IndisputableMonolith.NumberTheory.VectorCSymmetryOnlyNoGo
The module constructs an explicit toy completed-xi function depending only on the real part of s, with zeros exactly on the lines Re(s)=1/4 and Re(s)=3/4. Researchers developing Vector C cite it to test whether reflection and conjugation symmetries suffice for critical-line forcing. The construction proceeds by direct definition of the toy surface and verification that its zero set respects the required pairing invariants while lying off the line Re(s)=1/2.
claimLet $f$ be a function of the real part alone such that the zero set of the associated completed-ξ surface is precisely the union of the vertical lines Re$(s)=1/4$ and Re$(s)=3/4$. The surface then obeys the reflection symmetry $smapsto 1-s$ and conjugation symmetry while admitting zeros with Re$(s)neq 1/2$.
background
The module operates inside the NumberTheory domain and imports three upstream modules. CompletedXiSymmetry records the minimal functional-equation symmetry surface needed for Vector C, supplying reflection and conjugation invariants for zeroDeviation and zeroDefect but no forcing theorem. ZeroCompositionInterface isolates the abstract interface Vector C would require to convert a zero-location observable into a critical-line statement; it is classified as an alternate path. ZeroDoublingLaw encodes the concrete doubling recurrence satisfied by the defect observable: D(2t)=2D(t)^2+4D(t). The local setting is therefore a controlled toy model that isolates symmetry from additional structure.
proof idea
This is a definition module containing no proofs. It introduces the explicit objects toyXi, toyXi_reflection, toyXi_conjugation, toyCompletedXiSurface, and completedXiSurface_symmetry_only_no_go, then verifies by direct computation that the zero set lies on the stated vertical lines while the reflection and conjugation properties hold.
why it matters in Recognition Science
The module supplies a negative result that feeds the Vector C program: the symmetries recorded in CompletedXiSymmetry are insufficient by themselves to force zeros onto the critical line. It sharpens the alternate interface described in ZeroCompositionInterface by exhibiting the precise gap that must be closed by further structure such as the doubling law. The construction therefore clarifies why an additional observable beyond symmetry is required.
scope and limits
- Does not prove any positive critical-line forcing theorem.
- Does not apply to the actual completed xi function attached to the Riemann zeta.
- Does not classify all possible symmetry-preserving zero sets.
- Does not rule out other constructions that might combine symmetry with extra structure to recover the critical line.
depends on (3)
declarations in this module (12)
-
def
toyXi -
theorem
toyXi_reflection -
theorem
toyXi_conjugation -
def
toyCompletedXiSurface -
theorem
toyCompletedXiSurface_has_off_critical_zero -
theorem
completedXiSurface_symmetry_only_no_go -
theorem
zeroDeviationSet_neg_closed_not_enough -
structure
PureVectorCDoublingData -
theorem
pureVectorCDoublingData_of_zero -
theorem
pureVectorCDoublingData_offline_example -
theorem
pureVectorCDoublingData_requires_extra_input -
theorem
pureVectorCDoublingData_not_enough_for_critical_line