Independent research · quasi-steady wake aerodynamics

Nobody has measured the wake a Mooney flies through in formation, and this work has not either.

The Mooney Caravan is an annual mass formation arrival flown by real people. What wake field an aircraft inside it flies through, and what induced loads follow, has no published answer for this aircraft class. This program computes those loads under a quasi-steady assumption. It has never been compared against a measurement of a Mooney wake, and until it has, it says nothing to a pilot.

Register: Verification and cross-code comparison
Highest rung reached: R3, an external solver comparison
Quantity: C_l = L_roll/(qSb), quasi-steady

R0 to R2 are cleared. R3, the external solver comparison, has been exercised and holds a field withdrawn for violating conservation of circulation. R4, viscous CFD, has not been run. R5, measurement, has never been reached.

Limits

What this work does not say.

Real people fly this formation. A plausible-looking but unvalidated result could influence real decisions, so no handling, spacing, position or altitude-stagger advice follows from any of this.

Research, not operational guidance: No output of this program is advice about how to fly. This holds regardless of how confident a result appears.
Quasi-steady only: No handling, controllability or safety claim, at any rung, until a validated flight-dynamics model exists. It does not exist.
Verification is not validation: The program has never been compared against a measurement of a Mooney wake. Passing every check would still not make a single result validated.
Published guidance is context: Published wake-separation and formation guidance is recorded as background. It does not validate this program's models, and this program does not comment on it.
Attribution: Established results are attributed to their sources. The program claims integration and evidentiary discipline, not the underlying theory.
Forbidden outputs: Forbidden, not pending: no spacing, separation, position, altitude-stagger, sequencing or formation-geometry recommendation; no handling or controllability claim; no safety claim; no flight validation claim. Altitude staggering is an analysis variable here, never a recommendation.

Where a result would be most useful operationally is exactly where the fence is tightest, because that is the direction the harm runs.

The question

The problem

A three-aircraft element flies in fingertip, one lead and two wingmen, and behind each wing a pair of counter-rotating vortices is shed and left in the air. Every aircraft after the first flies through what the ones ahead of it left. The question is what that field looks like, and what rolling moment it puts on the wing that flies into it.

The formation's own documentation does not answer it. The 2019 briefing material treats wake turbulence as a qualitative pilot caution, and justifies inter-section spacing by compression absorption and traffic separation, not by any quantified vortex-decay or wake-avoidance model. There is no published Caravan wake-interaction model. That is what makes this a research question and not a reconstruction of doctrine. The formation's spacing rationale is documented as non-wake, so nothing computed here can be read as a check on it.

Two structural facts then decide everything downstream. The first is scale. A Mooney has a span of 10.70 m and a root circulation of about 20.5 m²/s, giving a circulation Reynolds number of roughly 1.3 × 10⁶. Almost the whole wake-vortex literature is transport-category, two orders of magnitude larger, and the lightest aircraft in the NASA AVOSS-validated decay corpus is still about five times above this value. Nondimensional structure transfers between scales. Empirical decay constants need not, and whether they transfer here is an open question.

Why the method is a vortex kernel and not a mesh

The second is range. Formation wake is a Lagrangian vortex-transport problem spanning roughly 7 to 210 span-lengths. Mesh CFD cannot carry that range affordably without dissipating the very cores that carry the signal. The program therefore owns a fast-time vortex kernel as its primary method and uses higher-fidelity solvers as comparison rungs, not as substitutes.

What the terms mean

Circulation Γ is the strength of a vortex, in m²/s. Wake age is time since the vortex was shed, which at a fixed airspeed is also a distance downstream. A core is the small radius over which a real vortex's swirl is finite rather than singular. Quasi-steady means the load is computed as if the encounter were frozen: there is no roll rate, no roll damping and no dynamics anywhere in the model.

The formation, to scale, with its one assumption marked

Geometry

The drawing below is a cross-plane cut, looking upstream, at true scale in both axes: the stated geometry, drawn to size. It is where the difference between a wingtip and a vortex core becomes visible.

core shedding +Γ
core shedding −Γ
σ_vpm = 1.123 m, the external solver's smoothing radius, drawn on the two adjacent wingtips because those are the shedding stations the conservation question is about
dashed dimension: assumed, not briefed
The dotted vertical is the lead's centreline. The port half of the element is its exact mirror and is not drawn.

Wingtip clearance 0.90 m Aft stagger 9.40 m Vertical step −1.50 m

Lateral offset: 11.60 m
Inner-core sep. d: 3.531 m
birth sep. / σ: 8.510
Wake-age offset: 0.1435 s

What the three inputs do

Wingtip clearance is the gap between the lead's starboard tip and the wingman's port tip. The briefed nominal is about 3 ft, 0.9 m, and it is swept from 0.9 to 5 m. Moving it moves the whole wingman, so it also moves the cores.

The cores are not at the tips. An elliptically loaded wing sheds its pair at ±b′/2, where b′ = (π/4)b = 8.4038 m, so each core sits 4.2019 m from its own aircraft's centreline and 1.15 m inboard of its own tip. At the briefed clearance the two adjacent tips are 0.9 m apart and the two adjacent cores are 3.196 m apart. Confusing the two is the easiest error to make on this geometry; I made a version of it and caught it.

Vertical step lifts or drops the wingman's whole plane. With the step present the inner-core separation becomes √(3.196² + 1.5²) = 3.531 m. It is drawn with a dashed dimension line because it is the only geometry input here not read off a document: it is graded low and filed as an assumption, no numeric step in evidence. The briefing says only "positive stepdown, spinner just visible".

Aft stagger is a streamwise offset and therefore cannot appear in a cross-plane cut at all, which is why the drawing does not change when you move it. What it changes is time. An aft wingman's wake is younger at the common encounter plane by stagger / V_TAS, which at the briefed 9.4 m and 65.507 m/s is 0.1435 s. The sign was checked in both the owned and the external solver's frames.

Birth separation over σ is the distance between the two adjacent shedding stations at the moment a particle is born, √(gap² + step² + stagger²), divided by the external solver's particle smoothing radius of 1.123 m. All three inputs move it, which is why the stagger slider changes this number while the drawing stays still: streamwise offset separates the two births even though it cannot appear in a cross-plane cut. It is reported because it is the parameter that orders every external-solver case by whether the simulation conserved circulation, and it is a statement about the numerics of that simulation and nothing else.

Without scripting the drawing does not render. At the briefed geometry: lateral offset 11.60 m, inner-core separation 3.196 m co-planar and 3.531 m with the assumed step, birth separation over σ = 8.51, wake-age offset 0.1435 s.

Formation geometry, graded by evidence

Parameter	Adopted nominal	Swept	Confidence	Category and source
Element composition	3-ship: lead, left, right	2 to 3 ship	high	2019 NTSB docket main brief, slides 3 to 5 and 10
Fingertip bearing	45°	40 to 50°	high	Main brief slide 22, plus the generic FFI guide
Wingtip clearance	0.9 m (3 ft)	0.9 to 5 m	high value, medium mapping	Main brief slide 22 diagram, read off a plan view
Aft stagger	9.4 m (31 ft, 0.85 b)	5 to 12 m	medium	Main brief slide 22 diagram, read off a plan view
Vertical step	1.5 m down (5 ft)	0 to 3 m	low, assumption	Inferred from "positive stepdown, spinner just visible". No numeric step in evidence
Element-lead interval	15 s	10 to 20 s	high	Main brief slide 20
Wingman interval	about 5 s	0 to 5 s	high	Main brief slide 20
Route position, the alternative to fingertip	2 ship-widths to 300 ft	2 to 4 wingspans	high	Main brief slides 23 to 24

The operating point

Span b: 10.70m
Effective span b′: 8.4038m
Area S: 15.5m²
Weight W medium: 1250kg
V_TAS: 65.507m/s
Density ρ medium: 1.0879kg/m³
Dyn. pressure q: 2334Pa
Circulation Γ₀: 20.468m²/s
Self-descent w₀: 0.3876m/s

Condition. Level cruise, quasi-steady, at 120 KIAS nominal with 105 to 130 swept, in a 2500 ft MSL block on a warm day: the density is taken at a 762 m pressure altitude and ISA + 13 K, which is an assumption and not the standard atmosphere, where 2500 ft would give 1.136. Weight and density are the two inputs graded medium, and between them they set every circulation on this page. The effective span is b′ = (π/4)b, the elliptic value. Indicated airspeed is never substituted for true airspeed.

Quantity of interest. The body-axis rolling-moment coefficient C_l = L_roll/(qSb), positive right-wing-down. It is not C_L and it is not a sectional c_l. Wake ages of 5 s or less in the external rung, and longer in the owned fast-time model.

What is missing from the evidence

Enroute inter-section distance beyond the 30 s takeoff offset is not published.
The numeric cruise vertical step is not published.
Cruise density altitude in warm summer conditions is bracketed rather than measured.
The physical mapping of the 3 ft and 31 ft diagram dimensions is graded medium and is treated as a self-consistent geometry, not a survey.
Element size is context-dependent: 3-ship in cruise, 2-ship under the single-runway agreement.

The model, and what it structurally cannot represent

Method

The chain is short enough to state in one breath. A wing carries weight, so it sheds a vortex pair at ±b′/2 with strength Γ₀ = W/(ρ V b′). The pair ages, descends and interacts. At a chosen wake age the field is evaluated across a plane. A following wing is walked through that plane and each strip of it sees a local upwash, hence a local change in angle of attack, hence a local lift increment. Integrating those increments about the centreline gives a rolling moment, and dividing by qSb gives C_l.

Quasi-steady

Every load here is quasi-steady: no roll rate, no roll damping, no dynamics. Phase 1 does not turn C_l into a roll rate because no defensible Mooney roll inertia or roll damping is available.

Two owned modes, run as each other's control

Mode A is isolated-generator superposition: each wake ages under the kinematic model and the field is the linear, source-tagged sum. Mode B is coupled multi-vortex transport: finite-core cross-plane point vortices with delayed births, integrated by RK4 with mutual induction, where the self-descent emerges from the induction, and which recovers Mode A when the coupling is switched off. The pair is the point: the robust signal in this program is not either curve, it is the divergence between them.

Standing caveat: Mode B is an inviscid, no-merge point-vortex transport. It overstates close-core interactions, because real viscous wakes merge and diffuse. Coupled tight-element magnitudes are upper-biased screening estimates, not demonstrated bounds.

The wing the external solver ran is not a Mooney wing

Every frozen external-solver campaign ran a rectangular flat surrogate: constant chord 1.4486 m, taper 1.0, zero dihedral, zero washout, unswept, root incidence 4.152° calibrated. The M20K reference wing has never been simulated. Separately, and independently of that, every Wave 2, Wave 3 and Wave 4 result uses M20J-derived span and area, 10.7 m and 15.5 m², while the reference aircraft is an M20K whose manual gives 10.9982 m and 16.238 m²: +2.79 % span and +4.76 % area. Phase 1 is the exception, and it matters because it is cited three times here: it hardcodes its own aircraft, 11.0 m and 16.3 m² at 1300 kg and an 1800 m pressure altitude, so its magnitudes are not poolable with the Wave numbers as though the two shared an aeroplane.

Neither fact is a defect in the runs. The results remain valid and reproducible for the model they were run with. They are not exact M20K-dimensional predictions, and nothing here should be read as one. The external rung itself is FLOWUnsteady's reformulated vortex particle method: a vortex-lattice wing shedding vortex particles onto a fast multipole backend, inviscid finite core, subfilter-scale model off, RK3, one particle per step, smoothing radius σ = λ V Δt at λ = 2. The production protocol is 700 steps at Δt = 0.008571 s, 6.0 s total, σ = 1.1230 m, 36 076 particles, 2087.6 s of wall time and 1.11 GiB peak.

What the model has no term for

Viscosity. Both owned modes and the external rung are inviscid.
Core merger and diffusion, which is what a viscous rung would be run to settle.
Turbulent decay, and therefore any decay constant borrowed from transport-category data.
Roll rate, roll damping, and every other dynamic quantity.
Compressibility, multi-surface loading and any leading-edge extension effect, which is why one attempted measurement anchor was closed as incomparable rather than adapted.

The ladder, and why a higher rung is not a better answer

Verification

Terms: code verification means the equations are solved right, and licenses no statement about physical fidelity. Solution verification means this particular solution has been refined to a demonstrated asymptote. Model validation means comparison against measurement of the real system, and comparison against another model, however expensive, is not it. Prediction means a validated model at an untested condition, which describes nothing in this program. Quarantine means a committed artifact that violates a necessary physical invariant, retained as evidence and forbidden from publication.

Rung	What it is	State	On what evidence
R0	Analytic identities and limiting cases	cleared	Point-vortex and Lamb-Oseen exact cases, elliptic-loading closed form, mirror antisymmetry, Γ-flip
R1	The owned fast-time kernel, the primary method	cleared	15/15 Phase 1, 24/24 Wave 2
R2	An independent cross-implementation reference, guarding against a self-consistent but wrong kernel	cleared	An independent N-vortex reference, sharing no law, no Biot-Savart sum and no integrator with the production kernel, agrees to about 1e-14
R3	External vortex-particle comparison	exercised, one field quarantined	Exercised across two campaigns; one committed field violates conservation of circulation and is withdrawn from every trusted contrast
Everything at or below this line is comparison. Nothing above it has been reached
R4	Viscous CFD or LES: merger, diffusion, turbulent decay	not run	Gated. It opens only on a measured deficiency that materially affects a quantity of interest, not on preference
R5	Experiment or flight measurement, the validation rung	never reached	No Mooney-scale wake measurement is known. Three separate attempts to borrow one each closed for a different recorded reason

A higher rung is a more expensive comparison. It is not automatically closer to the truth. The evidence for that is R3. It cost more than every rung below it combined, and it produced a field that violates Kelvin's theorem. The three cheapest checks available caught it, and two of them read data that was already on disk.

Admission to R3 is measured on facets, not on a checkmark

An external run is admitted on seven separately measured facets: artifact, provenance, source balance, field, applicability, case disposition and analysis eligibility. Each takes one of CLEAN, AMBIGUOUS, FAIL, UNAVAILABLE or NOT APPLICABLE. A run being complete never implies it is clean, and missing evidence yields NOT ASSESSABLE rather than a pass. That distinction is not administrative. It is exactly the distinction that caught the defect in the next chapter, on a run that had completed normally with intact files.

What is supported now, each finding with its bound

Evidence

Everything below rests on a surrogate wing that is not a Mooney wing, on dimensions taken from a different model in the family, and on an inviscid method at a rung that is comparison, not validation.

The inner counter-rotating pair rises rather than descending

An isolated vortex pair descends under its own induction. Inside a tight element the inner pair does not. In both resolved external runs the mean height of the inner structure goes from −0.63 m to +0.60 m over wake ages 1 to 5 s, and to +0.92 m at the finer resolution. The coupled owned model reproduces the motion, −0.49 m to +0.47 m. The isolated owned model gets the sign wrong, −1.13 m to −2.69 m.

Bound: direction only, not magnitude. One geometry, ages at or below 5 s, quasi-steady, inviscid. The magnitude is not refinement-converged, so the numbers above are lower estimates.
The vertical step is material in the one surviving clean contrast

Measured through the one surviving clean contrast, G11 − G01, at R = 15.14 / 15.66 / 8.91 across ages 1, 3 and 5 s. R is the size of the contrast divided by U_num, the local numerical discrimination scale measured between two resolutions at the baseline cell; the program's frozen bar for calling a difference material is R ≥ 2.

Bound: direction only. One geometry, one clean arm, ages at or below 5 s, inviscid, magnitude not refinement-converged. Part of the effect is the receiver becoming co-planar with the cores rather than the wake itself changing, because the receiver plane is held fixed while the sources move. U_num is local to the baseline cell and its transfer to the co-planar geometries is an assumption rather than a measurement.
The rule that the isolated model is always the closer proxy is false

Across the tested bracket the coupled Mode B runs 1.6 to 2.2 times the external solver's peak |C_l|, and the isolated Mode A is usually the closer proxy: ratios to the external solver are 2.2, 1.9 and 1.8 for Mode B against 1.9, 1.4 and 1.0 for Mode A at ages 1, 3 and 5 s. One clean co-planar cell reverses it, with Mode B closer at age 5 and R = 2.16.

Bound: the evidence for the reversal is a single unreplicated clean cell. The second reversal previously cited for this result was drawn from a quarantined field and is withdrawn. This is a one-scalar error proxy and is not a validation of either model.
In the clean co-planar cell there is one structure, not two inner cores

Both reduced models build an inner dipole by construction. In the clean co-planar external cell the verified core finder identifies a single structure of about 2Γ instead. That is a measured absence rather than a failed measurement, and it is corroborated by an independent particle-level method that shares none of the grid arithmetic.

Bound: one geometry, one resolution, one finder, inviscid, ages at or below 5 s. The corroborating method carries its own scale bound: at a linkage radius of half the smoothing radius the same cell shows two, so the statement holds at and above the smoothing scale. It is not a universal claim that the physical pair does not exist.
Accumulation down a long line saturates, in the owned model

Old wakes descend out of the way at about 5.8 m per 15 s interval, so exposure does not pile up linearly. Peak |C_l| reaches 95 % of its fifteen-element value after six to seven upstream elements, and total accumulation reaches 1.80 times the single-element value where a linear pile-up would give 15. Over a 23-element in-trail line the isolated peak varies only 4.1 % across four different hypotheses about where the section breaks fall.

Bound: owned model only, in a no-decay baseline, so descent alone is what caps it. There is no external-solver and no measurement evidence for any of this, and line accumulation is the largest evidentiary gap in the program. The coupled column for a line this long is coarse in timestep, non-convergent, and is not used.
The two owned models disagree about the shape of the tip-clearance response

Swept from 0.9 to 5.0 m at age 3 s, the isolated model rises monotonically and saturates, 0.0189 to 0.0238. The coupled model is non-monotonic, peaking near 2.0 to 2.95 m at 0.0328 and then declining to 0.0306. That is a qualitative disagreement, not a magnitude disagreement.

Bound: this is an open trigger, not a result. The external-solver clearance confirmation cases were evaluated as indicated and then not run, which is a budget-limited deferral and not a null finding. No clearance conclusion is drawn beyond the owned-model level, and none of this is a statement about how far apart aircraft should fly.

A tested handoff point, which is not a boundary

At route spacing, 2.5 spans or 26.75 m, the coupled and isolated models agree to within 2 to 15 % over ages 1 to 5 s, against a ratio rising to 1.86 at the tight fingertip. So the coupled correction is negligible there and large only in close formation. 2.5 spans is a tested screening handoff point and not a validated boundary; establishing a threshold would need a spacing sweep that has not been run.

The peak is not refinement-converged

The external solver's peak |C_l| climbs 7 to 13 % as the smoothing radius goes from 1.12 m to 0.79 m, with no demonstrated asymptote. Worse for attribution, that bracket moves the timestep, the smoothing radius and the particle count together, so the change cannot be assigned to any one of them. The converged follower moment is therefore unresolved, and every magnitude on this page is a lower estimate.

Withdrawn, quarantined, and without a verdict

The retractions

When evidence fails the claim is withdrawn in full, including every derived quantity. The states below are not interchangeable.

quarantined A committed artifact that violates a necessary physical invariant. Retained as evidence, forbidden from publication.
withdrawn A previously published claim retracted because its evidentiary basis failed. Not quiet removal: withdrawals stay visible.
no verdict The evidence does not distinguish the alternatives. This is not a null result. One unresolved arm is one arm.
narrowed The claim survives in altered form: a range instead of a value, or a ranking that must now be qualified by which arm it was measured on.
surviving Every cell the contrast rests on is field-admissible, so the reading stands, inside the bounds printed beside it. It carries the solid mark the cleared rungs carry.

The frozen factorial, and the cell that took it down

Four external-solver cases were run as a two by two over the vertical step and the aft stagger, all at the briefed 0.9 m tip clearance and at wake ages 1, 3 and 5 s. One of the four carries a field that cannot be physical.

Aft stagger 0 m

Aft stagger 9.4 m

Step −1.5 m

G00

step −1.5, stagger 0

The reused baseline, carried over from the previous campaign with its hash and a regression check.

G01

step −1.5, stagger 9.4

Field-admissible. Source balance 1e-4.

Step 0 m

G10

step 0, stagger 0

quarantined
Total circulation is conserved, so a cross-plane cut through a system of trailing vortex pairs must carry about zero net circulation. This cell carries about +67 m²/s, roughly 3.3 root circulations. The run completed normally, all 700 steps, and the files are intact.

G11

step 0, stagger 9.4

Field-admissible. Source balance 0.0000. The only clean co-planar cell, and it does a lot of work below.

The measurement, on the numbers a reader can check
What was measured	G01	G11	G10	Pass / fail boundary
Net circulation through the cut, ages 1 / 3 / 5 s, in m²/s. Must be about zero for a system of pairs	≈ 0	≈ 0	+66.56 +67.79 +67.06	\|·\| ≤ 2 pass ≥ 20 fail
Mirror antisymmetry residual, ages 1 / 3 / 5 s. A symmetric formation must produce an antisymmetric field	~0.007	~0.006	1.6845 1.5674 1.4456	≤ 0.10 pass ≥ 0.50 fail
Source balance r_p read from the solver's own particle dump, before any interpolation	0.0001	0.0000	0.6135	≤ 0.01 pass ≥ 0.10 fail
Banded circulation over the right inner region, y in [2, 12) m. Localises the break to one side	−2012	−11249	+12785	sign against G11

Three independent paths agree, and two of them share no arithmetic: a summed finite-difference vorticity over the grid interior, a closed line integral of velocity around the grid boundary agreeing with it to 0.071 m²/s or better, and, decisively, the solver's own pre-interpolation particle dump. So the defect is in the simulation, not in the cross-plane extraction and not in core identification. Two forensic replicates then settled what it is not: refining the smoothing radius reproduced it, which falsifies a resolution explanation, and a configuration-identical replicate reproduced it to about five significant figures, which retires run-to-run non-determinism. It is deterministic. The source-level cause remains undiagnosed.

What the quarantine costs, contrast by contrast

Contrast	What it isolates	State	Reading
G10 − G00	Vertical step, no stagger	withdrawn	Contains the quarantined cell
G11 − G01	Vertical step, stagger present	surviving	R = 15.14 / 15.66 / 8.91, material at every age. Direction only
G01 − G00	Aft stagger, step present	no verdict	R = 0.35 / 0.19 / 0.04, unresolved at every age
G11 − G10	Aft stagger, step removed	withdrawn	Contains the quarantined cell
Main effects and interaction	The whole factorial	withdrawn	Each of the vertical main effect, the stagger main effect and the interaction contains a term from the quarantined cell, at every age. Every derived row is marked untrusted by a mechanical check

Two published headline numbers went with them: a vertical main effect quoted at R = 8.1 to 13.7, and a claim that the vertical effect was about 6 to 11 times the stagger effect. Aft stagger is left with no verdict at all. That is not a finding that stagger does not matter: one unresolved arm is one arm.

The control parameter, reached from both directions

Instrumented runs at the production protocol, one factor moved per run. Every run here is co-planar and unstaggered, so the tip clearance and the adjacent-wing birth separation are the same length; in the factorial they are not, and the ordering parameter is the birth separation.
Run	What moved	gap	gap / σ	Final r_p
R1 R2 R3	Nothing. Independent repeats	0.90 m	0.801	0.61354
R4	Adjacent-wing separation only, by geometry	2.50 m	2.226	0.00034
R5	Smoothing radius only. The aircraft do not move	0.90 m	2.226	clean
R6	Wing processing order only	0.90 m	0.801	0.61354

This is a statement about the external simulation's numerics. It brackets a transition as failing at 1.15 smoothing radii and clean at 2.23 at production settings, and R4 and R5 reach that same ratio from opposite directions, one by moving the wings apart and one by shrinking the kernel while the aircraft stay where they are. That is what makes it a ratio rather than a distance. The parameter is confounded with particle overlap and is not a physical admissibility law. Nothing here is a statement about aircraft separation.

At the production timestep the conservation residual crosses the material level at step 26 and the structural level at step 46, identically across three independent repeats, and the break is confined to the two inner structures flanking the lead while the outboard tip pair never departs. Raising the adjacent-wing separation past a gap-to-smoothing-radius ratio of about 2.2 removes it entirely, by geometry alone, with every numerical parameter held at its production value. Swapping the order the wings are processed in changes nothing to five significant figures, including which side inverts, so a processing-order explanation is falsified.

Check-count is not coverage

246 checks passed on that field. Every one of them tested geometry contracts, operator equivalence, table arithmetic or artifact integrity. None of them tested whether a committed field satisfied a conservation law it must satisfy. The three cheapest checks in the program, net circulation, mirror antisymmetry and particle-sum conservation, each caught it immediately, and two of them read data that was already on disk. A figure is not validated because it looks physical, and residuals falling is not convergence.

On the owned side, one claim in fifty was withdrawn and stays withdrawn

A review adjudicated fifty reader-facing owned claims against two declared receiver dihedral bounds, under rules frozen before any result was read. One claim was withdrawn: Phase 1's vertical-symmetry claim, that the induced field is symmetric about the core plane. Under a flat receiver the rows above and below the plane are exactly equal. A dihedralled receiver is a shallow V, which samples above and below the core plane differently, so the symmetry was a property of the wing and not of the field. The break is 2.56 % on the full-span bound.

A later study solved both receiver mechanisms in one system and returned a favourable combined answer. The withdrawal did not come back. The most severe of the three axis statuses is taken, for exactly this reason: every arm is a declared sensitivity, no arm is the real wing, and an idealised joint answer cannot resolve a claim about unresolved real geometry.

Survived in altered form

Narrowed. "Peak |C_l| reaches 95 % of its terminal value by six upstream elements" becomes six to seven across the tested arms. The separate and more general statement, that a long line does not linearly pile up wake, is preserved unchanged.
Rankings qualified. Two published ranking tables each show strict reversals under the dihedral review and must now be quoted with the arm they were measured on. The identity of the largest-effect axis is preserved.
Retired. A comparative claim that the peak interaction exceeds the local coupled |C_l| flips on both bounds and is no longer repeated.
Withdrawn, numerical. A lower bound of more than 60 steps, because the micro-runs it rested on executed at timesteps of 0.300 and 0.100 s against a production 0.008571 s.

Where the evidence stops, and why the ceilings differ

Limits

Validation, solution verification and physics each bind at once, and a narrower one sits under them on the subject aircraft itself. Collapsing them into a single sentence about uncertainty would hide the useful part, which is that each would be lifted by a different kind of work.

Validation ceiling: the measurement rung has never been reached

No owned operator has been compared against any measurement of any real wake. Three attempts to borrow one each closed for a different recorded reason, and the three reasons are worth separating because they license different things. An F/A-18 dataset returned incomparable: compressibility at Mach 0.56 and 0.86 and multi-surface loading have no term at all in the owned equations, and extending the equations to force comparability is forbidden. A Cessna wind-tunnel dataset returned acquisition failed: seventeen routes attempted, eleven closed, both versions paywalled, which says nothing about that experiment's quality or its comparability. A NASA dataset returned insufficient: the measured field is published only as faired traces with no markers, merging through the core region at 600 dpi.

A negative result about the world, a negative result about a document, and a failure to obtain a document are three different things with three different licences. None of them is evidence that no anchor exists.
Solution-verification ceiling: no demonstrated asymptote

The peak of the primary quantity climbs 7 to 13 % under refinement with no asymptote shown, and the bracket that shows it moves three numerical parameters together, so the change cannot be attributed. The obvious control, separating the smoothing radius from the timestep, turned out to be unanswerable as posed rather than merely unexecuted: in the external driver the smoothing radius is a fixed multiple of the timestep and the inter-particle spacing is another, so the overlap ratio is identically that multiple and the three quantities carry only two degrees of freedom.

A later refinement ladder then found that the source-panel contribution is the dominant non-refinement error, and for both load quantities it exceeds half of the finest wake-refinement difference. A wake-only ladder cannot settle the magnitude. Source-panel refinement is the next numerical-design question, and it has not been run.
Physics ceiling: there is no viscosity anywhere

Both owned modes and the external rung are inviscid and cannot merge or diffuse a core. The coupled model's tight-element magnitudes are therefore upper-biased screening estimates, and that bias has never been bounded. Whether the inner structure merges is the question a viscous rung would exist to answer, and that rung has not been run.

This is the ceiling that most directly limits the topology result in chapter 05: one structure rather than two is a statement about an inviscid field at and above the smoothing scale, and a viscous field could differ.
Subject-aircraft ceiling: the wing itself is not resolved

The M20K dihedral convention, its reference plane and its spanwise application are unresolved, so every owned magnitude carries a declared-sensitivity envelope rather than a subject-aircraft value. Public documents cannot close it: the type certificate data sheet contains the word "dihedral" zero times across 65 pages and every M20 model, and the manufacturer's publication catalogue lists no drawing, repair, tooling or dimensional-inspection document for the type. What it does give is the name of the drawing index.

On top of this sit two receiver-side exposures that are never combined. Every owned load was computed with a flat receiver, which moves magnitudes by up to +22.8 %, and with a local strip closure carrying no aspect-ratio information, which moves every owned magnitude by about −35 %. They are never added and never averaged, because the interaction between them was measured directly at roughly half the geometry main effect and about fifty times its own numerical floor. Adding one percentage to the other would have been wrong by an amount comparable to one of them.

One further result of the closure work is worth stating because it cuts the other way. When the receiver closure was changed across 1131 cases and 82 739 receiver evaluations, every owned magnitude fell by about 35 % and every ranking, sign, zero crossing, peak station, attribution share and ratio survived exactly, with no withdrawals. That survival has a structural reason: for an elliptic chord the lifting-line modes decouple, so the change is exactly a common rescale for any incident field. It does not generalise to a rectangular or tapered chord, which is what the external solver ran.

What is still open, and what would move it

Unresolved

None of the following is planned work. Each item names what would advance it, so it is possible to see which ones unblock a rung and which only refine a number inside one.

The starboard asymmetry, which nothing on the candidate list explains

Two geometrically identical gaps, symmetric controls, and a processing-order swap that reproduced the original to five significant figures including which side inverts. Only the starboard inner structure inverts. It survived every control the onset study ran, and no candidate currently on the list accounts for it.

What would open a rung rather than refine a number

Source-panel refinement, which the admissible ladder identified as the dominant non-refinement error and could not settle from the wake side, so it is the next numerical-design question and it has not been run. A viscous rung, still required to settle whether the inner structure merges. And a measurement anchor that fits the owned equations' regime: unswept, incompressible, single-surface, high aspect ratio, with a published generator lift state and per-point moment uncertainty.

The one that a tape measure would close

The highest-leverage geometry unknown on the whole program is whether a printed 5.5° of dihedral means 5.5° per side or 5.5° total across both outward slopes. It is stated nowhere, and it is worth about 5.3 % of the induced rolling moment, roughly seven times the spread between the two bounds already tested, which is itself below the 2.0 % threshold the program calls minor. Over a 4 m baseline the two hypotheses differ by 384 mm of rise against 192 mm. A laser line and a tape measure across both wings would resolve it. That measurement has not been made.

And finishing it would leave every ceiling above exactly where it is. It is a good parallel task and a bad universal next step.

Recorded, none authorised

Deriving rather than bracketing the conservation defect, which needs per-operation instrumentation inside the external solver and is out of scope. External-solver confirmation of the tip-clearance disagreement. A superposition-error bound for a long line. Exercising the span-matched method anchor. A startup-similarity study ran and returned a startup patch implementation defect, with no candidate qualified and its conditional refinement ladder not authorised.

R0, R1 and R2 are cleared, R3 has been exercised and holds a field that had to be quarantined, and R4 and R5, the two rungs that would make any of it a statement about a real aeroplane, have never been reached. Within that reach sit the bounded findings of chapter 05, a factorial that was withdrawn in full, a claim that was retracted and stayed retracted under a later favourable result, and a contrast with no verdict.

Real people fly this formation. Until the measurement rung is reached, nothing here says anything to them about spacing, position, altitude stagger or handling.

Source. A private research repository. Every number on this page is read from a committed file in that repository: the program foundation document, the evidence register, the study reports for the external-solver campaigns, the forensic outcome document, the conservation-onset and refinement studies, and the owned claim status registry.

Figures. Both drawings were made for this page from the dimensions above.

Two numbers depart from a study report: the surviving clean arm is quoted as G11 − G01 at R = 15.14 / 15.66 / 8.91, following the forensic outcome, and the startup-similarity study is described as having run, following its committed results tables.