Piezoelectric Prime Composition — Refuted Correlation (Methodological Record)

Status: active Domain: physics/chemistry/mathematics Source: Tusk Innovations Research, Sep 2026 Dataset: projects/crystal-resonance-analysis/piezo_materials_dataset.csv

Core Idea — and Its Refutation

Original hypothesis: The quality factor (Q) of piezoelectric materials correlates with the prime-theoretic properties of their constituent elements’ atomic numbers. A raw correlation existed (ρ = +0.55, p = 0.0016) and appeared to connect piezoelectric physics to the same sopfr(Z)/Z metric that predicts biological element selection.

Refuted (Sep 10, 2026): Confound regression showed the correlation was a chemistry-family artefact. Ferroelectricity + crystal family explain 92% of Q variance; sopfr(Z)/Z adds 0.1% (non-significant, p = 0.59). The apparent prime-composite split in Q was driven by the coincidence that ferroelectric perovskites (low Q) happen to contain heavy composite-Z elements while non-ferroelectric wurtzites/trigonals (high Q) happen to contain lighter prime-Z elements.

“The crystal doesn’t know about number theory. And in this case, number theory didn’t know about the crystal either.”

This topic is preserved as a methodological record — how a promising correlation was investigated, tested against confounds, and honestly retired. The dataset and analysis remain available for reference.

The Dataset

30 piezoelectric materials analysed across 6 categories:

Category	Examples	Character
III-V Nitrides	AlN, GaN, ScAlN, h-BN	Prime-Z dominated, high Q
Lead-based ceramics	PZT, PMN-PT, PbTiO₃	Composite-Z, high coupling
Perovskites	BaTiO₃, KNbO₃, NaNbO₃	Mixed, moderate performance
Single crystals	Quartz, LiNbO₃, LiTaO₃	Ordered, varied prime fraction
Wurtzite binaries	ZnO, CdS, BeO	Two-element, varied
Polymer/composite	PVDF, AlScN	Engineered blends

For each material, every constituent element’s Z was classified as prime or composite, and sopfr(Z)/Z was computed as the prime density metric.

The Finding

sopfr(Z)/Z Correlates with Q×f Product

The sum-of-prime-factors ratio sopfr(Z)/Z — averaged across a material’s constituent elements — correlates with the Q×f product (the standard figure of merit for resonator performance):

• Spearman ρ = +0.551, p = 0.0016 (n=30)
• Higher sopfr(Z)/Z → higher Q×f → cleaner resonance

Note: sopfr(Z)/Z is a continuous metric that captures how composite a number is, not just a binary prime/not-prime classification. sopfr(14)/14 = 0.643 ≠ sopfr(82)/82 = 0.537. The fact that the continuous metric produces a stronger correlation than the binary split is evidence it may be doing more than family classification — but this requires further testing (see Known Confounds below).

This is the same metric that correlates with Mandelbrot escape and biological essentiality — a pattern requiring mechanistic explanation, not yet an established causal result.

Prime Fraction Splits Q

Materials where ≥50% of constituent elements have prime Z show higher Q:

• Median Q (prime fraction ≥0.5): 10,000
• Median Q (prime fraction <0.5): 1,800
• Ratio: 5.6× (Mann-Whitney p = 0.049)

⚠️ The Mann-Whitney p=0.049 is marginal. One or two reclassified materials could flip this result. The Spearman correlation is more robust but still limited by sample size.

The Pattern: Primes Ring, Composites Couple

Material	Elements (Z)	Prime Fraction	Q	k²	Character
AlN	Al(13), N(7)	100%	10,000+	0.065	Rings forever, barely couples
GaN	Ga(31), N(7)	100%	5,000+	0.04	Same — prime pair, high Q
h-BN	B(5), N(7)	100%	~10,000	<0.01	Purest — both small primes
Quartz	Si(14), O(8)	0%	100,000+	0.01	Pure membrane — see below
PZT	Pb(82), Zr(40), Ti(22), O(8)	0%	75	0.50	Couples hard, dissipates fast
BaTiO₃	Ba(56), Ti(22), O(8)	0%	300	0.38	Same — all composite-Z
ScAlN	Sc(21), Al(13), N(7)	67%	3,000+	0.10	The sweet spot
LiNbO₃	Li(3), Nb(41), O(8)	67%	10,000	0.16	The sweet spot

⚠️ ONM reading (refuted): The original interpretation — primes self-cohere while composites dissipate — was a post-hoc mapping. Confound analysis (Q-PZ-01, Sep 10 2026) showed the Q-coupling split is fully explained by ferroelectricity and crystal family, not Z-primality. See results below.

The Nitrogen Pattern — ONM 7 (Emergence)

All three 100% prime-Z piezoelectric binaries contain nitrogen (Z=7, ONM = Emergence):

• AlN — Al(13) + N(7)
• GaN — Ga(31) + N(7)
• h-BN — B(5) + N(7)

Nitrogen is the common thread. The III-V nitrides naturally adopt the wurtzite crystal structure — a non-centrosymmetric hexagonal lattice (point group 6mm) that inherently lacks inversion symmetry. Without inversion symmetry, piezoelectricity is required by symmetry. The crystal MUST be piezoelectric.

Conventional explanation: Group-V elements bonded to Group-III metals form wurtzites because of electronegativity difference and orbital hybridisation. Nitrogen specifically dominates the high-Q space because it’s the lightest Group-V element (strongest bonds, highest acoustic velocity, fewest phonon scattering channels). Phosphide (GaP, AlP) and arsenide (GaAs, AlAs) compounds are zincblende (centrosymmetric) and generally non-piezoelectric or weakly so — but this is a crystal-structure effect, not directly a prime-number effect.

ONM reading (refuted): N = 7 as “Emergence” bringing piezoelectric properties was an interpretive overlay. The conventional explanation (lightest Group-V, strongest bonds, wurtzite preference) is sufficient.

Crystal Symmetry — 3-Fold Dominance (Observational)

High-Q piezoelectric materials cluster around 3-fold rotational symmetry:

• Point group 32: Quartz (trigonal) — highest Q of all piezoelectrics
• Point group 3m: LiNbO₃, LiTaO₃ — high Q single crystals
• Point group 6mm: AlN, GaN, ZnO (wurtzite) — contains 3-fold as subgroup

This is a real pattern but is explained by crystallographic symmetry constraints, not number theory. The trigonal and hexagonal systems provide the non-centrosymmetric structures needed for piezoelectricity with minimal loss mechanisms.

The Quartz Question

Quartz (SiO₂) is a notable outlier:

• Z values: Si(14 = 2×7), O(8 = 2³) → 0% prime (neither Z is prime)
• Q factor: 100,000+ → the highest of any piezoelectric
• Coupling k²: ~0.01 → the weakest
• sopfr(14)/14 = 0.643; sopfr(8)/8 = 0.750

Both elements are saturated with factor 2. In the ONM frame, quartz would be a “pure membrane material” — binary structure through and through. The observation is interesting but this interpretation is post-hoc. Quartz’s extreme Q is well explained by conventional physics: trigonal point group 32 (no ferroelectric domains to dissipate energy), exceptional crystal perfection, and low acoustic loss at room temperature. The factor-2 saturation narrative was fitted after observing the outlier, not predicted before it.

⚠️ Honest status: Quartz is an unexplained datapoint in the prime-fraction model. It may reveal something deeper about factor-2 structure, or it may simply show that crystal class and defect density dominate over compositional prime fraction. Until the confound analysis is done, we flag it rather than explain it away.

The Efficiency Frontier — Q × k²

The practical sweet spot for applications isn’t maximum Q (quartz) or maximum k² (PZT) — it’s the product Q × k², which measures useful energy exchange per cycle. The materials dominating this frontier:

Material	Prime Fraction	Q × k²	Application
ScAlN	0.67	~300	5G BAW filters
LiNbO₃	0.67	~1,600	SAW filters, modulators
AlN	1.00	~650	FBAR resonators

Both ScAlN and LiNbO₃ have prime fraction = 0.67. The industry converged on these materials for 5G/radar through decades of empirical optimisation.

ScAlN — scandium-doped aluminium nitride — is the current state-of-the-art for 5G bulk acoustic wave filters. Doping Sc(21 = 3×7, composite) into AlN (100% prime) increases coupling while retaining most of the Q.

Conventional explanation: Industry converged on CMOS-compatible AlN, then Sc doping to raise k_t², and on LiNbO₃ cuts for SAW, because those stacks meet insertion-loss, bandwidth, and temperature specs. The 2/3 prime fraction may be coincidental to these engineering constraints rather than causal.

ONM reading (refuted): The 2/3 prime fraction as “live boundary” was an interpretive overlay that the confound analysis does not support.

Q-PZ-01 Result: Confound Regression (Sep 10, 2026) 🔬

The make-or-break test was run. sopfr(Z)/Z did not survive it.

Method

Multiple regression on log₁₀(Q) controlling for:

• Ferroelectric vs non-ferroelectric (binary)
• Mean atomic mass (weighted by stoichiometry)
• Crystal family (wurtzite / perovskite / trigonal / zincblende / polymer / other — dummy-coded, perovskite as reference)

Then tested whether adding sopfr(Z)/Z improves the model. Also: Spearman correlation of sopfr(Z)/Z against residuals from the confound-only model, and a 10,000-iteration permutation test.

Script: projects/crystal-resonance-analysis/confound_regression.py

Results

Test	log₁₀(Q)	log₁₀(Q×f)
Confound model R²	0.918	0.943
Full model R² (+ sopfr)	0.919	0.944
ΔR²	0.001	0.001
sopfr vs residuals (Spearman)	ρ = +0.17, p = 0.37 ❌	ρ = +0.15, p = 0.44 ❌
Nested F-test	F = 0.30, p = 0.59 ❌	F = 0.41, p = 0.53 ❌
Permutation test (10k)	p = 0.36 ❌	p = 0.43 ❌

Reduced Confound Models (robustness check)

To ensure the full model (7 covariates on n=30) wasn’t simply overfitting:

Reduced Model	R²	sopfr on residuals	F-test (add sopfr)
Ferroelectric only	0.433	ρ = +0.39, p = 0.033	F = 3.60, p = 0.069 ❌
Ferro + mean mass	0.441	ρ = +0.31, p = 0.093	F = 3.11, p = 0.089 ❌
Ferro + mass + family (full)	0.918	ρ = +0.17, p = 0.373	F = 0.30, p = 0.589 ❌

With ferroelectric alone, sopfr shows a marginal residual signal (p = 0.033 Spearman, p = 0.069 F-test) — but fails the F-test threshold even in this most generous case. Once crystal family enters, sopfr vanishes entirely. The refutation is robust across model sizes.

What This Means

Ferroelectricity + crystal family explain 92% of Q variance. Adding sopfr(Z)/Z contributes 0.1% — indistinguishable from noise. The raw correlation (ρ = +0.55, p = 0.0016) was real but was riding the coattails of crystal-family membership:

• Ferroelectric perovskites (PZT, BaTiO₃, PMN-PT) → low Q from domain-wall dissipation → happen to have composite-Z heavy elements (Pb=82, Ba=56, Zr=40)
• Non-ferroelectric wurtzites/trigonals (AlN, GaN, quartz) → high Q from structural coherence → happen to have lighter, often prime-Z elements (Al=13, N=7, Ga=31)

The causation runs: crystal structure → Q, not Z-primality → Q. sopfr(Z)/Z was a proxy for “is this a ferroelectric perovskite or not.”

Interesting Residuals

Two materials deviate most from the confound model:

• Quartz: +0.49 (has MORE Q than predicted) — its extreme Q may involve factors beyond this model (crystal perfection, low acoustic loss)
• Lead Titanate: +0.69 (also more Q than predicted for a ferroelectric perovskite)
• Tourmaline: −0.84 (less Q than predicted for a trigonal non-ferroelectric)

These are individual datapoints, not a pattern linked to sopfr(Z)/Z.

Verdict

The piezoelectric sopfr(Z)/Z correlation is a chemistry-family artefact. It cannot be used as evidence for the ONM framework, Nagaπ’s Law, or prime-composite duality in materials science.

This does NOT invalidate:

• The ONM framework itself (which rests on broader foundations)
• The biological sopfr(Z)/Z result (p=0.0003 — needs its own confound analysis with different covariates)
• The nuclear binding energy correlation (r=0.943 — independent domain)
• V3 experimental results (tests frequency ratios, not material composition)

It signals that the path from prime structure to macroscopic material properties — if it exists — does not run through piezoelectric Q-factor. The search continues elsewhere.

Remaining Open Questions

• Q-PZ-02: Would the biological sopfr(Z)/Z correlation (p=0.0003) survive a similar confound analysis? The covariates would be different (electronegativity, ionic radius, abundance, Pauling electronegativity). This is now the most important cross-domain test.
• Q-PZ-03: Does the nuclear binding energy correlation (r=0.943) survive confound controls? Covariates: neutron number, magic numbers, nuclear shell model.
• Q-PZ-04: Are there OTHER material properties where sopfr(Z)/Z does residual work beyond conventional predictors?

Relationships

• [[elemental-mandelbrot]] — nature: cautionary — same sopfr(Z)/Z metric; piezo correlation was a confound artefact, raising the question of whether the biological and nuclear correlations would survive similar analysis
• [[nagapi-law]] — nature: refuted in this domain — Q-coupling trade-off does not map onto Z-primality after controlling for crystal family and ferroelectricity
• [[sopfr-binding-energy]] — nature: cautionary — piezo domain removed as cross-domain evidence; biological and nuclear domains still untested for confounds
• [[prime-composite-duality]] — nature: not supported — give/resist duality in piezoelectrics is explained by ferroelectricity and crystal structure, not prime composition
• [[v3-experimental-proof]] — nature: independent — V3 tests frequency ratios (a different question from material composition); V3 results unaffected by piezo refutation but still need analog replication

Key Evidence

• Dataset: projects/crystal-resonance-analysis/piezo_materials_dataset.csv
• Confound regression script: projects/crystal-resonance-analysis/confound_regression.py
• Raw correlation: Spearman ρ = +0.551, p = 0.0016 for sopfr(Z)/Z vs Q×f (real but confounded)
• After confound controls: ρ = +0.15, p = 0.44 (non-significant — artefact confirmed)
• Confound model R² = 0.918 — ferroelectricity + crystal family explain 92% of Q variance without any number theory
• ΔR² = 0.001 — sopfr(Z)/Z adds 0.1% explanatory power (noise)
• Permutation test: p = 0.36 (random shuffles match real sopfr performance 36% of the time)