Structural identities, mode-level reformulation, and a three-term alternative for α−1 in the SO(3,3) matrix model on P3

A short companion note to the status paper [1], the spectral-gap analysis [2], and theIKKT verification [3]. We document four structural observations and one structural remark,supplementary to those papers, none of which modifies any of the empirical or pre-registeredclaims of [1] but each of which sharpens the structural reading. (i) The integer 13 thatappears at the smallest non-trivial level of the IKKT Burnside projection is the kissingnumber-plus-one in three dimensions; the operator algebra at this level has dimension 169,and the icosahedral lter reduces it to rank four. (ii) The obstruction documented in [1,§4.7] admits a reformulation from irrep-resolved class shifts to spin-resolved mode shifts,in which the system becomes underdetermined (4 unknowns for 3 constraints) but doesnot acquire a uniform solution; pentagonal dominance reappears at the mode level. (iii)The five-term Laurent expansion of [1] admits a three-term equivalent rearrangement withcoecients in the ring Z[φ, e, 1/120], exposing structural relations between the coefficientsc−1, c0 (collapsing into a Z[φ] element) and c2, c3 (collapsing into a φε2 term with internal (1−ε/2) correction). The rearrangement is presentationally compact, not deductively strongerthan the original form. (iv) Each coefficient of the formula admits a clean expression interms of the conjectured spectral determinants det′ρ2 , det′ρ3 , det′ρ4 ; in particular c3 = −det′ρ3exactly. We further record a structural remark: the factor (1−ε/2) in the second term has theform of the Riemann functional-equation reection s 7→ 1 − s applied to s = ε/2, extendingthe identication of 1/2 in c3 as the critical-line scalar [7] to a reection-symmetric structure.The factor is not the leading order of a logarithmic expansion. A derivation of this functionalequationform from the matrix model is not attempted. Throughout we preserve the tierseparation of [1]: Tier B identications remain Tier B, and the pre-registered predictionc4 = e2 is unchanged.