Canonical stress now follows from given normalized full current OPE rather than independent stress premise. Level1/E8 realization remains input. Invariant source operator identity, not physical selected intervention/time law, particle clock or3+2. Correlated source constructions are not independent experiments.
Read the exact result, assumptions and evidence
Result
With actual cocycle dual sign full240-root OPE gives sum_alpha alpha tensoralpha=60I8 and h_dual30. Thus T_Sug=(1/62)sum248:J^AJ_A:=(1/2)sum8:h²:=T_E8,c8. Same stress follows within same source fromr3 quartic Wick ladder. ForT_n=sum248J^A_-nJ_A_n full endpoint factors onC5 atn0..3=(134,103/2,5,1/2), exterior-defect Grams=(100,225/4,9,1/4)I5. FULL source vectors obey(T0/2+T1+T2+T3)f_i=124f_i: exterior pieces cancel exactly, not just compressions. No root/family branch or fitted rate.
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.
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 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.
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.
Geometric data reconstruction in given source, no physical minimization/real-state rule. Full complex state space has additional continuous minimacosθw1+i sinθw2; explicitly checked, never discarded. Overallantipode/permutation remain. No positivepolar transport or physical3+2 mark. Quartic code remains input; no new independent empirical confirmation or priority claim.
Read the exact result, assumptions and evidence
Result
Actual2+2 flatteningsH_i of normalized original quartics give ten intrinsic logical rank1 raysv_t in localr2 product, sumP_t=2I5, Tr(P_tP_s)=1/9. Configuration fixes projective real structure through antiunitary fixing all rays unique up to common phase. On intrinsic real projective locus, sum_t(v_t.w)^4>=2/5 has exactly12 oriented equality points, hence six axes with regular simplex Gram6/5I6−1/5J6. All1024 sign cases exhausted. OriginalN introduced ONLY AFTERWARDS matches literally w_jw_j^T=(6/5)N_pi(j)N_pi(j)^T without fitrotation. Previously separately used UNMARKED W6 simplex thus gains concrete common local field-operator origin.
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.
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 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.
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.
Mode components are not entire local zero modes or selected physical instruments. Matrixunit law only weight4, not all grades. Leadingr2 is not whole fourth-pole coefficient; retainr3 Taylor term. Wick contraction is field algebra, not executed measurement. ChosenM5 coefficients available, not physically selected. Endpoint map A→PAP is not multiplicative for full field algebra; M5 is realized by stated code-preserving weight4 family, not a representation of all source operations onC5.
Read the exact result, assumptions and evidence
Result
Actual local F_i(z)F_j#(w) has exactly five Wick contraction classesr0..4 with all Taylor terms. Leading weight8/6/4/2/0 Grams areI25,4I25,(18^1,8^9,0^15),4|I><I|,|I><I|. r1=sum_a g_ia conjugate(g_ja) contains actual erasure20; isolated3creation/3annihilation mode component actsEij tensorI4 on that grade3 carrier.4/4 componentsEij=P_i(u_-1)conjugate(P_j)(ubar_1)/sqrt(NiNj) on ENTIRE weight4 are exactly|f_i><f_j| plus0 on rest, without intermediate projection. Full local products permit nonscalar logical action and entireM5; old pureCartan-Weyl exclusions are not all-field exclusions.
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.
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 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.
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.
Existing level1 lattice/current realization, not rawP1 or physicalH_adm. Distinguish mode hull and Hilbert closure of smooth local fields. No exact10D/45D model or particle count; irreducible affine module differs from finite horizontal E8 decomposition.
Read the exact result, assumptions and evidence
Result
Four original Cartan annihilators take nonzero quartic to vacuum; all original root currents generate all E8 lattice charges, negative Cartan modes all oscillators. Algebraic mode hull ofC5 is exactlyV_E8; closed local smear hull isH_E8. Both twenties are included. Full character prod_m(1−q^m)^−8 sum_lambda q^(lambda²/2)e^lambda determines all horizontal grade decompositions; grades0..6 dimensions1,248,4124,34752,213126,1057504,4530744. Grade2=1+248+3875 checked by complete Freudenthal recurrence. Controlled grading preserves finite mode word above maximal intermediate weight; smooth local profiles retain controlled infinite Fourier tails.
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.
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 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.
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.
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.
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.
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.
Shared mathematical cause within the given core. The physical symmetric rule, origin of the core data, and an E8-current/fermion identification are not derived.
Read the exact result, assumptions and evidence
Result
P64 is characteristic as O2(G31). Fix_P64(Sym^4 V4) is exactly the actual five-code; the original codeword sums give its real intertwiner rather than a fitted embedding. The existing erasure intertwiner forces rho20(g)=rho5(g) tensor bar(rho4(g)) for the same words. The full fourth tensor power has Pauli fixed rank16, whereas the symmetric fourth power has rank5.
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.
Exact conditional internal shape selector and scoped control. No P1-derived field/potential, complete TFPT dynamical uniqueness, physical Higgs/gauge identification or T1–T8 closure.
Read the exact result, assumptions and evidence
Result
For every traceless Hermitian rank-five endomorphism with trX²=1, globally7/30≤trX⁴≤13/20. Exhaustive stationary-spectrum classification gives equality below only at2+3 and above at1+4. The even conjugation-invariant bounded-below controls -t+t²±trX⁴ therefore favor different splits, without an inserted preferred direction. The primitive spectral functional differentiates a supplied deformation; a positive profile does not imply a positive fourth-order shape term. Original X, trace constraint, deformation and sign remain to be derived.
The diagnostic grants X=X* on an unmarked complex rank-five carrier, trX=0 and trX²=1 for shape classification. It studies trX⁴ or -t+t²±trX⁴, t=trX². Field, trace constraint, contraction, sign, even symmetry and dynamics are not derived here from P1.
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.