hodge_implies_zero_charge
Assuming the RS Hodge conjecture, coarse-graining stability on a defect-bounded subledger forces zero total z-charge on the class whenever every J-cost minimal cycle carries zero charge. Researchers formalizing the Recognition Science translation of algebraic geometry cite this implication to check consistency of charge assignments under stability. The proof extracts the generating cycle list from the conjecture hypothesis, rewrites the sum equality, and simplifies to zero.
claimIf the RS Hodge conjecture holds, then for a defect-bounded subledger $L$ and coarse-graining stable class $cls$ on $L$, if $z_{charge}(cyc)=0$ for every J-cost minimal cycle $cyc$, it follows that $z_{charge}(cls)=0$.
background
A DefectBoundedSubLedger is a finite list of recognition events whose total J-cost satisfies $0$ to $phi^{44}$. A CoarseGrainingStableClass extends a cohomology class by the stability condition $z_{charge} leq L.defect$, ensuring survival under data-processing inequality. A JCostMinimalCycle is a recognition-closed subgraph whose cycle class obeys $z_{charge}=0$ or $z_{charge} leq 1$ (zero net defect).
proof idea
The term proof applies the RSHodgeConjecture hypothesis to $L$ and $cls$ to obtain a list of cycles together with the equality $cls.z_{charge} = sum$ of their charges. It rewrites the target equality and uses simp on the hypothesis that every cycle charge equals zero.
why it matters in Recognition Science
This theorem derives a necessary zero-charge condition from the RS Hodge conjecture, supporting the module's link to classical Hodge theory where nonnegative $z_{charge}$ encodes the $(p,p)$ type. The module proves the algebraic-to-Hodge direction and leaves the converse open; the present result supplies a consistency constraint on charges for any stable class generated by minimal cycles.
scope and limits
- Does not prove the RS Hodge conjecture.
- Requires the strict zero-charge assumption on all minimal cycles.
- Applies only inside defect-bounded finite subledgers.
- Does not address cycles with $z_{charge} leq 1$ rather than exactly zero.
- Yields no new physical predictions or numerical values.
formal statement (Lean)
154theorem hodge_implies_zero_charge (h : RSHodgeConjecture)
155 (L : DefectBoundedSubLedger) (cls : CoarseGrainingStableClass L)
156 (hzero : ∀ (cyc : JCostMinimalCycle L), cyc.cycle_class.z_charge = 0) :
157 cls.z_charge = 0 := by
proof body
Term-mode proof.
158 obtain ⟨cycles, hsum⟩ := h L cls
159 rw [hsum]
160 simp [hzero]
161
162/-! ## Part 4: Connection to Classical Hodge -/
163
164/-- The RS framework recovers the classical Hodge (p,p) type condition:
165 z_charge ≥ 0 is the RS version of the (p,p) Hodge condition.
166 Classical (p,p) classes have equal holomorphic and antiholomorphic degree. -/