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.
Only excludes these general composition/cyclicity conditions as sufficient algebra selectors. The new pair is not a same-center counterexample or full physical TFPT countermodel. Additional principles such as purification remain conditional, not P1-derived.
Read the exact result, assumptions and evidence
Result
The existing operational context-quotient/GNS result remains conditional on the original operation alphabet. Classical and quantum models obey general composition, causality and local tomography. C4 and L(M2) additionally share HS(M2), J and Omega, cyclic/separating faithfulness and order zero, yet only the latter is noncommutative; their centers differ. A coherent two-channel operation separates a restricted population protocol (double pulse:1 versus1/2), but source availability must first be derived.
Required assumptions
Explicit finite process and cyclicity controls
The controls compare finite classical and complex-quantum processes and C4 versus L(M2) on HS(M2), J(X)=X*, Omega=I/sqrt2. Both cyclic/separating, faithful and order zero; the C4/M2 centers differ. No complete P1/P2, OS, seam or minimality countermodel is asserted.
Do not identify source types
Observable algebra, OS reflection, state, collar operator, time evolution, carrier connection, family holonomy, and readout are distinct data until an original map relates them.
Excludes the finite rank/primitivity/channel-uniqueness implication in the stated covariant exterior class, stronger than the earlier nonminimal5+2 test. Does not establish multiple global original minimizers, equal S1 defect vectors, all anomalies or complete P1/P2 worlds. An original bosonic index and actual sector/vertex map could distinguish them but are not derived here.
Read the exact result, assumptions and evidence
Result
Rank5 is the first possible rank in the stated class: even ranks fail total parity, and at rank3 total degree5 is not a multiple of3. The same canonical volume tensor supplies nonzero primitive trilinears on E=C5 for1+4,2+3,3+2,4+1. Minimal covariant degree5 channel spaces modulo fermion exchange have dimensions1,2,1,1. Thus even unique primitive channel leaves1+4,3+2,4+1; with b>s,3+2 and4+1 remain. All four16-dimensional packets have Tr X=Tr X³=0 for X=(-s)I_V+bI_W.2652 integer group-action coefficients were checked exactly; no suitable coupling was fitted.
Required assumptions
Minimum rank in the covariant even-even-fundamental tensor test
Given E=V⊕W with b,s≥1, Hermitian metric and unit complex volume; G=S(U(V)×U(W)); two nontrivial positive even exterior fermion legs and one fundamental boson leg in W. Rank and split vary. Primitivity is in the holomorphic covariant invariant ring without independent conjugate/dual field variables. At rank5 restrict the same volume tensor vol(α∧β∧h) to all splits. No actual source vertex, equal complete S1 minimum defect or physical P1/P2 countermodel is assumed.
Do not identify source types
Observable algebra, OS reflection, state, collar operator, time evolution, carrier connection, family holonomy, and readout are distinct data until an original map relates them.
Excludes only full even internal CAR as coordinate algebra with J_H; positive Hodge signs conditional on rank/metric/volume. No physical CAR, J, KO6, spacetime or E=C5 origin derivation. Proper compatible subalgebras and originally derived symmetry breaking remain possible. Whole-space antiunitarity is not automatically the quaternionic W factor of a different triple.
Read the exact result, assumptions and evidence
Result
For the given E=C5, J_H²=1, J_HΓ=-ΓJ_H and (ΓJ_H)²=-1 hold on ΛE. Internal CAR is End(ΛE), its even part End(S+)⊕End(S-). J_H swaps both blocks and maps this full algebra onto itself; order zero would require commutativity. An explicit matrix-unit commutator has rank2. The full internal algebra is thus not a spectral coordinate algebra with this J_H. A selected left/right bimodule subalgebra remains missing. Reuse the existing SU5 no-selector: unmarked exterior algebra and trace balance select no3+2 projector.
Required assumptions
Given unmarked Hermitian rank-five carrier with volume
Given E=C5 with Hermitian metric and unit complex volume, F=ΛE, Γ=(-1)^degree and conjugate-linear Hodge J_H. Neither3+2, named particle/gauge modules nor Yukawas are inserted. Internal Clifford operators are not identified with physical local fermion fields; rank5 and volume are not newly source-derived.
Do not identify source types
Observable algebra, OS reflection, state, collar operator, time evolution, carrier connection, family holonomy, and readout are distinct data until an original map relates them.
Excludes only unique algebra reconstruction from these data in the declared product/domain setting. Not identical complete original tuples, full OS/P1/P2/minimality countermodels or a universal TFPT no-go. Additional algebra hypotheses need their own origin.
Read the exact result, assumptions and evidence
Result
On the same product skeleton A0=C∞(M)I and A1=C∞(M,Mn) acting on the left differ, with the same scalar center, identical D,J,Γ,B,R and fixed relative spectral/eta responses. Both obey order zero/one: JL_(b*)J^-1=R_b, [[D,L_a],R_b]=0. A1 mixes internal left indices, A0 does not. For n=2: fiber-span dimensions1/4, commutant dimensions16/4 and16 exactly vanishing left/right basis commutators. These operator/center/order conditions therefore do not reconstruct the multiplication algebra.
Required assumptions
Declared product skeleton for algebra identifiability
Given a compact spin collar with a local selfadjoint elliptic geometric boundary problem stable under smooth multipliers, or closed double with H1 domain. On H=L²(M,S)⊗HS(Mn(C)),n≥2: D=D0⊗I, B=B0⊗I, J=J0⊗(X→X*), Γ=Γ0⊗I, R=R0⊗I, with zero finite Dirac term. Compare scalar and left matrix insertions on the same H. No arbitrary APS-domain invariance, complete OS source, unit winding or minimal P1/P2 branch is claimed.
Do not identify source types
Observable algebra, OS reflection, state, collar operator, time evolution, carrier connection, family holonomy, and readout are distinct data until an original map relates them.
Excludes only the local invariant-theoretic inference under the stated polynomial interpretation. No P1/P2 countermodel or refutation of additionally justified global minimality or actual branch-tensor support; no new Lean build.
Read the exact result, assumptions and evidence
Result
For V=C5,W=C2, (V⊗W)⊗Λ4V⊗W→detV⊗detW is a nonzero S(U5×U2)-invariant tensor with even fermion degrees2/4. Centers and U1 charge force positive invariant total degrees(5m,2m); thus the tensor is primitive at carrier degree7. It refutes the local inference that r≥4 must factor in the invariant ring. The stronger Lean premise E≃(Λ²E)* remains valid and needs its own source derivation.
Required assumptions
Covariant invariant-ring primitiveness at variable carrier rank
For the local inference test: V=C^b, W=C², G=S(U(V)×U(W)), both fermion legs in Λeven(V⊕W). Primitiveness excludes invariant factors of smaller positive carrier degree in the covariant polynomial invariant ring. No global minimum over different sources/ranks or independently selected special tensor support is assumed.
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.
Literal source interface of the archived definition; not a refutation of the algebraic 3+2 statement for a supplied P2 carrier. Likewise (2G−I)|Ran(G)=I: full boundary data alone do not create an internal marking.
Read the exact result, assumptions and evidence
Result
Pprim=C Pgap implies Ran(Pprim)⊂Ran(C). Thus (2C−I)|H_car=I for a carrier embedded there, and E_−=0. Every isometric pullback W into Ran(C) also yields W*(2C−I)W=I. The later 3+2 polarization does not follow this way. The general lemma CQ=Q ⇒ (2C−I)Q=Q is kernel-checked. A different carrier factor needs an explicit different map.
Required assumptions
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.
Literal carrier embedding after positive compression
Read the archived phrase H_car after primitive admissibility compression literally as a subspace of Ran(Pprim)⊂Ran(C), then define internal polarization as (2C−I)|H_car. A different coefficient-factor interpretation instead requires its own embedding and internal involution.
Correction of the general geometric identification in the archived source, not rejection of P1/P2 or all seam realizations. A nonlocal R:=2C−I can be defined but thereby has no demonstrated geometric origin.
Read the exact result, assumptions and evidence
Result
The original source identifies R_tau=2P_C−I from reflection regularity. A compact product-Dirac counterexample with B=diag(A_1/4,−A_1/4) and local R=σx disproves that general inference: the deck lift on the two fixed boundary faces changes G, whereas 2G−I commutes with G. The principal-symbol comparison confirms the type distinction. R_alg=J_n(2C−I) is a reconstructed, generally nonlocal involution; its algebraic quarter turn does not derive charged quarter holonomy.
Required assumptions
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.
Local geometric lift on a Dirac boundary
The claimed geometric deck reflection acts at the fixed boundary through a local bundle lift over the same base reflection. Compare an ordinary positive-dimensional Dirac boundary with orthogonal Calderon projection; a nonlocal operator subsequently defined as 2P_C−I is not a pre-established geometric action.