On the Observable Ciphertext Properties of 46-Character HFGCS Broadcasts
A cohort analysis of the 46-character HFGCS broadcast band; one cohort (G2) at N = 7.
G2 N=7 · G3 N=1 · G4 N=1 · years 2022–2024
Abstract. We apply the probe battery of the 30-character format analysis, extended with an inter-position equality (IPE) probe, to the 46-character HFGCS broadcast cohort. After standard filtering, forward-fill group resolution, and first-broadcast itemisation, N = 7 unique messages remain, all in Group 2 (the Group 3 and Group 4 bands at this length each contain a single broadcast and are noted but not analysed). At N = 7 the probe battery operates at the floor of statistical power; findings are reported directly with power limits stated.
The first-order probe returns no significant symbol-level deviation: the aggregate M/5 pair sits at z = -0.86 and -1.84, neither crossing the per-symbol Bonferroni threshold for 32 tests (|z| > 3.16), and no other symbol in the distribution crosses threshold in either direction. The per-position and doublet probes are similarly null across all body positions.
The inter-position equality probe, however, returns a strong structural finding despite the small sample: pair (15, 23) shows msg[15] = msg[23] in every one of the 7 messages (z = +14.73), and the extended triple (15, 23, 27) is present in 5 of 7 messages. We refer to this triple as pattern A: a single character occupying three positions with fixed offsets of +8 and +12 from an anchor at position 15. A weaker outer equality at pair (17, 29) holds in 3 of 7 messages, but the neighbouring pairs (17, 25) and (25, 29) do not; we note this as a candidate secondary pattern without identifying it as established. Additional probes — internal repeats, doublets, compressibility — return null at this sample.
Pattern A is a genuine structural feature of the 46-character Group 2 cohort, surviving the per-family Bonferroni correction for all 1035 tested pairs by a wide margin. Its interpretation is left open here: the data does not distinguish between a fixed-field replication, an authenticator-internal coincidence, and other candidate generators. The result is observational.
In plain language. The 46-character HFGCS broadcast band has just 7 unique Group-2 messages in the dataset. That is a very small sample — not enough to detect most kinds of subtle patterns. The Group 3 and Group 4 bands at this length each have only one broadcast and cannot be analysed at all.
With only 7 messages, the standard symbol-frequency tests report nothing unusual. No letter is significantly over- or under-represented; no position shows an odd distribution; consecutive-character doublets behave as expected under uniform output.
But one test does fire: looking at every pair of positions for matched characters, we find that the character at position 15 is the same as the character at position 23 in every single one of the 7 messages. The pair (15, 27) also matches in 5 of 7 messages, and (23, 27) in 5 of 7. Taken together these three equalities describe a single character that appears at positions 15, 23, and 27 — the same character in all three places, in the majority of messages.
A weaker second pattern is hinted at: 3 of 7 messages have a matched character at positions (17, 29), but the intermediate pair (17, 25) and (25, 29) each hold in at most 1 message. At this sample size we can't separate a real secondary pattern from chance; we note it but don't claim it.
1.Introduction
The 46-character HFGCS broadcast band is sparsely populated: 11 raw broadcast-level observations survive standard filtering, all resolving to Group 2, Group 3, or Group 4 under forward-fill on the PR_GROUPS reference table. After first-broadcast itemisation, Group 2 retains N = 7 unique messages; Group 3 and Group 4 each retain a single message and are reported only for completeness.
We apply the probe battery established for the 30-character paper (the base paper in this project), extended with an inter-position equality (IPE) probe: first-order symbol marginals, per-position marginal z-scores, consecutive-character doublet rates, inter-position equalities at every pair of positions, length-≥3 internal-substring repeats, aggregate compressibility. At N = 7 the battery operates near the floor of statistical power; the only probe that returns a strong positive signal at this sample is the IPE scan.
2.Observation corpus
Standard filters: quality-flag exclusion of "\" and "*"; placeholder-character exclusion of "?", "_", "."; length-exact match to 46. Forward-fill group resolution is applied to every row. 11 raw broadcast-level observations remain in Group 2; first-broadcast itemisation on the message string retains N = 7 unique messages spanning calendar years 2022–2024. The Group 3 band contains 3 raw broadcasts resolving to a single unique message; the Group 4 band contains a single raw broadcast. These two singleton cohorts are excluded from the probe battery: every statistic reduces to either the value itself or an undefined standard deviation at N = 1.
Within the Group 2 cohort, no prefix is populous enough for per-cell within-prefix probes: the most common prefix appears at most twice across the 7 messages, far below the N ≥ 50 threshold used by the 30-character paper's within-prefix probes. §8 formally reports this and skips the per-prefix scans.
3.First-order symbol distribution
We test the null that each base32 symbol appears with uniform probability 1/32 at every body position. At N = 7 with 44 body positions carried, the body holds 308 characters; the expected per-symbol body count is 9.6. Per-symbol z-scores therefore discretise in steps of about 0.33σ.
Figure 1. Empirical frequency of each base32 symbol across body positions 3–46 of the 46-character Group 2 first-broadcast cohort (N = 7). Dashed line: uniform expectation 1/32 ≈ 3.125%. No symbol crosses the per-symbol Bonferroni threshold for 32 tests at α=0.05 (|z| > 3.16) in either direction. The largest deficit is 5 at z = -1.84; the largest excess is 6 at z = +2.09. At this sample size, every symbol's count falls within the chance envelope under uniform output.
symbol
count
pct
z
5
4
1.30%
-1.84
P
5
1.62%
-1.51
Z
6
1.95%
-1.19
7
6
1.95%
-1.19
I
7
2.27%
-0.86
M
7
2.27%
-0.86
Q
7
2.27%
-0.86
H
8
2.60%
-0.53
T
8
2.60%
-0.53
2
8
2.60%
-0.53
B
9
2.92%
-0.20
E
9
2.92%
-0.20
S
9
2.92%
-0.20
V
9
2.92%
-0.20
W
9
2.92%
-0.20
X
9
2.92%
-0.20
A
10
3.25%
+0.12
K
10
3.25%
+0.12
L
10
3.25%
+0.12
O
10
3.25%
+0.12
U
10
3.25%
+0.12
C
11
3.57%
+0.45
D
11
3.57%
+0.45
G
11
3.57%
+0.45
3
11
3.57%
+0.45
F
12
3.90%
+0.78
R
12
3.90%
+0.78
Y
12
3.90%
+0.78
J
14
4.55%
+1.43
N
14
4.55%
+1.43
4
14
4.55%
+1.43
6
16
5.19%
+2.09
Table 1. G2 body (positions 3–46) symbol counts, percentages, and Poisson z-scores against the uniform null. Rows sorted by z-score. No symbol crosses the per-symbol Bonferroni threshold. The M/5 pair, which carries a strong signal in the 30-character Group 1 cohort, is within the noise envelope here.
4.Position-wise bias profile
The per-position M/5 probe from the 30-character paper tracks Poisson z-scores for those two symbols at each body position. At N = 7 with per-position expectation 0.219, counts of 0, 1, 2 correspond to z-scores of roughly −0.48, +1.70, +3.87, so the per-position statistic is extremely coarse.
Figure 2. Per-position Poisson z-scores for symbols M and 5 across the 46-character Group 2 cohort. Solid horizontal line: uniform expectation (z = 0). Dashed red line: per-family Bonferroni threshold for 2·46 = 92 tests at α=0.05 (|z| > 3.46). At N = 7 the per-position signal is severely discretised; no position in either channel approaches the corrected threshold. The prefix positions 1–2 reflect the limited G2 PR distribution sampled at this length.
4.1Per-position all-symbol probe
The M/5 projection reports only two channels of a 32-channel probe. The full scan reports, for each body position, the largest |z|-score attained by any of the 32 symbols at that position.
Figure 3. Per-position max-|z| across all 32 symbols in the 46-character Group 2 cohort. Dashed red line: per-family Bonferroni threshold for 32·44 = 1408 body-cell tests at α=0.05 (|z| > 4.13). Dashed grey: uncorrected per-position α=0.05 for 32 symbols (|z| > 2.87). 1 body positions cross the Bonferroni threshold, and 25 positions attain max-|z| > 2.87; 70.4 such positions are expected under uniform output at this false-positive rate, so the signal is within chance for the uncorrected band. The single Bonferroni-crossing position (41, carrying O at count 3, z = +6.04) is an artefact of discretisation: at N = 7, a single count of 3 is all it takes to cross the Bonferroni threshold, and the position-wise false-positive rate is therefore not well-controlled at this sample size.
5.Doublet profile
We test the rate of consecutive-character doublets at each of the 45 position pairs (i, i+1). Under the uniform null the probability is 1/32 at each pair, so the expected doublet count is N/32 ≈ 0.219.
Figure 4. Doublet count per consecutive position pair in the 46-character Group 2 cohort. Dashed line: uniform expectation (0.22, virtually zero given N = 7). Most pairs show count = 0, with a handful of pairs at count = 1 (z = +1.70). No pair shows count ≥ 2. At this sample the doublet probe cannot distinguish structural suppression from chance-free uniformity; no trailing no-duplicate rule analogous to the 30-character Group 1 finding is visible, but neither can one be excluded from the data.
6.Inter-position equality probe
The consecutive-doublet probe of §5 tests only adjacent position pairs. Its generalisation tests every position pair (i, j) with j > i — including non-adjacent pairs — for character equality. At 46 characters there are 46·45/2 = 1035 such tests; the per-family Bonferroni threshold at α=0.05 is |z| > 4.06. Under the uniform null the expected match count at each pair is N/32 = 0.219.
Despite the small sample, the probe returns 5 pairs above the Bonferroni threshold. The ten strongest positive pairs are reported in Table 2.
pos a
pos b
gap
matches
exp
z
15
23
8
7
0.22
+14.73
15
27
12
5
0.22
+10.39
23
27
4
5
0.22
+10.39
17
29
12
3
0.22
+6.04
25
38
13
3
0.22
+6.04
4
8
4
2
0.22
+3.87
4
11
7
2
0.22
+3.87
7
19
12
2
0.22
+3.87
7
27
20
2
0.22
+3.87
8
32
24
2
0.22
+3.87
Table 2. Top ten inter-position equality pairs in the 46-character Group 2 cohort, by z-score. Expected count per pair: 0.219. Highlighted rows cross the Bonferroni threshold |z| > 4.06. The headline finding is pair (15, 23): every one of the 7 messages carries msg[15] = msg[23]. The pair (15, 27) and (23, 27) each carry the same-character equality in 5 of 7 messages. These three pairs together describe a single-character triple equality at positions (15, 23, 27) in the majority of the cohort.
Figure 5. Inter-position equality z-score heatmap across all 1035 position pairs (i, j) for the 46-character Group 2 cohort. Red cells: excess (pair carries more matches than expected under uniform); blue cells: deficit; pale cream: within the chance envelope. The matrix is dominated by the chance envelope; the three equalities at pairs (15, 23), (15, 27), (23, 27) appear as the deepest-red cells, with the candidate secondary pair (17, 29) visible at lower intensity.
6.1Pattern A and the candidate secondary pattern
Pattern A. The three Bonferroni-crossing pairs (15, 23), (15, 27), (23, 27) taken together describe a single character occupying three positions at fixed offsets from an anchor at position 15. The anchor pair (15, 23) holds in 7 of 7 messages (100%); the two extended pairs (15, 27) and (23, 27) each hold in 5 of 7 messages (71%). All three pairs individually exceed the per-family Bonferroni threshold by a wide margin. The pattern is position-locked to absolute offsets from the message start; whether this reflects a fixed-field replication, an authenticator-internal coincidence, or another generator is a question the data does not answer.
Candidate secondary pattern. The pair (17, 29) at count 3 of 7 also crosses the Bonferroni threshold, with a gap of 12 that matches the outer span of pattern A. However, the pairs (17, 25) and (25, 29) — which would be expected if position 17 anchored a second triple with the same +8 / +12 geometry as pattern A — each hold in at most 1 message and do not cross threshold. At N = 7 this is insufficient to decide between (a) a second real pattern that is sparser than pattern A, (b) an isolated (17, 29) equality with no wider structure, and (c) a chance-tail. We report the outer pair but do not claim the triple.
7.Internal-repeat structure
Applying the longest-internal-repeat probe — find the longest substring of length 3–10 that appears at two disjoint positions within each message — to the 46-character Group 2 cohort: 0 of 7 messages carry any repeat of length ≥ 3 anywhere. The null expectation at this length and N is approximately 0.22 messages, so the observed count is slightly below the chance rate. The IPE finding at pairs (15, 23, 27) would not be surfaced by this probe because the matching characters are single positions separated by fixed offsets, not contiguous substrings.
8.Within-prefix structural probes
The 30-character paper reports three within-prefix probes — pairwise positional mutual information, LZMA compressibility, and modular-difference structure across date-ordered consecutive messages — each requiring N ≥ 50 per prefix cell for credible z-scores. The 46-character cohort has no such cell: no single prefix accounts for more than two messages across the 7-message cohort. Per-prefix probes are skipped.
An aggregate whole-cohort compressibility probe still runs:
body bytes
obs ratio
null mean
null sd
z
308
1.0390
1.0218
0.0092
+1.87
Table 4. Aggregate LZMA compressibility of the body (positions 3–46) of the 46-character Group 2 cohort versus a 50-draw synthetic uniform base32 null of matched length. The observed ratio sits +1.87σ from the null mean — within the chance envelope at this small sample. The IPE finding does not produce a compressibility signature at N=7 because LZMA's dictionary-based approach does not capture character-level equality across fixed offsets without contextual repetition.
9.Synthesis
The 46-character Group 2 cohort carries one positive finding: a triple-equality structure at positions (15, 23, 27), with the anchor–copy-1 pair (15, 23) holding in 100% of the 7 messages and the full triple (15, 23, 27) holding in 5 of 7. All three pairs of the triple individually exceed the per-family Bonferroni threshold for 1035 IPE tests. We call this pattern A.
A candidate secondary pattern appears as a single Bonferroni-crossing outer pair at (17, 29), but the intermediate pairs (17, 25) and (25, 29) do not reach threshold. We note the outer pair as a finding of interest but do not claim a full triple at this sample size.
All other probes — first-order symbol marginals, per-position marginals, doublets, internal repeats, compressibility — return null. No Bonferroni-crossing symbol deviation, no per-position outlier survives correction for discretisation, no doublet suppression, no detectable internal repeats of length ≥ 3, and the compressibility ratio sits within the synthetic-uniform null envelope.
Figure 6. 46-character Group 2 observational region schematic. The 230-bit payload comprises the 10-bit prefix (P, positions 1–2, carrying the PR), a header region (H, positions 3–14, approximately 60 bits), three single-character anchor regions a₁ / a₂ / a₃ at positions 15 / 23 / 27 holding pattern A (the character at a₁ reappears verbatim at a₂ and a₃ in the majority of messages), and a trailing body region (T, positions 28–46, approximately 95 bits). The exact role of the repeated character is not inferred. The candidate secondary pair at (17, 29) is not separately depicted; its status is unresolved at N = 7.
9.1Interpretation and open questions
Pattern A is position-locked to absolute offsets from the message start (positions 15, 23, 27 are fixed integers, not fractions of the 46-character payload). This means the pattern describes a feature of the message format that begins at a fixed bit offset; it is not a tail-aligned constraint and not a fractional-of-length feature. Possible generators include a fixed-position field replicated three times, a coincidence between three outputs of a generator keyed on shared material, or a cryptographic-construction artefact; the ciphertext-only data here cannot distinguish between these.
The candidate secondary pair at (17, 29) has the same gap (+12) as the outer pair of pattern A. Whether it is part of a parallel triple, an isolated doublet, or chance is unresolved. Additional 46-character Group 2 observations, as they accumulate in the corpus, are the direct path to resolving this.
10.Conclusion
The 46-character HFGCS broadcast band is populated almost entirely by Group 2 traffic at N = 7 unique messages (the Group 3 and Group 4 bands each hold a single message). At this sample size the probe battery operates near the floor of statistical power for all marginal and positional probes, and those probes return null: no Bonferroni-crossing symbol deviation, no distinguishable per-position profile, no doublet suppression, no detectable internal repeats, no compressibility anomaly.
The inter-position equality probe returns a single clear finding: pattern A, a triple equality at absolute positions (15, 23, 27) with the anchor pair (15, 23) holding in 100% of the 7 messages. A weaker outer pair at (17, 29) crosses the Bonferroni threshold but is not completed by the intermediate pairs one would expect from a parallel triple. Pattern A is the one structural feature this cohort can be said to carry; further interpretation awaits either more observations or an independent channel of evidence.