VERIFICATION / VALIDATION REPORT

Evidence, including the inconvenient parts.

The numerical and bookkeeping tests below pass their stated criteria. The Falcon 9 CRS-5 comparison does not reproduce the published MECO state closely enough to claim vehicle-level trajectory validation. Both outcomes are shown because a validation page that hides failed fidelity is merely marketing with equations.

See measured results Read model limitations

Formal validation suites: 5 PASS
Atmosphere matrix: 9 / 9 within 0.05%
Default timestep: 0.2 s converged
Real-flight benchmark: Fidelity limitation found

01 / METHOD

Different claims need different evidence.

The project separates implementation verification from real-vehicle validation. A mathematically correct integrator can still drive an inaccurate vehicle model, and a good-looking trajectory can still be produced by broken bookkeeping. Each test therefore targets a narrower claim. The complete stdout from every validation run is preserved verbatim beneath its interpreted result, so summarized numbers can be audited against the actual executable output.

A / REFERENCEIndependent equations or tables

Atmosphere and orbital-state outputs are compared with references calculated outside the implementation under test.

Verified

B / NUMERICALIntegration behavior

RK4 is checked for conservation in a controlled two-body problem and for full-simulation timestep convergence.

Verified

C / SYSTEMMass and event integrity

Propellant depletion, burnout, separation, stage activation and final mass are checked against an analytical ledger.

Verified

D / EMPIRICALPublic flight benchmark

A Falcon 9 CRS-5 proxy is compared with NASA-published flight milestones. The MECO mismatch is treated as a limitation, not a pass.

Limitation found

Claim boundary

Passing these checks supports correctness of the tested implementation under its stated assumptions. It does not establish operational flight accuracy, certify the generic aerodynamic model, or turn approximate Falcon 9 inputs into flight telemetry.

02 / RESULTS

Validation status at a glance.

All values below are measured from the validation scripts included in the repository. Tolerances marked “project criterion” are acceptance thresholds chosen for implementation verification, not published uncertainty bounds.

Evidence	Measured result	Criterion	Interpretation	Status
U.S. Standard Atmosphere matrix	Max P error 0.013345%; max ρ error 0.012264%	≤ 0.05%	9 geometric-altitude reference cases within reporting tolerance	Pass
200 km circular orbit	7.788487985 km/s; 5301.004602 s; e = 0	Metric-specific numerical tolerances	Analytical two-body state reproduced to floating-point scale	Pass
RK4 energy conservation	Max energy drift 1.473889940×10⁻¹³%	≤ 0.01%	Negligible drift over the controlled 100-step test	Pass
Full-simulation timestep convergence	Worst 0.2 s vs 0.05 s difference 5.689672×10⁻⁴%	≤ 0.01%	Default 0.2 s step is converged for the tested example trajectory	Pass
Serial staging / mass ledger	26 / 26 checks; 494,000 kg → 5,000 kg	Exact logic + numerical bookkeeping tolerances	Mass and discrete events close correctly	Pass
Falcon 9 CRS-5 public-flight benchmark	MECO altitude error 10.87%; MECO Mach error 39.34%	Descriptive benchmark, no invented pass threshold	Current Falcon proxy is not vehicle-level trajectory validated	Limitation

03 / ATMOSPHERE

U.S. Standard Atmosphere reference matrix.

physics/atmosphere.js was compared against an independent U.S. Standard Atmosphere 1976 reference at nine geometric altitudes spanning sea level through the model's 86 km boundary.

RESULTPASS

Cases9 / 9

Max pressure error0.013345%

Max density error0.012264%

Reporting tolerance0.05%

Geometric altitude	Geopotential used	Model T	P error	ρ error	Status
0 km	0.000 km	288.150 K	0.000000%	0.001070%	Pass
5 km	4.996 km	255.676 K	0.000673%	0.000398%	Pass
11 km	10.981 km	216.774 K	0.001601%	0.000531%	Pass
20 km	19.937 km	216.650 K	0.003112%	0.002043%	Pass
32 km	31.840 km	228.490 K	0.005068%	0.003998%	Pass
47 km	46.655 km	269.684 K	0.007249%	0.006177%	Pass
51 km	50.594 km	270.650 K	0.007781%	0.006707%	Pass
71 km	70.216 km	216.846 K	0.010724%	0.009729%	Pass
86 km	84.852 km	186.946 K	0.013345%	0.012264%	Pass

WHY THESE ALTITUDES?

They cross every implemented lower-atmosphere region.

The set includes sea level, major layer boundaries and high-altitude points through 86 km. It is deliberately distributed across the piecewise lapse-rate model instead of checking only the easy sea-level anchor.

WHY GEOMETRIC + GEOPOTENTIAL?

The API takes geometric altitude; the standard equations use geopotential altitude.

The validation records both conventions so a reader can see the conversion rather than comparing an 86 km geometric input against an 86 km geopotential table row and manufacturing an error.

WHY 0.05%?

A project reporting threshold, not a USSA uncertainty claim.

The 0.05% criterion is intentionally much tighter than differences that would indicate a wrong layer, lapse rate or unit conversion while still allowing small differences from rounded constants/reference tabulation. The measured worst error is roughly one quarter of that limit.

ReferenceU.S. Standard Atmosphere, 1976 · NASA NTRS ↗validation/atmosphere-validation.js · source repository

RAW EXECUTION EVIDENCE View full terminal output · Atmosphere validation 68 lines

npm run validate:atmosphere Open raw .txt ↗


U.S. STANDARD ATMOSPHERE 1976 VALIDATION
========================================
Input altitude convention: GEOMETRIC altitude
Model under test: physics/atmosphere.js

    0 km geometric  ->  0.000 km geopotential
Layer: Troposphere
  Temperature  model=288.150000 K  ref=288.150000 K  error=0.000e+0%
  Pressure     model=101325.000000000 Pa  ref=101325.000000000 Pa  error=0.000e+0%
  Density      model=1.225012266 kg/m^3  ref=1.224999156 kg/m^3  error=0.001070%

    5 km geometric  ->  4.996 km geopotential
Layer: Troposphere
  Temperature  model=255.675543 K  ref=255.675543 K  error=7.686e-11%
  Pressure     model=54047.922628365 Pa  ref=54048.286146300 Pa  error=0.000673%
  Density      model=0.736431349 kg/m^3  ref=0.736428421 kg/m^3  error=0.000398%

   11 km geometric  ->  10.981 km geopotential
Layer: Troposphere
  Temperature  model=216.773513 K  ref=216.773513 K  error=2.101e-10%
  Pressure     model=22699.597314215 Pa  ref=22699.960739200 Pa  error=0.001601%
  Density      model=0.364799628 kg/m^3  ref=0.364801564 kg/m^3  error=0.000531%

   20 km geometric  ->  19.937 km geopotential
Layer: Tropopause / Isothermal Layer
  Temperature  model=216.650000 K  ref=216.650000 K  error=1.312e-14%
  Pressure     model=5529.139796870 Pa  ref=5529.311892900 Pa  error=0.003112%
  Density      model=0.088908099 kg/m^3  ref=0.088909916 kg/m^3  error=0.002043%

   32 km geometric  ->  31.840 km geopotential
Layer: Lower Stratosphere
  Temperature  model=228.489719 K  ref=228.489719 K  error=6.722e-11%
  Pressure     model=889.019356621 Pa  ref=889.064416900 Pa  error=0.005068%
  Density      model=0.013554609 kg/m^3  ref=0.013555151 kg/m^3  error=0.003998%

   47 km geometric  ->  46.655 km geopotential
Layer: Upper Stratosphere
  Temperature  model=269.684131 K  ref=269.684131 K  error=1.388e-10%
  Pressure     model=115.842715569 Pa  ref=115.851114000 Pa  error=0.007249%
  Density      model=0.001496428 kg/m^3  ref=0.001496520 kg/m^3  error=0.006177%

   51 km geometric  ->  50.594 km geopotential
Layer: Stratopause / Isothermal Layer
  Temperature  model=270.650000 K  ref=270.650000 K  error=0.000e+0%
  Pressure     model=70.452526483 Pa  ref=70.458009000 Pa  error=0.007781%
  Density      model=9.068406702e-4 kg/m^3  ref=9.069015000e-4 kg/m^3  error=0.006707%

   71 km geometric  ->  70.216 km geopotential
Layer: Lower Mesosphere
  Temperature  model=216.845911 K  ref=216.845911 K  error=1.086e-10%
  Pressure     model=4.479082590 Pa  ref=4.479563000 Pa  error=0.010724%
  Density      model=7.195819860e-5 kg/m^3  ref=7.196520000e-5 kg/m^3  error=0.009729%

   86 km geometric  ->  84.852 km geopotential
Layer: Upper Mesosphere
  Temperature  model=186.945908 K  ref=186.945908 K  error=1.008e-10%
  Pressure     model=0.373330471 Pa  ref=0.373380300 Pa  error=0.013345%
  Density      model=6.956966665e-6 kg/m^3  ref=6.957820000e-6 kg/m^3  error=0.012264%

SUMMARY
=======
Cases tested: 9
Maximum temperature error: 2.101e-10%
Maximum pressure error:    0.013345%
Maximum density error:     0.012264%
Reporting tolerance:       0.05%
Result: PASS

04 / ORBIT

200 km circular-orbit analytical benchmark.

The orbit module was given a mathematically circular planar state and compared with independently evaluated two-body equations using the same project Earth convention.

RESULTPASS

Circular speed7.788487985 km/s

Period5301.004602 s

Eccentricity0

Periapsis / apoapsis200 / 200 km

Radiusr = Rₑ + h

Circular speedv = √(μ / r)

PeriodT = 2π√(r³ / μ)

Specific energyε = −μ / (2r)

REFERENCE CONVENTIONS

Test altitude: 200 km
Earth radius: 6,371,000 m
Earth μ: 3.986004418×10¹⁴ m³/s²
Geometry: Spherical Earth, two-body planar orbit

The 200 km case is a simple low-Earth-orbit benchmark with a closed-form circular solution. It tests orbit.js implementation consistency, not the realism of a perfect spherical Earth at 200 km.

WHY THESE CONSTANTS?

The project radius is a rounded spherical mean-Earth convention. JPL lists a mean Earth radius of about 6371.0084 km. The gravitational parameter matches the IERS two-body convention of 398600.4418 km³/s². Because the validation intentionally uses the same Earth convention as the module, it verifies the orbital equations rather than independently validating the constants themselves.

JPL physical parameters ↗IERS Chapter 6 ↗

Quantity	Tolerance	Measured error	Why this tolerance?
Circular speed	1×10⁻⁶ m/s	0 m/s	Micro-metre-per-second numerical guardrail; not physical uncertainty.
Orbital period	1×10⁻⁶ s	0 s	Detects implementation disagreement while allowing floating-point arithmetic.
Specific energy	1×10⁻⁹% relative	1.228242×10⁻¹⁴%	Relative error avoids tying the criterion to the large absolute energy magnitude.
Angular momentum	1×10⁻⁹% relative	0%	Scale-independent comparison for a large-valued conserved quantity.
Eccentricity	1×10⁻¹⁰ absolute	0	The reference is zero, so relative error is not meaningful.
Semi-major axis / apses	1×10⁻³ m	≤ 9.313226×10⁻¹⁰ m	A 1 mm software tolerance is tiny at orbital scale yet comfortably above binary floating-point residue.

Numerical noteECMAScript Number uses IEEE-754 binary64 · ECMA-262 ↗validation/orbit-validation.js · source repository

RAW EXECUTION EVIDENCE View full terminal output · 200 km circular-orbit validation 98 lines

npm run validate:orbit Open raw .txt ↗

200 KM CIRCULAR-ORBIT VALIDATION
================================
Model under test: physics/orbit.js
Reference method: independent two-body circular-orbit equations
Reference Earth radius: 6,371,000 m
Reference Earth mu:     3.986004418e+14 m^3/s^2
Test altitude:          200 km

REFERENCE STATE
---------------
Radius:                 6571000.000 m
Circular speed:         7788.487984973 m/s  (7.788487985 km/s)
Orbital period:         5301.004602323 s  (88.350077 min)
Specific energy:        -30330272.546036 J/kg

MODEL OUTPUT
------------
Radius:                 6571000.000000 m
Altitude:               200000.000000 m
Speed:                  7788.487984973 m/s
Circular speed:         7788.487984973 m/s
Radial velocity:        0.000000e+0 m/s
Tangential velocity:    7788.487984973 m/s
Specific energy:        -30330272.546036 J/kg
Angular momentum:       51178154549.258614 m^2/s
Eccentricity:           0.000000e+0
Semi-major axis:        6571000.000000 m
Periapsis altitude:     200000.000000 m
Apoapsis altitude:      200000.000000 m
Orbital period:         5301.004602323 s
Basic classification:   BOUND_ORBIT
Target classification:  TARGET_ORBIT

CHECKS
------
PASS  Circular speed
      model=7788.487984973157
      ref=7788.487984973157
      error=0.000000e+0 m/s
      tolerance=1.000000e-6 m/s
PASS  Orbital period
      model=5301.004602322611
      ref=5301.004602322611
      error=0.000000e+0 s
      tolerance=1.000000e-6 s
PASS  Specific orbital energy
      model=-30330272.546035606414
      ref=-30330272.546035610139
      error=1.228242e-14 % relative
      tolerance=1.000000e-9 % relative
PASS  Specific angular momentum
      model=51178154549.258613586426
      ref=51178154549.258613586426
      error=0.000000e+0 % relative
      tolerance=1.000000e-9 % relative
PASS  Eccentricity
      model=0.000000000000
      ref=0.000000000000
      error=0.000000e+0 absolute
      tolerance=1.000000e-10 absolute
PASS  Semi-major axis
      model=6571000.000000000931
      ref=6571000.000000000000
      error=9.313226e-10 m
      tolerance=1.000000e-3 m
PASS  Periapsis altitude
      model=200000.000000000000
      ref=200000.000000000000
      error=0.000000e+0 m
      tolerance=1.000000e-3 m
PASS  Apoapsis altitude
      model=200000.000000000931
      ref=200000.000000000000
      error=9.313226e-10 m
      tolerance=1.000000e-3 m
PASS  Radial velocity
      model=0.000000000000
      ref=0.000000000000
      error=0.000000e+0 m/s
      tolerance=1.000000e-9 m/s
PASS  Tangential velocity
      model=7788.487984973157
      ref=7788.487984973157
      error=0.000000e+0 m/s
      tolerance=1.000000e-6 m/s
PASS  Basic classification = BOUND_ORBIT
PASS  200 km target orbit recognized

SUMMARY
=======
Reference circular speed: 7.788487985 km/s
Model circular speed:     7.788487985 km/s
Reference orbital period: 5301.004602 s
Model orbital period:     5301.004602 s
Eccentricity:             0.000000e+0
Periapsis altitude:       200.000000000 km
Apoapsis altitude:        200.000000000 km
Result: PASS

05 / RK4 ENERGY

Conservation in an isolated two-body problem.

The RK4 integrator was separated from thrust, drag, staging and mass flow. In ideal Newtonian two-body motion, specific mechanical energy should remain constant, so any drift is numerical rather than physical.

RESULTPASS

Step1 s

Steps100

Max energy drift1.473889940×10⁻¹³%

Allowed0.01%

0 s25 s50 s75 s100 s

07.37×10⁻¹⁴%2.46×10⁻¹⁴%2.46×10⁻¹⁴%2.46×10⁻¹⁴%

WHY ENERGY?

It should be conserved by the physical test problem.

With only conservative two-body gravity active, specific mechanical energy provides a direct diagnostic for integration error. There is no thrust or drag available to legitimately change it.

WHY 1 s × 100?

A short, deterministic implementation check.

The test is deliberately simple enough that a broken RK4 implementation cannot hide behind staging or guidance. Longer-duration and step-size behavior is addressed separately by the convergence study.

WHY 0.01%?

A conservative software guardrail.

The threshold was chosen to make materially drifting integration fail loudly. It is not an uncertainty claim for orbital prediction. The measured maximum drift is about 6.8×10¹⁰ times smaller than the threshold.

Secondary check: specific angular-momentum drift = 1.490752177×10⁻¹⁴%. Maximum energy drift occurred at step 20, so acceptance uses the maximum over the history rather than only the conveniently small final drift.

Implementationphysics/integrator.js · classical fixed-step RK4validation/energy-validation.js · source repository

RAW EXECUTION EVIDENCE View full terminal output · RK4 energy-conservation validation 50 lines

npm run validate:energy Open raw .txt ↗

RK4 TWO-BODY ENERGY-CONSERVATION VALIDATION
===========================================
Model under test: physics/integrator.js (RK4)
Force field: independent Newtonian two-body gravity
Atmosphere: OFF
Drag: OFF
Thrust: OFF
Mass flow: OFF

Earth radius used:       6,371,000 m
Earth mu used:           3.986004418e+14 m^3/s^2
Initial altitude:        200 km
Initial orbital speed:   7.788487985 km/s
RK4 time step:           1 s
RK4 steps:               100
Propagation duration:    100 s

ENERGY
------
Initial specific energy: -30330272.546035606414 J/kg
Final specific energy:   -30330272.546035613865 J/kg
Final difference:        -7.450580597e-9 J/kg
Final relative drift:    2.456483233e-14 %
Maximum relative drift:  1.473889940e-13 %
Maximum drift at step:   20 (t=20 s)

SECONDARY CONSERVATION CHECK
----------------------------
Initial angular momentum: 51178154549.258613586 m^2/s
Final angular momentum:   51178154549.258605957 m^2/s
Relative drift:           1.490752177e-14 %

SELECTED HISTORY
----------------
Step   0  t=  0 s  energy=-30330272.546035606 J/kg  drift=0.000000e+0 %
Step  25  t= 25 s  energy=-30330272.546035629 J/kg  drift=7.369450e-14 %
Step  50  t= 50 s  energy=-30330272.546035614 J/kg  drift=2.456483e-14 %
Step  75  t= 75 s  energy=-30330272.546035614 J/kg  drift=2.456483e-14 %
Step 100  t=100 s  energy=-30330272.546035614 J/kg  drift=2.456483e-14 %

ACCEPTANCE CRITERION
--------------------
Maximum energy drift must remain <= 0.01% over all 100 RK4 steps.
This tolerance tests numerical implementation/step-size behavior; it is not a physical-accuracy uncertainty for the real Earth.

SUMMARY
=======
Maximum energy drift:    1.473889940e-13 %
Allowed drift:           0.01000 %
Result:                  PASS

06 / CONVERGENCE

Does the full simulation settle as the timestep shrinks?

The same two-stage vehicle was simulated at five RK4 step sizes. Vehicle, guidance, aerodynamics, atmosphere and mission inputs were held constant; only solver.dt changed.

RESULTPASS

Timesteps1 → 0.05 s

Project default0.2 s

Fine comparison0.05 s

Worst default-vs-fine5.689672×10⁻⁴%

dt	Steps	Max altitude	Max speed	Mach @ 50 km	Max-Q	Apoapsis
1.00 s	356	293.242437 km	10.023551042 km/s	4.2654755	40.541461 kPa	28,621.120496 km
0.50 s	712	293.242539 km	10.023550481 km/s	4.2654354	40.553843 kPa	28,621.099992 km
0.20 s	1,778	293.242540 km	10.023550438 km/s	4.2654275	40.553665 kPa	28,621.098118 km
0.10 s	3,556	293.242536 km	10.023550473 km/s	4.2654273	40.553830 kPa	28,621.099529 km
0.05 s	7,112	293.242537 km	10.023550482 km/s	4.2654272	40.553896 kPa	28,621.099957 km

Max altitude8.89×10⁻⁷%

Max speed4.44×10⁻⁷%

Mach @ 50 km6.93×10⁻⁶%

Max-Q5.69×10⁻⁴%

Apoapsis6.43×10⁻⁶%

WHY 0.05 s AS REFERENCE?

It is a finer numerical comparison, not “truth.”

The 0.05 s run is four times finer than the project default and twenty times finer than the coarsest sweep. Convergence means successive refinements settle toward similar outputs.

WHY 0.01%?

A project-level numerical convergence criterion.

Selected continuous outputs from the 0.2 s default must remain within 0.01% of the 0.05 s run. This says timestep choice is not materially moving the tested outputs; it says nothing about real-world model accuracy.

WHY MACH @ 50 KM?

Maximum Mach is sampling-sensitive at the 86 km atmosphere boundary.

Above 86 km the current atmosphere model intentionally uses a vacuum approximation and Mach becomes undefined. A fixed 50 km interpolation is therefore a cleaner convergence metric than whichever sample happens to be the final sub-86-km point.

Stage 1 and Stage 2 burnout times differed from the 0.05 s run by only 1.48×10⁻¹¹ s and 4.27×10⁻¹¹ s. Final mass remained exactly 5,000 kg and the orbit classification remained BOUND_ORBIT.

ImplementationSame integrated vehicle; only solver.dt changedvalidation/convergence-validation.js · source repository

RAW EXECUTION EVIDENCE View full terminal output · Full-simulation timestep-convergence validation 110 lines

npm run validate:convergence Open raw .txt ↗

RK4 FULL-SIMULATION TIMESTEP-CONVERGENCE VALIDATION
===================================================
Model under test: integrated Analyzer simulation stack
Integrator:       classical RK4
Vehicle:          Example Two-Stage Vehicle
Guidance:         built-in generic assumed profile
Cd model:         built-in generic assumed Mach-Cd table
Only quantity changed between runs: solver.dt

Timesteps tested: 1, 0.5, 0.2, 0.1, 0.05 s
Project default:  0.2 s
Fine reference:   0.05 s

RUN RESULTS
-----------
 dt(s)   steps    maxAlt(km)   maxSpeed(km/s)   Mach@50km   maxQ(kPa)   peri(km)      apo(km)        ecc
  1.00     356    293.242437      10.023551042   4.2654755    40.541461   261.698795   28621.120496  0.681310386
  0.50     712    293.242539      10.023550481   4.2654354    40.553843   261.698852   28621.099992  0.681310227
  0.20    1778    293.242540      10.023550438   4.2654275    40.553665   261.698850   28621.098118  0.681310213
  0.10    3556    293.242536      10.023550473   4.2654273    40.553830   261.698848   28621.099529  0.681310223
  0.05    7112    293.242537      10.023550482   4.2654272    40.553896   261.698849   28621.099957  0.681310227

STAGE BURNOUT TIMES
-------------------
dt=1.00 s  S1=148.774056098 s  S2=355.536908098 s
dt=0.50 s  S1=148.774056098 s  S2=355.536908098 s
dt=0.20 s  S1=148.774056098 s  S2=355.536908098 s
dt=0.10 s  S1=148.774056098 s  S2=355.536908098 s
dt=0.05 s  S1=148.774056098 s  S2=355.536908098 s

MAXIMUM MACH DIAGNOSTIC (NOT AN ACCEPTANCE METRIC)
-------------------------------------------------
dt=1.00 s  maximum reported Mach=8.584980622
dt=0.50 s  maximum reported Mach=8.702895014
dt=0.20 s  maximum reported Mach=8.679156686
dt=0.10 s  maximum reported Mach=8.702894779
dt=0.05 s  maximum reported Mach=8.714793018
Maximum reported Mach is intentionally excluded from PASS/FAIL because Mach becomes undefined above the 86 km atmosphere-model boundary; the last sub-86-km sample therefore shifts with timestep.
Mach at a fixed 50 km geometric altitude is used instead.

DEFAULT dt = 0.2 s VS FINE dt = 0.05 s
---------------------------------------
PASS  Maximum altitude
      dt=0.2: 293242.539528143592
      dt=0.05:293242.536922068335
      relative difference=8.887098e-7 %
      tolerance=0.01000 %
PASS  Maximum speed
      dt=0.2: 10023.550437664569
      dt=0.05:10023.550482190221
      relative difference=4.442104e-7 %
      tolerance=0.01000 %
PASS  Mach at 50 km geometric altitude
      dt=0.2: 4.265427478088
      dt=0.05:4.265427182516
      relative difference=6.929494e-6 %
      tolerance=0.01000 %
PASS  Maximum dynamic pressure
      dt=0.2: 40553.664829958878
      dt=0.05:40553.895568308675
      relative difference=5.689672e-4 %
      tolerance=0.01000 %
PASS  Periapsis altitude
      dt=0.2: 261698.849793746136
      dt=0.05:261698.849114250392
      relative difference=2.596480e-7 %
      tolerance=0.01000 %
PASS  Apoapsis altitude
      dt=0.2: 28621098.117783337831
      dt=0.05:28621099.956778533757
      relative difference=6.425313e-6 %
      tolerance=0.01000 %
PASS  Eccentricity
      dt=0.2: 0.681310212518
      dt=0.05:0.681310226625
      relative difference=2.070606e-6 %
      tolerance=0.01000 %
PASS  Stage 1 burnout time
      dt=0.2: 148.774056097564 s
      dt=0.05:148.774056097549 s
      difference=1.475087e-11 s
      tolerance=0.200 s
PASS  Stage 2 burnout time
      dt=0.2: 355.536908097549 s
      dt=0.05:355.536908097592 s
      difference=4.274625e-11 s
      tolerance=0.200 s
PASS  Final mass consistency
      dt=0.2: 5000.000000000 kg
      dt=0.05:5000.000000000 kg
      difference=0.000000e+0 kg
      tolerance=1.000000e-6 kg
PASS  Orbit classification unchanged (BOUND_ORBIT)
PASS  Termination reason unchanged (allStagesComplete)

ACCEPTANCE CRITERION
--------------------
For the project default dt=0.2 s, selected continuous outputs must differ by <= 0.01% from the dt=0.05 s run.
Stage-burnout times must differ by <= 0.2 s, final mass must agree within 0.000001 kg, and classifications must remain unchanged.
These are numerical-convergence criteria only. They do not measure real-world launch accuracy.

SUMMARY
=======
Default timestep:                   0.2 s
Fine comparison timestep:           0.05 s
Worst default-vs-fine difference:   5.689672e-4 % (Maximum dynamic pressure)
Allowed continuous difference:      0.01000 %
Default integration steps:          1778
Fine integration steps:             7112
Result:                             PASS

07 / STAGING

Mass accounting and discrete-event integrity.

The staging implementation is checked twice: first against a controlled analytical ledger, then through the complete Analyzer event sequence. This prevents two compensating bookkeeping mistakes from receiving a ceremonial green check.

RESULTPASS

Checks26 / 26

Initial mass494,000 kg

Usable propellant460,000 kg

Final mass5,000 kg

INITIAL STACK494,000 kgPayload + both stages

−400,000 kgStage 1 usable propellant

S1 BURNOUT94,000 kgt = 148.774056 s

−25,000 kgStage 1 separation

POST-S169,000 kgt = 150.774056 s

−60,000 kgStage 2 usable propellant

S2 BURNOUT9,000 kgt = 355.536908 s

−4,000 kgStage 2 separation

PAYLOAD5,000 kgallStagesComplete

EVENT CHECKS

Stage 1 burnout → separation → Stage 2 burnout → separation
Configured 2 s Stage 1 coast delay reproduced
Zero-second Stage 2 delay reproduced
Stage activation flags and final completion state agree
Idealized separation preserves position and velocity

EDGE CASE

Residual propellant was tested separately.

A synthetic stage with 500 kg residual propellant verified that only usable propellant is burned; dry mass plus residual remains at burnout and is discarded together at separation.

The tiny mass/time tolerances are software-integrity thresholds. They do not imply milligram knowledge of a real launch vehicle or nanosecond-accurate physical separation.

Check type	Acceptance	Meaning
Mass ledger	≤ 1×10⁻⁶ kg absolute	Allows floating-point residue while making bookkeeping loss/gain fail immediately.
Configured event delay	≤ 1×10⁻⁸ s	Verifies event scheduling, not real separation timing uncertainty.
Separation state continuity	≤ 1×10⁻¹² per component	Tests the current idealization that separation changes mass/configuration only.
Order / flags	Exact logical agreement	Discrete state transitions have no meaningful “almost correct” interpretation.

Implementationphysics/staging.js + analyzer/simulation.jsvalidation/staging-validation.js · source repository

RAW EXECUTION EVIDENCE View full terminal output · Serial-staging mass & event validation 138 lines

npm run validate:staging Open raw .txt ↗

SERIAL-STAGING MASS & EVENT VALIDATION
======================================
Models under test: physics/staging.js + analyzer/simulation.js event orchestration
Vehicle:           Example Two-Stage Vehicle
Integrator:        classical RK4, dt=0.2 s (integrated portion)
Purpose:           mass-ledger and discrete-event integrity

ANALYTICAL MASS LEDGER
----------------------
Payload mass:                    5000.000 kg
Stage 1 dry mass:                25000.000 kg
Stage 1 usable propellant:       400000.000 kg
Stage 2 dry mass:                4000.000 kg
Stage 2 usable propellant:       60000.000 kg
Initial vehicle mass:            494000.000 kg
Stage 1 burnout mass:            94000.000 kg
After Stage 1 separation:        69000.000 kg
Stage 2 burnout mass:            9000.000 kg
Final mass after Stage 2 sep.:   5000.000 kg

INTEGRATED EVENT TIMELINE
-------------------------
t=148.774056097564 s  STAGE 1 BURNOUT
t=150.774056097564 s  STAGE 1 SEPARATION  discarded=25000.000000 kg
t=355.536908097549 s  STAGE 2 BURNOUT
t=355.536908097549 s  STAGE 2 SEPARATION  discarded=4000.000000 kg

CHECKS
------
PASS  Initial vehicle mass matches analytical stack sum
      model=494000.000000000 kg
      expected=494000.000000000 kg
      tolerance=1.000000e-6 kg
PASS  Stage 1 usable propellant bookkeeping
      runtime=400000.000000000 kg
      expected=400000.000000000 kg
PASS  Stage 1 consumes exactly its usable propellant
      consumed=400000.000000000 kg
      expected=400000.000000000 kg
PASS  Stage 1 is marked burned out with zero usable propellant
      remaining=0.000000e+0 kg
PASS  Stage 1 burnout stack mass matches analytical ledger
      model=94000.000000000 kg
      expected=94000.000000000 kg
PASS  Stage 1 separation discards dry + residual mass
      discarded=25000.000000000 kg
      expected=25000.000000000 kg
PASS  Stage 1 post-separation mass matches analytical ledger
      model=69000.000000000 kg
      expected=69000.000000000 kg
PASS  Idealized Stage 1 separation preserves position and velocity
      state continuity tolerance=1.000000e-12
PASS  Stage 2 becomes active after Stage 1 separation
      activeStageIndex=1
PASS  Stage 2 consumes exactly its usable propellant
      consumed=60000.000000000 kg
      expected=60000.000000000 kg
PASS  Stage 2 burnout stack mass matches analytical ledger
      model=9000.000000000 kg
      expected=9000.000000000 kg
PASS  Stage 2 separation discards dry + residual mass
      discarded=4000.000000000 kg
      expected=4000.000000000 kg
PASS  Final controlled mass equals analytical payload-only mass
      model=5000.000000000 kg
      expected=5000.000000000 kg
PASS  Final controlled runtime marks all stages complete
      allStagesComplete=true
PASS  Residual propellant remains in burnout mass and is discarded at separation
      usable propellant burned=9500.000000 kg (expected 9500)
      burnout mass=1600.000000 kg (payload + dry + residual = 1600)
      discarded mass=1500.000000 kg (dry + residual = 1500)
      final payload mass=100.000000 kg (expected 100)
PASS  Integrated staging event order is correct
      observed=stageBurnout:0 -> stageSeparation:0 -> stageBurnout:1 -> stageSeparation:1
      expected=stageBurnout:0 -> stageSeparation:0 -> stageBurnout:1 -> stageSeparation:1
PASS  Integrated Stage 1 burnout mass matches analytical ledger
      history mass=94000.000000000 kg
      expected=94000.000000000 kg
PASS  Stage 1 coast to separation consumes no additional propellant
      burnout mass=94000.000000000 kg
      pre-separation mass=94000.000000000 kg
      configured coast delay=2.000 s
PASS  Integrated Stage 2 burnout mass matches analytical ledger
      history mass=9000.000000000 kg
      expected=9000.000000000 kg
PASS  Integrated Stage 1 separation delay matches configuration
      burnout=148.774056097564 s
      separation=150.774056097564 s
      observed delay=2.000000000000 s
      configured delay=2.000000000000 s
      difference=1.136868e-13 s
      tolerance=1.000000e-8 s
PASS  Integrated Stage 2 separation delay matches configuration
      burnout=355.536908097549 s
      separation=355.536908097549 s
      observed delay=0.000000000000 s
      configured delay=0.000000000000 s
      difference=0.000000e+0 s
      tolerance=1.000000e-8 s
PASS  Integrated separation events report correct discarded masses
      Stage 1 event=25000.000000000 kg, expected=25000.000000000 kg
      Stage 2 event=4000.000000000 kg, expected=4000.000000000 kg
PASS  Integrated simulation final mass equals analytical final mass
      model=5000.000000000 kg
      expected=5000.000000000 kg
      tolerance=1.000000e-6 kg
PASS  Integrated total propellant consumption closes the mass ledger
      initial mass=494000.000000000 kg
      final mass=5000.000000000 kg
      discarded dry/residual mass=29000.000000000 kg
      inferred propellant burned=460000.000000000 kg
      expected usable propellant=460000.000000000 kg
PASS  Integrated final staging state marks all stages burned out and separated
      allStagesComplete=true
      separatedStageCount=2
PASS  Integrated simulation terminates because all stages complete
      terminationReason=allStagesComplete

ACCEPTANCE CRITERIA
-------------------
Mass ledger agreement:          <= 1.000000e-6 kg absolute difference
Configured event-delay error:   <= 1.000000e-8 s
Separation state continuity:    <= 1.000000e-12 absolute component difference
Event order / stage flags:      exact logical agreement

These are software bookkeeping/event-integrity criteria. They do not imply that real launch-vehicle stage-separation dynamics are modeled to these tolerances.

SUMMARY
=======
Checks passed:                   26/26
Initial mass:                    494000.000 kg
Total usable propellant:         460000.000 kg
Total discarded dry/residual:   29000.000 kg
Final mass:                      5000.000 kg
Stage separations:               2
Termination reason:              allStagesComplete
Result:                          PASS

08 / REAL FLIGHT

Falcon 9 CRS-5 public-flight benchmark.

This is the most important non-green result on the page. An approximate Falcon 9 v1.1 first-stage proxy is compared with rounded milestones published in NASA's CRS-5 mission press kit. It is a benchmark of the current model's vehicle fidelity, not a claim that the proxy reconstructs SpaceX telemetry.

Mach-1 timing0.878895%

LVA 69.384774 s
NASA ≈ 70 s

Close cross-check, not proof

MECO altitude10.868338%

LVA 71.305329 km
NASA ≈ 80 km

Substantial disagreement

MECO Mach39.343261%

LVA 6.065674
NASA ≈ 10

Major fidelity failure

EARLY ASCENT

Why is Mach-1 timing so close?

The proxy's approximate thrust-to-mass ratio, atmosphere and generic drag produce an early acceleration history that crosses Mach 1 near the rounded NASA timing. That is encouraging, but the reference itself is approximate and several imperfect assumptions can cancel. It should therefore be treated as a cross-check, not precise Falcon 9 validation.

MECO ALTITUDE

Why is the trajectory about 8.7 km low?

The altitude disagreement accumulates over the full first-stage ascent. The proxy does not contain CRS-5's exact mass loading, pitch program, throttle history or Falcon-specific aerodynamic coefficients. Those assumptions control how thrust is divided between vertical and horizontal acceleration and how the vehicle's acceleration changes as propellant is consumed.

Missing Earth rotation and launch-site geometry also change the inertial state, while the simplified propulsion model interpolates between thrust anchors rather than reproducing actual Merlin flight performance. The result is therefore best interpreted as a coupled vehicle-model fidelity error, not a single atmosphere or RK4 failure.

MECO MACH

Why is a 39.34% Mach error especially serious?

At 157 s the simulator reports only 1.787 km/s and Mach 6.066, while the NASA press kit describes the real vehicle as approximately Mach 10. The simulated vehicle is also lower, so its local speed of sound is different, but that atmospheric difference is not remotely large enough to turn Mach 6.1 into Mach 10.

The dominant signal is therefore that the proxy has accumulated too little vehicle speed by MECO. Likely contributors are the approximate mass state, assumed specific impulse and mass-flow history, simplified thrust evolution, generic guidance and generic aerodynamics. Because these effects interact, the benchmark does not assign fabricated percentages of the error to individual causes.

ERROR GROWTH

Why can early agreement coexist with late disagreement?

Trajectory error is cumulative. Small differences in mass, thrust, drag or thrust direction alter acceleration; that changes position and velocity; those changed states then alter the next timestep's gravity, atmosphere, drag and guidance conditions. Reasonable agreement at 70 s therefore does not guarantee agreement at 157 s.

ANumerical error

RK4 discretization / timestep effects. Independently tested above and tiny for the example convergence case.

BPhysics-model approximation

2D motion, generic Cd, simplified propulsion, assumed guidance and omitted rotation effects.

CVehicle-data uncertainty

Approximate Falcon masses, Isp, mission loading, throttle schedule and inaccessible flight-specific data.

PUBLISHED FLIGHT REFERENCES

Supersonic: ≈ 70 s
MECO: ≈ 157 s
MECO altitude: ≈ 80 km
MECO Mach: ≈ Mach 10

The NASA press kit reports Mach, not an explicit MECO velocity. The benchmark therefore compares Mach directly instead of inventing a “NASA velocity” using an arbitrary speed-of-sound convention.

NASA SpaceX CRS-5 mission press kit ↗

PROXY INPUT BOUNDARY

Sourced anchors: Approx. gross liftoff weight, 12 ft diameter, Merlin thrust anchors
Model assumptions: 400,000 kg aggregate usable propellant, 282/311 s Isp, generic Cd, generic open-loop guidance, 100% throttle

These assumptions are intentionally separated from sourced values. Public version-level documents are not sufficient to reconstruct an exact CRS-5 stage mass and flight-control history.

FAA Falcon Program environmental assessment ↗

Supported by this benchmark

The current proxy crosses Mach 1 near the rounded CRS-5 reference.
At 157 s it is materially low in altitude and very low in Mach.
The present Falcon proxy should not be described as vehicle-level trajectory validated.

Not supported by this benchmark

Assigning an exact fraction of the 39.34% mismatch to guidance, drag, mass, Isp or any single cause.
Claiming the atmosphere, RK4 or staging model is wrong solely because the Falcon proxy disagrees.
Treating rounded press-kit milestones as precision telemetry.

Benchmark codeNo PASS/FAIL threshold is invented for the real-flight comparison.validation/falcon9-validation.js · source repository

RAW EXECUTION EVIDENCE View full terminal output · Falcon 9 CRS-5 public-flight benchmark 251 lines

npm run validate:falcon9 Open raw .txt ↗

FALCON 9 CRS-5 PUBLIC-FLIGHT BENCHMARK + ERROR DIAGNOSIS
========================================================
Mission:           SpaceX CRS-5
Vehicle version:   Falcon 9 v1.1
Benchmark type:    approximate aggregate first-stage-ascent proxy
Result semantics:  descriptive comparison + sensitivity diagnosis; NOT formal pass/fail

PRIMARY SOURCES
---------------
SpaceX CRS-5 Mission Press Kit (NASA)
https://www.nasa.gov/wp-content/uploads/2018/07/spacex_nasa_crs-5_presskit-105.pdf
  Mission-contemporaneous source for ~70 s supersonic time, ~157 s MECO, 80 km / Mach 10 at MECO, Falcon 9 diameter, Merlin 1D thrust, and the statement that first-stage engines are gradually throttled near the end of first-stage flight.

Environmental Assessment for SpaceX Falcon Launches at KSC and CCAFS (FAA)
https://www.faa.gov/sites/faa.gov/files/space/environmental/nepa_docs/SpaceX_Falcon_Program_Final_EA_and_FONSI.pdf
  Later FAA summary source. It describes Falcon 9 v1.1 as ~1,100,000 lb gross lift-off weight, lists version-level propellant/thrust data, notes that propellant quantities vary with mission parameters, and states that Falcon trajectories are mission-specific.

NASA GSFC educational orbital-mechanics reference
https://pwg.gsfc.nasa.gov/Education/JAS1.htm
  Provides the useful reference that a due-east launch from Cape Canaveral near 28.5 deg N receives about 407 m/s of eastward inertial velocity from Earth's rotation. CRS-5 was not a simple due-east 2D launch, so 407 m/s is used only for a diagnostic sensitivity case.

U.S. Standard Atmosphere, 1976 (NASA NTRS)
https://ntrs.nasa.gov/citations/19770009539
  Reference family for the lower-atmosphere convention. LVA's atmosphere implementation is validated separately; here it is used only for a derived Mach/speed-of-sound diagnostic.

PUBLISHED FLIGHT REFERENCES
---------------------------
Supersonic time:
  value:    70
  source:   https://www.nasa.gov/wp-content/uploads/2018/07/spacex_nasa_crs-5_presskit-105.pdf
  location: Launch and Ascent, p. 5
  basis:    NASA states Falcon 9 reaches supersonic speed one minute and ten seconds after liftoff.
  precision:This is a rounded narrative flight anchor, not high-rate telemetry with a stated uncertainty.
MECO time:
  value:    157
  source:   https://www.nasa.gov/wp-content/uploads/2018/07/spacex_nasa_crs-5_presskit-105.pdf
  location: Launch and Ascent, p. 5
  basis:    NASA states first-stage MECO occurs approximately 157 seconds into flight.
  precision:The source explicitly says approximately; the benchmark therefore must not treat 157.000000 s as an exact telemetry truth.
MECO altitude:
  value:    80000
  source:   https://www.nasa.gov/wp-content/uploads/2018/07/spacex_nasa_crs-5_presskit-105.pdf
  location: Launch and Ascent, p. 5
  basis:    NASA states Falcon 9 is 80 km high at MECO.
  precision:The press kit reports 80 km as a rounded narrative value and does not provide an uncertainty interval.
MECO Mach:
  value:    10
  source:   https://www.nasa.gov/wp-content/uploads/2018/07/spacex_nasa_crs-5_presskit-105.pdf
  location: Launch and Ascent, p. 5
  basis:    NASA states Falcon 9 is traveling at 10 times the speed of sound at MECO.
  precision:Mach 10 is rounded and the source does not specify a local speed-of-sound convention or an explicit m/s velocity.

PUBLISHED VEHICLE ANCHORS USED BY BASELINE
------------------------------------------
Approximate gross liftoff mass:
  value:    approximately 1,100,000 lb
  source:   https://www.faa.gov/sites/faa.gov/files/space/environmental/nepa_docs/SpaceX_Falcon_Program_Final_EA_and_FONSI.pdf
  location: Section 1.2.2, p. 23
  note:     FAA describes Falcon 9 v1.1 gross lift-off weight as approximately 1.1 million pounds. This rounded value is used only as an aggregate initial-mass anchor.
Diameter:
  value:    12 ft
  source:   https://www.nasa.gov/wp-content/uploads/2018/07/spacex_nasa_crs-5_presskit-105.pdf
  location: Falcon 9 Overview / Advanced Technology, pp. 15-16
  note:     NASA CRS-5 press kit gives Falcon 9 diameter as 12 ft.
First-stage sea-level thrust:
  value:    9 engines x 654 kN each (147,000 lbf each)
  source:   https://www.nasa.gov/wp-content/uploads/2018/07/spacex_nasa_crs-5_presskit-105.pdf
  location: Falcon 9 Overview / Advanced Technology, pp. 15-16
  note:     NASA CRS-5 press kit states one Merlin 1D produces 654 kN at liftoff. This mission-contemporaneous source is preferred over later version-summary values.
First-stage high-altitude/vacuum thrust anchor:
  value:    9 engines x 716 kN each (161,000 lbf each)
  source:   https://www.nasa.gov/wp-content/uploads/2018/07/spacex_nasa_crs-5_presskit-105.pdf
  location: Falcon 9 Overview / Advanced Technology, pp. 15-16
  note:     NASA CRS-5 press kit states one Merlin 1D rises to 716 kN as it climbs out of the atmosphere.

PUBLIC-SOURCE CONSISTENCY WARNINGS
----------------------------------
FAA rounded v1.1 gross lift-off weight:  1,100,000 lb
FAA v1.1 version-table total propellant: 1,120,925 lbm
Propellant minus rounded gross value:    20,925 lb
Interpretation: these aggregate public figures cannot be combined as an exact CRS-5 mass ledger. The FAA itself notes that propellant quantities vary by mission parameters; the gross value is explicitly approximate.
Source: https://www.faa.gov/sites/faa.gov/files/space/environmental/nepa_docs/SpaceX_Falcon_Program_Final_EA_and_FONSI.pdf

NASA CRS-5 per-engine liftoff thrust: 147,000 lbf (654 kN)
Later FAA v1.1 summary thrust:        170,000 lbf per engine
Difference between public version summaries: 15.646 %
Interpretation: vehicle-version/public-summary values changed across documents. The benchmark therefore uses the mission-contemporaneous CRS-5 NASA thrust anchor rather than mixing later values into the baseline.
NASA source: https://www.nasa.gov/wp-content/uploads/2018/07/spacex_nasa_crs-5_presskit-105.pdf
FAA source:  https://www.faa.gov/sites/faa.gov/files/space/environmental/nepa_docs/SpaceX_Falcon_Program_Final_EA_and_FONSI.pdf

UNSOURCED / EXPLICIT MODELING ASSUMPTIONS
-----------------------------------------
Aggregate usable propellant reservoir: 400,000 kg
Isp sea level / vacuum:                282 s / 311 s
Aerodynamics:                          generic assumed Mach-Cd table
Guidance:                              built-in generic open-loop profile
Throttle:                              constant 100% in baseline
Earth rotation / launch latitude:      not modeled in baseline 2D ascent stack

BASELINE STATE AT NASA-PUBLISHED MECO TIME
------------------------------------------
Evaluation time:      157.000 s (imposed from rounded reference; not a model-predicted MECO time)
Model altitude:       71.305329 km
NASA altitude:        80.000 km (rounded narrative anchor)
Altitude difference:  -8.694671 km
Nominal percent diff: 10.868338 %

Model Mach:           6.065673929
NASA Mach:            10.000 (rounded narrative anchor)
Mach difference:      -3.934326071
Nominal percent diff: 39.343261 %

Model inertial speed: 1.787140068 km/s
NASA explicit speed:  not provided in the press kit

EARLY-ASCENT CROSS-CHECK
------------------------
Model Mach-1 crossing: 69.384774 s
NASA supersonic time:  ~70 s (rounded narrative anchor)
Timing difference:     -0.615226 s
Nominal percent diff:  0.878895 %

CONTROLLED ONE-AT-A-TIME SENSITIVITY RUNS
-----------------------------------------
These are DIAGNOSTIC perturbations, not claimed Falcon 9 uncertainties and not fitted parameters.
Mass/thrust/Isp: +/-5%; Cd: +/-25%; guidance: deliberately steeper/shallower assumed pitch programs.
The throttle-down and Cape-rotation cases are directionality tests, not CRS-5 reconstructions.

case                              alt(km)   dAlt(km)   speed(km/s) dSpeed    Mach       dMach      Mach1(s)   dMach1(s)
------------------------------------------------------------------------------------------------------------------
BASELINE                            71.305      0.000     1.787140 0.000000   6.065674   0.000000     69.385       0.000
Cd -25%                             71.525     +0.220     1.790715 +0.003575   6.086287  +0.020613     69.245      -0.140
Cd +25%                             71.087     -0.219     1.783566 -0.003574   6.045169  -0.020505     69.525      +0.140
Initial mass -5%                    82.254    +10.948     2.132353 +0.345213   7.632070  +1.566396     62.052      -7.333
Initial mass +5%                    61.203    -10.103     1.497168 -0.289972   4.783932  -1.281742     77.507      +8.122
Isp -5%                             74.784     +3.479     1.936489 +0.149349   6.684750  +0.619076     68.164      -1.221
Isp +5%                             68.333     -2.973     1.667149 -0.119991   5.554676  -0.510998     70.539      +1.154
Thrust -5%                          60.649    -10.656     1.482536 -0.304604   4.722468  -1.343206     77.988      +8.603
Thrust +5%                          81.757    +10.452     2.115151 +0.328011   7.551702  +1.486028     62.358      -7.026
More-vertical guidance              77.380     +6.075     1.731530 -0.055610   6.051137  -0.014537     69.383      -0.002
More-horizontal guidance            63.382     -7.923     1.905721 +0.118581   6.165573  +0.099899     68.631      -0.754
Approx. co-rotating Cape initial state   71.673     +0.367     2.122628 +0.335488   6.080098  +0.014424     69.337      -0.048
Illustrative late throttle-down     70.635     -0.671     1.637186 -0.149954   5.533243  -0.532431     69.385      +0.000

ERROR DIAGNOSIS A — MACH-1 CROSSING (~70 s)
--------------------------------------------
Observed nominal discrepancy: -0.615226 s (0.879 %).
Reference limitation: NASA gives a rounded narrative time ('one minute and ten seconds'), not a telemetry uncertainty interval.
Source: https://www.nasa.gov/wp-content/uploads/2018/07/spacex_nasa_crs-5_presskit-105.pdf

Sensitivity to +/-5% initial mass: -7.333 s / +8.122 s
Sensitivity to +/-5% thrust:       +8.603 s / -7.026 s
Sensitivity to +/-5% Isp:          -1.221 s / +1.154 s
Sensitivity to +/-25% Cd:          -0.140 s / +0.140 s
Conclusion: the 0.615 s agreement is smaller than the response to several plausible diagnostic perturbations. It is encouraging, but it cannot by itself validate the Falcon-specific mass/thrust/aero model; compensating errors can produce a close Mach-1 time.

ERROR DIAGNOSIS B — ALTITUDE AT 157 s
-------------------------------------
Observed nominal discrepancy: -8.694671 km (model low by 10.868 % relative to the rounded 80 km anchor).

High-sensitivity inputs in this proxy:
  Initial mass -5% / +5% -> altitude shift +10.948 / -10.103 km.
  Thrust -5% / +5%       -> altitude shift -10.656 / +10.452 km.
  These perturbations are not uncertainty claims, but they show that thrust-to-mass ratio is a dominant numerical lever. Public CRS-5 mass information is only approximate, and public version summaries are not internally precise enough to reconstruct the exact flight mass state.
  FAA source: https://www.faa.gov/sites/faa.gov/files/space/environmental/nepa_docs/SpaceX_Falcon_Program_Final_EA_and_FONSI.pdf

Material guidance sensitivity:
  More-vertical assumed profile   -> altitude shift +6.075 km.
  More-horizontal assumed profile -> altitude shift -7.923 km.
  FAA states Falcon trajectories are mission-specific; the public sources used here do not provide the CRS-5 pitch program. Guidance therefore materially affects altitude allocation and is a major unclosed fidelity term.
  FAA source: https://www.faa.gov/sites/faa.gov/files/space/environmental/nepa_docs/SpaceX_Falcon_Program_Final_EA_and_FONSI.pdf

Moderate mass-flow sensitivity:
  Isp -5% / +5% -> altitude shift +3.479 / -2.973 km.
  Baseline Isp is an explicit assumption. Because LVA derives mass flow from thrust/(Isp*g0), Isp changes the vehicle mass history and therefore acceleration.

Low sensitivity in this experiment:
  Cd -25% / +25% -> altitude shift +0.220 / -0.219 km.
  Approximate Cape co-rotation -> altitude shift +0.367 km.
  Therefore, within these perturbations, neither the generic Cd curve nor omitted rotation is large enough by itself to explain the ~8.7 km altitude shortfall.

Wrong-direction / masking effect:
  Illustrative late throttle-down -> altitude shift -0.671 km.
  NASA states the real first-stage engines were gradually throttled near the end of first-stage flight. The baseline incorrectly stays at 100%; adding an illustrative throttle-down reduces altitude further. Therefore missing throttle-down does NOT explain why baseline altitude is too low; constant full throttle actually biases the proxy slightly toward the reference and masks some discrepancy.
  NASA source: https://www.nasa.gov/wp-content/uploads/2018/07/spacex_nasa_crs-5_presskit-105.pdf

ERROR DIAGNOSIS C — MACH AT 157 s
---------------------------------
Observed nominal discrepancy: model Mach 6.066 vs rounded NASA Mach 10 (39.343 % nominal difference).

First separate vehicle-speed error from local speed-of-sound error:
  LVA speed of sound at model altitude (71.305 km): 294.632 m/s.
  LVA/USSA speed of sound at reference altitude (80 km):             282.537 m/s.
  Holding MODEL speed fixed but evaluating at 80 km gives Mach:       6.325.
  Original model Mach:                                                 6.066.
  Altitude/acoustic shift therefore changes Mach by only:              0.260.
  That is about 6.6 % of the total Mach gap under the LVA atmosphere convention.
  Conclusion: most of the Mach discrepancy is a vehicle-speed/trajectory discrepancy, not an atmosphere speed-of-sound discrepancy.
  Atmosphere reference family: https://ntrs.nasa.gov/citations/19770009539

Derived diagnostic only (NOT a NASA-reported velocity):
  Mach 10 at LVA's validated 80 km standard-atmosphere sound speed would correspond to ~2.825 km/s.
  Baseline model speed is 1.787 km/s, a derived gap of -1.038 km/s.

High-sensitivity inputs for Mach/speed in this proxy:
  Initial mass -5% / +5% -> Mach shift +1.566 / -1.282.
  Thrust -5% / +5%       -> Mach shift -1.343 / +1.486.
  This again identifies the exact thrust-to-mass history as a dominant unresolved proxy input. The public mass anchor is approximate, and public Falcon version thrust summaries differ across documents.

Moderate sensitivity:
  Isp -5% / +5% -> Mach shift +0.619 / -0.511.
  Isp is unsourced in the benchmark and directly controls modeled mass flow, so it is a material source of speed uncertainty.

Smaller sensitivity for the tested perturbations:
  More-vertical / horizontal guidance -> Mach shift -0.015 / +0.100.
  Cd -25% / +25%                  -> Mach shift +0.021 / -0.021.
  Approximate Cape co-rotation     -> Mach shift +0.014.
  Guidance strongly moves altitude but only modestly changes Mach in these two diagnostic profiles; Cd and co-rotation are much too small here to explain a 3.93-Mach shortfall alone.

Wrong-direction / masking effect:
  Illustrative late throttle-down -> Mach shift -0.532.
  Since the real vehicle throttled down near the end of first-stage flight, the baseline's constant 100% throttle should overstate late powered acceleration relative to a throttled case. It therefore cannot explain the low Mach; it partially hides the discrepancy.

Coupled-model conclusion:
  No single sensitivity perturbation tested here closes the Mach-10 gap. The benchmark is therefore most consistent with a combined fidelity problem: approximate mission mass/thrust state + assumed Isp/mass-flow history + non-Falcon guidance, with smaller contributions from generic aerodynamics and frame/rotation simplifications. These effects are nonlinear and MUST NOT be numerically added as an 'error budget'.

WHY THE MACH-1 MATCH DOES NOT CONTRADICT THE MECO MISMATCH
---------------------------------------------------------
At ~70 s the vehicle has experienced much less cumulative mass depletion and guidance evolution than at 157 s. The sensitivity runs show that small changes in thrust-to-mass and Isp can shift the Mach-1 time by seconds and later MECO-state quantities by much larger amounts. A close early crossing can therefore coexist with a large late-state discrepancy through cumulative model error and/or cancellation among assumptions.

SCIENTIFIC CLAIM BOUNDARY
-------------------------
SUPPORTED:
  - LVA reproduces the rounded ~70 s supersonic anchor within ~0.9% in this proxy.
  - At the imposed 157 s reference time, LVA is ~8.7 km below the rounded 80 km anchor and reports Mach ~6.07 instead of rounded Mach 10.
  - Controlled sensitivity runs show high dependence on thrust-to-mass ratio, material dependence on Isp/mass-flow and guidance, and much smaller dependence on the tested Cd and rotation perturbations.
  - A late throttle-down perturbation moves the result farther from the reference, so the baseline's missing throttle-down is not the cause of its low-Mach result.

NOT SUPPORTED:
  - assigning an exact percentage of the CRS-5 error to any one missing model feature;
  - treating the +/- perturbations as Falcon 9 uncertainty intervals;
  - claiming the NASA rounded anchors are exact telemetry;
  - claiming this proxy validates Falcon 9 flight guidance or aerodynamics.

SUMMARY
=======
Mach-1 crossing:      model=69.385 s, NASA~70 s, nominal diff=0.879 %
Altitude at 157 s:    model=71.305 km, NASA~80 km, nominal diff=10.868 %
Mach at 157 s:        model=6.066, NASA~10, nominal diff=39.343 %
Primary sensitivity finding: thrust-to-mass and mass-flow history dominate the tested late-state response; guidance materially controls altitude; tested Cd/rotation effects are smaller; throttle-down omission biases in the wrong direction to explain low Mach.
Result:               BENCHMARK + ERROR DIAGNOSIS COMPLETE (descriptive; no pass/fail threshold)

09 / STILL OPEN

Validation is not finished.

The evidence above closes several implementation questions, but it also identifies the next fidelity work instead of declaring victory because the page has enough green pixels.

OPEN 01

Vehicle-specific aerodynamics

The current Mach-Cd curve is generic. Drag cannot claim Falcon-level validation without a vehicle-specific aerodynamic reference or a defensible surrogate dataset.

OPEN 02

Falcon proxy reconstruction

Replace aggregate mass/Isp/guidance assumptions with better mission-specific public data where available, then rerun the same benchmark without tuning directly to the target outputs.