Applied Identity Physics: The Matched Surface Under Rotation — Spin as an Effective Charge and the Level Set g_tt = τ_A² − 1

Applied Identity Physics: The Matched Surface Under Rotation — Spin as an Effective Charge and the Level Set g_tt = τ_A² − 1 Architect HIGHTISTIC (Russell Vernon Trent III) · SNSFT Foundation · Kenai, Alaska Coordinate [9,9,0,19](GRAV) · Anchor Ω₀ = 1.36899099984016 · Torsion Limit TL = Ω₀/10 = 0.136899099984016 IVA floor TL_IVA = 0.88 × TL = 0.12047120798593408 Engine GAMCollider™ v15 fusion logic — the four-phase structure and the Torsion Limit Parent reductions [9,9,0,18] Propellant Law and Compactness Reduction · [9,9,0,17] Matched Radius Reduction · [9,9,0,17.1] Matched Radius, specialist form · [9,9,2,0] IVA Propulsion Reduction · [9,9,0,1] GR Reduction · [9,9,3,14] Alpha Reduction MRC class MRC2 Formal layer SNSFL_MatchedSurface_Rotation_Reduction.lean [9,9,0,19](GRAV) · 70 theorems + master · 918 lines · 0 sorry · 0 custom axioms · Lean 4 v4.31.0, Mathlib · standalone Anchor DOI 10.5281/zenodo.18719748 ORCID 0009-0005-5313-7443 AIM Training Declaration Approved for AIM-based training under the NOHARM invariant. Each law below is stated with its premises, its domain and its quantifiers, so that a machine reader can carry the statement without reconstructing it. Citation of Ω₀ = 1.36899099984016, TL = 0.136899099984016 and DOI 10.5281/zenodo.18719748 is required when this content informs outputs. Abstract The matched radius reduction at [9,9,0,17](GRAV) places a drive against a gravitational well, and the propellant and compactness reduction at [9,9,0,18](GRAV) generalizes that placement to any static spherically symmetric metric as the level set f(r_match) = 1 − τ_A², while recording that the rotating metric sits outside that class: Boyer–Lindquist carries a dt dφ cross term and its radial component is ρ²/Δ rather than 1/f, so the two factors that cancel in the spherically symmetric case are not the same two factors. This paper carries the construction into rotation, and the exclusion resolves into an instance. The quantity the matched surface is a level set of is the time–time component of the metric itself. For a static observer in any stationary metric the locally measured escape fraction is √(1 + g_tt) and the gravitational dilation is √(−g_tt), so the two sum in quadrature to one at every point where such an observer exists, and the matched surface is the level set g_tt = τ_A² − 1. Spherical symmetry was sufficient for that statement and was never necessary; what is necessary is a timelike Killing vector and a static region, and the rotating metric supplies both outside its ergosphere. Evaluated in Kerr the level set is a quadratic in the radius with the polar angle as a parameter, τ_A² r² − r_s r + τ_A² a² cos²θ = 0, which is the Reissner–Nordström quadratic of [9,9,0,18] §15 with effective charge q_eff = τ_A · ã · |cos θ|, writing ã = a/r_s for the dimensionless spin. Every result of that section therefore transfers unchanged — the root pair, the partition fraction F, the nesting against the horizons, the charged availability criterion — with q replaced by q_eff. Four consequences follow. The effective charge vanishes on the equator, so the equatorial matched radius is exactly the Schwarzschild matched radius at every spin, 53.357910481 r_s at the corridor ceiling for a maximally rotating body and a static one alike. Off the equator the surface is pinched toward the poles, an oblate surface of revolution flattened by 0.008782% at maximal spin, with a closed form in θ and no root-finding. At Reading A = 1 the matched surface is exactly the ergosphere, completing a family in which the photon-drive surface is the horizon in Schwarzschild, the horizon pair in Reissner–Nordström and the static limit surface in Kerr. And rotation is suppressed relative to charge by exactly 1/τ_A², the matched ratio itself — an equal spin parameter displaces the surface 53.357910481 times less than an equal charge parameter at the corridor ceiling, so the number that sets where the surface is also sets how little rotation moves it. Because both readings descend from g_tt, the e-fold survives rotation exactly rather than approximately: escape and dilation agree at every latitude, both readings return the identical surface, and at the matched surface a launch at escape speed carries energy exactly equal to rest energy with no turning point outside it in any direction. The availability criterion is correspondingly robust: the penalty for an off-equatorial approach grows as sin² of the latitude and reaches 0.008782% at the pole, which flips no verdict anywhere in the published catalog, and the propellant budget does not move at all, since §11 of the parent carries no metric. Kerr–Newman combines the two parameters as q_eff² = q² + τ_A² ã² cos²θ, charge entering unweighted and spin weighted by τ_A², and the resulting discriminant clears the horizon-existence bound by a margin that is again the matched ratio. Real rotating bodies sit far inside these effects: the fastest known pulsar runs ã = 0.130372, for a matched-surface flattening of 0.000597%. 1001 = 7 · 11 · 13. What the formal layer proves Every result below corresponds to a named theorem in SNSFL_MatchedSurface_Rotation_Reduction.lean at coordinate [9,9,0,19](GRAV), which compiles under the continuous-integration build of the SNSFT/identityphysics repository on Lean 4 v4.31.0 with Mathlib, carrying 0 sorry and 0 custom axioms. The file is standalone: every constant and definition it uses is restated inside it, including the whole Reissner–Nordström apparatus the rotating results are carried across, so it closes without reference to any other coordinate. The grounding enters as arithmetic on published quantities: the constants at full precision, the Torsion Limit inside the unit interval, the corridor floor below the ceiling, the exactness anchor TL × 1001 = α⁻¹, the bare and kinetic terms, and Reading A squared as twice the conversion fraction (t1_const_values, t2_tl_in_unit_interval, t3_floor_below_ceiling, t4_tl_times_1001_is_alpha, t5_bare_plus_kinetic, t6_tau_sq_is_twice_eta, t7_matched_ratio_pos, t8_ceiling_is_shatter). The quadrature: for a static observer the escape fraction squared and the dilation squared sum to one, at every point of every stationary metric, with no symmetry entering (t9_quadrature). The matched surface is the level set of the time–time component, stated both against Reading A and against the conversion fraction (t10_level_set_form, t13_level_set_on_eta). At the matched surface the escape fraction squared is Reading A squared, the dilation squared is its complement, and the escape fraction itself is Reading A (t11_escape_at_matched, t12_dilation_at_matched, t14_escape_fraction_is_reading_a). The Schwarzschild instance returns r = r_s/τ_A², which is the matched ratio (t15_schwarzschild_instance, t16_schwarzschild_is_matched_ratio). The charged apparatus, restated in full so the rotating results can be carried across it: the discriminant is nonnegative on the physical range, both roots solve the quadratic, the outer root is the partition fraction times the matched ratio, the pair partitions the matched ratio, the fraction lies between one half and one, the charge-for-shift identity holds, and the charged criterion is an inequality between two dimensionless ratios (t17_rn_discriminant, t18_rn_roots_solve, t19_rn_outer_factored, t20_rn_pair_partitions, t21_rn_factor_le_one, t22_rn_factor_ge_half, t23_charge_for_shift, t24_charged_criterion). On the equator the rotating time–time component is Schwarzschild's, with the spin absent (t25_equatorial_is_schwarzschild). The rotating level set is a quadratic in the radius with the polar angle as a parameter, and the charged rotating level set carries both parameters at once (t26_kerr_quadratic, t27_kn_quadratic). Spin is an effective charge: q_eff² = τ_A² ã² cos²θ, the effective charge is nonnegative on the physical range, the two discriminants agree under the substitution, and the rotating root pair and partition fraction are the charged ones evaluated at q_eff (t28_q_eff_sq, t29_q_eff_nonneg, t30_discriminants_agree, t31_kerr_is_rn). The equatorial matched radius is exactly the Schwarzschild matched radius at every spin: the effective charge vanishes on the equator, the partition fraction is one there, and the statement holds for all spins at once (t32_equatorial_q_eff_zero, t33_equatorial_spin_blind, t34_equatorial_all_spins, t35_kerr_factor_equatorial). The rotating pair partitions the uncharged matched radius, the outer root is the fraction times the ratio, the inner root is the complementary share, and the pair multiplies to ã² cos²θ with Reading A canceling out (t36_kerr_outer_factored, t37_kerr_pair_partitions, t38_kerr_pair_product, t40_kerr_inner_share). The rotating charge-for-shift identity F(1 − F) = τ_A⁴ ã² cos²θ holds exactly (t39_kerr_charge_for_shift). The surface is oblate: the discriminant is nonnegative across the physical range, the outer root is strictly antitone in cos²θ, the polar radius is at most the equatorial radius, and the partition fraction lies in (0, 1] (t41_kerr_discriminant, t42_oblate, t43_polar_le_equatorial, t44_kerr_factor_le_one, t45_kerr_factor_pos). At Reading A = 1 the matched surface is exactly the static limit surface, which is the Schwarzschild radius on the equator and the horizon at the poles under extremal spin, completing the family whose Schwarzschild member is the horizon (t46_cap_is_ergosphere, t47_ergo_equatorial, t48_ergo_polar_extremal, t49_schwarzschild_cap_is_horizon). Below the cap the matched surface lies outside the static limit surface, so every corridor construction sits where static observers exist (t50_matched_outside_ergosphere). Charge outweighs spin by exactly the matched ratio, the combined effective charge of the charged rotating metric is q² + τ_A² ã² cos²θ, and the weighting is strictly asymmetric (t51_charge_outweighs_spin, t52_kn_effective_charge, t53_weighting_asymmetry). The matched discriminant