Follow the published construction into its registered claims, then explore reviewed research questions, attempts and the connections that remain open.
Historical: common source, state and event selection
Outside the active mathematical research program since the user's23Oct2026 decision. Historical question: Do the original principles derive a common physical source together with its state, dynamical events, and physically observed fields?
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.
Analytic theorem for finite unitary Cartan-Weyl-only histories and original neutral code-Fock embedding. Not general nonunitary field products, root currents, interleaved stress/current histories, open processes with memory or all E8/P1 operations. No original P1 time law or new code-preservation axiom. Do not unite separate exclusions.
Read the exact result, assumptions and evidence
Result
Actual CCR fixes [J(V),J(W)]=i sigma I and exp(iJ(V))exp(iJ(W))=exp(−i sigma/2)exp(iJ(V+W)). Every finite ordered Cartan-Weyl history is phase times total displacement. Every nonzero total displacement turns any nonzero finite Fock polynomial into exponential nonpolynomial in Bargmann realization. Exactly code-preserving endpoint forces zero total displacement; neutral zero modes are trivial and only common phase remains. Entire stated class provides neither marked R5 nor3+2 selection. Nonzero individual profiles may cancel in total.
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 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 original Cartan-current insertion and specified Weyl class
In existing neutral Cartan-Fock module use full original current field and its actual CCR, all smooth real8-vector profiles with Vhat_-n=conjugate Vhat_n and h0=0. Ordered unitary classification uses only finite exp(iJ(V)) histories of these currents. Explicit mathematical class in given source net, not raw-P1-derived permission, field coupling, physical time or code=H_adm identification. Root currents, general field products and mixed stress histories are outside.
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.
All smooth Hermitian Cartan-current smearings in original neutral Fock module. Norm fractions are not measurement probabilities. Grade decomposition/code projectors are analysis only, no executed projection. No claim all higher field correlations are scalar or all other source fields excluded.
Read the exact result, assumptions and evidence
Result
For every smooth real8-vector profile with Vhat1=p.u+q.ubar, actual five-code obeys P J(V)²P=(3||q||²+||p||²+Tail)I5, Tail=sum_m>=2 m||Vhat_m||². Grade3 erasure squared norm is ||q||². Its fraction is at most1/3, strictly smaller for nonzero properly local profile with q!=0. Equality requires global rotating first harmonic; fixed real polarization has tighter1/4 bound. No profile produces only erasure output. Full first positive response Gram is scalar and selects no3+2 mark.
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 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 original Cartan-current insertion and specified Weyl class
In existing neutral Cartan-Fock module use full original current field and its actual CCR, all smooth real8-vector profiles with Vhat_-n=conjugate Vhat_n and h0=0. Ordered unitary classification uses only finite exp(iJ(V)) histories of these currents. Explicit mathematical class in given source net, not raw-P1-derived permission, field coupling, physical time or code=H_adm identification. Root currents, general field products and mixed stress histories are outside.
Existing E8_1 Cartan-Fock source and literal native embedding; no P1 pullback, code=H_adm identification or physical field assignment. No general ban on global P1 operations. New bridge connects previously checked carry to actual local OPE; prior checks are correlated groundwork.
Read the exact result, assumptions and evidence
Result
Actual current product obeys j_ubar_a(z)F_i(0)=G_ia(0)/z²+sum_k>=0 z^k/k! :partial^k j_ubar_a F_i:(0), without simple pole. Double pole is original erasure twenty; regular t_ia=ubar_a(-1)f_i contains exactly prior h5 primary r=t−7/85L_-2g+3/170L_-1²g, Gram78/85I20 and rho5 tensor conjugate4 action. [ubar_a(1),j_u_b(w)]=delta_ab also shows pure erasure is global Fourier mode, not an operator localized in a proper interval of given current net. OriginalP1 defines locality by support and A/A*-invariant restriction, not global compression.
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 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 original Cartan-current insertion and specified Weyl class
In existing neutral Cartan-Fock module use full original current field and its actual CCR, all smooth real8-vector profiles with Vhat_-n=conjugate Vhat_n and h0=0. Ordered unitary classification uses only finite exp(iJ(V)) histories of these currents. Explicit mathematical class in given source net, not raw-P1-derived permission, field coupling, physical time or code=H_adm identification. Root currents, general field products and mixed stress histories are outside.
All modes of original canonical source stress tensor
In given E8_1 lattice vacuum use exactly T_E8=(1/2)sum:h_A h_A:,c=8, without improvement/background charge. Examine all Virasoro modes and well-defined pure stress histories in original neutral Cartan-Fock space. This is a mathematically specified geometric transport class, not P1-selected physical history or time. It differs from prior class of all neutral weight2 ZERO MODES; do not combine exclusions. First erasure Ward action uses existing a_a=bar u_a(1), no extra source, root or family choice.
Pure original canonical stress histories in given source frame only; not all geometry/metric variations, affine KZ histories with other field insertions, noncanonical/nonstress fields or general P1 processes. No physical time identification or union with prior separate zero-mode exclusions.
Read the exact result, assumptions and evidence
Result
All modes/words of given canonical T_E8 act in generated code cycle as common Virasoro representation tensor I5. Every code compression is scalar, including excursions to higher grades and return; code-preserving unitary endpoint is a common phase only. No such pure stress history implements given nonscalar R5 or selects P2. U_conf=exp(i pi L0/2) agrees with native J on4/5/20 seed carriers, but acts as+i on newly forced weight5 twenty whereas native J there remains-i. Common Casimir phase does not repair this relative discrepancy.
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 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.
All modes of original canonical source stress tensor
In given E8_1 lattice vacuum use exactly T_E8=(1/2)sum:h_A h_A:,c=8, without improvement/background charge. Examine all Virasoro modes and well-defined pure stress histories in original neutral Cartan-Fock space. This is a mathematically specified geometric transport class, not P1-selected physical history or time. It differs from prior class of all neutral weight2 ZERO MODES; do not combine exclusions. First erasure Ward action uses existing a_a=bar u_a(1), no extra source, root or family choice.
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.
Unique orthogonal primary remainder relative to specified original erasure descendant block, not classification of all weight5 primaries. Mathematical decomposition, not executed projection. Neutral native twenty, not charged E8 F20 block or physical fermion/Higgs identification. First full/primary return Gram selects no3+2 mark.
Read the exact result, assumptions and evidence
Result
From [L_m,a_a]=-bar u_a(m+1), a_a L_-2 f_i=L_-2g_ia+t_ia with t_ia=bar u_a(-1)f_i. Original h3,c8 level2 Gram[[16,18],[18,84]] and overlap(1,0) uniquely give r_ia=t_ia-(7/85)L_-2g_ia+(3/170)L_-1²g_ia. These20 neutral weight5 primary states have Gram(78/85)I20 and exact original rho5 tensor bar rho4 action. Three existing native reflection lifts give60 literal intertwining checks. Full first Ward output Gram is19I20; code partial return is76I5 or312/85 I5 for primary remainder. No coupling or fitted field identification is chosen.
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 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.
All modes of original canonical source stress tensor
In given E8_1 lattice vacuum use exactly T_E8=(1/2)sum:h_A h_A:,c=8, without improvement/background charge. Examine all Virasoro modes and well-defined pure stress histories in original neutral Cartan-Fock space. This is a mathematically specified geometric transport class, not P1-selected physical history or time. It differs from prior class of all neutral weight2 ZERO MODES; do not combine exclusions. First erasure Ward action uses existing a_a=bar u_a(1), no extra source, root or family choice.
Given canonical E8_1 stress field and original neutral Cartan-Fock embedding. Conformal weights are not particle counts or physical time; no new irreducibility/gauge-group premise. Analytic all-word theorem; finite Fock checks only accompany it.
Read the exact result, assumptions and evidence
Result
Original J-positive mode1 polynomials are all Virasoro primary because u_a.u_b=0: L0=d,Lm>0=0. Actual4/5/20 carriers thus have(c,h,r)=(8,1,4),(8,4,5),(8,3,20). For all Virasoro descendant words Gram=G_(8,h)(lambda,mu)delta_ij. Generated source cycles factor as common unitary highest-weight representation modulo common nulls tensor actual multiplicity; native G31 acts on second factor. No R_neut branch, individual root or family omission is needed.
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 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.
All modes of original canonical source stress tensor
In given E8_1 lattice vacuum use exactly T_E8=(1/2)sum:h_A h_A:,c=8, without improvement/background charge. Examine all Virasoro modes and well-defined pure stress histories in original neutral Cartan-Fock space. This is a mathematically specified geometric transport class, not P1-selected physical history or time. It differs from prior class of all neutral weight2 ZERO MODES; do not combine exclusions. First erasure Ward action uses existing a_a=bar u_a(1), no extra source, root or family choice.
Same specified single-beta response family only. Norm fraction is not an instrument probability. Rank lower bound10 is not sufficiency of a10D realization, particle count, spacetime or3+2 derivation. No separate n=0 feedthrough, time-dependent maps or nonlinear models in excluded class; no wholesale transfer to other source histories.
Read the exact result, assumptions and evidence
Result
Complete exact feedback Gram Delta=B*B satisfies Delta>=5120I5 and has rank5. Every nonzero code input obeys sharp ||Pi Lpsi||²/||Lpsi||²<=9/44, equality exactly span(f2,f3); at least35/44 of squared response norm is exterior. Hankel/Gram block[[I,K],[K,H]] has exact rank10. Hence any fixed linear realization M_n=C A^n E for n>=0 with time-independent input/output maps needs at least10state dimensions. Closed5D continuation of these moments is excluded even after fixed input/output changes.
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.
Specified ordered response family of original single pair
Examine mathematically specified moments M_n=Pi L_beta^n Pi,n>=0, L_beta=o(W_beta)*o(W_beta), for same original doubled root(0,0,0,0,0,0,-2,-2). Pi/Q only split full amplitudes; no projection is executed. This tests composition/memory of the already determined source, not P1 selection of physical history, coupling, preparation or time. Full coherent multiroot processes must additionally retain nonneutral final charge sectors.
Fixed specified algebraic original single-pair response family. Given source functional is evaluable, but P1 physical event/time selection remains open. Neutral726 space is not complete for coherent multiroot cross terms.
Read the exact result, assumptions and evidence
Result
Original vertex formula gives complete192x726 single-beta S from neutral grade4 to charged grade4, with original Fock adjoint L=S* S. K=Pi L Pi,B=Q L Pi,D=Q L Q give exactly x_(n+1)=Kx_n+sum_(j=0)^(n-1)B*D^(n-1-j)Bx_j for code input. No projection is executed. Actual four-insertion response H=Pi L² Pi equals K²+B*B, not K². Example H00=81152/3 versus(K²)00=40144/9. Next memory term B*DB and Pi L³ Pi are also checked exactly from full source; all H entries independently reproduced in active Laurent/Heisenberg blocks.
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.
Specified ordered response family of original single pair
Examine mathematically specified moments M_n=Pi L_beta^n Pi,n>=0, L_beta=o(W_beta)*o(W_beta), for same original doubled root(0,0,0,0,0,0,-2,-2). Pi/Q only split full amplitudes; no projection is executed. This tests composition/memory of the already determined source, not P1 selection of physical history, coupling, preparation or time. Full coherent multiroot processes must additionally retain nonneutral final charge sectors.
Original sum and coherent profiles with unchanged moduli; no physically selected execution or code-closure theorem for arbitrary new weights. Prior complete positive-profile erasure trace gap remains separately valid. Other higher source histories are not excluded.
Read the exact result, assumptions and evidence
Result
For unchanged original unrenormalized root-input basis, Lsum=sum_beta S_beta* S_beta and <q,Lsum f0>=-2092/sqrt3 with leakage norm²>=4376464/3. All4 original family omissions give same exact witness. Independent targeted original-data check confirms35736-343776/4=-50208 for unnormalized witness. Full coherent S* S sum has additional charged cross terms, which vanish exactly in neutral q/f0 matrix element. Thus coherent profiles with |a_beta|=1 also fail code closure regardless of phases.
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.
The reviewed decision record says which approach can be reused, which failed in its scope and what a new attempt would have to change.
Insufficient for the goal
Polynomial degree as a physical source selector
The cited original seed definition assumes A_loc, tau_t, Theta, omega, [u_Sigma], and D_coll as data and shows only their collar completion. This cited evidence supplies no rule selecting the common physical source from a polynomial degree.
Consequence: Provide an explicit original-source rule mapping degree to source selection before assessing the strategy as a selector.
Order-four markings from given time and full charge
The whole family is completed by lattice saturation and exact classification, including an original-frame control.
Consequence: New knowledge requires a justified change to the original source operator or its field/sector map; further phases of the same family are already covered.
Use the downstream master functional as an original selector
The allowed source is already named in the zero/infinity barrier. A minimum on that restricted branch does not prove this upstream selection.
Consequence: Supply a selection rule computable from original data without a presumed B_min and prove its relation to the actual field/state/time operator.
Identify geometric deck lift with boundary polarization
The general implication from reflection regularity is disproved by a compact Dirac counterexample; carrier restrictions built on it need their actual operator map established again.
Consequence: Keep geometric lift, normal Clifford action and Calderon polarization distinct. For dynamics retain the full two-face relation and use the already verified inverse reader.
Determine the common original clock lift and event menu
Supply the actual common4/5/20 continuation of the existing coherent clock polynomial and the original-source event menu; jointly check normalization, erasure, products and full source time. Determine the complete4/5 generators or their currently free data from it.
Builds on: Reconstruct the actual common native operator object; Classify native rates and coherent residual channels
Resolved in scope
Reconstruct original mark and test native clock continuation
Consistently connect actual positive polar transport, unchanged four-clock and original decoder; test product lifts and the same native polynomial continuation on a decisive input.
Builds on: Reconstruct the actual common native operator object
Classify polar fibre and test existing native selection mechanisms
Completely classify actual clock-compatible marking fibre, characterize the original polar rule canonically and test existing moment parent/geometric product transport against the target operation.
Builds on: Reconstruct original mark and test native clock continuation
Retain actual original off-code states and compute feedback without fitted coupling or executed projection; decide whether same family admits closed five-dimensional linear continuation.
Builds on: Test full source-return code closure and erasure Ward terms
Derive actually original canonical geometric source histories including higher modes on4/5/20 and first forced erasure correction/native sector action without coupling choice; determine whether this class selects marked clock.
Builds on: Derive full source feedback and necessary memory
Check full local Cartan insertion and ordered product
Decide on actual original current field whether pure native erasure is local; derive regular carry and full local response and test stated ordered Cartan-Weyl class without fitted algebra.
Builds on: Derive original all-mode conformal/erasure bridge
Derive unmarked native simplex from local quartic product
Reconstruct a separately used native structure from full local product without its target coordinates; literal comparison only afterwards, physical selection separate.
Builds on: Determine full local source hull and unfiltered returns
Test original seam/state/event data on full field tensors
Determine actual event/state response on fulloriginalfieldtensors and whether given normalization/sectoraction yieldsoriginalpolar/P2 withoutnewselectionrule.
Builds on: Derive unmarked native simplex from local quartic product
Derive original P1 gap-scale term and test direct native spectral bridge
Evaluate explicit original source function without profile/measurement fitting and test smallest actually source-fixed3+2 bridge on full outputs. Keep physical overall selection separately open.
Builds on: Derive full original current/field products and original seam grading
The complete original boundary graph can determine B and, under CAR identification, the state jointly. The source must supply this graph, fiber comparison and scale. This replaces separate spectral/state searches; physical marking additionally needs the original local operator algebra. ReviewedKMSlocalcorrection:globalrotationmovesintervals,localTakesakitimedependsonrestrictedstate;deriveitsstate/netcomparisonfirst.
Original historical attempts linked to this question
These are original research records. Their reported verdicts are distinct from the reviewed results above.
e series source reconstruction 20261024
Reconstruct the canonical local E8 field completion from the already established marked D5+A3/C4 compiler glue; determine which original source and dynamics implications are actually proved.
Recorded verdict: PARTIAL
Historical inventory; not independently reviewed by this export.
Does the actual original seam representative derive a local mixed spinor weight-one current through quarter-twist fusion, and does the pasted scalar pair response identify that field?
Recorded verdict: PARTIAL
Historical inventory; not independently reviewed by this export.
Which marking and response memory follow from the original charge source, and can the derived four-grade factor close under the same Hamiltonian and actual clocks?
Recorded verdict: PARTIAL
Historical inventory; not independently reviewed by this export.
Does the existing signed neutral scalar/charge source generate a common time, state and ordered response law, and what information must an exact marked readout retain?
Recorded verdict: PARTIAL
Historical inventory; not independently reviewed by this export.
Does the proposed eightfold fold/multi-interval completion derive physical quotient locality and an E8 field extension from the same source while preserving the quarter phase, state and time?
Recorded verdict: EXCLUDED_IN_UNCHANGED_SOURCE
Historical inventory; not independently reviewed by this export.
Which full charged crossed-defect extensions of the fixed ordered CAR source preserve its marking, W4 composition and supplied time law, and which obligations remain for a joint source/state/event derivation?
Recorded verdict: PARTIAL
Historical inventory; not independently reviewed by this export.
Does the original source derive time, state and the internal 3+2 carrier together, or do the proposed Higgs/Yukawa repairs assume the missing field map?
Recorded verdict: PARTIAL_SOURCE_DERIVATION_NOT_CLOSED
Historical inventory; not independently reviewed by this export.
Can one D8-containing even-unimodular extension preserve the source charge/clock class, determinant-only packet response, unchanged fold time, and original vacuum requirement at once?
Recorded verdict: EXCLUDED_MARKED_FOLD_COMPLETION
Historical inventory; not independently reviewed by this export.
Which order-four markings are generated by the retained source time and full charge, and do they supply a physical common source without an inserted P5?
Recorded verdict: COMPLETE_INTERNAL_CHARACTER_CLASSIFICATION_PHYSICAL_SOURCE_OPEN
Historical inventory; not independently reviewed by this export.
Does the actual P1/P2 seam determine a common field, state and time map with the quarter phase retained, and what exact operator conditions permit reconstruction?
Recorded verdict: CONDITIONAL_SOURCE_RECONSTRUCTION_NO_P1_DERIVATION
Historical inventory; not independently reviewed by this export.