Established within the stated premises
finding:source-cubic-and-quarter-select-even-theta-and-spin-half-turn-lift
Original cubic and quarter select spin line and half-turn lift. S0=O(q0+q1+q2−K_C) is even,S0²=K_C,and equivalent to p definition. c has no nonzero fixed vector on Jac(C)[2],so S0 is unique c-invariant theta characteristic among16. Q preserves S0;at marked fixed point q0,Q*eta=i eta. Any compatible spin lift alpha has alpha^4=−I,order8. beta=alpha composed Q*alpha is order4 lift of h=Q²,independent of alpha->−alpha,with local phase+i. Previously open S0,beta selection is closed by existing data.
Scope: Spin bundle of actual genus2 source,not honest C2 linearization. Subsequent finding:spin-rigidified-theta-module-intertwines-full-native-heisenberg-clock-package closes complex module bridge for Wq. Physical choice of inverse point norm and state remains open.
Required premises and original source excerpts
- Recovered genus2 cover with retained coefficients
Retain exactly the older three-sheet cover C:w³=(z²-1)/(z²+1), eta=z(1-w³), eta²=1-w⁶, six Weierstrass points p_a=omega^a,q_a=-omega^a, given z base and clock markings. Seek actual flat coefficient lines for existing v117 monodromy, not mere permutation equivalence or an assumed field intertwiner.
premise:recovered-genus-two-cover-with-retained-signed-coefficients - Unchanged A0* Fuchsian equation and actual C4 clock
Fix exactly v115 A0*, U=diag(1,i,-i), residues U^k A0* U^-k at i^k and regular base point z=0. Retain base rotation, positive meridian on t=z⁴ and simultaneous fiber basis change. No new Riemann Hilbert solution or D4 reflection lift is assumed.
premise:actual-original-a0star-fuchsian-connection-and-c4-clock - 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.
premise:typed-source-structures
Established within the stated premises
finding:source-cubic-matches-native-pauli-translations-after-context-target-correction
Source cubic matches native Pauli translations after correcting context target. v783 distinguishes code labels L/(1+i)L from P_nat=N64/mu4;full Sp4 modules admit no linear intertwiner. On P_nat original sigma is fixed-point-free with polynomial(x²+x+1)²,exactly as c on Jac(C)[2]. Phi with columns IY,YZ,ZY,YI is explicit symplectic intertwiner. Among35 planes exactly5 are sigma invariant,exactly2 symplectic and orthogonal,selecting same unordered triple pair natively. Exactly18 c-equivariant symplectic markings remain. Earlier0-versus2 fixed-space conflict concerns only wrong code/context target.
Scope: This lemma covers exact symplectic bridge. Full central and four-dimensional bridge for Wq is closed in subsequent finding:spin-rigidified-theta-module-intertwines-full-native-heisenberg-clock-package. Older v784/v796 objects and exclusions retain scope.
Required premises and original source excerpts
- Recovered genus2 cover with retained coefficients
Retain exactly the older three-sheet cover C:w³=(z²-1)/(z²+1), eta=z(1-w³), eta²=1-w⁶, six Weierstrass points p_a=omega^a,q_a=-omega^a, given z base and clock markings. Seek actual flat coefficient lines for existing v117 monodromy, not mere permutation equivalence or an assumed field intertwiner.
premise:recovered-genus-two-cover-with-retained-signed-coefficients - Unchanged A0* Fuchsian equation and actual C4 clock
Fix exactly v115 A0*, U=diag(1,i,-i), residues U^k A0* U^-k at i^k and regular base point z=0. Retain base rotation, positive meridian on t=z⁴ and simultaneous fiber basis change. No new Riemann Hilbert solution or D4 reflection lift is assumed.
premise:actual-original-a0star-fuchsian-connection-and-c4-clock - Native Pauli translations and frozen Gaussian phase class
Native theta-translation target is P_nat=N64/mu4 from v783 in frozen four-component frame,with actual sigma and J=iI. Code quotient L/(1+i)L is the distinct duality-twisted context module. G31 has center mu4;full Clifford matrix group has center mu8 and additional coset zeta8 G31. Retain these concrete actions and phases.
premise:fixed-native-pauli-translation-target-and-gaussian-phase-class - 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.
premise:typed-source-structures
Established within the stated premises
finding:joint-source-clock-relations-have-one-native-clifford-gauge-class
Joint clock relations have exactly one native Clifford gauge class. Under explicit module condition beta->J=iI,no matrix in any64-element G31 fiber over the9 admissible Q actions squares to+iI;in zeta8 G31 each fiber has8. Original relation alpha sigma alpha^(-1)=sigma^(-1) leaves exactly alpha and−alpha. Explicit alpha=zeta8(XI)U has alpha²=J,alpha⁴=−I,and joint group order24. Actual C_G31(sigma) has order72 and is transitive on all9 sign classes. Thus one gauge class remains for fixed sigma,J,Pauli system.
Scope: Finite matrix construction with fixed central character. Geometric bridge for Wq is now separately proved by finding:spin-rigidified-theta-module-intertwines-full-native-heisenberg-clock-package. Conjugacy class is not automatically physical gauge;P1/P2 anchors,state and readout excluded.
Required premises and original source excerpts
- Recovered genus2 cover with retained coefficients
Retain exactly the older three-sheet cover C:w³=(z²-1)/(z²+1), eta=z(1-w³), eta²=1-w⁶, six Weierstrass points p_a=omega^a,q_a=-omega^a, given z base and clock markings. Seek actual flat coefficient lines for existing v117 monodromy, not mere permutation equivalence or an assumed field intertwiner.
premise:recovered-genus-two-cover-with-retained-signed-coefficients - Unchanged A0* Fuchsian equation and actual C4 clock
Fix exactly v115 A0*, U=diag(1,i,-i), residues U^k A0* U^-k at i^k and regular base point z=0. Retain base rotation, positive meridian on t=z⁴ and simultaneous fiber basis change. No new Riemann Hilbert solution or D4 reflection lift is assumed.
premise:actual-original-a0star-fuchsian-connection-and-c4-clock - Native Pauli translations and frozen Gaussian phase class
Native theta-translation target is P_nat=N64/mu4 from v783 in frozen four-component frame,with actual sigma and J=iI. Code quotient L/(1+i)L is the distinct duality-twisted context module. G31 has center mu4;full Clifford matrix group has center mu8 and additional coset zeta8 G31. Retain these concrete actions and phases.
premise:fixed-native-pauli-translation-target-and-gaussian-phase-class - 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.
premise:typed-source-structures
Established within the stated premises
finding:spin-rigidified-theta-module-intertwines-full-native-heisenberg-clock-package
Spin-rigidified theta space is identified as complete complex Heisenberg and clock module. For L=O(2Theta),N={g in G(L):g⁴=1} is canonical order64 group with mu4 center. Every symplectic Phi lifts to N≅N64;Stone-von Neumann identifies Wq=H0(L) tensor Nq^(-1) with native C4 module. c lift has no remaining Pauli shift because1−Sigma invertible. alpha gamma alpha^(-1)=gamma^(-1) and trace(Ad gamma)=1 force trace(gamma)=1,hence exactly gamma->sigma without cube-root phase. Same intertwiner sends alpha to fully classified joint Clifford conjugacy class.
Scope: Complex four-dimensional module closure for explicitly constructed Wq including full central action. Physical source selection of one inverse Nq copy remains open. Does not derive integral Gaussian E8 lattice,physical metric,W8 dynamics,or state rule.
Required premises and original source excerpts
- Recovered genus2 cover with retained coefficients
Retain exactly the older three-sheet cover C:w³=(z²-1)/(z²+1), eta=z(1-w³), eta²=1-w⁶, six Weierstrass points p_a=omega^a,q_a=-omega^a, given z base and clock markings. Seek actual flat coefficient lines for existing v117 monodromy, not mere permutation equivalence or an assumed field intertwiner.
premise:recovered-genus-two-cover-with-retained-signed-coefficients - Unchanged A0* Fuchsian equation and actual C4 clock
Fix exactly v115 A0*, U=diag(1,i,-i), residues U^k A0* U^-k at i^k and regular base point z=0. Retain base rotation, positive meridian on t=z⁴ and simultaneous fiber basis change. No new Riemann Hilbert solution or D4 reflection lift is assumed.
premise:actual-original-a0star-fuchsian-connection-and-c4-clock - Native Pauli translations and frozen Gaussian phase class
Native theta-translation target is P_nat=N64/mu4 from v783 in frozen four-component frame,with actual sigma and J=iI. Code quotient L/(1+i)L is the distinct duality-twisted context module. G31 has center mu4;full Clifford matrix group has center mu8 and additional coset zeta8 G31. Retain these concrete actions and phases.
premise:fixed-native-pauli-translation-target-and-gaussian-phase-class - 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.
premise:typed-source-structures