Start with a physical or mathematical question. See what is known, what has been tried, why an approach worked or failed, and which connection remains open.
Analytic finite saturation theorem plus separate all-mode theorem inside the explicit normal HS class. Finite checks do not prove all-mode saturation. The old interaction control is only an algebraic countercheck,not an admitted P1 coupling or new model study. No primitive law or physical odd-field/clock/reflection map is derived or freely chosen. For an actually nonlinear P1 half action,retain its operator contributions instead of forcing free saturation.
Read the exact result, assumptions and evidence
Result
For arbitrary faithful even rho,the actual canonical cross coherencemj=Tr(sqrt(rho)aj*sqrt(rho)aj) obeys0<=mj<=sqrt(cj(1−cj)). Equality holds exactly when sqrt(rho)aj=qj aj sqrt(rho),qj=sqrt((1−cj)/cj). On full finite CAR,saturation for all modes in a covariance eigenbasis is equivalent to a quasifree product density. On the existing normal all-mode HS carrier,actual all-mode saturation givesMij=qjCij,then full projection covariance and quasifreeness without uncontrolled density inverses. The already given H_int control has both deficits14745600/82945²>0 and half-modular factors2/32 and8/128 depending on spectator occupation. Positive standard doubling and its quarter clock remain,but projection coherence fails. The original property still to be tested is therefore a linear half-modular action on the actual odd fields,not merely existence of positive doubling.
Required assumptions
Normal even sea CAR source with canonical graded HS doubling
For the algebraic source test use the already given sea Fock/CAR carrier,with faithful even normalized traceclass density rho and its standard HS space,Omega=sqrt(rho),L_A X=AX,R_A X=XA,Gamma X=Pi X Pi. Gamma is the canonical GNS implementer of source parity fixing Omega. Partner fields are represented in the fixed mode-index chart by b_j=−Gamma R_aj;this assumes no physical second P1 boundary. The finite saturation theorem covers arbitrary faithful even finite densities;the separate all-mode route requires the actual normal HS carrier,a complete covariance eigenbasis and saturation in every mode. For the free elliptic connection,rho is the already derived original thermal density,not a newly chosen geometry or temperature. TomitaJ,OSTheta,classical doubling and physical time remain distinct until actually mapped.
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.
Minimal common holomorphic carrier of expressly given root coordinates and flat coefficient lines. No new physical worldsheet,CAR state or full compiler field map. Geometric clock here inverts deck generator;it is not separate commuting compiler order12 clock.
Read the exact result, assumptions and evidence
Result
For D:H¹²=f(z),w=H⁴,g=H³ and actual unramified V4 cover Yq/C,function fields are linearly disjoint over C. D/C is totally C4 ramified at four points and has no nontrivial unramified intermediate. X=Norm(D times_C Yq) has degree16 over C,degree48 over z,group C12 times_C3 A4 isomorphic A4 times C4,genus41. Minimality holds among all connected common covers:g and w determine H=w/g;trivializing three lines forces Yq. Common C3 quotient is identified once. Accepted quarter lifts combine diagonally with order4.
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.
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 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.
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.
Excludes exactly these packets for fixed v117 clock and target marking. Does not refute their mathematical existence. Equal dimension,A4 group and genus5 alone do not identify a flat source map.
Read the exact result, assumptions and evidence
Result
Initial natural packet O(p_a-q_a) is nontrivial unramified V4 two-torsion on C, but v_a=z u_a gives V4 field over w sphere ramified at all six Weierstrass points, genus3. Actual target orders are4,3,4 rather than4,3,2. Positive triangle O(p_a-p_(a+1)) gives2,3,4 with fixed target marks. Both are different marked connections. Comparing imported T and B_k matrices would have been circular; actual function field and ramification test decide.
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.
Only literal base inversion of unchanged A0* connection. Actual finite C4 monodromy is positively closed by old triangle proof. No exclusion of other base maps or other parabolic D4 objects. Same number or abstract group does not replace common clock marking.
Read the exact result, assumptions and evidence
Result
For A0* local exponent analysis forces every rational z->1/z lift into finite polynomial ansatz. Transformation equation G prime=A(z)G+z^-2G A(1/z) has only zero solution. Actual C4 quotient connection has isotropy orders4 and2 at0 and infinity, exchanged by base inversion. It is not marked D4 pullback with orders4 at both, despite abstract S4 generator pairs occurring.
Required assumptions
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.
Local theorem in fixed marked C2 and massless free annulus with fixed spin/connection,canonical boundary transport and source-independent integrable weights. No assertion for full ramified equator graphs,other bundle insertions,bulk sea response or primitive P1 admission. QGEO.REALIZE.01 remains C/O.
Read the exact result, assumptions and evidence
Result
Four marked branch points have one complex modulus lambda. The original cyclic Q acts by lambda to1/lambda;at lambda=-1 its tangent action is minus1,with no invariant modulus. The explicit curve family y²=x⁴+a x²−1 confirms this by F_a(ix)−F_a(x)=−2ax². Thus rho nonzero is excluded from the existing supplied P1\mu4 parity compiler. Combined with the proved correct conformal boundary trace,fixed spin/connection and given free quantization,the whole normalized transported Cauchy/Wick source is unchanged by nearby compatible geometry changes. Source-independent averaging adds no connected fermion four-point correction. Existing clock-given square rigidity is reused;its primitive physical derivation is not claimed.
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.
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.
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 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.
Finitely tested property of the expressly chosen learning rule. Frozen response is not the full Jacobian;no general attractor theorem,no identification with physical time or an original TFPT Clock.
Read the exact result, assumptions and evidence
Result
Nine prescribed beta/eta pairs around the original settings were executed5000 steps each. Three reach numerical rest,six retain motion. At beta=.25,changing eta=.15 to.075 changes frozen-response periods from11201/64540 to18551/106907 iterations. For all9 states,the observed frozen120x120 response reproduces the next1000 nonlinear steps with relative error<8e-14. Normal and-OO runs produce identical result bytes.
Required assumptions
Explicitly chosen recursive SOM rule on60 native signal cells
Given solely for the authorized model study is the60x35 complex update of RECURSIVE-SOM-AUDIT.txt section1,with native K60,fixed neighbors,row normalization,Euclidean winners,batch means and chosen c,beta,eta. The code basis is readout only. This update and its initial alignment are not primitive TFPT laws.
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.
Existing even E8 lattice, Gaussian J and sign lifts
Given existing even unimodular E8 with actual J(x0,x1,...)=(-x1,x0,...) and J²=-I, and the original ordered sign lattice cocycle. Sign automorphisms have kernel Hom(L,F2). Retain original integral ordered basis and generator phases; no enlargement to U1 characters except explicitly defined scalar pushouts.
Finite original root lines,given J/metric and four native registers; no novelty claim for the old parent. First nonuniform projective moment does not exclude arbitrary nonlinear recognition. No physical time,dissipation,preparation,P1,3+2,clock or48c3^4 selection; E8 and J are not derived.
Read the exact result, assumptions and evidence
Result
The actual60 complex E8/J root lines obey exactly M1=I4/4,M2=Pi_sym2/10,M3=Pi_sym3/20 and40M4=Pi_C+Pi_sym4. Extracting40M4−Pi_sym4 without a five-dimensional target literally recovers the original code projector. The preexisting parent I−20M4 has spectrum0^5,(1/2)^30,1^221. Rebuilding the identity from the original Gaussian roots gives byte-identical normal/-OO results and two firing negative controls. Algebraically(20M4)^n=Pi_C+2^(−n)(Pi_sym4−Pi_C) for n>=1.
Required assumptions
Existing even E8 lattice, Gaussian J and sign lifts
Given existing even unimodular E8 with actual J(x0,x1,...)=(-x1,x0,...) and J²=-I, and the original ordered sign lattice cocycle. Sign automorphisms have kernel Hom(L,F2). Retain original integral ordered basis and generator phases; no enlargement to U1 characters except explicitly defined scalar pushouts.
Actual clock/complex-reflection subgroup and native quartic space
R=diag(-1,1,1,1) on Gaussian C4 is the complex reflection of root line(1+i)e0; G0=<J,R>=C4×C2. P5 is the existing five-dimensional symmetric quartic subspace in Sym4(V4), W6=exterior²V4. R is not a real codim1 E8 Weyl reflection; P5 is not the full fourth tensor power.
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.
Pure neutral W8 subalgebra in the given charge sources. Does not identify entire CAR modules, charged seam operators, geometric supports or the full marked clock tuple.
Read the exact result, assumptions and evidence
Result
The actual root automorphism g=-M R M^-1 with coordinate reversal R maps d_1/4 to d_3/4, fixes W8 and preserves N. Its Fock intertwiner therefore identifies every separated pure W8 correlation function in both real charge modules. The exact two-point calculation verifies this explicitly. W8 alone cannot select these original quarter labels.
Required assumptions
Fixed octic completion in the existing current space
In the supplied positive c8 current space,seek W8 only in the declared17 dimensional space:H8=sum_roots(alpha.h_-1)^8-30Q11^4 and the16 weight8 Q_ab monomials. H8 leading coefficient is1;root zero mode and L1=L2 must vanish. This specifies a mathematical existence ansatz,not an extra physical source or preparation rule. Uniqueness beyond this space is not 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.
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.
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 repair with existing covers. Identity of normal S2 and original punctured source base is unproved. No freely chosen double cover as P1 selection. Full relative C12 monodromy retains its own global criterion.
Read the exact result, assumptions and evidence
Result
An actually supplied degreeq pullback to the same normal base sends class2 to2q mod4. Every even degree removes this topological obstruction,smallest positive degree2;on compact S2 only branched or as a general degree map. Existing C2,C6,C12 do this only with proved same base and bundle map. Parity does not select C12. A given nonflat smooth connection remains nonflat under branched pullback. Sequence1 to Z4 to pi1(K/N) to N to1 explains why trivial N or G31 monodromy can hide class2.
Required assumptions
Native Cartan four in whole conjugated source
Given original Cartan h_C,H+=ker(J-i),and whole native clock chat with chat^9=Jhat,chat^4=sigmahat. Existing inner conjugacy transports the entire tuple and Cartan to kappa. H+ is not identified with SU4 family factor. In current W_m orientation,+i64 is its dual branch. No preferred conjugator is selected.
Old joint class2 bundle on the same normal sphere
Given natural lift iota_m,sum m=-2,acts over the old common normal S2 with(c1(E3),c1(E2))=(0,1). For every m,its K=(Spin10 x SU4)/Z4 principal bundle has class2 in pi1(K)=Z4. Proposed choice is an everywhere smooth whole Cartan subspace in its fixed kappa sector. No identity with separate punctured flavor base is assumed.
Actual degree pullback to precisely the same normal base
Conditional cover connection assumes an actual degreeq map f:Sigma to S2 from a closed oriented surface,pulling back exactly the joint K bundle. Connection statement assumes a smooth branched cover. Existing C2,C6,C12 qualify only after proving identity of base and bundle map.
Existing marked cyclic cover tower and compact elliptic images
Use only existing C_n:u_n^n=x^4-1 with fixed x base,deck Delta_n and already forced compiler clock R12=Q12 Delta12^4=(ix,omega s),Q12=(ix,s). Compare compact algebraic curves over C and pullback function fields. A comparison of punctured local systems must retain its removed points,peripheral loops,spin and field transports.
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.
Topological exclusion only for whole Cartan-subspace choice in fixed quarter sector over precisely the same primitive normal base. Needs no connection. Does not exclude single vectors,arbitrary rank4 carriers,punctured bases or allowed defect and degeneration sites. Previous parallel-clock theorem remains valid. No splitting of old sign-lift extension claimed.
Read the exact result, assumptions and evidence
Result
Pointwise centralizer of whole Cartan four is T^J=F2^4;setwise normalizer N has order16*46080=737280. An everywhere smooth Cartan-subspace choice reduces the K_C bundle to N. On simply connected S2 this would trivialize it,contradicting old class2 in Z4 for every natural m. Separately,a whole Cartan four parallel under the same K connection forces curvature0.
Required assumptions
Native Cartan four in whole conjugated source
Given original Cartan h_C,H+=ker(J-i),and whole native clock chat with chat^9=Jhat,chat^4=sigmahat. Existing inner conjugacy transports the entire tuple and Cartan to kappa. H+ is not identified with SU4 family factor. In current W_m orientation,+i64 is its dual branch. No preferred conjugator is selected.
Old joint class2 bundle on the same normal sphere
Given natural lift iota_m,sum m=-2,acts over the old common normal S2 with(c1(E3),c1(E2))=(0,1). For every m,its K=(Spin10 x SU4)/Z4 principal bundle has class2 in pi1(K)=Z4. Proposed choice is an everywhere smooth whole Cartan subspace in its fixed kappa sector. No identity with separate punctured flavor base is assumed.
Existing natural joint lift with given split connection
Assume a GIVEN unitary E=E3+E2 and GIVEN block U3xU2 connection A_E. Use only the natural iota_m with sum m=-2 completely classified on22September and their derived connection A_Wm=rho_LambdaEven(A_E)+M_m tr A_E. This does not select A_E from P1;an independent nonabelian family connection is not silently 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.