Start with a physical or mathematical question. See what is known, what has been tried, why an approach worked or failed, and which connection remains open.
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.
Read the exact result, assumptions and evidence
Result
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.
Required assumptions
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.
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.
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.
Observable algebra, OS reflection, state, collar operator, time evolution, carrier connection, family holonomy, and readout are distinct data until an original map relates them.
Literal archived globally positive kernel on finite measures
v42section01631..148 andv45metrology00560..178:finite nonzerononnegative nu on[0,infty),Tnu(dt)=intK(s,t)dnu(s)/intpsi(s)dnu(s) dt,global lowerboundK>=m>0,Lebesguetargetdt,finitepositivedenominator. Scope exactlythiscombination,notactuallydifferentoriginalfinite-support/reference/integrablekernel.
Only shared leading distances with the stated equal weighting. Different q/p spectra, unequal weights, full mode sums and actual sector-dependent original C6 residue packets with additional length-six windings remain outside the exclusion. Frobenius exponents are not a boundary clutch without an actual operator dictionary.
Read the exact result, assumptions and evidence
Result
Original diagonal TOTAL lengths Lu=(7,3,0),Ld=(7,5,2),Le=(8,5,3) have sorted gap ratios3/4,3/2,2/3; separate numeric left/right lists are not supplied. A common single-leg list requires one ratio. For equal-weight same-list two-leg readers, normalize x=(0,1,t): all six relative S3 permutations are degenerate or give only t-1 and its reciprocal. No arbitrary real triple reproduces all three original rows. Entire class is analytically decided.
Required assumptions
Do not identify source types
Observable algebra, OS reflection, state, collar operator, time evolution, carrier connection, family holonomy, and readout are distinct data until an original map relates them.
Common leading triple distances with equal two-leg weights
Restricted candidate class only: arbitrary shared real triple x, sector positive scales/offsets and permutations, single leg or equal-weight same-list two legs. Not a P1 axiom. Different lists/weights, full sums and extra operator histories lie outside.