The golden ratio (\(\varphi \approx 1.618\)) 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 \(\varphi\)-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 \(H/\varphi - T\)showed the most consistent group separation (Distance-to-Diameter Ratio \(=0{.}546\); Fisher Ratio \(=0{.}561\)), with all three pairwise diagnostic comparisons reaching \(p<0{.}0001\) (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, \(H/\varphi - T\)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 \([0{.}835,\,0{.}873]\); CNTRL vs. rest: 0.911, 95% CI \([0{.}894,\,0{.}927]\); MCI vs. rest: 0.736); the Likelihood Ratio Test confirmed that \(H/\varphi - T\)contributes information not captured by SO, age, and education combined (\(\Lambda =121{.}91\), \(p<0{.}0001\)), 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 \(= 0.911\) [0.894, 0.927] for healthy-control identification using the full multivariate model. As a secondary descriptive observation, group means of \(H/\varphi - T\)approximated the Fibonacci values F(5), F(7), F(9); a permutation-based null model (B \(=10{,}000\), \(\hat{p}<0{.}0001\)) 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 \(\varphi\)-convergent sequences, although the algebraic link between \(H/\varphi - T\)and \(\varphi\) 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.
