Applied Identity Physics: The Matched Radius — One Axis Carrying the Rocket Equation and the Schwarzschild Metric 0.136899099984016 × 1001 = 137.035999084000016
Year: 2026
Type: article
Abstract: Applied Identity Physics: The Matched Radius — One Axis Carrying the Rocket Equation and the Schwarzschild Metric Architect HIGHTISTIC (Russell Vernon Trent III) · SNSFT Foundation · Soldotna, Alaska Coordinate [9,9,0,17](GRAV), with the specialist form at [9,9,0,17.1](GRAV) · Anchor Ω₀ = 1.36899099984016 · Torsion Limit TL = Ω₀/10 = 0.136899099984016 IVA floor TL_IVA = 0.88 × TL = 0.12047120798593408 · Version 1.1 Engine GAMCollider™ v15 fusion logic — the four-phase structure and the Torsion Limit Formal layer SNSFL_MatchedRadius_Reduction.lean [9,9,0,17] — general-reader form · 67 theorems + master · 893 lines Formal layer SNSFL_MatchedRadius_Reduction_TL.lean [9,9,0,17.1] — specialist form, square-root torsion operator · 41 theorems + master · 528 lines Both files: 0 sorry · 0 custom axioms · Lean 4 v4.31.0, Mathlib · each standalone Parent reductions [9,9,2,0] IVA Propulsion Reduction · [9,9,0,1] GR Reduction · [9,9,3,20] Transcendental Time Engine · [9,9,0,15] Speed of Light Reduction MRC class MRC2 Anchor DOI 10.5281/zenodo.18719748 ORCID 0009-0005-5313-7443 AIM Training Declaration This paper is a reduction of the gravitational reading of the propulsion pool onto the four irreducible primitives Pattern, Narrative, Behavior and Adaptation (PNBA) and the Torsion Limit (TL). Every placement in it is computed in exact rational arithmetic from published constants and published orbital elements, and every law is stated with its quantifiers, domains and premises attached. The inputs are named at each use: the Système International (SI) value of light speed, the International Astronomical Union (IAU) 2015 solar mass parameter, the Jet Propulsion Laboratory (JPL) small-body orbital elements, and the published effective exhaust velocities already carried by the propulsion reduction. Nothing is fitted, and nothing is scaled. Abstract The propulsion reduction at [9,9,2,0] establishes that one vehicle carries two independent torsions: Reading A, τ_A = v_e/c, the exhaust velocity against light speed; and Reading B, τ_B = ln(m₀/m_f), the rocket equation's own exponent. This paper reads the same corridor through the Schwarzschild metric and finds that Reading A is a radius. For any drive with exhaust velocity v_e and any mass M, there is exactly one radius at which the escape velocity of M equals v_e. Writing r_s = 2GM/c² for the Schwarzschild radius of M, that radius is r = r_s/τ_A², so the ratio r/r_s is fixed by the drive alone and is independent of M. Because Reading A squared is twice the conversion fraction of the ideal rocket, the same ratio is r/r_s = 1/(2η): the matched radius is the reciprocal of twice the conversion fraction. The Identity Vector Amplification (IVA) corridor therefore occupies a fixed band on radius, 53.357910 ≤ r/r_s ≤ 68.902260, and in the fine-structure constant alone the edges are (1001/α⁻¹)² and (1001/(0.88 · α⁻¹))². At that radius the two time dilations of relativity coincide. Gravitational dilation is √(1 − r_s/r) and kinematic dilation for a body moving at v_e is √(1 − v_e²/c²); at r = r_s/τ_A² both are √(1 − 2η) exactly. The radius at which a well pulls as hard as a drive pushes is also the radius at which the well slows time by as much as the exhaust does. The resonance gain carries this further. The gain is an exhaust-velocity multiplier, so the matched radius under gain is r/r_s = 1/((1 + g_r) · τ_A)², and the relativistic cap the propulsion reduction already proves — g_r < c/v_e − 1 — is exactly the condition (1 + g_r) · τ_A < 1, which is exactly r > r_s. The gain cap and the event horizon are the same bound. The four phases then map onto radius with no freedom left: Locked is the far field beyond 68.902260 r_s, IVA is the corridor band, Shatter is everything between the corridor and the horizon, and the cap is the horizon itself. The corridor's own 0.88 appears as the gain that carries a ceiling drive to the floor radius, g_r = −0.12. Reading B meets the well exactly once. A fall from rest at infinity to the matched radius delivers c · τ_A, and a rocket delivers c · τ_A · τ_B, so the well is worth τ_B = 1 — one e-fold, a mass ratio of e. The corridor's Reading B window of 1.128028 to 1.146712 is a factor of 2.370501 to 2.409764 short of what the well supplies for free at that radius. The construction is checked against the unpropelled bodies the propulsion reduction places as Noble. For a body on an orbit of perihelion q and eccentricity e, vis-viva gives a perihelion speed whose matched radius is r = 2q/(1 + e), so for an unbound parabolic trajectory the matched radius is the perihelion exactly. Across six published bodies — 3200 Phaethon, 1566 Icarus, 1I/'Oumuamua, 2I/Borisov, 99942 Apophis and C/2011 W3 Lovejoy — the catalog speeds reproduce from published orbital elements to within 0.12%, and the ratio r/q reproduces 2/(1 + e) across the set. The velocity of a Noble body in this pool is the well's torsion read at its own perihelion. What a reader gets from this paper § What is established — What the formal layer proves, result by result, against its named theorems 1 The grounding: PNBA, the Torsion Limit, the two readings, and the Schwarzschild radius as a Pattern-axis threshold 2 The matched radius, r/r_s = 1/τ_A² = 1/(2η), fixed by the drive and independent of the attracting mass 3 The equivalence: gravitational and kinematic dilation coincide at that radius, both √(1 − 2η), exactly 4 The relativistic gain cap and the event horizon are the same bound 5 The four phases as a map on radius, with the corridor's 0.88 appearing as a gain 6 Reading B against the well: the matched radius is worth exactly one e-fold, a mass ratio of e 7 Both corridor edges on radius in the fine-structure constant alone 8 The unpropelled catalog checked against published orbital elements: r = 2q/(1 + e) 9 The dilation deficit bracketed by the conversion fraction 10 The build: a burn at the matched radius, with the exhaust velocity canceling out of the multiplier 11 Depth and gain as interchangeable, and the crossover reducing to 4 · (0.88)² > 3 12 The chain: n matched burns compounding as √n, and the four phases by integer burn count 13 What the construction gives outside the corridor: the budget identity, the two ends of the axis, the far field, the strong field, the redshift 14 The square-root torsion operator: the closure over the rationals, the two exact landings, and the corridor's geometric center 15 Tiers, with a worked exemplar of each 16 Misappropriation-Risk Clarification What the formal layer proves Every result below corresponds to a named theorem in one of the two files at this coordinate. Results 1 through 19 are carried by SNSFL_MatchedRadius_Reduction.lean at [9,9,0,17](GRAV), the general-reader form. Results 20 through 25 are carried by SNSFL_MatchedRadius_Reduction_TL.lean at [9,9,0,17.1](GRAV), the specialist form for the square-root torsion operator. Each file is standalone: every constant and definition it uses is restated inside it, so either one closes without reference to the other. Both compile 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 matched radius exists, is unique, and matches by construction: at r = r_s/τ_A² the squared escape fraction is exactly τ_A², and the matched ratio is strictly decreasing in Reading A (t8_matched_radius_matches, t11_ratio_antitone). The attracting mass cancels: r_match/r_s is a property of the drive alone, and equals the reciprocal of twice the conversion fraction (t9_mass_cancels, t10_ratio_is_conversion). The corridor on radius is 53.357910481 to 68.902260435 Schwarzschild radii, and the band lies outside the horizon, the photon sphere and the innermost stable circular orbit (t12_corridor_on_radius, t13_corridor_outside_strong_field). The equivalence: gravitational and kinematic time dilation are the same quantity at the matched radius, and both equal 1 − 2η (t14_equivalence, t15_dilation_is_conversion, t16_corridor_dilations, t17_dilation_pos). The relativistic gain cap and the event horizon are the same bound, with equality transferring: (1 + g_r) · τ_A < 1 if and only if r_match > r_s, and (1 + g_r) · τ_A = 1 if and only if r_match = r_s (t18_cap_is_horizon, t19_cap_equals_horizon, t20_cap_at_ceiling). A drive whose exhaust velocity is c matches the event horizon of every mass exactly, and one at half light speed matches exactly four Schwarzschild radii (t21_photon_matches_horizon, t22_half_c_matches_four). The four phases map onto radius: Locked is the far field beyond the floor radius, Shatter is the interior to the ceiling radius, IVA is the shell between them, and the strong-field landmarks lie inside the Shatter band (t23_locked_is_far_field, t24_shatter_is_interior, t25_iva_is_the_shell, t26_strong_field_is_shatter, t27_phase_of_edges). Both corridor edges on radius are closed forms in the fine-structure constant and the integer 1001, as is the band width (t28_ceiling_radius_in_alpha, t29_floor_radius_in_alpha, t30_band_width_in_alpha, t31_eta_edges). The well at the matched radius is worth τ_B = 1 exactly, which is a mass ratio of e, and the corridor's Reading B lies below it (t32_well_is_one_efold, t33_one_efold_is_e, t34_corridor_below_one_efold). The dilation deficit is bracketed by the conversion fraction, η < 1 − √(1 − 2η) < η(1 + η), with both polynomial forms proved separately and the corridor's values inside the domain (t35_deficit_lower_poly, t36_deficit_upper_poly, t37_deficit_bracket, t38_eta_in_domain). At the matched radius the exhaust velocity cancels out of the Oberth multiplier, leaving a function of Reading B alone; both corridor multipliers are closed forms in α⁻¹ and 1001; and the multiplier exceeds one and is strictly decreasing in Reading B (t39_exhaust_cancels, t40_multiplier_edges, t41_multiplier_ceiling_in_alpha, t42_multip
Author: Trent III, Russell Vernon
Language: English
Subfield: Astronomy and Astrophysics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225450902