Original60reflection class, ordered punctures, marked central generator and simultaneous inner conjugation. General theorems use their explicitly stated hypotheses.
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Result
Central orbifold H2 decides lifting; based lifts form an H1torsor and unbased lifts require centralizer quotient. Cone power defects are complete on oriented punctured quotient surfaces. The existing determinant class on C_m*C_mwith opposite parity lifts exactly for8|m, then16lifts; an original order8matrix example is given.
Required assumptions
Active purely mathematical marked-seam program
Within the stated mathematical scope, marked pairs, lattice transport systems, reflections, clocks and cohomology are independent mathematical objects. Earlier origin/completion tasks are historical, not active. Distinguish new definitions, derived theorems and optional markings.
Original G31reflections and central determinant class
The60original reflections generate G31of order46080. Use exactly E=Z4xG31with determinant cocycle from original holonomy proof§4; distinguish E,effective diagonal quotient and noncentral lattice lifts. Mark z.
Ordered G31reflection quadruples and chosen quarterclock
U=P1minusmu4, four ordered positive meridians in the60reflection class with product1. Chosen clocks have exact order4and conjugate meridians cyclically. Equivalence is simultaneous inner conjugation; central relaxations are separately defined relation contracts.
Cohomology of specified local systems on four-punctured sphere; no identification of differently dimensional representations or invariants of every bundle/bordism.
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Result
Existing inclusions P5->Sym4(V4) and exterior²V4->V4⊗V4 are G0-equivariant. J versus JR gives complex H1 dimensions8/8 onV4,12/15 on exterior²V4,15/14 onP5. Integral E8 system with fourJ has H0=0,H1=Z16⊕(Z/2)^4,Hk=0 fork>=2; Smith form1^4,2^4. Local coefficients distinguish marked systems of equal clock order.
Required assumptions
Existing even E8 lattice, Gaussian J and sign lifts
Given existing even unimodular E8 with actual J(x0,x1,...)=(-x1,x0,...) and J²=-I, and the original ordered sign lattice cocycle. Sign automorphisms have kernel Hom(L,F2). Retain original integral ordered basis and generator phases; no enlargement to U1 characters except explicitly defined scalar pushouts.
Four ordered punctures with fixed quarter-clock projection
X is CP1 minus ordered(1,i,-1,-i), positive meridians product1, fixed rho(gamma_i)=J. Classify all sign lifts into full preimage Gamma of<J>, modulo simultaneous Gamma conjugation with optional prescribed peripheral conjugacy classes. No mapping-class, puncture permutation or additional spatial reflection-equivariance quotient.
Actual clock/complex-reflection subgroup and native quartic space
R=diag(-1,1,1,1) on Gaussian C4 is the complex reflection of root line(1+i)e0; G0=<J,R>=C4×C2. P5 is the existing five-dimensional symmetric quartic subspace in Sym4(V4), W6=exterior²V4. R is not a real codim1 E8 Weyl reflection; P5 is not the full fourth tensor power.
Full cocycle and global trivialization on G0 only; no splitting decision for full G31 or W(E8).
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Result
For actual G0=<J,R>=C4×C2 the full8×8factor table of the original ordered lattice lift is computed. Its Hom(L,F2) class vanishes: b=(0,86,6,80,136,223,142,217), ordered1,R,J,JR,J²,J²R,J³,J³R, satisfies b(g)+g.b(h)+c(g,h)=b(gh) on64pairs. All512cocycle triples checked.
Required assumptions
Existing even E8 lattice, Gaussian J and sign lifts
Given existing even unimodular E8 with actual J(x0,x1,...)=(-x1,x0,...) and J²=-I, and the original ordered sign lattice cocycle. Sign automorphisms have kernel Hom(L,F2). Retain original integral ordered basis and generator phases; no enlargement to U1 characters except explicitly defined scalar pushouts.
Actual clock/complex-reflection subgroup and native quartic space
R=diag(-1,1,1,1) on Gaussian C4 is the complex reflection of root line(1+i)e0; G0=<J,R>=C4×C2. P5 is the existing five-dimensional symmetric quartic subspace in Sym4(V4), W6=exterior²V4. R is not a real codim1 E8 Weyl reflection; P5 is not the full fourth tensor power.
Central quotient extension with trivial coefficient action, distinct from split noncentral extension over<J> and the G0 factor.16classes describe the Z-induced family, not all H2(F2^5,U1).
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Result
Z(Gamma)=F2^5 and Gamma/Z=F2^5, with full central cocycle kappa((u,e),(a,f))=(ea,ef). Restriction vanishes exactly on subgroups in e=0; otherwise a nontrivial square obstructs an order2 lift.32central characters yield16marked U1 classes,15nonzero: pure clock carry is a U1 coboundary, other classes are distinguished by commutators.
Required assumptions
Existing even E8 lattice, Gaussian J and sign lifts
Given existing even unimodular E8 with actual J(x0,x1,...)=(-x1,x0,...) and J²=-I, and the original ordered sign lattice cocycle. Sign automorphisms have kernel Hom(L,F2). Retain original integral ordered basis and generator phases; no enlargement to U1 characters except explicitly defined scalar pushouts.
Complete only for declared ordered quarter-clock/sign-lift stratum under Gamma conjugation; not full G31/W(E8) reflection moduli or additional equivariant structures.
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Result
Gamma=F2^8 semidirect C4 has order1024. Peripheral classes C_u over J are indexed by F2^4. Four ordered classes admit product1 iff sum ui=0. Normal forms(x,y) modulo translation(u1+u0,u2+u0) give256classes for diagonal profiles,128otherwise. Total526336;4096orbits size16 with stabilizer C4×C2^4 and522240size32 with stabilizer C2^5. Image-order census4:16,8:720,16:14880,32:80640,64:430080; no surjection.
Required assumptions
Existing even E8 lattice, Gaussian J and sign lifts
Given existing even unimodular E8 with actual J(x0,x1,...)=(-x1,x0,...) and J²=-I, and the original ordered sign lattice cocycle. Sign automorphisms have kernel Hom(L,F2). Retain original integral ordered basis and generator phases; no enlargement to U1 characters except explicitly defined scalar pushouts.
Four ordered punctures with fixed quarter-clock projection
X is CP1 minus ordered(1,i,-1,-i), positive meridians product1, fixed rho(gamma_i)=J. Classify all sign lifts into full preimage Gamma of<J>, modulo simultaneous Gamma conjugation with optional prescribed peripheral conjugacy classes. No mapping-class, puncture permutation or additional spatial reflection-equivariance quotient.
Declared definitions and derived standard extension/gluing algebra, not a classification of all object groups.
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Result
Marked lattice transport functors with explicit reflection/clock markings define a category, products and boundary gluing under inverse-conjugate holonomies. Ordered generator phase eta gives the full factor c(g,h)(y)=eta_h(x)eta_g(hx)/eta_gh(x), x=(gh)^-1y. It is Hom(L,F2)-valued with coefficient action, not generally scalar or central.
Required assumptions
Active purely mathematical marked-seam program
Within the stated mathematical scope, marked pairs, lattice transport systems, reflections, clocks and cohomology are independent mathematical objects. Earlier origin/completion tasks are historical, not active. Distinguish new definitions, derived theorems and optional markings.
Existing even E8 lattice, Gaussian J and sign lifts
Given existing even unimodular E8 with actual J(x0,x1,...)=(-x1,x0,...) and J²=-I, and the original ordered sign lattice cocycle. Sign automorphisms have kernel Hom(L,F2). Retain original integral ordered basis and generator phases; no enlargement to U1 characters except explicitly defined scalar pushouts.
The concrete element/orbit and arithmetic statements are available. No fresh kernel run, nor a newly checked single Lean theorem about all abstract subgroups or lattice isomorphisms. The general subgroup conclusion also uses finite group theory; concrete E8 basis embedding and physical carrier selection are separate.
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Result
In Fin4 x Fin4 with qZ=(5x^2+3y^2) mod8, named Lean declarations classify all six isotropic elements and four odd-coordinate isotropic generators. Explicit orbit identities give H1 and H2; klein_not_isotropic gives the nonisotropic Klein witness, spinor_swap_exchanges swaps the two orbits, and unique_halfway_stage classifies the only nonzero isotropic order-two element (2,2). Index and sector arithmetic are additional separate declarations.
Required assumptions
Specified finite carrier discriminant form
The model is El=Fin4 x Fin4 with componentwise addition modulo4 and qZ(x,y)=(5x^2+3y^2) mod8. Its identification with a concrete physical carrier is not inferred without premises.
Algebraic uniqueness under the stipulated Pascal/half-spinor exhaustion condition. Original source and proof text reviewed; no fresh Lean kernel run. AX.P2.01 remains a declared physical interface; the equation does not select itself from an original source.
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Result
CarrierRankUniqueness.carrier_rank_pascal_unique states and proves for every natural g that 2^g=g^2+g+2 iff g=5. Its proof combines the small cases with pascal_growth for all g>=6. This is unbounded; PascalLadder has a separate bounded K=2 statement. For g=5, N_fam=3 and g+N_fam=8 are computed.
Required assumptions
Pascal exhaustion condition as input
g is natural; the stipulated equation is 2^g=g^2+g+2. The equation itself is the additional carrier exhaustion rule.
Conditional representation theorem without novelty claim, not efficient or local physical simulation. History/preparation dependence and growing memory are allowed; Bell, noncontextuality or resource witnesses under independently justified extra conditions are not invalidated. Protocol direction is assumed, not physical causality derived.
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Result
Every finite normalized prefix-consistent response family with past marginals independent of future choices has a finite classical stochastic realization: preparation/full history as memory, conditional ratios as transitions, append the next choice/report pair. The construction also reproduces adaptive protocols. It was checked exactly on all requested D tables through horizon4. Finite sequential disturbance and order dependence alone do not exclude unrestricted classical history models.
Supply finite preparation/action/report sets, finite horizon and nonnegative normalized sequential probability tables. Summing out the last report recovers prefixes and earlier marginals do not depend on future choices. Impose no memory, locality, stationarity or preparation-noncontextuality bounds on the representation.
Standard residual representation consequence and explicit counterexamples, not a new ontology. Distinguishability, repeatability, context reproduction and serial availability are separate assumptions. Separate L1/L2 from L3.
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Result
The separate benchmark assumptions give at least two continuation-distinguishable classes and preserve the selected report label under uninterrupted repetition. This largely restates the substantive repeatability premise rather than explaining physical origin. A test can preserve its report while incrementing a separately readable modulo3 counter; result repetition therefore forces no complete idempotence, preservation of all correlations, metric time or eternal material memory. Possible reports alone select no numerical weights.
Required assumptions
Explicit repeatable-distinction benchmark
Recognizable reports, reproducibility of a declared context and available serial continuations are separate commitments. The sharp benchmark additionally has an immediately result-repeatable test and two preparations with different certain reports. These conditions are stronger than mere observability; probability, free erasure, re-preparation, replica control or physical time are not assumed to follow.
Three countermodels, not an exhaustive classification of operational worlds. I/D assume probabilities/convexity; their numbers are predictions, not new measurements or physical dimensions. All-context equality follows analytically from linearity, not the finite word check.
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Result
Three explicit single-system theories satisfy the same sharp repeatability benchmark and are operationally inequivalent: R has two classes without mixing, I a classical simplex, D a square with nonunique decompositions. XZX preserves the first X report with probability1 in I and1/2 in D; D identifies both diagonal mixtures under all generated protocols while I distinguishes them by parity. All30 X/Z words through length4 were checked exactly. D has a classical hidden two-bit implementation of its generators; not all classical models are excluded.
Required assumptions
Explicit repeatable-distinction benchmark
Recognizable reports, reproducibility of a declared context and available serial continuations are separate commitments. The sharp benchmark additionally has an immediately result-repeatable test and two preparations with different certain reports. These conditions are stronger than mere observability; probability, free erasure, re-preparation, replica control or physical time are not assumed to follow.
Three explicit operational countermodels
R has deterministic preparation/read traces and continued partial-identity branches only, no primitive mixing or internal re-preparation. I is the classical two-bit simplex with probabilities and nondisturbing joint access. D is the square with explicit positive X/Z branches, square symmetries, preparations, mixtures and finite serial/adaptive closure. These selected rules define counterworlds rather than necessary physics.
Refinement of already reviewed diagnosis and explicit counterworld comparison of new classes; no refutation of all TFPT and no closure of a physical goal.
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Result
The checked origin attempts leave operation alphabet, state/correlation family and intervention rule unspecified. This motivates a methodology change but does not prove global underdetermination of all complete P1/P2 models: scalar/matrix comparisons change A, while A, state and time belong to the full source tuple. The four independent classes force neither a five-direction carrier,3+2,E8,quartic nor TFPT clocks. Compare TFPT only after their characterization.
Required assumptions
TFPT for later comparison only
The explicit research reset studies independent starting classes. E8, five,3+2, SM algebra, Yukawa data, Hilbert space, complex amplitudes and TFPT clocks are not inputs. Preserve existing TFPT results. After at most three failed origin attempts on unchanged premises change at least one fundamental premise or direction; coding bugs do not count. State assumptions, necessary consequences and newly explained properties.