Natural Fermat deck has H10 multiplicities(0,2,1),H1(0,3,3);scalar cubic twists never give(1,1,1). Existing C12 Phi12 bridge has rank4 with(0,2,2). Same abstract C3 class or existing rank4 intertwiner does not replace a common marked rank3 dictionary.
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.
Fixed three-slot permutation and original compiler comparison
Test the explicitly proposed fixed three-slot P_nat candidate from v417 against the P_* pullback of the v117 pair in the existing September dictionary. Q or G remain fixed on the same named comparison carrier. This coordinate identification is the tested ansatz, not an established physical identification of all native E8 representations or clock carriers.
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.
Excludes only automatic single smooth quarter domain from named raw flat connection and ambient continuity. No ban on an originally derived other real trace,joint boundary source or field map. Full holomorphic kernel in F1 remains established;earlier RP on a smooth annulus cut remains separately valid.
Read the exact result, assumptions and evidence
Result
Supplied detours give T_out=I,T_in=P,without a free U2 choice. Common raw flat meridians(P,−P,P,−P) split into two characters exchanged by Q. Both radially pulled-back scalar operators have spectrumZ with multiplicity2. Literal ramified pullback is(1+z²)partial_z/(2z);smooth ambient continuity has only one isotropic trace dimension instead of maximal2. Moving-tangent spin phases±iP are not a circle boundary condition. No single deck character remains Q invariant. A closed real trace with C_base=T C_E T*,0<=C_base<=I still needs proof.
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.
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.
Exact holomorphic half-form Green kernel on named four-mark C2 source. Uniqueness concerns underlying spin-line class,not an extra Q linearization phase. No automatic positive equal-time CAR covariance,smooth quarter circle or physical time/action normalization. Raw coefficient monodromy remains order2.
Read the exact result, assumptions and evidence
Result
Partition{1,−1}|{i,−i} is exactly the supplied Q fixed nontrivial spin-line class by divisor D_A and H0 test. With g⁴=(x²−1)/(x²+1),S_alpha=sqrt(dx dz)[g(x)/g(z)+g(z)/g(x)]/[2(x−z)] is the actual normalized Szego kernel. Its complete two-sheet matrix has constant unitary diagonalization to diag(g(x)/g(z),g(z)/g(x))/(x−z),without an undetermined elliptic remainder. dlogg=x dx/(x⁴−1) has alternating residues±1/4;four central spin meridians square to−I and multiply toI. Q exchanges both character lines. Pole,spin divisor,conjugate-sheet cancellation and H0(alpha)0 prove the global identity;theta3 numerical checks are only diagnostics.
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.
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.
Counterexample only to force selection from finite endpoint values and heat positivity. No replacement P1 or exclusion of a fully specified transfer,dynamical chi,quantum measure or restricted geometry paths.
Read the exact result, assumptions and evidence
Result
Two strictly positive normalized four-atom heat measures have exactly the same F(0),F(A),F(B),S(0)=.8675847696814872,but opposite derivatives±.0027432473040925574 on the same common scale path. Rational null relations,prime logarithms and an interval certificate prove this beyond rounding. Recursing equal endpoint data cannot reconstruct the difference. Finite stationarity for every positive heat profile forces equal nonzero squared spectra and a constant scale orbit. The already fixed scalar transferT(q)=(1-q)^(-5/4)keeps its unique derivative;the actual geometry insertion and complete source terms remain required.
Required assumptions
Finite data test of positive heat-channel values
Algebraic test on C2:R=diag(1,-1),A=2I,B=sigma_x,D=A+B,chi1,common scaleD(s)=exp(-s)D and fixed eta phase. Compare normalized positive finite heat measures with identical endpoint operator values for a necessary data sufficiency test. Neither is asserted to be P1 or the same fully geometry-dependent transfer channel.
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.
Typed variation identities;no Pin comparison identification with D_rel,no automatic odd-profile admissibility,no asserted infinite trace contract or quantum completion.
Read the exact result, assumptions and evidence
Result
Exactly deltaA=-({h_plus,A}+{h_minus,B})/2 and deltaB=-({h_plus,B}+{h_minus,A})/2. Even/odd relative heat responses and chi variation are derived;moving measure,R,eta and domain terms remain. Original reference source-independence does not imply geometry-independence. Independent noncommuting matrix differentiation with odd profile and chi agrees to3.1e-92.
Required assumptions
Original reflection split with declared geometry variation
Original pair A=(D+RDR)/2,B=(D-RDR)/2 on a declared common domain with fixed R and density-transport variation deltaD=-{h,D}/2. Trace identities hold finitely or under an actually proved differentiable relative-trace contract. Moving chi,mu,R,eta and domains remain separate. Geometric admissibility is not inferred.
Original admissibility and graded-source interfaces
v42 defines P_adm through primitive seam/Cauchy,gap,color and determinant conditions; v31 closure assumes an RP CAR kernel without identifying it with the old circle. v42 exp(W_k[J]) is a formal parent integral; spectral action, positive geometric marginal and full graded source are not equated. The separate eta term of the master functional is retained. Seek an original local image, not global Hilbert equivalence.
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 state witness whose positivity is not inferred from finite regulators. No exclusion of other originally derived P1 states,admissible subalgebras,or properly transported current/time maps. Q0 conditioning is a mathematical comparison,not claimed physical selection. Purification leaves these marginal expectations unchanged.
Read the exact result, assumptions and evidence
Result
Exact telescoping of the actual Fermi covariance gives omega(J_-m J_m)=m/(exp(4pi m)-1),hence omega_Q0(J_-1 J_1)=0.00000348735451780811656...>0 versus0 in the old Hardy state. Q0 conditioning removes only the theta factor. Every nonzero real smooth mean-zero profile supported in a proper interval obeys omega_ell(Wf)/omega_Hardy(Wf)=exp[-sum_m m|fm|²/(exp(4pi m)-1)]<1. States thus differ already on every such local neutral current net under the unchanged original current dictionary.
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.
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.
Complete free current source on existing elliptic midpoint annulus under expressly given Gaussian quantization. b4pi belongs toHalpha,2pi to2Halpha,32pi toKalpha=Halpha/8;the same density operator. No physical temperature,primitive P1 measure,marked equator-graph/collar intertwiner,or interacting Wick theorem. Native unbounded limits retain old domains/smearing/normal-order hypotheses.
Read the exact result, assumptions and evidence
Result
The actual labelled Calderon/Szego covariance gives C_r=(1+exp(4pi(r+alpha)))^-1 and rho=Z^-1 exp(-4pi Halpha),Halpha=N+Q²/2+(alpha-1/2)Q per quarter. The full current functional is omega(Wf)=Theta_alpha(f0)/Theta_alpha(0) exp[-1/2 sum_m m coth(2pi m)|fm|²],with exact original BCH phase for every finite ordered Weyl word. Theta retains all charges. The explicit normal representation is left action on the rho^(1/2) cyclic space;its subspace for currents alone,the full HS space for full CAR. Previously reconstructed neutral vertex products give the same Wick field correlations in this state. Finite fermion determinants and grouped1/2/4/8 pair comparisons independently check the analytic derivation;normal and-OO are byte-identical.
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.
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.
Excludes only the natural scalar/spin direct connection as the actual original P2 connection,including scalar-twist repairs. Does not exclude other originally derived nonabelian connections on the common bundle. Separate exact v117 matrix properties from numerical ODE identification.
Read the exact result, assumptions and evidence
Result
Natural induced connection on Vell has residues0,1/2,1/2 at each ramification mark,monodromy1,−1,−1,polynomial(lambda−1)(lambda+1)^2,order2,trace−1. Original P2 target has residues0,1/3,2/3,spectrum1,omega,omega²,polynomiallambda³−1,order3,trace0. Thus no local connection intertwiner exists. Nonzero scalar twists preserve the repeated eigenvalue and cannot repair it. Natural direct sum is also reducible,while stated original <U,M0> is irreducible. v499 already separates the Z2 pillowcase from the Z3 weight cover. Another connection on the same bundle remains additional data to derive.
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 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.
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.