Constructing a symplectic structure that preserves the ordinary symmetries and the topological invariance for topological Yang-Mills theory, it is shown that the Kodama (Chern-Simons) state traditionally associated with a topological phase of unbroken diffeomorphism invariance for instantons, exists actually for the complete topological sector of the theory. The case of gravity is briefly discussed.Comment: Introduction and Conclusions have been extended, some appropriate references have been added. 8 pages. submitted to physics letters

Reinforced concrete (RC) buildings with masonry-induced soft-storey irregularities exhibit extreme seismic vulnerability, a critical risk often underestimated by conventional code-based design. Standard equivalent static methods typically fail to capture the intense concentration of seismic demand at the flexible ground level, leading to unconservative designs that do not meet performance objectives. This research proposes a corrective linear-static methodology to address this deficiency. A new Equivalent Lateral Force profile (ELF1) was developed, derived from modal analyses of 235 representative soft-storey archetypes. This profile was integrated with a more realistic response modification coefficient (Rᵢ₁ = 5.04), which was determined to be 37% lower than the normative R-factor (R=8) prescribed by code. Nonlinear static (pushover) analyses confirmed that the conventional design resulted in "irreparable" damage (mean GDI = 0.82). In contrast, redesigning the structure using the proposed ELF1 and Rᵢ₁ methodology successfully mitigated damage concentration, upgrading the structural performance to a "repairable" state (mean GDI = 0.52). Finally, Incremental Dynamic Analysis validated the approach; the redesigned structure satisfied FEMA P695 collapse prevention criteria, achieving an Adjusted Collapse Margin Ratio (ACMR) of 2.10. This study confirms the proposed ELF1 method is a robust and practical design alternative that accurately accounts for soft-storey mechanisms within a simplified linear framework.

Reinforced concrete buildings with masonry-induced soft-storey irregularities exhibit extreme seismic vulnerability, a critical risk often underestimated by conventional code-based design. Standard equivalent static methods typically fail to capture the intense concentration of seismic demand at the flexible ground level, leading to unconservative designs that do not meet performance objectives. This research proposes a corrective linear–static methodology to address this deficiency. A new Equivalent Lateral Force profile (ELF_i1) was developed, derived from modal analyses of 235 representative soft-storey archetypes to accurately account for stiffness heterogeneity. This profile was integrated with a realistic response modification coefficient (R_i1 = 5.04), determined to be 37% lower than the normative R-factor (R = 8) prescribed by code. Nonlinear static analyses confirmed that conventional design resulted in “irreparable” damage (mean Global Damage Index = 0.82). In contrast, redesigning the structure using the proposed ELF_i1 and R_i1 methodology successfully mitigated damage concentration, upgrading structural performance to a “repairable” state (mean Global Damage Index = 0.52). Finally, Incremental Dynamic Analysis validated the approach; the redesigned structure satisfied FEMA P695 collapse prevention criteria, achieving an Adjusted Collapse Margin Ratio (ACMR) of 2.10. This study confirms the proposed method is a robust and practical design alternative for soft-storey mechanisms within a simplified linear framework.