Follow the published construction into its registered claims, then explore reviewed research questions, attempts and the connections that remain open.
Interlocking clocks
The published foundation in this area
23 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.
TFPT uses several distinct algebraic clocks and transfers: Coxeter period 30, phase automorphisms, an order-12 compiler action, and separate relaxation factors. Their mathematical periods do not by themselves define physical duration.
Starting structure in the Atlas orientation
Known structures include finite actions: Coxeter number 30, μ4 on phase data, an order-12 compiler action, and separate relaxation rates.
Which connection is sought
The open work is a common source derivation, event interpretation, and a parent realizing one time rule for matter, boundary, and instruments.
How it connects
Clock actions organize E8/code readouts and event sequences; algebraic orders are neither measured seconds nor proofs of dynamics.
Complete operator and local-net theorem in the GIVEN original cut-cover CAR realization. Support is the two-preimage region, not a single cover arc. The later eightfold fold, cross-arc CAR transporters and original P1 support/Theta/collar/reference map do not follow from this scalar current-net theorem alone. No time interpretation of product parameters or physical gate closure.
Read the exact result, assumptions and evidence
Result
The old stress formula gives J_n=2(L_n^(3/4)−L_n^(1/4)) on the same CAR generators for every n including0. The comparison within the existing cut-cover source is not free: u_cov W0=W1 and u_cov W1=W0 u_base give U=[[0,u_base],[I,0]],rank[C,U]1 and phase-independent adjoint Fock transport AdGamma(U). The first arrow preserves factor seas and Wick ordering. Hence on factor0 J(f)=2(AdGamma(U)^−1 T_(3/4)(f)−T_(1/4)(f)). Existing self-adjointness/energy bounds and the stable unitary Trotter sequence show that this transported stress pair generates exactly the local U1 current net on every base interval. The full winding remains charged and is not globally vacuum-preserving because of the second arrow.
Required assumptions
Existing full cut-cover CAR source with actual winding and sea
The already constructed NS double-cover source Hcov=W0K⊕W1K is given, with common negative sea m<0 and original winding U=[[0,u_base],[I,0]], u_base e_m=e_(m+1). Quarter stress fields retain their original CAR coefficients and Wick ordering. Base intervals carry two cover preimages. This assumes the mathematical source realization and does not derive its physical P1 field map or descent into a fixed Pin collar domain.
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.
Only exact energetic selection of the ENTIRE original five-code on the same four registers. Arbitrary Hermitian terms and frustration allowed. Does not exclude one selected code state,ancillas,thermodynamic systems,effective four-body interactions,open dynamics or approximate selection. Marginal equality is for the two mixed states,not all pure code inputs. No physical-origin derivation.
Read the exact result, assumptions and evidence
Result
rho_C=Pi_C/5 andrho_S=Pi_sym4/35 have exactly equal reductions on all4+6+4 subsets of sizes1,2,3; each three-register reduction isPi_sym3/20. Their global trace distance is6/7. Every Hermitian at-most3-local Hamiltonian on the same four registers therefore has equal mean energy. If the ENTIRE codeC has minimal energyE0, positivity ofH−E0I forces the full35-dimensional symmetric space to be ground as well. An exact five-dimensional code ground space is impossible. The existing4-local moment parent attains the necessary interaction order.
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.
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.
Excludes generation by Lhat_n=Lcal_(8n)/8+constant alone. Does not exclude the full original stress source,the old F8 overlap or P1. Introduces no extra nonstress source or physical gate.
Read the exact result, assumptions and evidence
Result
In the folded c8 space the Virasoro cycle at N level1 is span(d.h_-1 Omega). The actual first W8 response has perpendicular component v_perp=grad_tan H8(d),norm²51311452775625/9345848836096 in both quarters. All16 O8-invariant quantum Q_ab corrections contribute only radially at this level. An independent240-root enumeration reconstructs the whole first coefficient and one-point value from all17 old ansatz coefficients. The entire smoothed response is outside the closed Lhat cycle. This restricted exclusion is consistent with the positive full original c1 reconstruction.
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.
Complete mathematical cyclicity in the GIVEN original source; concrete application of standard representation theory, no general novelty claim. No actual local P1 map, specified single physical intervention or local-net equality. A matching c1,h1/32 P1 stress module would conditionally give the cyclic state intertwiner but still needs original identification.
Read the exact result, assumptions and evidence
Result
For the same original quarter CAR source,p=alpha-1/2=plus or minus1/4 and c1 lowest weight h1/32. This is not a Kac weight m²/4. The injective Virasoro Verma map and equal partition characters prove at each grade that the full original stress cycle generates the entire fixed relative-charge sector. All W8 response components and their positive-semigroup smoothing are therefore in it. At original grade8 the first response is explicitly reconstructed by22 literal CAR stress words: all rational coefficients,residual0,full rank22 modulo three primes,and the identical complete Gram from independent c1 Sugawara calculation. All original modes and intermediate states are retained.
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.
Targeted original-source review of the named claims, not an exhaustive archive theorem or rejection of positive compiler/microscopic results. Charged E8 extension and stronger typeIII exhaustion are not imposed as prerequisites for neutral W8.
Read the exact result, assumptions and evidence
Result
Reviewed originals retain the old circle, concrete microscopic neutral CAR field route and conditional E8 net extensions. v366 sets inclass=True and tests supplied copy counts and target constants; v464 uses a chosen sin(k) chain. v469 gives the extension route under its subnet hypotheses. v994 explicitly labels C7 cited/open; v1030 explicitly leaves microscopic generators and local word/adjoint tails unproved. The current SEAM.EQUIV.01 parent is open and its MMST branch conditional. The earlier complete Dirac vector cocycle cancellation is retained in its Lagrangian/CAR scope. None of these passages constructs the actual marked local P1 W8 image.
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.
Exact dynamics in the same source/time contract. Does not exclude finite approximations or finite native observer codes; does not fix absolute physical time.
Read the exact result, assumptions and evidence
Result
For every ell>=1 the connected spectral norm is a_ell=sum_(r=1)^min(ell,8) binom(ell+r-1,2r-1) A_r >=ell A1>0. Smoothing by the existing positive generator exp(-epsilon N) makes the insertion norm finite and only multiplies these spectral contributions by exp(-2epsilon ell). For every epsilon>0 the cyclic time subspace remains infinite dimensional. No finite spectral-support model with this same time reproduces the exact two-point response.
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.
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.
Sufficient conditional current route and necessary test within that route. Actual P1 image and full target comparison remain open. A single Weyl pair is insufficient; charged seam fields and closed collar/reference domains require additional original transport. No universal P1 no-go claim.
Read the exact result, assumptions and evidence
Result
On a proper interval choose real compactly supported chi, kappa=(1/(2pi))integral(chi_prime)^2, f=chi and g=-2chi_prime/kappa. The original level1 current cocycle is c(f,g)=-2i; the Weyl commutator is exp(2ist)I. A normal unital local current embedding must preserve this identity. If it also preserves state, time, reflection and original eight-arc support, GNS gives an isometry onto its cyclic image. On that image the same point-split products and counterterms fix W8. A global unitary onto the entire P1 space is unnecessary for this subfield. Reviewed original passages give C_seed from an already supplied local seed to boundary data and assume the positive CAR kernel; they do not supply the concrete reverse current embedding.
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.
Excludes only the unshifted vacuum replacement with this same W8 and time. No general exclusion of P1, other operators or other actually derived sources.
Read the exact result, assumptions and evidence
Result
The supplied quarter source has positive W8 spectral norm A1=93237295693374937125/15629680934138150912 at N level1. The unshifted vacuum has exactly A8 X^8 and no response at levels1 through7. No intertwiner can identify this same W8 insertion, state and time generator between those specified data. On the folded neutral sector N=(Lcal0^alpha-1/32)/8=K_alpha and H0=8N. The first shifted response has H0 grade8; the homogeneous unshifted W8 state has H0 grade64.
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.
Exact theorem in the given positive neutral CAR current source with the already fixed W8. No P1 target operator, measured value or absolute time calibration.
Read the exact result, assumptions and evidence
Result
For both original quarter charges, z^8 w^8 <W8(z)W8(w)> is exactly sum_(r=0)^8 A_r X^r, X=q/(1-q)^2. Bosonic exchange, homogeneity and the sole pole at q=1 of order at most16 prove completeness. Nine rational Fock norms determine all coefficients without a fit. A0 is the squared old one-point value, A8=65028096000/43, and an independent stress Ward test forces A7=2(85/256)A8=43182720000/43. All A_r are positive. Regular and optimized checker outputs are byte-identical.
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.
Excludes only index-one and quarter-compatible single-line pullback in this normal spin factor. Actual normal-base identity and original field transport are absent;an actually existing further tensor factor may compensate the projective cocycle. Does not exclude original bosonic doublet count or whole TFPT. Does not identify Calderon projector with deck projector.
Read the exact result, assumptions and evidence
Result
On old E:y²=x⁴-1,the spin Dirac twisted by p*O1 has ordinary complex index2 for every spin. In q-fixed even spin alpha=O(T-O),u=(y+i)/x²,kernel is span(1,u). Genuine deck and quarter lifts have D=[[0,-1],[1,0]],Q=a diag(1,-1),so D²=-I and QD=-DQ independently of scalar phases. Quarter exchanges the two deck eigenlines;no common rank1 kernel sector.
Required assumptions
Existing elliptic curve with a specified normal spin factor
On the actually existing E:y²=x⁴-1 with p=x,delta:(x,y)->(x,-y),q:(x,y)->(ix,y),compare normal spin Dirac operators twisted by p*O1. In the q-fixed nontrivial spin alpha=O(T-O),O=(0,i),T=(0,-i),transport the two-dimensional holomorphic kernel with genuine spin lifts. No identity of this curve base with the original normal base or of delta with the Calderon reflector is assumed.
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.