Φ

Executive Summary

category	Φ	weight	Percentile Rank	distribution
Overall	3.18	100%	7.6%
surfaces	2.69	30%	8.7%
edges	2.49	25%	7.8%
corners	0.68	25%	0.6%
alignment	7.89	20%	90.5%

Executive Summary | Φ scores are derived from multi-angle photometric imaging; higher scores indicate a surface closer to mint condition. All statistical tests are performed against reference cards in the most immaculate condition (i.e., GEM MT, Pristine, Black Label; n=100+). Overall: composite Φ score of surfaces (30%), edges (25%), corners (25%), and alignment (20%). Percentile rank: the card’s score expressed as a percentage of the simulated market population (n = 10,000). Distribution: histogram of the simulated reference population with the card’s score marked in red.

---

Table of Contents

---

Figure 1: Lambert Test (Front Surface)

Figure 1 | The Lambert algorithm detects dents, creases, delamination, and other surface irregularities. All reflectance data are measured using proprietary imaging technology. a) Raw reflectance signal across the full card surface. Grid columns (A–F) and rows (1–6) denote spatial zones in millimetre coordinates; grayscale encodes reflectance magnitude (low–high).
b) Reflectance signal adjusted relative to a signal baseline obtained from gem mints (n ≥ 100).
c-d) Reflective signal as a function of severity, assessed by significance (c) and intensity (d). Evaluated card (red, solid); mean gem mint reference set (black, solid); mean heavily played/damaged reference data (black, dotted). y-axis: count (#) or survival. x-axis: magnitude of irregularity. Wasserstein distance (w) and associated signifiance (p.value) to each reference set was calculated using a leave-one-out permutation.
e) Percentage of flagged reflections per spatial zone (surface, four edges, and four corners).
f) Deviation from the gem mint references per spatial zone. Red points indicate Bonferroni-adjusted significance (p < 0.05, adjusted); black indicates nominal significance (p < 0.05, unadjusted).

sig (auc)	sig (w)	res (auc)	res (w)	dropped	sig+res score	region	integrity	Φ score	contribution
0.75	0.36	0.46	0.41	sig (w)	0.54	surface	1.00	0.54	10%
						edge	0.81	0.44	10%
						corner	0.89	0.48	10%

Table 1 | Summary of Lambert test statistics and calculation of scores. Exponential-decay scores for significance (Fig. 1c) and response (Fig. 1d) profiles derived from wasserstein distance to gem mint references. Direction and magnitude are determined by weighted mean or AUC, each anchored to a pipe-specific gem mint baseline and decay constant λ, gated by significance of departure from reference (p < 0.05). The lowest of the four component scores is dropped before computing the SIG+RES composite. Integrity (Fig. 1f; x-axis: log₁₀[z]) is applied as a per-region spatial penalty multiplier, down-weighted at marginal significance. Φ score is the product of SIG+RES and Integrity per region (max 10). Contribution is each region’s fractional weight in the final grade.

---

Figure 1b: Pendulum Test (Front Surface)

Figure 1b | The Pendulum algorithm detects foil scratches and anomalies in holographic foil elements. All reflectance data are measured using proprietary imaging technology. Performed only on foil (shiny) cards. Measurements inconsistent with holographic gem mint reference cards are flagged as irregular reflections. See Figure 1 legend for panel descriptions; downstream statistical framework is identical.

sig (auc)	sig (w)	res (auc)	res (w)	dropped	sig+res	region	integrity	Φ score	contribution
3.28	4.40	3.23	3.62	res (auc)	3.77	surface	1.00	3.77	20%
						edge	0.34	1.30	10%
						corner	0.31	1.18	10%

Table 1b | Summary of Pendulum Statistics. See Table 1 legend for column descriptions. Φ scores are calculated per region using Pendulum test metrics only (max 10).

---

Figure 2: Centering & Alignment

Figure 3 | Centering and alignment measured at sub-millimetre precision from proprietary multi-angle reflectance imaging. Results expressed as L/R ratio, T/B ratio, and centroid displacement — the geometric centre of the printed image area relative to the card centre ([0.00, 0.00] = perfect centering).
Top: Aligned card image and schematic of measured border widths (mm ± SE); diagonals indicate centroid estimation. Uncertainties reflect inner print edge detection error.
Bottom left: Centroid displacement (mm) from card centre; crosshair indicates measured position. Centroid is robust to inner print rotation and inter-grader variability, and is corrected for perspective and optical distortion.
Bottom right: L/R and T/B centering ratios reported as best- and worst-case ranges across measurement uncertainty.

region	metric	close to perfect	alignment score	contribution
centroid	(-0.083, 0.097)	No	8.22	10%
lr	[48.3/51.7]	No	7.67	5%
tb	[48.1/51.9]	No	7.46	5%

Table 3 | Summary of Centering and Alignment Statistics. Metric reports the measured value for each region (centroid displacement in mm; L/R and T/B as border ratios). Alignment scores reflect proximity to perfect centering. Cards whose margin of error overlaps 49/51 (or 51/49) receive a perfect alignment score. Contribution (%) indicates each region’s weight in the overall final grade.

---

Figure 3: Back Surface

Figure 4 | Edge and corner integrity of the card back assessed via 2D image processing. Infractions are detected by comparison against gem mint reference datasets. Left: Integrity-adjusted image (greyscale). Right: Unrolled edge strips (top, bottom, left, right) and corner detail views; false colour encodes infraction intensity (white = higher intensity). Survival curves for edge and corner infraction severity; evaluated card (red, solid) compared to gem mint reference (black, dotted). Distributional distance (w) to gem mint is quantified with permutation-derived p-values.

region	res (auc)	Φ score	contribution
edge	9.00	0.70	5%
corner	0.05	0.70	5%

Table 4 | Summary of Back Integrity Statistics. Exponential-decay scores for card back edge and corner derived from wasserstein distance to gem mint references (Fig. 4, bottom). Direction and magnitude are determined by AUC, each anchored to a gem mint baseline and decay constant λ. Φ score is the product of card back edge and corner (max 10). Contribution is each region’s fractional weight in the final grade.

---

Methodology & Disclosure

All photometric data are empirically measured using in-house proprietary imaging technology. Topographic characteristics of surface, edge, and corner zones are analyzed using the Lambert and Pendulum (holographic foils only) algorithms. The back surface is not surface scanned, and relies on statstiatical analysis of 2D imaging data. Φ.exe does not use any AI-based technologies or image recognition. Centering is measured at sub-millimetre precision with homographic alignment correction. All scores are derived from an empirically measured reference population (gem mint, pristine, 10+) and are fully reproducible and auditable. Sincere gratitude to the collectors who generously lent their cards to build the gem mint/pristine dataset.

Questions: [email protected]