Applied Identity Physics: The Propellant Law at the Matched Radius — A Saving Bounded by e and the Compactness Criterion C > 2η
Year: 2026
Type: article
Abstract: Applied Identity Physics: The Propellant Law at the Matched Radius — A Saving Bounded by e and the Compactness Criterion C > 2η Architect HIGHTISTIC (Russell Vernon Trent III) · SNSFT Foundation · Soldotna, Alaska Coordinate [9,9,0,18](GRAV) proposed · 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 [9,9,0,17] 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 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) establishes that Reading A of a drive, τ_A = v_e/c, names a radius: for any exhaust velocity and any attracting mass there is exactly one radius at which the local escape velocity equals that exhaust velocity, the ratio is r/r_s = 1/τ_A² = 1/(2η), and the mass cancels. That reduction states the result forward — fix the mass ratio, get the speed. This paper states it inverse and then asks where the maneuver exists at all. Inverse, the propellant law. Fixing the target speed rather than the mass ratio inverts the chain law exactly: τ_B = √(1 + (v_∞/v_e)²) − 1, so the mass ratio a burn at the matched radius requires is exp(√(1 + (v_∞/v_e)²) − 1) against the ideal rocket's exp(v_∞/v_e). The ratio of the two is the propellant saving, and it has a closed form: S(x) = e · exp(x − √(1 + x²)) with x = v_∞/v_e. Because x − √(1 + x²) is strictly negative and strictly increasing to zero, S is strictly increasing and strictly bounded above by e, which it approaches as the burn grows. The bound is not a coincidence of the algebra: the limit is e¹, and the 1 in that exponent is the τ_B = 1 that the matched radius well is worth. The one e-fold the well supplies is the one e-fold of propellant the vehicle stops carrying. At the small-burn end the cost ceases to be exponential altogether — τ_B → x²/2, so the propellant fraction grows quadratically in the target speed rather than exponentially. The corridor edges realize 1.497759263 and 1.469548301, which is 55.0995% and 54.0617% of the available e, and the fraction captured has its own closed form: x = (t² − 1)/(2t) with t = ln(S/e), giving 50% of e at x = 0.374774, 90% at x = 4.692931 and 99% at x = 49.744556. Where the maneuver exists, the compactness criterion. A matched radius is only reachable if it lies outside the attracting body. Writing C = r_s/R for the standard compactness of a body of Schwarzschild radius r_s and physical radius R, the condition r_match > R reduces with no residue to C > τ_A² = 2η: a drive can use its matched radius at a body if and only if the body is more compact than the drive is efficient. Two fractions from unrelated domains — the gravitational compactness of an object and the rest-mass conversion fraction of a rocket — are compared directly, and the comparison is the availability test. The corridor's thresholds are the exact rationals TL² = 73208451470443787388498001/3906250000000000000000000000 = 0.018741363576434 and TL_IVA² = 8858222627923698274008258121/610351562500000000000000000000 = 0.014513311953590, standing in the ratio 625/484 = 1/0.88², the corridor width factor squared. Against the published catalog the criterion partitions cleanly: the Sun sits at C = 4.245005 × 10⁻⁶ and is excluded by four orders of magnitude, the densest white dwarfs reach only 1.279742 × 10⁻³, and every neutron star and black hole clears the threshold by at least 18×. The corridor maneuver is a compact-object maneuver, and the arithmetic says so without appeal to engineering. The dual reading, and the empty band. Inverting the criterion, every body names a drive: τ_surface = √C is the Reading A of the drive whose matched radius is exactly that body's surface. Classifying τ_surface across the catalog places every ordinary star and every white dwarf in Locked and every neutron star and black hole in Shatter, with the IVA corridor corresponding to C ∈ [0.014513311953590, 0.018741363576434) — a 1.4 M☉ body of radius 220.6 to 284.9 km, which is between a white dwarf and a neutron star, where no stable configuration exists. The IVA band on compactness is the collapse gap, so the criterion admits no marginal object in either direction. The construction was never Schwarzschild-specific. For any static, spherically symmetric metric written ds² = −f(r)c²dt² + dr²/f(r) + r²dΩ², the same two factors of f cancel in the locally measured escape speed, giving v_local = c√(1 − f(r)) with no expansion taken. The matched radius is therefore the solution of f(r_match) = 1 − τ_A² = 1 − 2η — the radius at which the metric function equals the squared time-dilation factor. That reframes the parent reduction's equivalence result: the agreement of gravitational and kinematic dilation at the matched radius is not a property of one distinguished radius, it is the equation that locates the radius, and the Schwarzschild r = r_s/τ_A² is the case where the level set happens to be linear. The condition is quadratic in Reissner–Nordström and cubic in Schwarzschild–de Sitter, where the cosmological term contributes 4.8829 × 10⁻⁴¹ of τ_A² at a solar-mass matched radius. The class is exactly those metrics in which g_tt and g_rr are reciprocal with no cross term; Kerr is stationary rather than static and carries a dt dφ cross term, so a rotating body's matched radius is a separate construction. The charged case has a closed form. Reissner–Nordström gives the quadratic τ_A² r² − r_s r + r_Q² = 0, so charge splits the matched radius into the pair r± = r_s[1 ± √(1 − 4q²τ_A²)]/(2τ_A²) with q = r_Q/r_s. Across the physical domain — sub-extremal charge and Reading A in (0,1) — the inner root lies inside the outer horizon and the outer root outside it, with no exception in a scan of 5,049 pairs, so exactly one root is a place a vehicle can occupy and the maneuver survives at every sub-extremal charge. The outer root is the Schwarzschild matched radius times [1 + √(1 − 4q²τ_A²)]/2, minimized at extremal charge where it becomes [1 + √(1 − 2η)]/2, so the maximum inward shift over the whole charge range is (1 − √(1 − 2η))/2 — the parent reduction's own dilation deficit, halved. For the corridor that is 0.4707501464% at the ceiling and 0.3641589160% at the floor. At extremal charge with Reading A at its supremum the pair merges exactly at the horizon, r_s/2, carrying the parent's photon-drive statement into the charged metric. Combined. The saving depends on Reading B alone and the availability depends on the body alone, so magnitude and availability are fully decoupled: a corridor drive saves the same 1.497759263 at a 1.4 M☉ neutron star, a 10 M☉ black hole and Sagittarius A*, while the matched radius itself moves from 220.6 km to 1,575.8 km to 6.776 × 10⁸ km. The resonance gain trades against availability in the other direction: since the gain multiplies Reading A, the criterion becomes C > ((1 + g_r)τ_A)², so a gain of 1.5 tightens the requirement by 6.25× and a gain of 4 excludes a 1.4 M☉ neutron star outright. At the relativistic cap the requirement goes to C → 1, where only a horizon qualifies — the same identification of cap with horizon that [9,9,0,17] carries, now read as an availability statement. 1001 = 7 · 11 · 13. What a reader gets from this paper § What is established 1 Grounding: the Torsion Limit, the two readings, the four phases, the matched radius 2 The inversion: fix the target speed, get Reading B 3 The propellant law, and the saving bounded strictly by e 4 Why the bound is e: the well's e-fold is the propellant's e-fold 5 The small-burn end, where the cost stops being exponential 6 How much of e a given burn captures, in closed form 7 The compactness criterion C > 2η 8 The criterion against the published catalog 9 The dual reading: every body names a drive 10 The empty band: IVA on compactness is the collapse gap 11 Magnitude and availability decoupled 12 What the resonance gain costs in availability 13 A worked build at a neutron star 14 The matched radius as a level set of the metric function, and the equivalence as its definition 15 Reissner–Nordström: the matched radius as a pair, and the charge correction bounded by the dilation deficit 16 Tiers, with a worked exemplar of each 17 Misappropriation-Risk Clarification 1. Grounding The full grounding is carried in the parent papers. It is stated here once in short, so this paper stands on its own. 1.1 The Torsion Limit The Torsion Limit. The Universal Torsion Limit is TL = 0.136899099984016 and the Sovereign Anchor Constant is Ω₀ = TL × 10 = 1.36899099984016. Nothing in this paper sets it. It enters at full corpus precision from the inverse fine-structure constant, by direct subtraction: Napkin math, corpus precision Value 1/α 137.035999084000016 Subtract one TL 137.035999084000016 − 0.136899099984016 = 136.899099984016 What is left 136.899099984016 = TL × 1000, exactly Bare + kinetic 136.899099984016 + 0.136899099984016 = 137.035999084000016 Rounded to published digits 137.035999084 = CODATA 2018 Term Value QED reading PNBA reading Bare: TL × 1000 136.899099984016 The bare electron contribution Pattern capacity at the electromagnetic scale Kinetic: TL × 1 0.136899099984016 Radiative corrections, a renormalized series External forcing F_ext at Layer 0, one exact term Total: TL × 1001 137.0359990
Author: Trent III, Russell Vernon
Language: English
Subfield: Astronomy and Astrophysics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225577360