IndisputableMonolith.NumberTheory.VectorCSymmetryOnlyNoGo
This module constructs an explicit toy completed-ξ function that depends only on the real part of its argument. Its zeros lie exactly on the vertical lines Re(s)=1/4 and Re(s)=3/4. The construction satisfies the reflection and conjugation symmetries recorded in CompletedXiSymmetry yet places zeros off the critical line. Researchers examining Vector C would cite it to separate symmetry data from critical-line forcing. The argument is by direct definition and zero-set verification.
claimLet $f(s)$ be a completed-ξ-style function depending only on $Re(s)$. The zero set of $f$ is exactly the union of the lines $Re(s)=1/4$ and $Re(s)=3/4$.
background
The module imports CompletedXiSymmetry, which records the minimal functional-equation symmetry surface needed for Vector C and supplies reflection and conjugation invariants for zeroDeviation and zeroDefect. It also imports ZeroDoublingLaw, whose defect observable obeys the doubling recurrence $D(2t)=2D(t)^2+4D(t)$, and ZeroCompositionInterface, the abstract interface that would convert a zero-location observable into a critical-line theorem. The local setting is an alternate, non-primary path that tests whether symmetry data alone suffices.
proof idea
This is a definition module, no proofs. It introduces the toy function by explicit construction, states its zero set, and records that the construction obeys the imported symmetries while admitting off-critical zeros.
why it matters in Recognition Science
The module supplies a concrete counterexample to symmetry-only forcing and therefore feeds ZeroCompositionInterface. It shows that the pairing data from CompletedXiSymmetry is insufficient without further structure such as the doubling law. It touches the open question of what additional input is required to obtain an unconditional critical-line result in the Vector C setting.
scope and limits
- Does not claim that symmetry forces all zeros onto the critical line.
- Does not classify all possible zero sets compatible with reflection symmetry.
- Does not interact with the primary RS forcing chain T0-T8.
- Does not derive any mass formulas or RS-native constants.
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