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.
Excludes direct or spectral identification of these concrete kernels with existing mark, not every joint source history or mixed operator algebra.
Read the exact result, assumptions and evidence
Result
None of194 nonzero individual returns commutes with original P2. Sum spectrum is3581,4191,3419²,3790 and commutes with tau but not sigma or P2. Exact witnesses [Ksum,sigma]12=772 and [Ksum,P2]01=(488sqrt2-61sqrt3)/5. Its spectral functions therefore do not give original P2; all4 supplied family variants leave sum unchanged.
Required assumptions
Given local E8_1 vacuum representation and explicit cocycle
Positive level-one E8 lattice vacuum representation in orthonormal D5+D3 coordinates, declared simple-root cocycle, level-one Heisenberg Ward rules and internal quarter character v=(0^5,1/2^3). Insertions are separated. No raw-P1 pullback or physical time law is assumed or derived from this representation.
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.
Actual positive Cartan quartic Fock embedding
Use original Construction-A roots with physical metric dot/2 and u_j=(h_2j-i h_2j+1)/sqrt2 as J=+i basis. Canonically embed the actual five Sym^4 code support sums as neutral grade4 Heisenberg polynomials in the given E8_1 lattice vacuum. No physical preparation follows.
Original neutral root-pair moment fields
Use existing E8_1 neutral_source_response reader:64 chirally selected roots, original omission of one SU4 family weight line, B48=12T_E8+R_neut, AR=4P_A2-8P_U1. Family omission is an explicit existing input, not derived here from P1; neither it nor the R port or physical coupling is newly selected.
Original displayed J-equivariant root isometry
Use existing S=_SIMPLE and T=_TARGET_SIMPLE from cartan_source._glue_deck_character_certificate: O=T S^-1/sqrt2 maps actual Construction-A roots r/sqrt2 to original Chevalley roots r/2, preserving metric and J. This is the existing concrete map, not a newly fitted embedding; physical P1 selection stays separate.
Original positive adjoint two-insertion response
Compute o(W_beta), W_beta=E_beta,0 R_neut, with unchanged original real transverse v_beta. Ordered return is positive Fock Gram o(W)*o(W) between code or original trimer states without intermediate projection. Allow W=sum a_beta W_beta for arbitrary complex input coefficients; distinct lattice charges force nonnegative weights |a_beta|². Nonadjoint returns, other complex transverse fields, longer histories or physical instruments are not thereby selected.
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.
Existing E8_1 vacuum, original code embedding and given R_neut branch. Return amplitudes are not selected physical channels.
Read the exact result, assumptions and evidence
Result
With existing isometry O and full vertex-mode action, all194 nonzero original W_beta give positive definite nonscalar rank5 returns on actual code. Unchanged sum over original root-current input basis is diag(3581,4191,3419,3419,3790), identical for all4 original family omissions. No intermediate projection or fitted coupling is inserted.
Required assumptions
Given local E8_1 vacuum representation and explicit cocycle
Positive level-one E8 lattice vacuum representation in orthonormal D5+D3 coordinates, declared simple-root cocycle, level-one Heisenberg Ward rules and internal quarter character v=(0^5,1/2^3). Insertions are separated. No raw-P1 pullback or physical time law is assumed or derived from this representation.
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.
Actual positive Cartan quartic Fock embedding
Use original Construction-A roots with physical metric dot/2 and u_j=(h_2j-i h_2j+1)/sqrt2 as J=+i basis. Canonically embed the actual five Sym^4 code support sums as neutral grade4 Heisenberg polynomials in the given E8_1 lattice vacuum. No physical preparation follows.
Original neutral root-pair moment fields
Use existing E8_1 neutral_source_response reader:64 chirally selected roots, original omission of one SU4 family weight line, B48=12T_E8+R_neut, AR=4P_A2-8P_U1. Family omission is an explicit existing input, not derived here from P1; neither it nor the R port or physical coupling is newly selected.
Original displayed J-equivariant root isometry
Use existing S=_SIMPLE and T=_TARGET_SIMPLE from cartan_source._glue_deck_character_certificate: O=T S^-1/sqrt2 maps actual Construction-A roots r/sqrt2 to original Chevalley roots r/2, preserving metric and J. This is the existing concrete map, not a newly fitted embedding; physical P1 selection stays separate.
Original positive adjoint two-insertion response
Compute o(W_beta), W_beta=E_beta,0 R_neut, with unchanged original real transverse v_beta. Ordered return is positive Fock Gram o(W)*o(W) between code or original trimer states without intermediate projection. Allow W=sum a_beta W_beta for arbitrary complex input coefficients; distinct lattice charges force nonnegative weights |a_beta|². Nonadjoint returns, other complex transverse fields, longer histories or physical instruments are not thereby selected.
Actual first algebraic source variation in given E8_1 vacuum;194 complex root responses are not particle counts or selected events. Leakage existence is orthogonally covariant; numerical code matrices require actual source isometry. Second responses, joint return histories, physical history, P1 source and marking selection remain undetermined.
Read the exact result, assumptions and evidence
Result
Original vertex formula gives E_beta,0 T_AR=(v_beta.h(-1))e_beta with v_beta=-AR beta+(beta^T AR beta)beta/2 and beta.v_beta=0. All240 original Chevalley roots have exact norm distribution0:46,14:96,30:32,32:60,64:6, hence194 nonzero source responses without fitted coefficients. Their weight-two zero mode on quartic f has PBW component [f(b)v_beta(-3)-(D_c f)(b)beta(-3)/3]e_beta. Basepoint freedom and transversality prove leakage of each individual nonzero direction; compression vanishes by lattice charge. Four original family omissions have R-sum0, without deriving a physical averaging rule.
Required assumptions
Given local E8_1 vacuum representation and explicit cocycle
Positive level-one E8 lattice vacuum representation in orthonormal D5+D3 coordinates, declared simple-root cocycle, level-one Heisenberg Ward rules and internal quarter character v=(0^5,1/2^3). Insertions are separated. No raw-P1 pullback or physical time law is assumed or derived from this representation.
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.
Actual positive Cartan quartic Fock embedding
Use original Construction-A roots with physical metric dot/2 and u_j=(h_2j-i h_2j+1)/sqrt2 as J=+i basis. Canonically embed the actual five Sym^4 code support sums as neutral grade4 Heisenberg polynomials in the given E8_1 lattice vacuum. No physical preparation follows.
Original neutral root-pair moment fields
Use existing E8_1 neutral_source_response reader:64 chirally selected roots, original omission of one SU4 family weight line, B48=12T_E8+R_neut, AR=4P_A2-8P_U1. Family omission is an explicit existing input, not derived here from P1; neither it nor the R port or physical coupling is newly selected.
Complete exclusion of this neutral local weight-two zero-mode class including temporary code exits. Do not unite with separate inner E8 exclusion: mixed root/quadratic histories, individual mode roundtrips and higher fields remain outside. R_neut itself is not zero as a field.
Read the exact result, assumptions and evidence
Result
The full lattice-neutral weight-two space is8 derivatives plus36 symmetric quadratic fields. On original quartic code o(T_A)=dGamma(A), compression=(trA)/2 I5. Exact leakage sum is4||Ac-trA I8/8||²+(5/2)||Aa||²; real Gram relative to matrix HS metric has spectrum0^1,4^15,(5/2)^20 and kernel only A=aI. Original R_neut compresses0 with universal leakage bounds240..384, B48 compresses48I and B64 acts64I. Any sequence solely of these neutral zero modes acts by U8 on mode1; a code-preserving endpoint forces the U4 stabilizer and logical phase-G31 action, excluding marked R5.
Required assumptions
Given local E8_1 vacuum representation and explicit cocycle
Positive level-one E8 lattice vacuum representation in orthonormal D5+D3 coordinates, declared simple-root cocycle, level-one Heisenberg Ward rules and internal quarter character v=(0^5,1/2^3). Insertions are separated. No raw-P1 pullback or physical time law is assumed or derived from this representation.
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 positive Cartan quartic Fock embedding
Use original Construction-A roots with physical metric dot/2 and u_j=(h_2j-i h_2j+1)/sqrt2 as J=+i basis. Canonically embed the actual five Sym^4 code support sums as neutral grade4 Heisenberg polynomials in the given E8_1 lattice vacuum. No physical preparation follows.
Original neutral root-pair moment fields
Use existing E8_1 neutral_source_response reader:64 chirally selected roots, original omission of one SU4 family weight line, B48=12T_E8+R_neut, AR=4P_A2-8P_U1. Family omission is an explicit existing input, not derived here from P1; neither it nor the R port or physical coupling is newly selected.
Solely neutral local weight-two zero modes
Consider energy-preserving zero modes o(v)=v_(wt(v)-1) of all Hermitian lattice-charge-zero weight-two fields in given vacuum and temporal sequences solely of these generators. Grant the mathematical class without claiming P1-derived physical intervention permission, time or projections. Alternating root-current operations, individual K48 mode roundtrips, nonneutral fields and higher mode products are different classes.
Actual J-positive canonical Cartan Fock embedding in given lattice vacuum and compact inner E8 zero-mode operations. Not an exclusion of higher modes, nonlinear mode products, other derived field maps, non-automorphism interventions or all P1 physics; no physical preparation or selection of the embedding proved.
Read the exact result, assumptions and evidence
Result
The original lattice vertex law gives root PBW component f(b)alpha(-3)ealpha/3. The actual five quartics have no common projective zero; all240 root zero modes therefore leak from the neutral code. Distinct charge sectors prevent cancellation. The stabilizer has Lie algebra Cartan8 and H0=T8; its components normalize T8 and preserve the actual H+ four-plane. The full native U4 code-stabilizer theorem restricts its logical action to phase-G31, outside which R5 lies. Arbitrary inner-group sequences returning to code are included in the exclusion. L0 acts as4I5; conformal time and Casimir shift do not enlarge the logical action.
Required assumptions
Given local E8_1 vacuum representation and explicit cocycle
Positive level-one E8 lattice vacuum representation in orthonormal D5+D3 coordinates, declared simple-root cocycle, level-one Heisenberg Ward rules and internal quarter character v=(0^5,1/2^3). Insertions are separated. No raw-P1 pullback or physical time law is assumed or derived from this representation.
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 positive Cartan quartic Fock embedding
Use original Construction-A roots with physical metric dot/2 and u_j=(h_2j-i h_2j+1)/sqrt2 as J=+i basis. Canonically embed the actual five Sym^4 code support sums as neutral grade4 Heisenberg polynomials in the given E8_1 lattice vacuum. No physical preparation follows.
Compact inner E8 current operation class
Consider exponentials of compact weight1 current zero modes and arbitrary sequences preserving the given code at the endpoint. Optional conformal L0 propagation commutes with them. Nonlinear Hamiltonian polynomials, higher modes and non-automorphisms are not identified with or excluded as this class.
Excludes this literal charged four-current identification with unchanged tensor intertwiner, not the actual Cartan four-space or every derived composite field map.
Read the exact result, assumptions and evidence
Result
Under literal C4/register identification all three source clock branches act as -I20 on three C currents but as R5 tensor I4 on F20. The actual W intertwiner has squared HS residual48. Every traceless Q(H) already acts0 on C and its products but nontrivially on F. Thus finite cover agreement is not a full field-algebra map.
Required assumptions
Given local E8_1 vacuum representation and explicit cocycle
Positive level-one E8 lattice vacuum representation in orthonormal D5+D3 coordinates, declared simple-root cocycle, level-one Heisenberg Ward rules and internal quarter character v=(0^5,1/2^3). Insertions are separated. No raw-P1 pullback or physical time law is assumed or derived from this representation.
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.
Original coupled determinant event lift
Compare all60 original events with h=exp(-i pi/4)r and A5=-rho5(r) on the coupled finite SU4 cover. This is not the unchanged order2 G31 word or a selected physical pulse.
Literal charged-current/register identification
The tested dictionary identifies the four original charged C_a currents with the unchanged native four-register and requires the actual erasure tensor intertwining under gl5 source operations. This is a tested hypothesis, not a P1-derived fact.
Given original E8_1 vacuum source and coupled finite lift; no P1 source selection, physical time, instrument, polar minimization or physical field identification.
Read the exact result, assumptions and evidence
Result
Actual mixed Chevalley brackets give Q(H)|F20=H tensor I4, Q(H)|C4=trH/3 and kappa=trHK-trH trK/9. Noncommuting E+ responses exist. Separated Ward responses distinguish original polar and countermarks. Clock-current norm25/18 is constant throughout CP1 and selects no point. Three gl5 lifts of given R5 have full orders12,4,12; a genuine order4 source clock exists. All60 actual finite event charts match the native erasure twenty-action on the coupled determinant cover.
Required assumptions
Given local E8_1 vacuum representation and explicit cocycle
Positive level-one E8 lattice vacuum representation in orthonormal D5+D3 coordinates, declared simple-root cocycle, level-one Heisenberg Ward rules and internal quarter character v=(0^5,1/2^3). Insertions are separated. No raw-P1 pullback or physical time law is assumed or derived from this representation.
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.
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.
Original coupled determinant event lift
Compare all60 original events with h=exp(-i pi/4)r and A5=-rho5(r) on the coupled finite SU4 cover. This is not the unchanged order2 G31 word or a selected physical pulse.
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.
Given original code/decoder and logical operation. No selection of H, logarithm branch, physical time normalization or actual intervention permission. Unitary class, not all CPTP instruments. Native group action differs from the decoder by an explicit gauge factor whose physical compensation remains open.
Read the exact result, assumptions and evidence
Result
The actual decoder D(A)=W(A tensor I4)W* is an injective corner *-homomorphism. Thus D(A)D(B)=D(AB), product preservation of unitary complement extensions and exp(-itD(H))=D(exp(-itH))+I-Pi follow without a new composition rule for given H. Every unitary lift with full branchwise erasure compatibility is necessarily and sufficiently R tensor V4, forced by the commutant rather than fitted. On C64 an arbitrary unitary44-complement block remains.
Required assumptions
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 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.
Unitary continuation with full branchwise erasure action
Consider unitaries T on the actual erasure carrier satisfying T(rho tensor I4/4)T*=RrhoR* tensor I4/4 for every logical state rho and a given logical unitary R. This is a declared compatibility class, not a physical instrument rule derived from P1.
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.