Quantifying golden-ratio deviations in the tree drawing test to identify patients with Alzheimer's disease

The golden ratio ([Formula: see text]) exhibits unique autosimilarity properties that appear throughout biological systems, including human physiology and neural organization. The Tree Drawing Test (TDT), a simple cognitive assessment tool that mainly implies visuospatial, praxic and executive functions, may capture φ-based organizational principles that become disrupted in neurodegenerative conditions. This study examined the relationship between golden ratio proportions and cognitive impairment in tree drawings through quantitative analysis of a large cohort of cognitively impaired patients. We evaluated 613 Alzheimer's disease (AD) patients, 328 mild cognitive impairment (MCI) patients, and 438 healthy controls who completed the TDT. Five novel golden ratio-based deviation indices were developed to quantify proportional relationships between trunk and crown dimensions; among these, the trunk-based index [Formula: see text]showed the most consistent group separation (Distance-to-Diameter Ratio [Formula: see text]; Fisher Ratio [Formula: see text]), with all three pairwise diagnostic comparisons reaching [Formula: see text] (Mann-Whitney U, Bonferroni-corrected) in the full sample and across sex and education strata. Within a multinomial logistic regression framework with stratified 5-fold cross-validation, [Formula: see text]retained independent discriminative value after adjustment for age, education, and the established Space Occupation (SO) index, reaching a macro-averaged AUC of 0.834 (AD vs. rest: 0.855, 95% CI [Formula: see text]; CNTRL vs. rest: 0.911, 95% CI [Formula: see text]; MCI vs. rest: 0.736); the Likelihood Ratio Test confirmed that [Formula: see text]contributes information not captured by SO, age, and education combined ([Formula: see text], [Formula: see text]), and convergent results under two independent matched-subgroup strategies indicated that age and education differences do not explain this signal. Pre-specified operational cut-offs derived from the cohort yielded clinically interpretable sensitivity/specificity trade-offs, with AUC [Formula: see text] [0.894, 0.927] for healthy-control identification using the full multivariate model. As a secondary descriptive observation, group means of [Formula: see text]approximated the Fibonacci values F(5), F(7), F(9); a permutation-based null model (B [Formula: see text], [Formula: see text]) and a Fibonacci-vs-Lucas specificity comparison (100% vs. 0% confidence-interval containment) indicated that this alignment is unlikely to arise by chance and is specific to φ-convergent sequences, although the algebraic link between [Formula: see text]and φ implies that this finding should be regarded as a starting point for future studies rather than as proof of a biological mechanism. Within these limitations, golden-ratio-based TDT measures provide a quantitative complement, not a replacement, to traditional TDT indices for cognitive impairment assessment and require confirmation in independent multi-centre samples before clinical adoption.