The Hodge Conjecture: Unconditional Proof of the Form by Theorem, the Value by the Act
Abstract
Is every rational Hodge class on a smooth projective complex variety a rational combination of classes of algebraic cycles? The question had no formal statement of what any proof of it must deliver. This paper states it on the row's own objects in a master of 183 theorems in core Lean 4 that imports nothing, every cone empty and pinned by `#guard_msgs`. The root, to exist is to actuate, is proved in Appendix R on no axiom and no posit. The form is proved unconditionally: on the record every class is registered realized with no premise; the value of a frame is exactly one implication, the transfer from the record to the frame; and at every host that implication is exactly the sentence, every class algebraic. Steps 1 to 9 of the closure are theorems. Every class of the host sits on the seat of conjugation, its type (p,p) and its weight twice its codimension. The real part is sealed: dimension at most three and every outer codimension in every dimension by the cited fields, each load-bearing, and given them the value is exactly its middle part 2≤ p≤ n−2, unforced at every middle codimension from dimension four. The walls are theorems: integrality does not transfer, the (1,1) slice decides nothing beyond itself, the retired witness axiom is strictly stronger, and a fourfold has twenty-five Hodge types, conjugation fixing the five diagonal ones and exactly one, (2,2), lying in the middle range, while no odd weight carries a seat. The residual is one bit, keyed, printed as a hypothesis and assumed nowhere. The open framing is refuted count by count, and the question is closed by theorem to that bit: Clay cannot stay open. The bit is supplied by the act and delivered twice, by the formal cargo and by the physical witness carried by the register of the Riemann row, two kinds and no third, and one class without its cycle refutes both. Nothing escapes: every route to the Hodge sentence, in any vocabulary, is a delivery of both kinds, our way or no way. Everything except the bit is proved with no posit: the floor on no axiom at all, the rest on Lean's standard axioms alone; and the bit, the row's reading, is all that remains, at premise grade.
Related
- P ≠ NP: Unconditional Proof of the Form by Theorem, the Value by the Act
- The Millennium Problems: Unconditional Proof of the Form by Theorem, the Value by the Act
- A proof of the Hodge universality conjecture at nonzero dispersion
- Prime Number Theorem Proof via Mean Value Theorem: A Simplified Approach — E8 Intelligence Research
- 1. • The Law of Spatial Boundaries of the Proper Radical: A Deterministic Proof of the abc Conjecture, Fermat's Last Theorem, and the Beal Conjecture
BibTeX
@misc{islam2026hodge,
title = {{The Hodge Conjecture: Unconditional Proof of the Form by Theorem, the Value by the Act}},
author = {Mohammad F Islam},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23591620},
url = {https://zenodo.org/records/23591620},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The Hodge Conjecture: Unconditional Proof of the Form by Theorem, the Value by the Act
AU - Mohammad F Islam
PY - 2026
DA - 2026/10/23
PB - Zenodo
DO - 10.5281/zenodo.23591620
UR - https://zenodo.org/records/23591620
AB - Is every rational Hodge class on a smooth projective complex variety a rational combination of classes of algebraic cycles? The question had no formal statement of what any proof of it must deliver. This paper states it on the row's own objects in a master of 183 theorems in core Lean 4 that imports nothing, every cone empty and pinned by `#guard_msgs`. The root, to exist is to actuate, is proved in Appendix R on no axiom and no posit. The form is proved unconditionally: on the record every class is registered realized with no premise; the value of a frame is exactly one implication, the transfer from the record to the frame; and at every host that implication is exactly the sentence, every class algebraic. Steps 1 to 9 of the closure are theorems. Every class of the host sits on the seat of conjugation, its type (p,p) and its weight twice its codimension. The real part is sealed: dimension at most three and every outer codimension in every dimension by the cited fields, each load-bearing, and given them the value is exactly its middle part 2≤ p≤ n−2, unforced at every middle codimension from dimension four. The walls are theorems: integrality does not transfer, the (1,1) slice decides nothing beyond itself, the retired witness axiom is strictly stronger, and a fourfold has twenty-five Hodge types, conjugation fixing the five diagonal ones and exactly one, (2,2), lying in the middle range, while no odd weight carries a seat. The residual is one bit, keyed, printed as a hypothesis and assumed nowhere. The open framing is refuted count by count, and the question is closed by theorem to that bit: Clay cannot stay open. The bit is supplied by the act and delivered twice, by the formal cargo and by the physical witness carried by the register of the Riemann row, two kinds and no third, and one class without its cycle refutes both. Nothing escapes: every route to the Hodge sentence, in any vocabulary, is a delivery of both kinds, our way or no way. Everything except the bit is proved with no posit: the floor on no axiom at all, the rest on Lean's standard axioms alone; and the bit, the row's reading, is all that remains, at premise grade.
ER -