Excludes the specified moment parent, its code algebra with sigma/tau and this geometric product-path parallel transport. Not a general exclusion of combined local drives plus parent, collective code paths, extra dynamics, ancillas or every physical adiabatic protocol.
Read the exact result, assumptions and evidence
Result
The existing Hnat obeys Hnat V=0, hence exp(-itHnat)V=V for every t; every marking direction has equal ground energy. Alg(Hnat|Code,sigma,tau) is scalar on E+. Reusing the existing Berry theorem and one-register erasure gives V*X_jV=tr(X)I5/4 at each site. All local-product code paths have scalar connection; for a closed code path the previously classified full product stabilizer forces phase-S6 holonomy, outside which R5 lies.
Required assumptions
Actual native G31 event representation
Use the actual original Hamming/E8/J matrices and60 reflections, their four-dimensional action, characteristic P64=O2(G31), and existing S6 quotient. These are accepted mathematical core data, not a physical source derived from P1 in this round.
Symmetric quartic and actual erasure intertwiner
Use Sym^4 of the native four-representation, the actual five codeword sums, and the existing one-register erasure intertwiner. The symmetric tensor rule and its physical selection are not silently inferred from the full tensor product.
Existing logical clock and charge marking
Retain the existing logical labels weak0/1, color2/3/4, source point5 and Y5,R5,sigma5. Equivalence under their stabilizer is not automatically an equivalence of lifts commuting with the original native four/P1 clock.
Actual Clifford identification and finite-overgroup theorem
Retain the actual concrete identification of <G31,zeta8 I4> with complex Clifford X2. Apply NRS6.5 only to the proved finite determinant slice; a group-order match alone does not suffice.
Actual original moment parent
Use the actual native fourth root moment with 40M4=Pi5+Psym4 and Hnat=I-(Pi5+Psym4)/2. Add no new marking-dependent Hamiltonian. Physical identification as a time generator is not inferred merely from matrix existence.
Geometric parallel transport of local-product code paths
Consider moving code subspaces T(t)RanV, T(t)=tensor_j Uj(t), Uj(0)=I, returning to the original code. Use the Grassmann/Berry connection induced by the actual Hilbert inner product. No general collective code path, additional internal dynamics or physical adiabatic generator is assumed or excluded.
Completeness only for the stated operator-compatibility class. Countermarks are not full P1 sources and need not obey the original polar rule. Unique distance minimization does not prove a physical law minimizing that distance or matrix positivity from reflection positivity.
Read the exact result, assumptions and evidence
Result
The entire stated native mark class is P=F-cc*, [c] in CP1(E+), or RP1 for real projectors. Every point gives the same quarter spectrum and square; q families are natively S3fix-equivariant. The native q/sigma stabilizer cannot remove the parameter. Original cpol=-sqrt(3/5)e0+sqrt(2/5)e1 and c=e0 differ exactly: HS squared4/5, marked-ray responses2/5 versus0. Uq is also the uniquely nearest isometry to actual orthonormalized Dq, with Frobenius cost7-2sqrt6.
Required assumptions
Actual native G31 event representation
Use the actual original Hamming/E8/J matrices and60 reflections, their four-dimensional action, characteristic P64=O2(G31), and existing S6 quotient. These are accepted mathematical core data, not a physical source derived from P1 in this round.
Symmetric quartic and actual erasure intertwiner
Use Sym^4 of the native four-representation, the actual five codeword sums, and the existing one-register erasure intertwiner. The symmetric tensor rule and its physical selection are not silently inferred from the full tensor product.
Original clock/simplex without an imposed polar mark
Retain actual normalized W6 simplex, sigma=(234), tau=(01), q=5. Consider ALL Hermitian rank2 projectors P with RanP inside Fixsigma, [P,tau]=0 and one positive/one negative tau eigenvalue on RanP. The original polar rule is a comparison point, not an imposed condition of this compatibility class.
This excludes exactly the stated native polynomial continuation with the fixed decoder P220, not every coherent common-clock lift. The logical decoder does not prove original physical permission for the collective intervention.
Read the exact result, assumptions and evidence
Result
The existing three-register decoder derives encoded R5 tensor I4. Substituting actual sigma20 and all four native tau20 into the same R polynomial with original decoder marking gives unitarity and the correct square but fails common erasure action. The pure input rho=(I-F_sigma5)(I+s23)/2 yields output-difference eigenvalues +1/4,+1/4,-1/4,-1/4 and16 zeros: trace distance1/2 for all four lifts.
Required assumptions
Actual native G31 event representation
Use the actual original Hamming/E8/J matrices and60 reflections, their four-dimensional action, characteristic P64=O2(G31), and existing S6 quotient. These are accepted mathematical core data, not a physical source derived from P1 in this round.
Symmetric quartic and actual erasure intertwiner
Use Sym^4 of the native four-representation, the actual five codeword sums, and the existing one-register erasure intertwiner. The symmetric tensor rule and its physical selection are not silently inferred from the full tensor product.
Original simplex and positive polar rule
Use the actual W6 simplex and raw metric, unchanged native three-clock and original positive polar prescription at one of its three marked fixed points. The prescription and its physical selection are not derived from P1.
Original decoder and same native quarter polynomial
Fix original erasure decoder D(A)=W(A tensor I4)W* and its mark P220=P2 tensor I4. Substitute actual sigma20=sigma5 tensor bar(sigma4) and all four native reflection lifts of original tau5 into the existing R polynomial. This tests that continuation, not all coherent lifts.
Classification of code-preserving product unitaries and stated classical mixtures. Collective operations, ancillas, general LOCC, postselection and protocols leaving the code temporarily are not excluded.
Read the exact result, assumptions and evidence
Result
All six actual code-projector pair marginals are (I+Swap)/4. Every code-preserving U1 tensor U2 tensor U3 tensor U4 therefore has projectively equal factors. The identical local stabilizer is K=U(1)G31 by leakage Gram32I15, a finite determinant slice and NRS6.5. R5 lies projectively outside logical S6, excluding every such product lift. Rank-one Choi also excludes convex native-word-channel mixtures reproducing Ad_R5 on all inputs.
Required assumptions
Actual native G31 event representation
Use the actual original Hamming/E8/J matrices and60 reflections, their four-dimensional action, characteristic P64=O2(G31), and existing S6 quotient. These are accepted mathematical core data, not a physical source derived from P1 in this round.
Symmetric quartic and actual erasure intertwiner
Use Sym^4 of the native four-representation, the actual five codeword sums, and the existing one-register erasure intertwiner. The symmetric tensor rule and its physical selection are not silently inferred from the full tensor product.
Existing logical clock and charge marking
Retain the existing logical labels weak0/1, color2/3/4, source point5 and Y5,R5,sigma5. Equivalence under their stabilizer is not automatically an equivalence of lifts commuting with the original native four/P1 clock.
Actual Clifford identification and finite-overgroup theorem
Retain the actual concrete identification of <G31,zeta8 I4> with complex Clifford X2. Apply NRS6.5 only to the proved finite determinant slice; a group-order match alone does not suffice.
Positive polar rule and physical interpretation remain inputs. One mark orbit does not prove every anisotropic source invariant or select a physical state. The mark remains distinct from all15 matching-response projectors.
Read the exact result, assumptions and evidence
Result
In consistent original N coordinates P2(q)=F_sigma-c_q c_q*, c_q=[n2+n3+n4-3(sqrt(6)-1)n_q/5]/sqrt(3). For q=5 this is the actual existing mark without SM charges. All three clock-fixed points form one mark orbit under actual native reflection lifts commuting with sigma4. N=O C^T, O=-rho_C(reverse), transports all labels together.
Required assumptions
Actual native G31 event representation
Use the actual original Hamming/E8/J matrices and60 reflections, their four-dimensional action, characteristic P64=O2(G31), and existing S6 quotient. These are accepted mathematical core data, not a physical source derived from P1 in this round.
Symmetric quartic and actual erasure intertwiner
Use Sym^4 of the native four-representation, the actual five codeword sums, and the existing one-register erasure intertwiner. The symmetric tensor rule and its physical selection are not silently inferred from the full tensor product.
Original simplex and positive polar rule
Use the actual W6 simplex and raw metric, unchanged native three-clock and original positive polar prescription at one of its three marked fixed points. The prescription and its physical selection are not derived from P1.
Classification of this explicit residual class, not all sources. No original physical permission for the coherent Kraus events or selection of gamma.16-record equivariance is sectorwise/on the block algebra, not on extra End(H29) coherences. G31 covariance does not prove covariance under arbitrary coherent clock polynomials.
Read the exact result, assumptions and evidence
Result
CPTP and Tr4 Phi(X)=Tr4 X for all joint X force Phi=id5 tensor D by the multiplicative domain; factorization is not a fitted ansatz. Native G31 covariance forces D=Dq with -1/15<=q<=1. Continuous semigroups have q(t)=exp(-gamma t), gamma>=0. The original central J-character projector and Paulis realize this in the coherent block-algebra menu with common normalized branchwise erasure records; the gauge correlator is q(t).
Required assumptions
Actual native G31 event representation
Use the actual original Hamming/E8/J matrices and60 reflections, their four-dimensional action, characteristic P64=O2(G31), and existing S6 quotient. These are accepted mathematical core data, not a physical source derived from P1 in this round.
Symmetric quartic and actual erasure intertwiner
Use Sym^4 of the native four-representation, the actual five codeword sums, and the existing one-register erasure intertwiner. The symmetric tensor rule and its physical selection are not silently inferred from the full tensor product.
Complete logical nondisturbance and native covariance
A CPTP residual channel Phi on V5 tensor bar(V4) preserves Tr4 Phi(X)=Tr4 X for ALL joint X and is covariant under the native G31 product representation. This is stronger than preservation only of erasure states rho tensor I4/4 and defines an explicit conditional process class.
Extended coherent source-algebra operation menu
For realization of the residual channel, unitary elements of the represented group C*-algebra may serve as Kraus events. This mathematical menu does not assert physical permission derived from P1. The block algebra or each sector is considered; extra coherences between inequivalent representations are not automatically accessible observables.
Explicit operator identification and one logical equivalence class, not six independent physical options. A common native-four/P1 clock continuation and the twenty-lift of a specified coherent R polynomial are not thereby derived.
Read the exact result, assumptions and evidence
Result
For each of the six remaining F20 representatives actual native C,G satisfy C^5=G^4=I and GCG^-1=C^2. Simultaneous intertwiners to the original v419 carrier have solution dimension1 and rank4. The six representatives form ONE orbit of the existing logical stabilizer, already S2×C3 acts regularly. R5 is not a native group word by an exact centralizer test; the compiler clock instead maps to an order3 operator on the quartic code. The canonical cyclotomic trace metric is carried isometrically into the native four-metric by normalized intertwiners, leaving an overall phase.
Required assumptions
Actual native G31 event representation
Use the actual original Hamming/E8/J matrices and60 reflections, their four-dimensional action, characteristic P64=O2(G31), and existing S6 quotient. These are accepted mathematical core data, not a physical source derived from P1 in this round.
Symmetric quartic and actual erasure intertwiner
Use Sym^4 of the native four-representation, the actual five codeword sums, and the existing one-register erasure intertwiner. The symmetric tensor rule and its physical selection are not silently inferred from the full tensor product.
Existing logical clock and charge marking
Retain the existing logical labels weak0/1, color2/3/4, source point5 and Y5,R5,sigma5. Equivalence under their stabilizer is not automatically an equivalence of lifts commuting with the original native four/P1 clock.