Excluded in the stated class
finding:geometric-square-clock-is-not-theta-v117-quarter-generator
On actual tau=i torus,Q:z->iz induces modularS on theta squares. It has projective order2 with eigenvalue ratio1:1:−1. U_v117 and its exactly conjugate normalizedT have projective order4 and three distinct eigenvalues. Thus no invertible linear or projective intertwiner sendsQ toU_v117. Positive joint dictionaryT->U andST->M remains valid.
Scope: Excludes only unchanged geometricQ mark in actual theta-square representation. No refutation of an originally derived family/sewing transport explicitly retaining distinct paths and central phases. No general exclusion of elliptic flavor sources.
Required premises and original source excerpts
- Original P2 order-three cusp class and stable bundle anchor
Target the original P2 Deligne class with cusp weights0,1/3,2/3 and holomorphic anchor O(-2)+O(-1)^2. Exact v117 M0_EXACT has polynomiallambda³−1,trace0,order3;its joint representation with U is irreducible. Keep these exact matrix properties distinct from the numerical ODE identification there. Test whether the natural scalar/spin pushforward of existing C2 realizes this actual connection;physical selection of the direct sum as flavor module is not presumed.
premise:original-p2-order-three-cusp-and-stable-anchor-target - Theta-square modular action with distinct geometric Q
Test standard S/T action on three theta squares as a representation of a marked elliptic family, retaining weight and central phases. Geometric Q:z->iz at fixed tau=i is the S homology change, not an already supplied T moduli path. Do not assume this family action has the same origin as spatial monodromy of fixed four-punctured P2.
premise:theta-square-modular-transport-with-distinct-geometric-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:canonical-conformal-boundary-trace-fixes-existing-elliptic-cauchy-state
The conformal Dirac law and boundary density give V_sigma=e^((d-1)sigma/2), which exactly cancels the harmonic-spinor factor:V_sigma Tr psi_sigma=Tr psi0. Thus V_sigma P_sigma V_sigma*=P0 for the orthogonal closed Cauchy-data projection. Both marked faces and regional supports are preserved. In common bulk L2 the required trace is T_sigma=e^-sigma_boundary/2 Tr for Dtilde=e^-sigma/2 D e^-sigma/2. Quarter/fold are transported afterwards. The existing labelled marginal and full free current source are identical for all compatible metrics on the same complex curve. An unweighted raw trace produces an exactly detected spurious covariance change.
Scope: Analytic smooth massless conformal Cauchy-data theorem, not a regulator-limit claim. The free state consequence uses the existing CAR quantization premise. Newly metric-defined stress, spectra, resolvents, APS projections, full marked graph traces, P1 origin and physical time are not identified; previous nonzero Schur/sea responses remain intact.
Required premises and original source excerpts
- Free chiral Cauchy quantization of the canonical elliptic annulus
Treat the dividing half of the given square curve in its canonical flat conformal representative as a product collar. Compute the Cauchy graph of Dplus=partial_t+B_Sigma,B_Sigma=−i partial_s with NS boundary,then interpret it using the expressly free chiral CAR construction as a quasifree two-boundary state. The labelled second-face compression is the covariance;the other is complementary. This Gaussian interpretation does not derive the original relative spectral-action measure,P1 time or EM/Quillen response.
premise:free-chiral-cauchy-quantization-of-canonical-elliptic-annulus - Original quarter CAR source with fixed fold and current dictionary
The already supplied single-channel CAR source with actual quarter domains alpha=1/4,3/4, negative-D sea, positive original excitation time, unitary eightfold F and actual channel diagonalization is fixed. M,O,U+ and the conformal NS point-splitting reference are the same original dictionary data; the single delta counterterm follows from the kernel difference. Even CAR current composites retain full arc correlations, charges, domains, all modes, original time and the existing subtraction prescription including finite derivative counterterms. No E8 root/spinor extension is assumed for this neutral current subalgebra. Identity with the original P1 net, Theta, collar and reference sector is not supplied. Real profiles and pulse strengths are intervention data.
premise:original-quarter-car-fold-current-subalgebra - 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 - Conformal metric family of the same marked massless annulus
Study the existing smooth midpoint-annulus source on the same complex square curve with fixed spin, cut and face markings and massless geometric chiral Dirac operator. g_sigma=exp(2sigma)g for arbitrary smooth real sigma, with canonical spin lift and bulk/boundary L2 transport derived from volume densities. On the fixed complex curve this includes all Hermitian metrics. Connection, mass and complex structure do not vary. An unchanged original reflection/clock requires invariant profiles; otherwise transport the symmetry as well. No identification with the unresolved full marked equator graph, primitive P1 measure or physical time. The averaging lemma expressly retains the given free conditional CAR quantization, source-independent integrable total weights and nonzero normalization.
premise:same-marked-elliptic-annulus-massless-conformal-metric-family
Established within the stated premises
finding:conformal-factor-averaging-preserves-given-free-boundary-wick-words
After the derived boundary transport, each given conditional free CAR word on the same marked curve is independent of sigma. Its identical normalized source functional therefore factors out of any source-independent integrable total geometric weight of nonzero normalization, including determinants, reference, Jacobian and anomalies. The connected fermion four-point defect stays exactly zero, as do all higher Wick rules. No particular geometric measure is selected or discarded. Pure conformal geometry variance is therefore not evidence of an interacting boundary source. With other varying data, integrating out this factor may change their effective weight; their covariance requires a separate test.
Scope: Only the given conditional free quantization and correctly transported CAR observables; no Gaussian theorem for original S_prim/P1 measure and no assertion that connected current correlations vanish. Z_ref[0] cancels in Z_rel[J]/Z_rel[0], not automatically from geometric forces. The next decisive four-field test requires an original covariance-changing variation or higher fermion vertex.
Required premises and original source excerpts
- Free chiral Cauchy quantization of the canonical elliptic annulus
Treat the dividing half of the given square curve in its canonical flat conformal representative as a product collar. Compute the Cauchy graph of Dplus=partial_t+B_Sigma,B_Sigma=−i partial_s with NS boundary,then interpret it using the expressly free chiral CAR construction as a quasifree two-boundary state. The labelled second-face compression is the covariance;the other is complementary. This Gaussian interpretation does not derive the original relative spectral-action measure,P1 time or EM/Quillen response.
premise:free-chiral-cauchy-quantization-of-canonical-elliptic-annulus - Original quarter CAR source with fixed fold and current dictionary
The already supplied single-channel CAR source with actual quarter domains alpha=1/4,3/4, negative-D sea, positive original excitation time, unitary eightfold F and actual channel diagonalization is fixed. M,O,U+ and the conformal NS point-splitting reference are the same original dictionary data; the single delta counterterm follows from the kernel difference. Even CAR current composites retain full arc correlations, charges, domains, all modes, original time and the existing subtraction prescription including finite derivative counterterms. No E8 root/spinor extension is assumed for this neutral current subalgebra. Identity with the original P1 net, Theta, collar and reference sector is not supplied. Real profiles and pulse strengths are intervention data.
premise:original-quarter-car-fold-current-subalgebra - 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 - Conformal metric family of the same marked massless annulus
Study the existing smooth midpoint-annulus source on the same complex square curve with fixed spin, cut and face markings and massless geometric chiral Dirac operator. g_sigma=exp(2sigma)g for arbitrary smooth real sigma, with canonical spin lift and bulk/boundary L2 transport derived from volume densities. On the fixed complex curve this includes all Hermitian metrics. Connection, mass and complex structure do not vary. An unchanged original reflection/clock requires invariant profiles; otherwise transport the symmetry as well. No identification with the unresolved full marked equator graph, primitive P1 measure or physical time. The averaging lemma expressly retains the given free conditional CAR quantization, source-independent integrable total weights and nonzero normalization.
premise:same-marked-elliptic-annulus-massless-conformal-metric-family
Established within the stated premises
finding:retained-seam-defect-has-derived-causal-geometric-response-with-contact
On the same source R_h(s,t)=Tr(DeltaC_s(t)partial_sD_s) and exactly partial_sR_h|0=k<h²>+integral_0^t chi_hh. Full mode sum weights are(E±k)^3/(16E),chi(0)0,chi_prime(0)=-k<h_prime g_prime>;it converges for smooth profiles. At t1 the retained collar-preserving probe gives contact+.3861727422,causal part-.4054682615,total-.0192955193. Direct commutator error<3.6e-13,independent full-family derivative after extrapolation<3.3e-8. Omitting contact gives error.386. Actual400-step machine computes the same relative response;129/257-mode difference<5.4e-7.
Scope: Exact relative response form of the given source;no assertion of separately defined infinite vacuum quadratic operators or geometry-change Fock implementers. Long curves finitely tested. No identification with primitive action variation,selected history,native quartic words or physical time.
Required premises and original source excerpts
- Retained zero-branch defect on a common Pin geometry history
The original normal-zero branch k=pi/2,b1,c+i carries the retained positive quarter defect Pi. Compare the given pure D0 sea C0 and C0+Pi,both evolved by the same D_s=exp(-s h/2)D0 exp(-s h/2). The profile and tested step/pulse histories are named diagnostic inputs. Relative geometry response is Tr(U Pi U* partial_sD_s). It is not defined into the primitive P1 action response or a thin local seam map.
premise:retained-zero-branch-defect-on-common-pin-geometric-history - Given flat Moebius Dirac boundary family and its full sea
The comparison fixes D=−i(sigma1 partial_x+sigma2 partial_w),Psi(x+1,−w)=c sigma1 Psi(x,w),c=±i,sigma1 Psi|boundary=b Psi|boundary,b=±1,and the canonical full two-dimensional negative D spectral polarization. This specified primary-literature family is not identified with all possible original P1 collars or covariances.
premise:given-flat-mobius-dirac-boundary-family-and-full-sea - Controlled conformal normal geometry of the given Pin Dirac source
For the authorized source study,the same given Dirac comparison D0 on its fixed(c,b)Pin/boundary domain is varied by smooth real even profiles h:g_e=exp(2e h)g0,D_e=exp(-e h/2)D0 exp(-e h/2)after unitary density transport. Profiles/amplitude are explicitly chosen diagnostic interventions,not a P1-selected history. Original pure D0 sea remains initial state;thermal Green quantization stays separate.
premise:given-pin-source-conformal-normal-geometry-probes - 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:same-final-pin-geometry-retains-history-while-relative-index-factor-is-fixed
For the two actual commonly evolved states P_t=U C0 U*,Q_t=P_t+Pi_t and exactly(I-zQ_t)(I-zP_t)^-1=I-zPi_t,determinant1-z,without requiringC_t-C0 inL1. After50 pulse steps and50 steps with exactly restored D0,the t1 relative readouts differ:no pulse-.4739486483,positive+.01-.4701253569,negative-.01-.4780397086. Recursion matches direct evolution to4.8e-14. Causal kernel predicts pulse derivative.3956885828,finite symmetric difference.3957175880. Complete historical sector weight remainsQ_full=e^-2alpha R(g,alpha);its geometry response at fixed alpha comes fromR and is not calculated by the index.
Scope: Exact transported pair identity on the full given source;pulse values from a finite129-mode run with chosen duration/amplitude. No cyclic cutoff shift as an index proof. Full P1 measure/reference Ward identification remains open;the relative CAR response is not substituted into delta logR without proof.
Required premises and original source excerpts
- Retained zero-branch defect on a common Pin geometry history
The original normal-zero branch k=pi/2,b1,c+i carries the retained positive quarter defect Pi. Compare the given pure D0 sea C0 and C0+Pi,both evolved by the same D_s=exp(-s h/2)D0 exp(-s h/2). The profile and tested step/pulse histories are named diagnostic inputs. Relative geometry response is Tr(U Pi U* partial_sD_s). It is not defined into the primitive P1 action response or a thin local seam map.
premise:retained-zero-branch-defect-on-common-pin-geometric-history - Given flat Moebius Dirac boundary family and its full sea
The comparison fixes D=−i(sigma1 partial_x+sigma2 partial_w),Psi(x+1,−w)=c sigma1 Psi(x,w),c=±i,sigma1 Psi|boundary=b Psi|boundary,b=±1,and the canonical full two-dimensional negative D spectral polarization. This specified primary-literature family is not identified with all possible original P1 collars or covariances.
premise:given-flat-mobius-dirac-boundary-family-and-full-sea - Controlled conformal normal geometry of the given Pin Dirac source
For the authorized source study,the same given Dirac comparison D0 on its fixed(c,b)Pin/boundary domain is varied by smooth real even profiles h:g_e=exp(2e h)g0,D_e=exp(-e h/2)D0 exp(-e h/2)after unitary density transport. Profiles/amplitude are explicitly chosen diagnostic interventions,not a P1-selected history. Original pure D0 sea remains initial state;thermal Green quantization stays separate.
premise:given-pin-source-conformal-normal-geometry-probes - 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 - Historical SF sector action and explicit relative normalization
fullv27/v28 set J1=SF,J=d SF,d2,SF1 in S_def(alpha)=S_def0+alpha J. Review concerns the intended defect-only relative reweighting and its extension to complete measure-compatible sectors Z_s=e^(-alpha d s)Z_s^0(alpha),Z0=Z0^0(alpha). Constancy of R=Z_s^0/Z0^0 is an explicit condition, not assumed from SF.
premise:historical-sf-sector-action-and-relative-normalization
Established within the stated premises
finding:full-projector-covariance-and-positivity-force-all-car-words
For an arbitrary positive CAR state with full covariance C=P=P*=P²,GNS zero norms force a(f)Omega=0 onkerP and a*(f)Omega=0 onranP. CAR normal ordering uniquely determines all finite words and hence the full state as the pure Fock/Slater state omegaP. Quasifreeness,gauge invariance and zero anomalous pairs follow instead of being assumed. This is the charged application of Araki1970 Lemma4.3. A positive ensemble mean with projection covariance P forces that same covariance and state almost surely;complex weights are outside this ensemble argument.
Scope: Exact infinite CAR-state theorem under actual full projection covariance,not a new general CAR theorem and requiring no extra Gaussian premise. GNS uniqueness gives no implementability in an arbitrarily prechosen old Fock representation. A geometric Cauchy projection is not automatically presented as a physical covariance;P1 and its real joint field carrier remain to be identified.
Required premises and original source excerpts
- Test a positive joint CAR source against the full geometric projection
Consider an arbitrary normalized positive state on the specified joint charged two-face CAR algebra,without assuming quasifreeness or U1 invariance. The geometric target is the existing closed chiral annulus graph P_A with both marked faces,W=1/2 and original NS B_Sigma. Actual identity C_source=P_A is an explicit condition of the uniqueness theorem and an unresolved P1 source equation,not a fact inferred from geometry or RP. A finite error test applies only on a genuinely P_A reducing CAR block. Complete Dirac vector source,OS Padm and physical time are not identified with this chiral scalar target. No freely chosen purification supplies missing physical cross correlations.
premise:positive-joint-car-source-tested-against-full-geometric-projector - 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:actual-two-face-projector-reduces-free-source-test-to-cross-coherence
The original closed annulus graph has P_A=[[I-C_A,X_A],[X_A,C_A]],C_A=(I+exp(2WB_Sigma))^-1,X_A=(2cosh(WB_Sigma))^-1 and X_A²=C_A(I-C_A). Its full kernel is a projection,a single face is not. An actual positive P1 CAR source with this joint kernel would therefore inherit all previous free current/Weyl/native-composite correlations under the old domain/subtraction dictionary without an extra Gaussian premise. The missing physical cross equation is omegaP1(aL*(f)aR(g))=<f,X_Ag>. Three positive source controls share the same face marginals;dephased and product states even share the full diagonal covariance but have four-point values0 andp(1-p). On an actual finite P reducing block,delta=Tr[(I-P)C]+Tr[P(I-C)] gives ||rho-rhoP||1<=2sqrt(delta) for arbitrary positive states. All256 ordered four-field words are independently checked;normal/-OO byte-identical.
Scope: Sufficient identification test for the existing chiral scalar two-face source;no physical purification freely selected from C_A,no complete Dirac vector source or new time. Nonprojection marginals still require higher-word comparison. Quantitative error only inside the tested reducing block;no global control from finite samples. Original P1 measure,joint closed real graph trace/domain and physical cross equation remain open. Prior four-field requirement is sharpened only for the full projection case.
Required premises and original source excerpts
- Test a positive joint CAR source against the full geometric projection
Consider an arbitrary normalized positive state on the specified joint charged two-face CAR algebra,without assuming quasifreeness or U1 invariance. The geometric target is the existing closed chiral annulus graph P_A with both marked faces,W=1/2 and original NS B_Sigma. Actual identity C_source=P_A is an explicit condition of the uniqueness theorem and an unresolved P1 source equation,not a fact inferred from geometry or RP. A finite error test applies only on a genuinely P_A reducing CAR block. Complete Dirac vector source,OS Padm and physical time are not identified with this chiral scalar target. No freely chosen purification supplies missing physical cross correlations.
premise:positive-joint-car-source-tested-against-full-geometric-projector - 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
Excluded in the stated class
finding:retained-marked-quarter-compiler-has-no-local-geometric-wick-correction
Four marked branch points have one complex modulus lambda. The original cyclic Q acts by lambda to1/lambda;at lambda=-1 its tangent action is minus1,with no invariant modulus. The explicit curve family y²=x⁴+a x²−1 confirms this by F_a(ix)−F_a(x)=−2ax². Thus rho nonzero is excluded from the existing supplied P1\mu4 parity compiler. Combined with the proved correct conformal boundary trace,fixed spin/connection and given free quantization,the whole normalized transported Cauchy/Wick source is unchanged by nearby compatible geometry changes. Source-independent averaging adds no connected fermion four-point correction. Existing clock-given square rigidity is reused;its primitive physical derivation is not claimed.
Scope: Local theorem in fixed marked C2 and massless free annulus with fixed spin/connection,canonical boundary transport and source-independent integrable weights. No assertion for full ramified equator graphs,other bundle insertions,bulk sea response or primitive P1 admission. QGEO.REALIZE.01 remains C/O.
Required premises and original source excerpts
- Existing C2 group with untwisted raw flat spin pullbacks
On the existing marked parity curve E:y²=x⁴−1,study h=(x,−y),Q=(ix,y),kappa=(bar x,bar y) and flat square-lattice spin characters. The explicit raw one-particle calculation uses the untwisted line alpha with chi(A)=chi(B)=−1. Complex-linear pullbacks U and anti-linear Real structures j are distinct;no fitted U1 phase. This is not the normal twist alpha tensor p*O1 with its two-dimensional kernel. An original P1 Spin/Pin or Calderon transport is not presumed.
premise:existing-c2-group-with-untwisted-raw-flat-spin-pullbacks - Free chiral Cauchy quantization of the canonical elliptic annulus
Treat the dividing half of the given square curve in its canonical flat conformal representative as a product collar. Compute the Cauchy graph of Dplus=partial_t+B_Sigma,B_Sigma=−i partial_s with NS boundary,then interpret it using the expressly free chiral CAR construction as a quasifree two-boundary state. The labelled second-face compression is the covariance;the other is complementary. This Gaussian interpretation does not derive the original relative spectral-action measure,P1 time or EM/Quillen response.
premise:free-chiral-cauchy-quantization-of-canonical-elliptic-annulus - Conformal metric family of the same marked massless annulus
Study the existing smooth midpoint-annulus source on the same complex square curve with fixed spin, cut and face markings and massless geometric chiral Dirac operator. g_sigma=exp(2sigma)g for arbitrary smooth real sigma, with canonical spin lift and bulk/boundary L2 transport derived from volume densities. On the fixed complex curve this includes all Hermitian metrics. Connection, mass and complex structure do not vary. An unchanged original reflection/clock requires invariant profiles; otherwise transport the symmetry as well. No identification with the unresolved full marked equator graph, primitive P1 measure or physical time. The averaging lemma expressly retains the given free conditional CAR quantization, source-independent integrable total weights and nonzero normalization.
premise:same-marked-elliptic-annulus-massless-conformal-metric-family - 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:changed-elliptic-modulus-has-derived-cauchy-response-and-neutral-fourfield-variance
Under the expressly changed modulus-family premise,both real structures and free Moebius quotient persist,but Q exchanges rho and minus rho. Actual primitive periods give I(rho)=2K((2−rho)/4),r=I(−rho)/I(rho),W=r/2,tau=ir and r_prime(0)=2Gamma(3/4)²/Gamma(1/4)². Free boundary covariance is c_n,alpha=[1+exp(4pi r(n+alpha))]^-1,alpha1/4,3/4. Its first variation is fixed and trace class;the original global holomorphic spin kernel independently confirms a nonzero variation. For actual lowest opposite NS modes,any positive shared geometry law gives exactly kappa4=−Var(c). Diagnostic averaging at rho plus/minus.1 yields−8.133849897e−6 with positive normalized two-mode density. In the retained original fixed-Q compiler rho0 disables this mechanism.
Scope: Exact conditional free Cauchy/CAR family calculation and finite70-digit checks;not an admitted original compiler branch,derived P1 law or variance,common physical time or spacetime interaction. ModularS differs fromT;complex structure differs from flat U1 Quillen moduli;holomorphic spin kernel is not directly positive boundary covariance.
Required premises and original source excerpts
- Explicitly changed marked elliptic modulus family with transported spin
Consider E_rho:y²=x⁴+i rho x²−1 for real |rho|<2 with the continued original nontrivial spin partition,both original dividing real structures and free Moebius glide. Expressly continue free chiral Cauchy quantization on the two-face annulus normalized by its primitive A cycle. Q and the original inversion act between rho and minus rho fibers;their original fixed-fiber action is expressly relaxed. The mixing formula conditionally assumes an arbitrary positive normalized source-independent law on this new family. Neither admission in the fixed C2 compiler nor P1 measure support,variance,full marked graph domain,physical time or an interacting source is derived.
premise:real-marked-elliptic-complex-structure-family-with-transported-spin - 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:actual-equatorial-half-form-trace-fixes-orthogonal-hardy-range
Actual measure is |dz|=|dx|/(|c||y|). Closed half-form trace is L2 unitary to |dx|. The two bounded Cauchy idempotents gP_Hg^-1 and g^-1P_Hg, |g|²=|tan(theta)|^1/2, have exactly closed range H² by outer cyclic factors. Their orthogonal range projector is P_H⊕P_H. Integrable branch singularities do not remove H1 vertex coupling. Intrinsic boundary generator has variable speed 4K(-1)sqrt(|sin2theta|), failing constant base-winding Ward relation.
Scope: Actual full equatorial trace for supplied metric. No equality with old quarter state, positive annulus KMS state or physical time. Preserve smooth annulus results and original W8 Hardy comparison.
Required premises and original source excerpts
- Existing C2 group with untwisted raw flat spin pullbacks
On the existing marked parity curve E:y²=x⁴−1,study h=(x,−y),Q=(ix,y),kappa=(bar x,bar y) and flat square-lattice spin characters. The explicit raw one-particle calculation uses the untwisted line alpha with chi(A)=chi(B)=−1. Complex-linear pullbacks U and anti-linear Real structures j are distinct;no fitted U1 phase. This is not the normal twist alpha tensor p*O1 with its two-dimensional kernel. An original P1 Spin/Pin or Calderon transport is not presumed.
premise:existing-c2-group-with-untwisted-raw-flat-spin-pullbacks - Original full four-mark inversion and distinct equatorial Real slots
The supplied marked base has x->ix and the inversion x->1/x already present in v512. Compare their actual lift to E:y²=x⁴−1,retaining the original four-marked base circle Gamma=|x|1. v510/v519 distinguish pointwise equatorial x->1/bar x from the OS family x->mu bar x acting along Gamma. These operations must not be identified in a source comparison. The old free carrier is named;no general physical P1 equality is presumed.
premise:full-original-four-mark-inversion-and-equatorial-real-slots - Ramified holomorphic half-form trace of the given C2 spin line
Retain actual E:y²=x⁴−1, its already supplied Q fixed nontrivial NS NS spin-line class alpha and normalized Szego kernel. Holomorphic trace includes sqrt(dz/dx), while ordinary flat coefficient p_*alpha remains separate. No extra Q linearization phase, positive equal-time CAR covariance or smooth circle domain is assumed.
premise:ramified-holomorphic-half-form-trace-of-existing-c2-spin - Actual L2 half-form trace on full marked equator
Retain supplied square elliptic curve, flat metric and NS spin line. On the full ramified equator sqrt(dz/dx) transports actual L2 norm to |dx|. Keep holomorphic kernel, orthogonal analytic range, positive CAR covariance and H1 vertex domain distinct. No new channel metric or free vertex condition.
premise:actual-elliptic-half-form-l2-trace-on-full-marked-equator - 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
Excluded in the stated class
finding:actual-holomorphic-quarter-cauchy-idempotents-are-not-positive-car-covariances
In actual L2 measure gP_Hg^-1 and g^-1P_Hg are not self-adjoint. Range and kernel have exact nonzero overlap integral |tan(theta)|^1/2 exp(-2i theta)dtheta/(2pi)=-1/sqrt2. An additionally chosen channel metric could orthogonalize them but would change the source.
Scope: Excludes only unchanged oblique idempotents as CAR covariance in actually induced measure. Orthogonal analytic range is constructively fixed; no general exclusion of positive elliptic sources.
Required premises and original source excerpts
- Existing C2 group with untwisted raw flat spin pullbacks
On the existing marked parity curve E:y²=x⁴−1,study h=(x,−y),Q=(ix,y),kappa=(bar x,bar y) and flat square-lattice spin characters. The explicit raw one-particle calculation uses the untwisted line alpha with chi(A)=chi(B)=−1. Complex-linear pullbacks U and anti-linear Real structures j are distinct;no fitted U1 phase. This is not the normal twist alpha tensor p*O1 with its two-dimensional kernel. An original P1 Spin/Pin or Calderon transport is not presumed.
premise:existing-c2-group-with-untwisted-raw-flat-spin-pullbacks - Original full four-mark inversion and distinct equatorial Real slots
The supplied marked base has x->ix and the inversion x->1/x already present in v512. Compare their actual lift to E:y²=x⁴−1,retaining the original four-marked base circle Gamma=|x|1. v510/v519 distinguish pointwise equatorial x->1/bar x from the OS family x->mu bar x acting along Gamma. These operations must not be identified in a source comparison. The old free carrier is named;no general physical P1 equality is presumed.
premise:full-original-four-mark-inversion-and-equatorial-real-slots - Ramified holomorphic half-form trace of the given C2 spin line
Retain actual E:y²=x⁴−1, its already supplied Q fixed nontrivial NS NS spin-line class alpha and normalized Szego kernel. Holomorphic trace includes sqrt(dz/dx), while ordinary flat coefficient p_*alpha remains separate. No extra Q linearization phase, positive equal-time CAR covariance or smooth circle domain is assumed.
premise:ramified-holomorphic-half-form-trace-of-existing-c2-spin - Actual L2 half-form trace on full marked equator
Retain supplied square elliptic curve, flat metric and NS spin line. On the full ramified equator sqrt(dz/dx) transports actual L2 norm to |dx|. Keep holomorphic kernel, orthogonal analytic range, positive CAR covariance and H1 vertex domain distinct. No new channel metric or free vertex condition.
premise:actual-elliptic-half-form-l2-trace-on-full-marked-equator - 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
Excluded in the stated class
finding:full-equatorial-checkerboard-free-algebra-fails-fixed-reflection-positivity
The two components of p^-1(|x|<1) lie on opposite positive and negative time halves for each fixed S or hS reflection. Actual NS Green kernel gives a globally U1 neutral even normal-ordered word :barpsi(f+)psi(g-): with strictly negative reflected norm for compact supports in both faces. Analytic interior witness is independent of boundary and vertex choices.
Scope: Only full globally neutral even free CAR algebra on the union under a fixed reflection. No automatic claim about local gauge invariance in original admissible algebra. Separate parity/gauge per face is additional data. Positive annulus source remains valid.
Required premises and original source excerpts
- Existing C2 group with untwisted raw flat spin pullbacks
On the existing marked parity curve E:y²=x⁴−1,study h=(x,−y),Q=(ix,y),kappa=(bar x,bar y) and flat square-lattice spin characters. The explicit raw one-particle calculation uses the untwisted line alpha with chi(A)=chi(B)=−1. Complex-linear pullbacks U and anti-linear Real structures j are distinct;no fitted U1 phase. This is not the normal twist alpha tensor p*O1 with its two-dimensional kernel. An original P1 Spin/Pin or Calderon transport is not presumed.
premise:existing-c2-group-with-untwisted-raw-flat-spin-pullbacks - Original full four-mark inversion and distinct equatorial Real slots
The supplied marked base has x->ix and the inversion x->1/x already present in v512. Compare their actual lift to E:y²=x⁴−1,retaining the original four-marked base circle Gamma=|x|1. v510/v519 distinguish pointwise equatorial x->1/bar x from the OS family x->mu bar x acting along Gamma. These operations must not be identified in a source comparison. The old free carrier is named;no general physical P1 equality is presumed.
premise:full-original-four-mark-inversion-and-equatorial-real-slots - Ramified holomorphic half-form trace of the given C2 spin line
Retain actual E:y²=x⁴−1, its already supplied Q fixed nontrivial NS NS spin-line class alpha and normalized Szego kernel. Holomorphic trace includes sqrt(dz/dx), while ordinary flat coefficient p_*alpha remains separate. No extra Q linearization phase, positive equal-time CAR covariance or smooth circle domain is assumed.
premise:ramified-holomorphic-half-form-trace-of-existing-c2-spin - Actual L2 half-form trace on full marked equator
Retain supplied square elliptic curve, flat metric and NS spin line. On the full ramified equator sqrt(dz/dx) transports actual L2 norm to |dx|. Keep holomorphic kernel, orthogonal analytic range, positive CAR covariance and H1 vertex domain distinct. No new channel metric or free vertex condition.
premise:actual-elliptic-half-form-l2-trace-on-full-marked-equator - 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:genuine-d4-orbifold-branch-has-unique-irreducible-v117-class
J_D4=1/2-(x⁴+x^-4)/4 generates C(x)^D4. For genuine unitary SU3 descent on(infinity,2,4), c=diag(1,i,-i), regular cubic M=a², four half-meridian branches have traces0,2,2omega,2omega². Trace2 is impossible; complex branches are reducible. Unique survivor is a³=1,p=(1/2,1/4,1/4), the v117 matrix orbit. For genuinely linearized stable parabolic object canonical MS descent follows from uniqueness of adapted metric. Treat concrete A0* connection separately.
Scope: Complete theorem in exactly specified genuine D4 class. Same abstract matrix class does not identify different marked base paths. Do not infer full D4 connection symmetry from C4 rotation. No physical P1 origin.
Required premises and original source excerpts
- Original P2 order-three cusp class and stable bundle anchor
Target the original P2 Deligne class with cusp weights0,1/3,2/3 and holomorphic anchor O(-2)+O(-1)^2. Exact v117 M0_EXACT has polynomiallambda³−1,trace0,order3;its joint representation with U is irreducible. Keep these exact matrix properties distinct from the numerical ODE identification there. Test whether the natural scalar/spin pushforward of existing C2 realizes this actual connection;physical selection of the direct sum as flavor module is not presumed.
premise:original-p2-order-three-cusp-and-stable-anchor-target - Genuine stable unitary D4 orbifold descent
The expressly specified stable determinant-trivial parabolic rank3 branch has genuine mark-preserving D4 linearization with Coxeter fiber c=diag(1,i,-i) and regular cubic meridian M=a². Apply Mehta Seshadri to this object. Do NOT identify the concrete A0* ODE with it merely from C4 rotation.
premise:genuine-d4-unitary-stable-parabolic-orbifold-descent - 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
Excluded in the stated class
finding:wall-weight-additivity-does-not-exclude-stable-v117-parabolic-bundle
Section011 row-sum argument assumes complete flag splitting along three global BG summands. W_wall is then direct sum of three parabolic degree-zero lines, a different reducible wall point. Four exact v117 meridians generate irreducible A4 order12; compact kernel cover has genus5. H0(E)=V^A4=0 and degE=-4 give exactly O(-2)+O(-1)^2. MS object is stable. Equal weight sets do not identify relative flags.
Scope: Excludes universal wall forcing and identification of W_wall with irreducible v117. Declared split A0* raw-flag test is separately exactly stable; without concrete Levelt/Birkhoff lattice its affine flags are not automatically ODE Deligne extension. No paper edits.
Required premises and original source excerpts
- Original P2 order-three cusp class and stable bundle anchor
Target the original P2 Deligne class with cusp weights0,1/3,2/3 and holomorphic anchor O(-2)+O(-1)^2. Exact v117 M0_EXACT has polynomiallambda³−1,trace0,order3;its joint representation with U is irreducible. Keep these exact matrix properties distinct from the numerical ODE identification there. Test whether the natural scalar/spin pushforward of existing C2 realizes this actual connection;physical selection of the direct sum as flavor module is not presumed.
premise:original-p2-order-three-cusp-and-stable-anchor-target - 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:recovered-september-c4-triangle-closes-original-a0star-monodromy
The28Sept synthesis already contains analytic identification of unchanged v117 ODE: t=z⁴ and diag(1,z,z³) give three Fuchsian points of orders4,3,2; sole infinity resonance coefficient is exactly zero. Alternative frame diag(1,z,z^-1) gives semisimple exponents -1/2,0,-1/2 without resonance. Image is faithful S4 rotation representation; positive t0 meridian is U^-1. Marked pair is exactly v117 up to simultaneous basis change and remaining diagonal basis-sign convention. Diagonal0,-i/2,+i/2 follows analytically. Original four meridians have A4 image; U extends to S4. No new ODE fit needed.
Scope: Reused older analytic proof independently checked now. Closes ODE group identification incorrectly treated again as numerical-only. No new origin derivation selecting A0*, no complete field/state/time isomorphism. Physical gates remain open.
Required premises and original source excerpts
- Original P2 order-three cusp class and stable bundle anchor
Target the original P2 Deligne class with cusp weights0,1/3,2/3 and holomorphic anchor O(-2)+O(-1)^2. Exact v117 M0_EXACT has polynomiallambda³−1,trace0,order3;its joint representation with U is irreducible. Keep these exact matrix properties distinct from the numerical ODE identification there. Test whether the natural scalar/spin pushforward of existing C2 realizes this actual connection;physical selection of the direct sum as flavor module is not presumed.
premise:original-p2-order-three-cusp-and-stable-anchor-target - 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
Excluded in the stated class
finding:actual-a0star-connection-does-not-have-rational-base-inversion-lift
For A0* local exponent analysis forces every rational z->1/z lift into finite polynomial ansatz. Transformation equation G prime=A(z)G+z^-2G A(1/z) has only zero solution. Actual C4 quotient connection has isotropy orders4 and2 at0 and infinity, exchanged by base inversion. It is not marked D4 pullback with orders4 at both, despite abstract S4 generator pairs occurring.
Scope: Only literal base inversion of unchanged A0* connection. Actual finite C4 monodromy is positively closed by old triangle proof. No exclusion of other base maps or other parabolic D4 objects. Same number or abstract group does not replace common clock marking.
Required premises and original source excerpts
- 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:actual-v117-sign-lines-are-negative-weierstrass-triangle-two-torsion
On C:eta²=1-w⁶ the correct lines are L_a=O(q_a-q_(a+1)),q_a=-omega^a. Kummer roots h_a²=(w+omega^a)/(w+omega^(a+1)) multiply to1. V4 cover Zq of w sphere ramifies only at three q points and has genus0. Actual quotient coordinate t=((1+w³)/(1-w³))² gives degree24 with uniform ramification orders4,3,2 over0,1,infinity. Thus Zq is regular S4 triangle cover of same marked v117 class. Fiber product with z⁴=t has ordinary A4 image; intersection field is C(z²), since S4 has no C4 quotient. Over C this is exactly unramified genus5 V4 kernel cover. Hence pi*E_v117 decomposes into three expressly written unitary two-torsion lines. Equality follows from actual Kummer cover and target marks, not fitted target matrices.
Scope: Exact flat coefficient lift with actual marked finite monodromy. No intertwiner of primitive CAR fields,state,positive time or support; no implicit selection of joint C2 NS field linearization. Older source already supplied curve and point intertwiner; line identification is separately proved here.
Required premises and original source excerpts
- Original P2 order-three cusp class and stable bundle anchor
Target the original P2 Deligne class with cusp weights0,1/3,2/3 and holomorphic anchor O(-2)+O(-1)^2. Exact v117 M0_EXACT has polynomiallambda³−1,trace0,order3;its joint representation with U is irreducible. Keep these exact matrix properties distinct from the numerical ODE identification there. Test whether the natural scalar/spin pushforward of existing C2 realizes this actual connection;physical selection of the direct sum as flavor module is not presumed.
premise:original-p2-order-three-cusp-and-stable-anchor-target - 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 - 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 - 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
Excluded in the stated class
finding:antipodal-and-positive-sign-packets-fail-actual-marked-v117-ramification
Initial natural packet O(p_a-q_a) is nontrivial unramified V4 two-torsion on C, but v_a=z u_a gives V4 field over w sphere ramified at all six Weierstrass points, genus3. Actual target orders are4,3,4 rather than4,3,2. Positive triangle O(p_a-p_(a+1)) gives2,3,4 with fixed target marks. Both are different marked connections. Comparing imported T and B_k matrices would have been circular; actual function field and ramification test decide.
Scope: Excludes exactly these packets for fixed v117 clock and target marking. Does not refute their mathematical existence. Equal dimension,A4 group and genus5 alone do not identify a flat source map.
Required premises and original source excerpts
- Original P2 order-three cusp class and stable bundle anchor
Target the original P2 Deligne class with cusp weights0,1/3,2/3 and holomorphic anchor O(-2)+O(-1)^2. Exact v117 M0_EXACT has polynomiallambda³−1,trace0,order3;its joint representation with U is irreducible. Keep these exact matrix properties distinct from the numerical ODE identification there. Test whether the natural scalar/spin pushforward of existing C2 realizes this actual connection;physical selection of the direct sum as flavor module is not presumed.
premise:original-p2-order-three-cusp-and-stable-anchor-target - 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 - 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 - 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
Excluded in the stated class
finding:common-cyclic-twelve-cover-does-not-trivialize-v117-unramified-sign-lines
Common cyclic fiber-product cover D:h¹²=(z²-1)/(z²+1) transports g=h³ and w=h⁴. Over C it is h⁴=w, totally ramified at two simple zeros and two simple poles of w. Its unique quadratic intermediate h² with square w is also ramified. Therefore D neither contains nor trivializes any of three nontrivial everywhere unramified actual v117 lines L_a. Geometric Q lift (z,h)->(iz,i/h) transports underlying curves but does not replace missing coefficients or a common field state.
Scope: Only specified cyclic mu12 root cover as automatic sign or field completion. Older general C12 monodromy/intertwiner contract is unchanged. Does not exclude other common higher covers or actual transported field maps.
Required premises and original source excerpts
- Existing C2 group with untwisted raw flat spin pullbacks
On the existing marked parity curve E:y²=x⁴−1,study h=(x,−y),Q=(ix,y),kappa=(bar x,bar y) and flat square-lattice spin characters. The explicit raw one-particle calculation uses the untwisted line alpha with chi(A)=chi(B)=−1. Complex-linear pullbacks U and anti-linear Real structures j are distinct;no fitted U1 phase. This is not the normal twist alpha tensor p*O1 with its two-dimensional kernel. An original P1 Spin/Pin or Calderon transport is not presumed.
premise:existing-c2-group-with-untwisted-raw-flat-spin-pullbacks - Ramified holomorphic half-form trace of the given C2 spin line
Retain actual E:y²=x⁴−1, its already supplied Q fixed nontrivial NS NS spin-line class alpha and normalized Szego kernel. Holomorphic trace includes sqrt(dz/dx), while ordinary flat coefficient p_*alpha remains separate. No extra Q linearization phase, positive equal-time CAR covariance or smooth circle domain is assumed.
premise:ramified-holomorphic-half-form-trace-of-existing-c2-spin - 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 - 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 - 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:negative-sign-packet-has-source-selected-quarter-clock-lift
Original geometric lift Q(z,w,eta)=(iz,w^-1,-i eta/w³) preserves q orbit and satisfies Q²=(−z,w,−eta). Kummer roots obey Q*h_a=epsilon_a omega/h_(-a-1),product epsilon_a=1. Actual local marked inertia t proportional s⁴ at q_0 forces epsilon_0/epsilon_2=−1 and epsilon_1=−1. The two inverse lifts are V4 frame-conjugate. Determinant-one coefficient representation fixes the remaining line with+1,square has two minus signs,spectrum1,i,−i. Explicit simultaneous basis change identifies U; positive quotient monodromy is contravariantly U^-1. C4 equivariance of existing coefficient lift is closed.
Scope: Equivariant flat coefficient lines of unchanged v117 source with actual clock direction. No selection of CAR field quantization or state,no identification of modularT with geometricQ,no new base inversion symmetry.
Required premises and original source excerpts
- Original P2 order-three cusp class and stable bundle anchor
Target the original P2 Deligne class with cusp weights0,1/3,2/3 and holomorphic anchor O(-2)+O(-1)^2. Exact v117 M0_EXACT has polynomiallambda³−1,trace0,order3;its joint representation with U is irreducible. Keep these exact matrix properties distinct from the numerical ODE identification there. Test whether the natural scalar/spin pushforward of existing C2 realizes this actual connection;physical selection of the direct sum as flavor module is not presumed.
premise:original-p2-order-three-cusp-and-stable-anchor-target - 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 - 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 - 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
Open questions preserve their achieved subresults
Each research problem has its own scope. A resolved task does not close its whole physical parent. Historical tasks remain explicitly historical in the source record.
Future quotient and positive GNS reconstruction
What is determined by complete future behavior, and what additional positive word-kernel source is needed to reconstruct a cyclic operator realization?
- Resolved in scope: Future quotient from complete rRead result
- Resolved in scope: Conditional GNS reconstruction from supplied KRead result
- open: Select complete K from the original source
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?
Seam connection and family action
Does the joint discrete theorem determine a concrete connection and a family coupling compatible with the determinant seam action?
- Resolved in scope: Separate classes from actual transportRead result
- open: Derive the common connection from the original source
Physical time and complete transfer
Do the original reflection/projector structures determine the full physical time generator and required field transfer?
- Resolved in scope: Reuse the sign-projector counterexampleRead result
- Resolved in scope: Reuse conditional full-transfer reconstructionRead result
- open: Identify the physical complete TFPT transfer
- Resolved in scope: Reuse complete collar reconstructionRead result
- Resolved in scope: Classify homogeneous GKLS embedding of the shared register familyRead result
- Resolved in scope: Decide every homogeneous GKSL logarithm of the fixed M4 channelRead result
Flavor ratio and absolute amplitude scale
Which flavor quantity is fixed on the derived discrete stratum, and which requires the continuous holonomy anchor?
- Resolved in scope: Reuse the derived flavor ratioRead result
- open: Determine the absolute holonomy amplitude
Alpha: derived terms, unique root and physical EM identification
Where do the factors and closed alpha equation come from, which provenance steps are proved, and which precise physical EM identification remains open?
- Resolved in scope: Separate root calculation from source selectionRead result
- open: Derive the actual EM functional variation and Thomson identification
- Resolved in scope: Establish the unique global positive rootRead result
Geometric flow and gravity normalization
Do common matter-only observations derive geometric horizon flow and the area response from the original gravity source?
- Resolved in scope: Record scope of fixed original functionalRead result
- open: Derive geometric modular response and area coefficient
Clocks and complete neutral source response
Which complete quadratic response can the actual clock images reconstruct?
- Resolved in scope: Reconstruct all neutral quadratic matrix directionsRead result
- open: Derive physically admissible probe access
Clock action on charge, time and quarter transition
How does the actual Gaussian J act on the complete marked source?
- Resolved in scope: Construct original J lift with complete time/charge actionRead result
- open: Derive physical meaning of the frame orbit
Which responses distinguish charged time action?
Which data separate sources with identical complete neutral vacuum response?
- Resolved in scope: Determine twist blindness completelyRead result
- Resolved in scope: Reconstruct twist from charged responsesRead result
- open: Derive actual charged source probes
Which seam action distinguishes native opposite charges?
Can existing native charge fields produce the quarter-twist time response of the seam?
- Resolved in scope: Check native fields and response discriminatorRead result
- open: Derive actual marked seam/state action
Constructive seam search: operators, state and source compatibility
What does the complete phase search decide, and which additional state information prevents a false source solution?
- Resolved in scope: Finite search closed, actual winding and parity equivalenceRead result
- Resolved in scope: Positive finite-word state extensionRead result
- Resolved in scope: E8 current group conditionally determines the P signRead result
- Resolved in scope: The same shift functional hides distinct ground statesRead result
- Resolved in scope: No phase hit realizes the unchanged sourceRead result
Which partner actually belongs to the original seam?
Which charge, state type and time action belong to the root partner and original sheet pairing?
- Resolved in scope: Concrete pair operator; source neutrality is not hypercharge neutralityRead result
- Resolved in scope: Fixed quarter time splits the color/weak multipletRead result
- Resolved in scope: Explicit conjugation for the existing sheet pairingRead result
- Resolved in scope: Actual partner map between quarter domainsRead result
- Resolved in scope: Bilinear sheet pairing is not Hermitian kernel selectionRead result
- Resolved in scope: Actual quarter conjugation transports the supplied markingRead result
- Resolved in scope: No time-stabilizing orthogonal repair of the fixed markingRead result
- Resolved in scope: Spinor16 is a state sector, not a 16-real one-particle fieldRead result
Automated construction of operator maps and sources
Which tools generate operator maps, learn counterexamples, and can search for the full positive source?
- Resolved in scope: Calibrate new search without supplying the target mapRead result
- open: Synthesize a joint field, marking and transfer dictionary
- open: Search for the positive common source kernel itself
- Resolved in scope: Automatically construct, certify and distinguish finite kernelsRead result
- Resolved in scope: Repair search conditions against source types and collar controlsRead result
- Resolved in scope: Reconstruct multi-field sewing of the canonically identified boundary sourceRead result
- Resolved in scope: Retain absolute reference amplitude in searchRead result
- Resolved in scope: Integrate supplied free source including boundary response and scalarRead result
- Resolved in scope: Convert the full finite CAR kernel into a link functionalRead result
- Resolved in scope: Close original recursive word reconstruction with memoryRead resultRead result
- Resolved in scope: Retain whole original caller through scale and readoutRead resultRead result
- Resolved in scope: Execute actual shared Higgs source with full matter words and local RPRead resultRead result
- Resolved in scope: Reduce whole constant original Higgs spectral fiberRead result
- open: Resolve original geometric Higgs residue sign and field map
- Resolved in scope: Evaluate local original Higgs heat identification and whole finite consumerRead resultRead result
- Resolved in scope: Execute whole variable bound Higgs reference and original scale mapRead resultRead result
- Resolved in scope: Execute whole inner Higgs scale/source and normal kernel transportRead resultRead result
- Resolved in scope: Execute given whole superconnection and normal spectral readerRead resultRead result
- Resolved in scope: Determine own-admissibility and whole total-chiral spectral readersRead resultRead result
- Resolved in scope: Derive whole scale-energy map,normal profile and relative flowRead resultRead result
- Resolved in scope: Execute original Green energy producer and mixed scale readoutRead resultRead result
- Resolved in scope: Match current action argument and common Higgs/gravity weightsRead resultRead result
- Resolved in scope: Execute whole original curvature consumers and Higgs radial sourceRead resultRead resultRead result
- Resolved in scope: Execute original whole four-ghost source and geometric type testRead resultRead result
- Resolved in scope: Execute original moving normal kernel derivative and whole source responseRead result
- Resolved in scope: Decide fixed original O1 source and normal operator placementRead result
- Resolved in scope: Compare whole Higgs mixture and covariance reader on invariant four-field witnessRead result
- Resolved in scope: Test exact bounded cutoff on whole ordinary Higgs volumeRead result
- Resolved in scope: Test whole original circle joint mode and IR Higgs normalizationRead result
- Resolved in scope: Test original O1 mode covariance against whole Higgs-law classRead result
- open: Reconstruct original localization from the whole process
- Resolved in scope: Test bare native-code word compression against actual localityRead result
- Resolved in scope: Decide original geometric local-time class and execute actual interval readerRead resultRead result
- Resolved in scope: Execute actual regional two-face partner and supported charged wordRead resultRead result
- Resolved in scope: Execute original color marginal, modular field law and complete class readerRead resultRead resultRead result
- Resolved in scope: Reconstruct common Higgs law from whole original quarter source towerRead resultRead resultRead result
- Resolved in scope: Execute original record action on Doily and ordered whole source responseRead resultRead resultRead resultRead result
- Resolved in scope: Locally implement original record process and test unchanged exterior sourceRead resultRead resultRead result
- Resolved in scope: Execute original elliptic insertion,whole source and causal responseRead resultRead resultRead result
- Resolved in scope: Evaluate actual joint Higgs and color group sourceRead resultRead result
- Resolved in scope: Execute whole original graph color reader with typed original center selectorRead resultRead result
- Resolved in scope: Execute original color fixed source and first inherited grading testRead resultRead result
- Resolved in scope: Execute actual original color-spinor CAR four-word and complete spin-sector sewingRead resultRead result
- Resolved in scope: Decide actual color admissibility of density operations and coordinate regionsRead resultRead result
- Resolved in scope: Execute actual whole seam-operation class and two-face responseRead resultRead result
- Resolved in scope: Execute entire original C6 word expansion with source amplitudeRead resultRead result
- Resolved in scope: Execute original collective Higgs force source and entire mean dualRead resultRead resultRead result
- Resolved in scope: Execute original all-mode force with actual time and whole sourceRead resultRead result
- Resolved in scope: Execute original local space/time force source and causal liftRead resultRead result
Origin reverse selection, field dictionary and original P1 kernel
What does reverse selection of P1/P2 give for concrete local fields and multi-field responses, and which identification with the original P1 kernel has actually been proved?
Observable reality: minimal conditions and counterworlds
Which operational conditions are independently required, which inequivalent models satisfy them, and which observable distinctions remain for selection?
Formal carrier core: unbounded Pascal uniqueness and finite glue
Which carrier and glue statements are already present in the papers and Lean sources, and which physical selection does not follow?
- Resolved in scope: Classify the unbounded Pascal proofRead result
- Resolved in scope: Classify the formal finite glue scopeRead result
Active mathematical program: marked seam monodromies and lattice lifts
Which marked seam/lattice transport geometries, lift classes and local-coefficient invariants arise from existing clock/reflection/quartic data? First complete case: sign lifts of fourJ representation on CP1 minusμ4.
- Resolved in scope: Define marked seam category and factor formulaRead result
- Resolved in scope: Completely classify quarter-clock four-puncture liftsRead result
- Resolved in scope: Determine central class and all subgroup restrictionsRead result
- Resolved in scope: Determine full G0 cocycle and classRead result
- Resolved in scope: Determine native4/5/6local cohomologyRead result
- Resolved in scope: Classify unrestricted G31 reflection monodromiesRead resultRead result
- Resolved in scope: Determine lift class beyond G0Read result
- open: Classify additional reflection/clock equivariance and mapping-class quotients
- Resolved in scope: Equivariant lifting and divisibility by eightRead result
- Resolved in scope: All62clock obstructions and central transport variantsRead result
- Resolved in scope: Actual group comparison and central pushout exclusionRead result
- Resolved in scope: Complete invariants and explicit collisionsRead result
- Resolved in scope: Compare E6 and E7 slices with the actual quarter clockRead resultRead resultRead resultRead result
- Resolved in scope: Classify original character-valued G31monodromy liftsRead resultRead result
- Resolved in scope: Determine E7-pair action on all original lattice fieldsRead result
- Resolved in scope: Completely connect E6/E7 cuts to the original compilerRead result
- Resolved in scope: Compare common covers and actual clock carriersRead result
- Resolved in scope: Prove shared representation consequence for48,4/3 and5/4Read result
- Resolved in scope: Completely decide joint compiler-field-cover constructionRead resultRead resultRead resultRead result
- Resolved in scope: Reconstruct canonical local compiler field source from C4Read resultRead result
- Resolved in scope: Eliminate redundant sector and multiplicity assumptionsRead result
- Resolved in scope: Classify native equivariant markingsRead result
- Resolved in scope: Classify pure quarter clock mapping class quotientRead result
- Resolved in scope: Determine complete cut algebra and lossless tetrahedron transformRead resultRead resultRead resultRead result
- Resolved in scope: Completely compare scalar glue and actual native compiler classRead resultRead resultRead resultRead result
Common clock sectors, quartic geometry and operational reconstruction
Which previously separate data of the original TFPT core follow from a common clock and representation geometry, and which complete finite measurement classification follows without asserting physical source selection?
Automatic joint origin hypotheses and source responses
Which simple joint rules connect the existing P1 source chain,double cover,information and compiler,and which whole candidate classes can be decided automatically using actual source consequences?
- Resolved in scope: Decide even-matter word class and check actual AI operator expressionsRead resultRead result
Existing TFPT derivations and correction of repeated gap diagnoses
Which common source/reconstruction chain already exists in the old original papers, and which purported global gaps arose from reducing P1 or choosing an optional CAR route?
Complete physical Hamiltonian source with Gauss sector and thermal word limit
Which whole source is constructed on the original local Hamiltonian route, how do physical constraints and thermal words follow, and how is its actual IR response distinguished from the original covariant geometric source branch?
Original TFPT source versus DET16 mirror vertex
Does the same original TFPT source produce the concrete DET16 generator and whole responses,or is a computable parent supplied as extra input?
Whole original source and canonical vacuum attachment
Which vacuum joins are already in the whole original source,and how do H ground source and R104 field/Gauss source join completely without second state choice or parent change?
Craig chiral measure and the original whole TFPT source
Which concrete operator and Berezin measure map preserves the original TFPT source when connecting Craig overlap measure?
- open: Compare original4D fermion Hessian and measure line with Craig
Cyclic quantum universe and original Genesis connection
Does original TFPT dynamics yield the exact cyclic quantum return in the September/October press lead and a geometric Genesis transition?
- open: Test periodicity and Genesis response on the same whole source
Physical constants as joint constraints on the TFPT source
Which independent constants constrain the original common source, and can one relative heat product supply gravity and vacuum energy together?
- open: Evaluate original volume, curvature and gauge responses together