FINDINGS.md

Complete Findings

All validated discoveries and negative results from 77 investigations.

Validated Positive Findings (170)

Methodology Validation

#	Finding	Effect Size	Investigation
1	All hash functions (MD5, SHA-1, SHA-256, SHA-3, BLAKE2, SHA-512) are geometrically indistinguishable from random	d ≈ 0, 0 significant metrics	1d/hashes.py
2	21 geometries collapse to ~6 independent dimensions	r > 0.85 clusters	Redundancy analysis
2a	GARCH(1,1) returns now detectable vs IID normal (was negative)	29 sig; Symplectic:windowed_area_cv d=3.6, Lorentzian:causal_order d=2.5	1d/negative_reeval.py
2b	markov_mixing (spectral gap of value-transition matrix) is non-redundant	F=5025, max |r|=0.902 with framework; noise>0.5, sine<0.1	1d/markov_explore.py
2c	memory_order (order-2 vs order-1 Markov log-likelihood) is orthogonal to framework	max |r|=0.573; Lorenz>0.3, noise<0.2; genuinely new axis	1d/protocol_orthogonality.py

Cryptography

#	Finding	Effect Size	Investigation
3	ECB mode massively detectable with structured plaintext	d = 19-146	1d/ciphers.py
4	All ECB block ciphers (AES, DES3, Blowfish) produce identical geometric signatures	E8 = 2 roots for all	1d/ciphers.py
5	Stream ciphers (AES-CTR, ChaCha20, RC4) look random	d ≈ 0	1d/ciphers.py
6	AES becomes indistinguishable from random at exactly 4 rounds	40 metrics at R=1, 0 at R=4	1d/reduced_aes.py
7	LSB stego weakly detectable via LSB-correlation extraction	d = 1.06 (Fisher trace), 1 sig metric at 100% rate	1d/stego.py
8	PVD steganography massively detectable at raw byte level	42 sig metrics (natural_texture), detectable at 10% rate	1d/stego_deep.py
9	Spread spectrum stego detectable at raw byte level	25 sig metrics (natural_texture), detectable at 25% rate	1d/stego_deep.py
9a	Matrix embedding (Hamming syndrome coding) invisible to all geometries	0 sig at all rates, both carriers	1d/stego_deep.py
9b	Bitplane extraction does NOT improve stego detection (8x sample reduction)	0 sig across all techniques with bitplane	1d/stego_deep.py
9c	Delay embedding amplifies detectable stego signal	PVD: 16 sig via DE2 at 50% rate	1d/stego_deep.py
9d	2D co-occurrence matrix detects PVD and SS via spatial metrics alone	PVD: 13/15, SS: 14/15; tension_mean d=−5.6	2d/stego_bitmatrix.py
9e	2D diff grid amplifies LSBMR from near-invisible to solidly detected	14 sig (up from 2–6 in 1D); Wasserstein d=−1.36	2d/stego_bitmatrix.py
9f	2D representation choice is critical — co-occurrence Σ=27, diff grid Σ=24, bitplane tiled Σ=0	binary grids kill SpatialField metric variance	2d/stego_bitmatrix.py
9g	8 spatial geometries (80 metrics) nearly 4x PVD/SS detection on co-occurrence	PVD: 13→49, SS: 14→52; every geometry contributes unique detections	Framework 2D battery
9h	HodgeLaplacian saturates co-occurrence detection (8/9 metrics); SpectralPower adds 7/8	Laplacian energy, Poisson recovery, spectral slope are new detection axes	Framework 2D battery
9i	10 diverse 2D field types: all 45 pairwise distinguished by 8 spatial geometries	min 45/80 sig (Stripes vs Spirals), max 78/80 (Checkerboard vs Voronoi)	2D field type validation

PRNG Testing

#	Finding	Effect Size	Investigation
10	E8 catches RANDU from raw bytes	d = -19.89	1d/prng.py
11	E8 catches glibc LCG from raw bytes	d = -15.00	1d/prng.py
12	Tropical geometry is the second-best PRNG detector	d = -8	1d/prng.py
13	RANDU detectable at ALL bit planes	d > 10 for most planes	1d/prng.py

RNG Quality Testing (10 generators, systematic)

#	Finding	Effect Size	Investigation
13a	Self-check (urandom vs urandom) returns 0 sig — validates methodology	0/131	1d/rng.py
13b	CRYPTO (os.urandom, SHA-256 CTR) and GOOD (PCG64, SFC64) = 0 sig	0/131	1d/rng.py
13c	RANDU massively detected: 44 sig metrics, Penrose index_diversity d=-37	44/131	1d/rng.py
13d	Middle-Square massively detected: 78 sig metrics	78/131	1d/rng.py
13e	Geometric metrics outperform standard: RANDU 44 vs 4, Middle-Square 78 vs 7	11x, 11x	1d/rng.py
13f	Delay embedding reveals XorShift128 (0→2 sig via Thurston height_drift)	DE2 amplification	1d/rng.py
13g	RANDU detectable at just 500 bytes (41 sig)	N=500	1d/rng.py
13h	Middle-Square detectable at 500 bytes (72 sig)	N=500	1d/rng.py
13i	E8, Higher-Order Stats, Torus T², Ammann-Beenker are top PRNG weakness detectors	heatmap	1d/rng.py
13j	Penrose index_diversity is single most discriminating metric	d=-37 (RANDU)	1d/rng.py
13k	Klein Bottle linear_complexity detects XorShift32 as GF(2)-linear	d>1000; LC=0.125 vs ~1.0 for all other sources; Berlekamp-Massey LFSR saturates at 32 bits	1d/gf2_explore.py
13l	Klein Bottle rank_deficit orthogonal to linear_complexity (r=-0.035)	XorShift=0.067 (suspiciously full-rank), random≈0.03, Thue-Morse=0.875; max |r|=0.85 with framework	1d/gf2_explore.py

Chaotic Systems

#	Finding	Effect Size	Investigation
14	10/10 chaotic maps detected as non-random	varies	1d/chaos.py
15	45/45 pairwise chaotic map distinctions	varies	1d/chaos.py
16	Each chaotic map has a unique geometric fingerprint	Fisher: 17/45 pairs	1d/chaos.py
17	Lorenz uniquely destroys Penrose 5-fold symmetry	d = 60	1d/chaos.py
18	Rossler shows extreme Tropical linearity	d = 266	1d/chaos.py
18a	Standard map (K=6) detected as non-random despite uniform mixing	44 sig; Predictability:sample_entropy d=−46, Cayley:spectral_gap d=−17	1d/negative_reeval.py
18b	Arnold cat map detected as non-random despite Anosov mixing	19 sig; Predictability:sample_entropy d=−86, SpectralGraph:spectral_dim d=+42	1d/negative_reeval.py

Biology

#	Finding	Effect Size	Investigation
19	All DNA sequence types have distinct geometric signatures	291 significant findings	1d/dna.py
20	Microsatellites are the most detectable DNA feature	d = -18.3 (E8)	1d/dna.py
21	Viral DNA uniquely caught by Sol geometry	d = 6.2	1d/dna.py
22	Protein-coding DNA caught by S2xR sphere concentration	d = 7.2	1d/dna.py
23	E. coli vs shuffled E. coli: geometry sees DNA ordering	7 significant metrics	1d/dna.py
23a	Globular proteins distinguishable from IDPs (53 sig metrics)	Symplectic d=+2.55, HOS d=+2.37	1d/proteins.py
23b	Both globular and IDP sequences non-random vs uniform AA	glob=37, IDP=40 sig	1d/proteins.py
23c	Hydrophobicity encoding far superior to ordinal/MW for protein discrimination	53 vs 25 vs 13 sig	1d/proteins.py
23d	Delay embedding tau=4 is peak among delays (alpha-helix period ~3.6)	38 sig at tau=4 vs 37,35	1d/proteins.py
23e	Glob-vs-IDP signal is overwhelmingly compositional, not sequential	glob vs shuffled = 1 sig only	1d/proteins.py

Neural Networks

#	Finding	Effect Size	Investigation
24	All NN weight types distinguishable from random	57-71 significant metrics	1d/nn_weights.py
25	Backdoored weights detected by 15 metrics	d = 7.12 (Wasserstein)	1d/nn_weights.py
26	Dense vs conv weights distinguished	d = 11.9 (Cantor)	1d/nn_weights.py

Quantum Geometry

#	Finding	Effect Size	Investigation
26a	Coherence (interference) massively detectable vs decoherent sum	100 sig metrics, d ≈ 4000 (Clifford)	1d/quantum_geometry.py
26b	Box, Oscillator, Hydrogen, and Quantum Walk states distinct	16.8 avg sig metrics	1d/quantum_geometry.py
26c	Energy levels (n) produce distinct geometric fingerprints	12-17 sig vs ground state	1d/quantum_geometry.py
26d	Wavepacket dispersion detected as geometric evolution	11-14 sig metrics across time	1d/quantum_geometry.py

Memory Geometry

#	Finding	Effect Size	Investigation
26e	Arrays, Linked Lists, Hash Tables, and Trees have distinct topologies	77.5 avg sig metrics	1d/memory_geometry.py
26f	Sequential vs Random memory traces are massively distinguishable	99 sig metrics	1d/memory_geometry.py
26g	Memory fragmentation (Array vs Linked List) detectable via geometry	80 sig metrics, d ≈ 57	1d/memory_geometry.py
26h	Data structure density (load factor) affects geometric signature	97 sig metrics	1d/memory_geometry.py

Esoteric Code (Esolangs)

#	Finding	Effect Size	Investigation
26i	Malbolge is NOT random (distinguishable from high-entropy ASCII)	52 sig metrics, d ≈ -30	1d/esoteric_code.py
26j	Whitespace syntax creates "invisible" geometric structure	63 sig metrics vs white noise	1d/esoteric_code.py
26k	Brainfuck loop depth (recursion) is geometrically measurable	67 sig metrics (flat vs deep)	1d/esoteric_code.py
26l	Zalgo text corruption saturates geometric distance	107 sig metrics at max intensity	1d/esoteric_code.py

Binary Analysis

#	Finding	Effect Size	Investigation
26a	x86-64, ARM64, WASM, and Java highly distinct	91.0 avg sig metrics	1d/binary_anatomy.py
26b	Machine code has strong sequential structure	47-49 sig metrics vs shuffled	1d/binary_anatomy.py
26c	Packed data distinguishable from encrypted data	33 sig metrics	1d/binary_anatomy.py
26d	Control-flow flattening (obfuscation) massively detectable	84 sig metrics vs raw x86	1d/binary_anatomy.py
26e	Embedded high-entropy payloads (shellcode) detectable	61 sig metrics	1d/binary_anatomy.py

Compression

#	Finding	Effect Size	Investigation
27	bz2 output distinguishable from zlib/lzma	d = 7.75	1d/compression_algos.py
28	Compressed structured data ≠ random	d = 128 (coverage)	1d/compression_algos.py

Collatz Sequences

#	Finding	Effect Size	Investigation
29	All 7 Collatz encodings detected as non-random	73-103 significant metrics	1d/collatz.py
30	21/21 pairwise Collatz encoding distinctions	d = 7.4 (hailstone_small vs hailstone_large)	1d/collatz.py
31	Hailstone sequences carry sequential structure (destroyed by shuffling)	coverage ratio 0.31	1d/collatz.py
32	Parity sequence is purely marginal (survives shuffling)	all ratios ≈ 1.0	1d/collatz.py
33	3n+1 vs 5n+1 variant geometrically distinguishable	71 sig, d = 12.8	1d/collatz.py
34	Starting number magnitude affects geometric signature	90-98 sig metrics across ranges	1d/collatz.py
35a	Sharp phase transition at k=1→2 in (2k+1)n+1 family	86 sig (k=1) → 35 sig (k=2)	1d/collatz_deep.py
35b	Tropical slopes match theory exactly: log₂(3) for 3n+1	μ=1.5855 vs theory 1.5850	1d/collatz_deep.py
35c	Optimal delay embedding at τ=2, monotonic decrease beyond	78 sig at τ=2, 42 at τ=21	1d/collatz_deep.py
35d	U-shaped bitplane signal: both LSB and MSB carry structure	LSB=77, MSB=68 sig metrics	1d/collatz_deep.py
35e	Ordering matters most at mod 4 (not mod 2 or higher)	63 metrics (mod 4) vs 40 (mod 2)	1d/collatz_deep.py
36a	Syracuse encoding of v₂ sequence is the richest Collatz representation	105 sig metrics	1d/collatz_deep2.py
36b	Mean v₂(3n+1) = 2.001, exactly matching Geometric(1/2) theory	empirical vs theoretical	1d/collatz_deep2.py
36c	45 metrics are convergence-specific (sig for 3n+1, vanish for 5n+1)	45 vs 3 divergence-specific	1d/collatz_deep2.py
36d	Fisher, Heisenberg, Sol, Spherical, Wasserstein are 100% convergence-aware	all sig metrics vanish at k=2	1d/collatz_deep2.py
36e	Composition order matters: DE2→BP0 beats BP0→DE2 (86 vs 75 sig)	exceeds both individual baselines	1d/collatz_deep2.py
36f	37 metrics track drift rate across (2k+1)n+1 with |r| > 0.8	kurt_mean r=+0.986	1d/collatz_deep2.py

Prime Numbers

#	Finding	Effect Size	Investigation
37a	All 7 prime encodings detected as non-random	49-100 significant metrics	1d/primes.py
37b	21/21 pairwise encoding distinctions	min 71 sig (last_digit vs binary), max 109 sig	1d/primes.py
37c	Prime gaps distinguishable from Cramér random model	55 sig metrics	1d/primes.py
37d	Prime gaps distinguishable from semiprime gaps	80 sig metrics	1d/primes.py
37e	All 7 encodings have ordering-dependent structure	orig vs shuf: 12-83 sig	1d/primes.py
37f	Gap geometry evolves with prime size	all 6 range pairs: 68-82 sig	1d/primes.py
37g	gap_pairs is richest encoding (100 sig vs random)	Torus chi2 d=441, Lorentzian d=307	1d/primes.py
38a	52 metrics detect pure primality (sig vs both Cramér AND random)	2-adic mean_distance |d|=128	1d/primes_deep.py
38b	Heisenberg and Algebraic geometries are 100% Cramér-sensitive	all sig metrics also distinguish from Cramér	1d/primes_deep.py
38c	Model hierarchy closes the gap: Cramér=54 → Even-gap=37 → Sieved=30 → Dist-matched=14	progressive improvement	1d/primes_deep.py
38d	14 metrics are pure sequential correlation in prime gaps	Lorentzian causal_order d=10.5	1d/primes_deep.py
38e	Delay embedding does NOT amplify real-vs-Cramér signal	47-52 sig across τ=1-10 vs 54 raw	1d/primes_deep.py
38f	Cramér gap shrinks with scale: 75 (1K) → 54 (1M)	31 always-sig, 61 scale-dependent	1d/primes_deep.py
38g	14 dist-matched survivors confirmed: pure sequential correlation in prime gaps	Lorentzian d=10.5, HOS:perm_entropy d=4.2	1d/primes_deep2.py
38h	Prime gaps are time-irreversible (4 sig forward vs reversed)	Spiral:growth_rate d=+4.01 is strongest asymmetry detector	1d/primes_deep2.py
38i	57% linear, 43% nonlinear sequential structure in prime gaps	IAAFT: 6 sig (nonlinear), block-shuffle: 3 sig	1d/primes_deep2.py
38j	mod6=1 vs mod6=5 gaps geometrically distinct (Lemke Oliver–Soundararajan)	46 sig; Projective ℙ² d=−5.90 top discriminator	1d/primes_deep2.py
38k	Both residue classes retain independent sequential structure	mod6=1: 9 sig, mod6=5: 7 sig vs dist-match	1d/primes_deep2.py
38l	7/14 survivors persist at ALL scales (1K through 1M) — fundamental	Lorentzian (3), Aperiodic (3), HOS:perm_entropy (1)	1d/primes_deep2.py
38m	Sieved Cramér model has lowest KL divergence (0.011) but HIGHEST framework detection (83 sig)	Transition structure and geometric structure are orthogonal	1d/prime_protocol.py
38n	Prime gap state machine dissolves with scale: determinism 0.28→0.14, H_cond 2.88→3.42 bits (10³→10⁷)	Transition structure is a finite-range artifact, not fundamental	1d/prime_protocol.py

Number Theory

#	Finding	Effect Size	Investigation
39a	All 8 arithmetic function encodings detected as non-random	50-105 sig metrics	1d/number_theory.py
39b	Even degenerate μ(n) (3 values) and λ(n) (2 values) strongly detected	μ=78, λ=71 sig metrics	1d/number_theory.py
39c	All 28 pairwise encoding distinctions	min 75 (μ vs λ), max 118 (ζ vs d)	1d/number_theory.py
39d	Mertens function has 37 metrics beyond random walk model	E8 diversity d=4.62, Clifford regularity d=-3.79	1d/number_theory.py
39e	d(n) has 85 metrics beyond distribution-matched model	Cantor max_gap d=64.7; massive sequential correlation	1d/number_theory.py
39f	Ω(n) has 88 metrics beyond Erdős–Kac normal approximation	Heisenberg xy_spread d=-54.8	1d/number_theory.py
39g	Zeta zero spacings have 90 metrics beyond Wigner surmise (GUE)	Fisher jeffreys d=-24.3, Torus entropy d=22.5	1d/number_theory.py
39h	ALL 8 encodings are ordering-dependent	d(n) strongest (57 orig vs shuf), λ weakest (8)	1d/number_theory.py
39i	d(n) and Ω(n) detection stable across 4 decades of scale	99-107 sig across 1K-1M	1d/number_theory.py

Continued Fractions

#	Finding	Effect Size	Investigation
40a	All 5 constants' CF coefficients detected as non-random	√3=100, √2=96, π=89, e=87, ln2=86 sig	1d/continued_fractions.py
40b	π CF coefficients indistinguishable from iid Gauss-Kuzmin — geometry confirms "almost all" applies to π	0 sig vs GK surrogates	1d/continued_fractions.py
40c	ln 2 shows weak sequential signal beyond GK distribution	11 sig vs GK	1d/continued_fractions.py
40d	e strongly distinguishable from GK (deterministic [1,2k,1] pattern)	65 sig vs GK	1d/continued_fractions.py
40e	π and ln 2 have zero ordering dependence (orig vs shuffled)	π=0, ln2=0 sig	1d/continued_fractions.py
40f	√3 ordering = 48 sig (period-2 destroyed by shuffling); √2 = 0 (constant, perfect control)	validates methodology	1d/continued_fractions.py
40g	π vs ln 2 = only 4 sig pairwise (both look GK-like)	weakest pair among 10	1d/continued_fractions.py
40h	Algebraic vs transcendental CFs: 96-100 sig pairwise	√2 vs π=98, √3 vs π=100	1d/continued_fractions.py
40i	Khinchin's constant: π geo_mean=2.6624 ≈ K=2.6854, ln 2 geo_mean=2.6266	both approach K	1d/continued_fractions.py
40j	√3 delay embedding peaks at τ=2, matching its period-2 CF structure	τ=2 → 103 sig (vs 101 at τ=1)	1d/continued_fractions.py

Unsolved Problems: Goldbach's Comet

#	Finding	Effect Size	Investigation
41a	Goldbach g(2n) massively non-random	86 sig metrics vs random	1d/unsolved.py
41b	Hardy-Littlewood prediction captures most but not all structure	17 sig metrics beyond HL (E8, HOS lead)	1d/unsolved.py
41c	g(2n) has strong sequential correlations (destroyed by shuffling)	51 sig vs shuffled	1d/unsolved.py
41d	g(2n) sequential structure beyond marginal distribution	56 sig vs distribution-matched	1d/unsolved.py
41e	HL predicts g(2n) with r=0.992 but systematic underestimate (+19 residual)	mean residual = +19.07	1d/unsolved.py
41f	Goldbach structure robust across scales (n=100 to n=200K)	17-32 sig vs HL across ranges	1d/unsolved.py

Preprocessing

#	Finding	Effect Size	Investigation
40	Delay embedding gives 52x improvement for lag detection	d: 0.17 → 9.1	Delay embedding study
41	FFT and raw analysis are complementary (17+20 exclusive pairs)	varies	Spectral study
42	kurt_mean (4th order) is independent from all 23 geometries	max r = 0.22	Higher-order study
43	Permutation entropy: 33x better for recurrence detection	d = 250 vs 7.5	Higher-order study

2D Spatial Field (8 geometries, 80 metrics)

#	Finding	Effect Size	Investigation
44	Ising model: all temperatures vs random distinguished	varies	2d/ising.py
45	Ising: multiscale_coherence_4 peaks near T_c	scale-free structure	2d/ising.py
46	Reaction-diffusion: 15/15 morphology pairs	d = 97 (tension_std)	2d/reaction_diffusion.py
47	Percolation: 28/28 probability pairs	d = 261 (n_basins)	2d/percolation.py
48	Cellular automata: 14/15 rule pairs	d = 78.5 (anisotropy)	2d/cellular_automata.py
49	ECB penguin detected in 2D, CBC/CTR invisible	d = -283	2d/ecb_penguin.py
50	Maze algorithms: 15/15 pairs	d = 108.9	2d/mazes.py
51	Wave equation: 15/15 source configs	d = -496	2d/wave_equation.py
52	Voronoi tessellations: 10/10 point process pairs	d = -154	2d/voronoi.py
53	Growth models: 3/3 (DLA/Eden/random)	d = -178	2d/growth_models.py
54	Sandpile SOC: 5/6 pairs, convergence detected	d = 261	2d/sandpile.py
55	Lenia continuous CA: 15/15 configs	d = 249	2d/lenia.py
56	Near-identical rules detected: GoL ≈ HighLife, Kruskal ≈ AldousBroder	d ≈ 0	Various 2D

Structure Atlas (179 sources, 16 domains)

#	Finding	Effect Size	Investigation
57	179 data sources from 16 domains mapped into 233-metric structure space	8.9 effective dimensions, PC1+2 = 40.0%	1d/structure_atlas.py
58	Cross-domain twins: EEG Eyes Closed ↔ Bearing Outer (d=0.084), NASDAQ ↔ Accel Stairs (d=0.16), Kepler Non-planet ↔ Accel Sit (d=0.17)	cosine distance in z-scored metric space	1d/structure_atlas.py
58a	Synthetic DNA ↔ DNA Chimp (d=0.001) — synthetic generator essentially nails real chimp DNA	closest pair across all 179 sources	1d/structure_atlas.py
58b	fBm H=0.3 ↔ fBm H=0.7 (d=0.002) — two Hurst exponents geometrically indistinguishable	despite Hurst parameter being a key discriminator in 1-on-1 tests	1d/structure_atlas.py
59	DNA (synthetic and real) forms isolated cluster (d=0.001-0.05 within, d>0.4 to everything else)	4 bio sources, uniquely distinctive	1d/structure_atlas.py
60	White Noise ≈ AES Encrypted (d=0.074) — pseudo-random cluster confirmed	Cluster 5: chaos + noise + AES + gzip + Pi	1d/structure_atlas.py
61	Surrogate decomposition: ECG Supraventr. most sequential (87/233 metrics disrupted by shuffling)	White Noise/AES = 0 disrupted	1d/structure_atlas.py
61a	Pi (base 256) has more metrics disrupted by rolling (69) than by shuffling (45)	positional structure beyond sequential autocorrelation	1d/structure_atlas.py
61b	Chaos maps are time-asymmetric: Henon (26 rev), Logistic (25), Tent (23) metrics disrupted by reversal	consistent with iterative map's forward direction	1d/structure_atlas.py
62	Financial returns time-asymmetric at 12K bytes (Nikkei: 18, NYSE: 17, NASDAQ: 15 metrics disrupted by reversal)	markets have geometric time arrow	1d/structure_atlas.py
63	54% of all nearest-neighbor pairs (90/168) cross domain boundaries	structure space reflects universal motifs, not domain	1d/structure_atlas.py
64	Multi-scale: ECG Normal detectable at all scales (92-101 sig at 256-8192 bytes), DNA Human rock-solid (90-98)	NYSE: 50→87 at 8192 (financial needs longer windows), White Noise: 0 at all scales	1d/structure_atlas.py

Bearing Fault Diagnosis

#	Finding	Effect Size	Investigation
65	CWRU bearing faults: all 4 conditions (Normal/Ball/Inner/Outer) distinguished	pairwise: 39-99 sig metrics	1d/bearing_fault.py
66	Mandelbrot/Julia fractal metrics are inner-race specialists	9/10 fractal metrics sig for Inner (d=1.9-5.5), 3/10 for Outer	1d/bearing_fault.py
67	Mandelbrot interior_fraction distinguishes Inner from Normal (0.60→0.90)	d=5.5	1d/bearing_fault.py

Mathematical Constants

#	Finding	Effect Size	Investigation
68	Base-256 digits of Pi/e/Phi/Sqrt2 indistinguishable from each other but ALL differ from white noise	0 sig pairwise, 84-85 sig vs random	1d/math_constants.py
69	CF taxonomy: every pair of constants' CF terms massively distinguishable	32-118 sig pairwise (6 constants)	1d/math_constants.py
70	CF(Pi) vs Gauss-Kuzmin i.i.d. = only 5 sig — barely detectable sequential structure	consistent with Pi CF being "almost all"	1d/math_constants.py
71	CF(e) vs shuffled = 69 sig — deeply patterned transcendental	[2,1,2,1,1,4,1,1,6,...] pattern	1d/math_constants.py
72	CF(ln2) is the most distinctive CF (118 sig vs algebraics)	more distinctive than CF(Pi)	1d/math_constants.py
73	Same constant (Pi), 4 representations produce wildly different fingerprints	52-106 sig pairwise (base-256/base-10/binary/CF)	1d/math_constants.py
74	Algebraic boundary: CF(Sqrt3)=52 sig vs shuffled (period-2), CF(Sqrt2/Sqrt5/Phi)=0 (constant)	CF(e)=69, CF(Pi)=9, Gauss-Kuzmin=0	1d/math_constants.py

Sorting Algorithms

#	Finding	Effect Size	Investigation
75	Memory access traces of sorting algorithms are geometrically distinguishable	merge/quick/heap/radix/bubble/insertion	1d/sorting_algorithms.py
76	Pivot strategy (quicksort) is near-null (4-7 sig) — low-level choices barely visible	algorithm choice >> implementation detail	1d/sorting_algorithms.py

Music Theory

#	Finding	Effect Size	Investigation
77	Musical intervals and scales detectable via geometry	harmonic structure fingerprinted	1d/music_theory.py

Network Protocols

#	Finding	Effect Size	Investigation
78	Network protocol byte streams have distinct geometric signatures	protocol fingerprinting	1d/network_protocols.py

Neuroscience / EEG

#	Finding	Effect Size	Investigation
79a	EEG spectral peaks show reproducible geometric lattice alignment across 109 subjects	per-subject excess enrichment +0.305, t=13.89, p<0.0001; 101/109 positive	1d/eeg_phi.py
79b	φ (golden ratio) is NOT uniquely preferred as a spectral scaling ratio — ranks 5th of 12 ratios tested	excess: φ=+0.308, 5/3=+0.327, π=+0.554, 2=+0.446, e=+0.357	1d/eeg_phi.py
79c	Claimed f₀=7.5 Hz does not reach 95% significance vs phase-rotation null	p>0.05; Schumann 7.83 Hz shows higher excess (+0.384 vs +0.266)	1d/eeg_phi.py
79d	2D (f₀, r) joint heatmap: global optimum at (4.94, 3.92), far from claimed (7.5, φ)	score at optimum +0.913 vs +0.342 at claim	1d/eeg_phi.py
79e	Per-band: only gamma (31.8–51.4 Hz) significant (p=0.024); alpha (p=0.277) and high beta (negative enrichment) fail	φⁿ claim predicts ALL bands	1d/eeg_phi.py
79f	φ structure survives IAAFT surrogation → spectral property, not nonlinear phase coupling	p=0.127	1d/eeg_phi.py
79g	Noble-position test (u=1/φ=0.618): marginal, does NOT reach significance	p=0.060 (w=0.05), p=0.075 (w=0.15)	1d/eeg_phi.py
79h	φ rank improves to #1/12 under noble-position metric (from #5/12 at u=0.5)	but p=0.071 — trend, not evidence	1d/eeg_phi.py
79i	Kuiper omnibus: phase non-uniformity is NOT f₀-specific (phase-rotation p=0.614)	alpha dominance explains all non-uniformity	1d/eeg_phi.py
79j	Phase target sweep: actual enrichment peak at u=0.575, between both theoretical predictions	neither PKL (u=0.5) nor noble (u=0.618) is correct target	1d/eeg_phi.py
79k	D4 rescan with noble metric: optimum at (4.33, 4.00), still far from claimed (7.5, φ)	new metric does not rescue claimed parameters	1d/eeg_phi.py
79l	FOOOF extraction: phi enrichment weakens (p=0.066→0.352), φ rank #1→#3, per-subject t=8.05→1.99	effect is extraction-method-dependent	1d/eeg_phi.py
79m	Subject-averaged PSD FOOOF: p=0.581 (732 peaks, 6.7/subject)	standard FOOOF practice yields non-significant result	1d/eeg_phi.py
79n	Alpha-only FOOOF per-ch, u=0.618, w=0.15: p=0.025 — sole significant combination tested	Kuiper p=0.513; φ rank #4/12; one point in a large analytical space	1d/eeg_phi.py
79o	FOOOF parameter sensitivity: p=0.28–0.44 across all settings on averaged PSDs	stable across parameter choices	1d/eeg_phi.py
79p	Phase target sweep differs between methods: FOOOF u=0.880, medfilt u=0.575	enrichment pattern is extraction-dependent	1d/eeg_phi.py
79q	Bonn dataset (non-motor-imagery): phi enrichment absent in healthy resting EEG	Eyes Closed p=0.64, Eyes Open p=0.78; φ rank #4/12 with negative excess	1d/eeg_phi.py
79r	Seizure shows strongest (non-significant) Bonn phi trend — opposite of prediction	Seizure p=0.13 (medfilt), p=0.12 (FOOOF); healthy should be strongest if real	1d/eeg_phi.py
79s	Motor-imagery objection refuted: phi absent in Bonn clinical EEG (no motor tasks)	Kuiper phase-rot p=0.36 on healthy pooled; dataset choice is not the issue	1d/eeg_phi.py
80a	All 8 EEG classes (5 Bonn + 3 PhysioNet) massively non-random	146-178/235 sig metrics vs shuffled; self-check 0/235	1d/eeg_geometry.py
80b	Seizure EEG has massive nonlinear structure (survives IAAFT)	71/235 sig vs IAAFT; healthy EO=24, eyes closed=9	1d/eeg_geometry.py
80c	PhysioNet resting EEG has ZERO nonlinear structure — entirely spectral	Occipital=0, Frontal=0, Central=0 vs IAAFT	1d/eeg_geometry.py
80d	PhysioNet brain regions (occipital/frontal/central) geometrically indistinguishable	Occ vs Frontal=0 sig, Occ vs Central=1 sig	1d/eeg_geometry.py
80e	Bonn vs PhysioNet classes highly distinguishable (dataset effect dominates)	42-146 sig pairwise across datasets	1d/eeg_geometry.py
80f	Seizure most discriminable from normal resting (Central)	134 sig metrics (Seizure vs Central)	1d/eeg_geometry.py
80g	All 44 geometries detect structure in at least 1 EEG class	1449 total detections; E8, Recurrence, Multifractal lead	1d/eeg_geometry.py
80h	Resting EEG neighbors in structure space: ARMA, Pink Noise, Bearing Normal	linear stochastic processes; confirms D3 zero-nonlinear result	1d/eeg_geometry.py
80i	Seizure EEG neighbors: Bearing Outer, Kuramoto Oscillators	nonlinear dynamical systems; consistent with ictal dynamics	1d/eeg_geometry.py
81a	Signature distance matrix has massive spatial structure	77/80 spatial metrics sig vs shuffled; d up to +76	2d/meta_geometry.py
81b	Metric correlation field has spatial structure	71/80 spatial metrics sig; block-diagonal by geometry family	2d/meta_geometry.py
81c	Mean 15 sig metrics pairwise across 16 domains	binary and bio most distinguishable; geophysics/quantum least	2d/meta_geometry.py
81d	Top discriminating geometry: Cantor Set (F=8.80), metric: mean_gap (F=21.7)	gap structure varies most across domains	2d/meta_geometry.py
81e	Bottom discriminators: quasicrystal geometries (F=1.6–2.0)	QC metrics uniform across domains — most data lacks QC structure	2d/meta_geometry.py
81f	Effective dimensionality 10.3; 11 SVs above Marchenko-Pastur; 90% at dim 25	metric space genuinely high-dimensional, not reducible	2d/meta_geometry.py
81g	Fibonacci Word ≈ L-System Algae (d=0.0025) — framework detects substitution equivalence	both are Lindenmayer systems; geometric identity validates framework	2d/meta_geometry.py
81h	AES ≈ Gzip ≈ Bzip2 ≈ LCG ≈ White Noise cluster (d≈0.01)	maximal entropy sources geometrically indistinguishable — correct	2d/meta_geometry.py
81i	Random Telegraph most isolated source (NN d=0.65)	Poisson-switched binary has no geometric peer in atlas	2d/meta_geometry.py
81j	LOO 5-NN domain classification: 47.5% (vs 6.25% random)	framework captures domain structure but geometric clusters ≠ provenance labels	2d/meta_geometry.py
81k	Bio 100% classified, number_theory 0%, geophysics 0%	bio has coherent geometric signature; number_theory/geophysics too heterogeneous	2d/meta_geometry.py

Astrophysics

#	Finding	Effect Size	Investigation
82a	GW150914 detected with massive significance in ASD-whitened strain (no template)	H1: 91 Bonferroni-sig metrics (α=2×10⁻⁴), L1: 58	1d/gravitational_waves.py
82b	54 metrics coincident in both detectors, all same-sign — zero opposite-sign	21 geometry families; sign coherence rules out artifacts	1d/gravitational_waves.py
82c0	Top discriminators: Mandelbrot escape_time_variance, HOS kurt_max, Zariski nonsep_fraction	H1 d=−14.9, L1 d=+20.3 (kurt_max); p < 10⁻⁵⁰	1d/gravitational_waves.py
82c1	Chirp geometric profile: low entropy, narrow vocabulary, high kurtosis, smooth phase-space	entropy/vocabulary negative d, kurtosis/Zariski/symplectic positive d	1d/gravitational_waves.py
82c2	Raw strain produces null (max |d|=1.0) — correct negative control	seismic noise dominates; whitening essential	1d/gravitational_waves.py
82c	Artificial SETI signals detectable at SNR -13 dB	0.05 linear SNR	1d/seti.py
82d	Exotic geometry outperforms spectral features for Spread Spectrum/Chaos	64 sig vs 1 sig (spectral) @ SNR 0.5	1d/seti.py
82e	Geometry detects 7/7 artificial signal types in "Cocktail" noise	Colored + RFI	1d/seti.py

AI / Text

#	Finding	Effect Size	Investigation
83a	AI vs Human text massively distinguishable at raw byte level (UTF-8)	112/252 sig metrics, d > 6.0	1d/llm_text_detection.py
83b	100% classification accuracy (LOO 5-NN) on 16KB windows	60/60 correct	1d/llm_text_detection.py
83c	Fractal geometry is top discriminator: AI text is "smoother"	Mandelbrot interior d=+6.09, escape_time d=+5.86	1d/llm_text_detection.py
83d	Higher-Order Statistics (Kurtosis) discriminates AI text	d = +5.62 (AI distribution more peaked)	1d/llm_text_detection.py

Negative Results (27)

These are equally important — they define the boundaries of what geometric analysis can and cannot do.

#	Finding	Implication
1	AES-CTR indistinguishable from random across ALL geometries, preprocessings, higher-order stats, and bit planes	Fundamental limit; AES works
2	MT19937 passes (0 sig after metric pruning; was 1 borderline)	Byte-level structure undetectable
3	XorShift128 undetected raw (0 sig), but DE2 reveals 2 metrics (note: XorShift32 now detected by Klein Bottle linear_complexity, see #13k)	Thurston height_drift exposed by delay embedding; 128-bit state too large for BM at 4K bytes
4	MINSTD passes (0 sig after metric pruning; was 1 borderline)	Byte-level LCG structure undetectable
4a	SFC64 and PCG64 completely indistinguishable from os.urandom	Modern PRNGs are geometrically random
5	RC4 stream looks random (even 24-bit key)	Stream cipher output is geometrically indistinguishable
6	Standard map ≈ random RECLASSIFIED → Positive #18a (44 sig with new Predictability/Cayley geometries)	Mixing ≠ random; deterministic skeleton detected
7	Arnold cat map ≈ random RECLASSIFIED → Positive #18b (19 sig with new Predictability/SpectralGraph geometries)	Anosov mixing ≠ random
8	LSB steganography invisible (0 sig after metric pruning; was 1 borderline)	Byte-level changes too small for all geometries
8a	Matrix embedding completely invisible to all geometries (confirmed 1D and 2D)	d=0.00 across 131 metrics, 4 representations, all rates
8b	Bitplane extraction does not improve stego detection	8x sample reduction destroys statistical power
9	GBM indistinguishable from IID normal	Random walk is geometrically random
10	Ornstein-Uhlenbeck ≈ IID normal	Mean-reversion too subtle for byte encoding
11	Random vs encrypted is the hardest classification boundary	Both designed to look random
12	Multi-scale analysis doesn't improve general classification	69.5% < 79.5% baseline
13	Cepstrum mostly useless after uint8 quantization	Information crushed
14	GARCH hard to detect after byte quantization RECLASSIFIED → Positive #2a (29 sig with Symplectic/Lorentzian/G2)	CDF-encoded returns detectable; volatility clustering has geometric signature
15	GoL ≈ HighLife	Near-identical rules (differ only in B6)
16	Sandpile 10k ≈ 50k iterations	Both at SOC steady state
17	Kruskal ≈ Aldous-Broder maze generation	Both produce uniform spanning trees
18	π CF coefficients indistinguishable from iid Gauss-Kuzmin (0 sig, 0 ordering)	Geometry confirms "almost all" applies to π
19	π digits indistinguishable from random in base-256 AND base-10 (0 sig vs random, 0 vs shuffled, 0 across positions 0K-48K, 0 under delay embedding τ=1-5)	Strong geometric evidence for normality of π
19a	e and √2 digits also indistinguishable from random (same battery: all 0 sig)	Evidence extends to e and √2
20	Protein sequential ordering is negligible — globular vs shuffled = 1 sig, IAAFT = 0 for both classes	Glob-vs-IDP is compositional (AA frequencies), not sequential
21	EEG φⁿ lattice not supported — φ ranks #5/12 at u=0.5, #1/12 at u=1/φ (p=0.06); Kuiper omnibus p=0.614; FOOOF extraction weakens signal (p=0.35); one (method×band×window×target) combination reaches p=0.025 but fails Kuiper	Real spectral organization exists but is not robustly φ-specific across extraction methods or omnibus tests
21	CF(Sqrt2) vs shuffled = 0 sig — constant sequence shuffled is identical (1d/math_constants.py)	Validates methodology (not a failure)
22	Base-10 digits of Pi/e/Phi/Sqrt2 all indistinguishable from each other (0 sig) (1d/math_constants.py)	Normal number digits = i.i.d. uniform
23	GF(256) Reed-Solomon syndrome metrics dead — syndrome_weight F=0.6, syndrome_entropy r=1.000 with 2-adic:mean_distance; positive control failed (sub-block syndromes non-zero even for codewords) (1d/syndrome_explore.py)	GF(256) algebra is orthogonal to real-valued structure by design
24	D16 (16D) and Leech lattice (24D) redundant with existing framework — D16_unique_roots r=0.983 with D4 Triality, Leech_unique_roots r=0.981 with Boltzmann; Leech_lattice_dist r=0.924 closest to passing (1d/leech_explore.py)	Higher-D lattice projections remeasure the same distributional properties E8/D4 already capture
25	Protocol informatics metrics mostly redundant — 4/5 killed: determinism_idx r=0.935, time_asymmetry r=0.919, state_compression ≡ determinism_idx, nw_alignment r=0.955 (1d/protocol_orthogonality.py)	State machine metrics on byte sequences largely recaptured by existing dynamical/entropy metrics
26	Markov kl_from_iid dead — r=0.996 with existing metrics (essentially identical to entropy_rate) (1d/markov_explore.py)	KL divergence from IID already measured by Information geometry
27	NW sequence alignment is noise — real/shuffled score ratio = 1.00 at all prime gap ranges; no conserved motifs (1d/prime_protocol.py)	Needleman-Wunsch alignment inapplicable to non-biological sequences

Key Takeaway

The framework detects genuine structure with large effect sizes (d = 7-266) while producing zero false positives on validated random sources (self-check: 1/200 at Bonferroni, within expected FPR). The AES-CTR negative result confirms that the methodology is honest — geometries report "no structure" when encryption is working correctly. A 2026-03-05 re-evaluation of all 25 negative results (negative_reeval.py) reclassified 3 former negatives as positives: the Standard Map and Arnold Cat Map — previously indistinguishable from random — are now detected by Predictability and Cayley/SpectralGraph geometries (44 and 19 sig respectively), and GARCH(1,1) returns are now detectable via Symplectic and Lorentzian geometries (29 sig). The remaining 22 negatives held, and 4 borderline detections (MT19937, MINSTD, LSB stego, protein ordering) dropped to 0 sig after metric pruning — the framework got both sharper and cleaner. The Structure Atlas investigation (structure_atlas.py) mapped 199 data sources from 16 domains into the 200-metric structure space and found it has 9.2 effective dimensions (PC1+2 = 39.1%) — the framework's 200 metrics (across 54 geometries) span a compact but sufficient space to classify diverse real-world data. 54% of all nearest-neighbor pairs cross domain boundaries, revealing that the framework captures universal structural motifs rather than domain-specific artifacts. Cross-domain twins (EEG Eyes Closed ↔ Bearing Outer d=0.084, NASDAQ ↔ Accel Stairs d=0.16) share geometric profiles despite having nothing in common physically.

The 2D spatial geometry battery (8 geometries, 80 metrics) demonstrates that genuinely different mathematical lenses — differential geometry (Surface), algebraic topology (Persistent Homology 2D), complex analysis (Conformal 2D), integral geometry (Minkowski Functionals), scaling analysis (Multiscale Fractal 2D), Hodge theory (Hodge-Laplacian), and spectral analysis (Spectral Power 2D) — each contribute unique discriminative power. On stego co-occurrence matrices, PVD detection jumps from 13/15 (Spatial Field alone) to 49/80 (all 8 geometries). On 10 diverse field types, all 45 pairs are distinguished.

The deep Collatz investigations (collatz_deep, collatz_deep2) demonstrate a particularly striking application: five specific geometry families (Fisher, Heisenberg, Sol, Spherical, Wasserstein) detect convergence-specific structure in 3n+1 that categorically vanishes in divergent variants. The convergence mechanism has a geometric character — information-geometric, nilpotent, solvable — that is absent, not merely attenuated, in divergent maps.

The deep prime gap investigations (primes_deep.py, primes_deep2.py) trace the sequential structure of prime gaps to its roots. Of 131 metrics, 14 survive distribution-matching — detecting ordering that no marginal model explains. These 14 decompose into 57% linear (autocorrelation) and 43% nonlinear (IAAFT-surviving) components. Prime gaps are temporally irreversible (4 sig metrics forward vs reversed), consistent with the sieve's inherent directionality. Residue classes mod 6 are massively geometrically distinct (46 sig), with both classes retaining independent sequential structure — the Lemke Oliver–Soundararajan bias made geometric. Seven of the 14 survivors persist from primes near 1K through 1M, establishing them as fundamental features of prime distribution rather than small-prime artifacts.

The number theory investigations reveal that classical limit theorems leave substantial geometric structure unexplained. The Mertens function M(n) — whose random-walk behavior is equivalent to RH — has 37 metrics beyond what a random walk with matching step probabilities can produce. Zeta zero spacings differ from the GUE/Wigner prediction in 90 metrics. Even the divisor function d(n) has 85 metrics of sequential correlation destroyed by shuffling. These results suggest that exotic geometries detect multiplicative number-theoretic structure that standard probabilistic models do not capture.

The continued fractions investigation provides a striking validation of Khinchin's theorem: π's CF coefficients are geometrically indistinguishable from iid Gauss-Kuzmin samples (0 sig, 0 ordering dependence), confirming that "almost all" applies to π as far as 131 exotic geometric metrics can detect. ln 2 shows a faint crack (11 sig vs GK), while algebraic constants are trivially distinguishable. The √2 control (constant sequence, 0 ordering sig) and random self-check (0 sig) validate the methodology.

The EEG golden ratio investigation (eeg_phi.py) demonstrates the framework's value as a hypothesis-testing tool for external claims. Using 109 subjects from PhysioNet and a phase-rotation permutation null, the investigation found that while EEG spectral peaks do show reproducible geometric organization (101/109 subjects positive, t=13.89), the specific claim of φⁿ lattice structure anchored at f₀=7.5 Hz is not supported. The golden ratio ranks 5th among 12 tested scaling ratios, the claimed f₀ is not the optimal anchor, and a 2D joint (f₀, r) heatmap shows the claimed parameters sitting in a mediocre region of the landscape. A follow-up (D9) tested the alternative hypothesis that enrichment should be measured at u=1/φ=0.618 (the noble position from mode-locking avoidance theory) rather than u=0.5 (band centers). This is theoretically motivated and more favorable to φ — it improves φ's ratio ranking from #5 to #1 of 12. But the effect remains marginal (p=0.06), a Kuiper omnibus test shows the phase non-uniformity is entirely explained by the peak frequency distribution rather than f₀-specific alignment (phase-rotation p=0.614), and the (f₀, r) parameter space still does not favor the claimed values. The lesson: proper null models (phase-rotation instead of uniform) and comparison against alternative hypotheses (multiple ratios, not just φ) are essential for claims of specific mathematical organization in biological data.

The RNG quality testing investigation (rng.py) provides a clean validation story: 10 generators spanning a quality gradient from cryptographic to historically broken. CRYPTO/GOOD generators return 0 significant metrics (self-check urandom vs urandom also 0), while RANDU (44 sig) and Middle-Square (78 sig) are massively detected. Geometric metrics outperform standard statistical tests by 11x on the worst generators. Delay embedding newly reveals XorShift128 (undetected raw), and RANDU is detectable from just 500 bytes. Different weaknesses have distinct geometric fingerprints — Penrose quasicrystal metrics detect lattice structure (d=-37), while Higher-Order Statistics catches nonlinear correlations.

The unsolved problems investigation (unsolved.py) addresses two famous open questions. For normality of π: digits of π, e, and √2 are completely indistinguishable from random bytes across 131 geometric metrics, in both base-256 and base-10, at four different digit positions (0K-48K), and under delay embedding at τ=1-5. Combined with the CF result (π passes iid Gauss-Kuzmin), this is comprehensive geometric evidence for normality. For Goldbach's comet: g(2n) is massively structured (86 sig vs random), with strong sequential correlations (51 sig vs shuffled). The Hardy-Littlewood prediction captures most structure (r=0.992) but 17 metrics detect patterns beyond HL — primarily via E8 Lattice and Higher-Order Statistics. This gap persists across scales from n=100 to n=200K.

The gravitational wave investigation (gravitational_waves.py) applies the full 252-metric framework to LIGO GW150914 strain data, comparing a 200 ms event window against 500 background windows with Bonferroni correction (α = 2×10⁻⁴). In ASD-whitened data, 91 metrics are Bonferroni-significant in H1 and 58 in L1. Of these, 54 are significant in both detectors with perfect sign agreement — zero opposite-sign coincidences across 21 geometry families. Top effects reach d = −14.9 (Mandelbrot escape_time_variance, H1) and d = +20.3 (HOS kurt_max, L1). The coherent geometric portrait of a chirp: low entropy (predictable), narrow bigram vocabulary (narrow-band), high kurtosis (concentrated merger burst), smooth phase-space orbit (deterministic), high symplectic area variation (frequency sweep). Raw strain produces a null (max |d| = 1.0) — the correct negative control confirming that whitening is essential and the framework isn't hallucinating structure in seismic noise. This is template-free burst detection: the framework identifies structural anomaly without knowing what a gravitational wave looks like.