Results

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Observations only. Interpretation belongs in the thesis chapter and in limitations; mixing them here is how a result becomes hard to reuse.

All figures are from the primary element set (10,734 elements, median epoch 2026-08-29 21:17:42 UTC, seed 20271002) unless stated otherwise.

Headline numbers

Quantity	Value	Uncertainty	Source run
Worst horizontal displacement under single-beam spoofing	$1.51 \times 10^{-10}$ m	Worst case over 384 attempts, not a mean	E3
Largest uniform offset applied while achieving that	22.6 km of range	—	E3
Position resolution $\delta$ at $\sigma_\tau = 10$ ns	3.1 m	Empirical RMS 3.1 m against CRLB 3.1 m	E2
Estimator efficiency	0.97 to 1.02	Ratio of empirical RMS to CRLB across four decades of $\sigma_\tau$	E2
Satellites visible above 25°	68 median	Range 56 to 82 over 96 epochs	E1
HDOP, six angularly diverse links	1.02 median	Range 1.00 to 1.12	E1
HDOP, six highest-elevation links	2.73 median	Range 1.74 to 6.01	E1
Relay detection, $M \geq 5$ at $q = 2$	100%	Over 12 sampled epochs	E4
Relay detection, $M = 3$ at $q = 2$	67%	Over 12 sampled epochs	E4
False positives at 10 ns, up to 1000 terminals/km²	0%	300-terminal honest population	E5
False positives at 100 ns, 100 terminals/km²	28.3%	300-terminal honest population	E5

Resolution against the analytic bound

$\sigma_\tau$	CRLB $\delta$	Empirical RMS	Ratio
1 ns	0.31 m	0.30 m	0.97
10 ns	3.1 m	3.1 m	1.00
100 ns	30.9 m	31.4 m	1.02
1 μs	308.6 m	302.7 m	0.98

False positives against legitimate density

$\sigma_\tau$	$\delta$	1/km²	10/km²	100/km²	1000/km²
10 ns	3.0 m	0%	0%	0%	0%
100 ns	30.1 m	0%	0%	28.3%	99.3%
1 μs	301 m	25.7%	99.7%	100%	100%

Across ground sites

Site	Latitude	Visible	HDOP	$\delta$ at 10 ns	FP at 100/km²
Singapore	1.4°N	34	1.04	3.1 m	0%
Washington DC	38.9°N	70	1.02	3.1 m	0%
Grand Forks	47.9°N	71	1.02	3.0 m	0%
Tromsø	69.7°N	20	1.10	3.3 m	0%
Sydney	33.9°S	67	1.01	3.0 m	0%

Degradation without simultaneous observation

Measured as $\Delta_{\max} = c\,s / \omega_{\mathrm{LOS}}$ , the displacement an adversary can fabricate before a constrained clock model rejects it.

Assumed oscillator	Stability	90° pass	30° pass
Rubidium / GPSDO	$10^{-10}$	2.3 m	4.1 m
OCXO	$10^{-8}$	228 m	411 m
TCXO	$10^{-6}$	22.8 km	41.1 km
Uncompensated XO	$10^{-5}$	228 km	411 km

Figures

Registered under artifacts and referenced by identifier. Identifiers are provisional; these are not yet promoted out of the project working directory.

ID	Caption	Produced by
FIG-001 (provisional)	Relay detection rate versus relay size at $\sigma_\tau = 10$ ns, for quota 1, 2 and 5	E4, via pgfplots reading `eval-relay.csv` at compile time

Tables

ID	Caption	Produced by
TABLE-001 (provisional)	Empirical resolution against the Cramér–Rao bound	E2
TABLE-002 (provisional)	False-positive rate versus legitimate terminal density	E5
TABLE-003 (provisional)	Geometry across five ground sites	E8

Against the hypotheses

Hypothesis	Prediction	Observed	Standing
HYP-001	Estimator attains $\delta = c\,\sigma_\tau\cdot\mathrm{HDOP}$	Ratio 0.97 to 1.02 across four decades	supported
HYP-002	Uniform offset produces zero horizontal displacement	Worst case $1.51\times10^{-10}$ m over 384 attempts spanning a full orbital period	supported
HYP-003	Relay of $M$ identities detected once $M$ exceeds quota	Total from $M=5$ at $q=2$ ; 67% at $M=3$	supported above a floor at $M \approx q+3$

Unexpected observations

Things that were not predicted and are not fully explained. Recording them is how the next research question gets found.

Link selection dominates measurement precision. Choosing six satellites for angular diversity rather than elevation improves median HDOP by 2.7×, from 2.73 to 1.02, and collapses its spread from 1.74–6.01 to 1.00–1.12. This was not a hypothesis; it was found while checking whether the obvious selection rule was sensible. It costs nothing, since it changes which links to measure rather than how many.
The result is latitude-invariant even where the constellation is sparse. Visible count varies more than 3× across the five sites, from 20 at Tromsø to 71 at Grand Forks, yet HDOP stays within 1.01 to 1.10 and $\delta$ within 3.0 to 3.3 m. Six well-separated links appear to be sufficient, and even the sparsest site offers six. Why the tail is this flat is not established.
The naive selection degrades further on independent data. On the cross-check element set the top-elevation HDOP tail reaches 8.90 against 6.01 on the primary set, while the diverse selection is unchanged at 1.02 median. The bad rule is not merely worse but less stable across datasets.
Small relays sit at a detection floor that was not predicted. At $M = 3, q = 2$ detection is 67%, not near-total. The floor appears to be where relay size approaches the quota and density estimation loses its signal, but the boundary has not been characterised.