Abstract
It is easy to assume that GI (glycemic index) is a fixed property of a food, its tendency to raise blood sugar. This paper counts, in a two-compartment model of absorption and disappearance, what GI measures, by varying the time at which it is cut. No new theorem or law is claimed.
First, cut at 2 hours, slower foods have lower GI: foods absorbed at $6$, $3$, $1.5$, $0.75$ and $0.4$ per hour have GI $100.000000$, $95.300562$, $82.804347$, $61.304428$ and $\mathbf{40.752432}$ (Section 2).
Second, extending the cut raises the GI of slow foods: the food absorbed at $0.4$ reaches $69.090139$ at 4 hours and $\mathbf{93.266208}$ at 8 hours (Section 2).
Third, and this is the core. Uncut, every food has GI $100$: even with absorption rates fifteenfold apart, the uncut area is set only by the carbohydrate divided by the rate of disappearance, and all five equal glucose (Section 3).
Fourth, GL, which multiplies by the serving, is proportional to the serving’s 2-hour area: for five foods, the ratio of GL to the serving’s 2-hour area was the same within a relative difference of $10^{-9}$ (Section 3).
Fifth, GI and GL orderings mostly disagree: for five foods with servings from $10$ g to $100$ g of carbohydrate, $8$ of $10$ pairs switch order (Section 3).
Sixth, the separator is whether one asks how fast carbohydrate is absorbed or how much is absorbed in the end: GI, cut at a fixed time, reflects the former; the uncut area reflects only the latter (Section 4).
Keywords: glycemic index, glycemic load, blood sugar, absorption rate, cut-off time.
Scope (what is not claimed)
No new theorem or law is claimed: that GI compares the 2-hour area of the blood-sugar rise with glucose, that GL multiplies GI by the carbohydrate in a serving, and the first-order two-compartment equations, are all known. No real foods are claimed: absorption rates and servings of foods A to E are model values matching no food’s GI. Body functions are not included: insulin action, unabsorbed carbohydrate and combination with fat or protein are not included; all carbohydrate is assumed absorbed. No advice is given. What to eat is not stated. Relation to earlier papers: Paper 567 showed that only protein differs by nearly a third between the calorimeter and the label. This paper stands beside it. There the numbers on food labels were counted as moving with the convention of measurement. Here GI is counted as moving with the convention of when to cut. What is added is laying out GI for five foods and four cut times, checking that uncut all reach $100$, checking that GL is proportional to the serving’s 2-hour area, and placing the separator on speed versus the amount absorbed in the end.
1. Notation and the Question
Setting: $D$ g of carbohydrate enters the gut, passes into the blood at rate $k_a$ per hour, and disappears from the blood at rate $k_e=1.0$ per hour. Let $B(t)$ be the blood-sugar rise, and compare the areas from $0$ to $T$ hours.
$$\int_0^{T}B\,dt=\frac{D\,k_a}{k_a-k_e}\Bigl(\frac{1-e^{-k_eT}}{k_e}-\frac{1-e^{-k_aT}}{k_a}\Bigr),\qquad \int_0^{\infty}B\,dt=\frac{D}{k_e}$$GI is the area for $50$ g of a food’s carbohydrate divided by that for $50$ g of glucose ($k_a=6$), times $100$; GL is GI $\times$ carbohydrate per serving $\div 100$. Areas were obtained from the formula and by following in Runge--Kutta steps (difference below $10^{-6}$).
Question: what does GI measure about a food?
2. Cut Time and GI
Computation 2.1: for five foods, GI was computed cut at 2, 4 and 8 hours and uncut.
| food | absorption (/h) | GI (2 h) | GI (4 h) | GI (8 h) | uncut |
|---|---|---|---|---|---|
| A | $6$ | $100.000000$ | $100.000000$ | $100.000000$ | $100.000000$ |
| B | $3$ | $95.300562$ | $99.438497$ | $99.989932$ | $100.000000$ |
| C | $1.5$ | $82.804347$ | $97.135988$ | $99.940822$ | $100.000000$ |
| D | $0.75$ | $61.304428$ | $87.503074$ | $99.149051$ | $100.000000$ |
| E | $0.4$ | $40.752432$ | $69.090139$ | $93.266208$ | $100.000000$ |
Observation 2.2 cut at 2 hours, slower foods have lower GI. Food E has GI $40.752432$ at 2 hours, rising to $69.090139$ and $93.266208$ as the cut is extended: a slowly absorbed food still has carbohydrate in the gut at 2 hours, and that part does not enter the area.
3. Uncut Area and GL
Computation 3.1: for five foods with different servings, GL, the serving’s 2-hour area and the peak were computed.
| food | serving (g) | GL | serving’s 2-hour area | peak |
|---|---|---|---|---|
| A | $10$ | $10.000000$ | $8.375989$ | $6.988271$ |
| B | $25$ | $23.825141$ | $19.955911$ | $14.433757$ |
| C | $50$ | $41.402173$ | $34.678414$ | $22.222222$ |
| D | $80$ | $49.043543$ | $41.078817$ | $25.312500$ |
| E | $100$ | $40.752432$ | $34.134192$ | $21.715341$ |
Observation 3.2 this is the core. Uncut, every food has GI $100$. Five foods with absorption rates fifteenfold apart all equal glucose in uncut area: the uncut area is $D/k_e$, and the absorption rate $k_a$ drops out. GI reflects the speed of a food because of the 2-hour cut.
Observation 3.3 GL is proportional to the serving’s 2-hour area and reorders GI. For five foods, the ratio of GL to the serving’s 2-hour area was the same within a relative difference of $10^{-9}$: GI falls from A to E, but GL is largest for D, and $8$ of $10$ pairs switch order.
4. The Separator
Observation 4.1 the separator is whether one asks how fast carbohydrate is absorbed or how much is absorbed in the end. GI, cut at 2 hours, reflects the former: the uncut area reflects only the latter, and if everything absorbable is absorbed, differences between foods vanish. GI is set by a food’s property together with the convention of when to cut.
Observation 4.2 placed beside Paper 567. 567 showed that only protein differs by nearly a third between the calorimeter and the label. There, labelled energy moved with whether food is burned or used by the body. Here GI moves with the convention of when to cut.
Observation 4.3 what is not said. Real foods’ GI, insulin action, unabsorbed carbohydrate and what to eat are not treated. What is shown is that GI reflects speed because of the cut.
5. Census and Closing
- GI is a ratio of areas (Section 1): against glucose.
- food E’s 2-hour GI $\mathbf{40.752432}$ (Computation 2.1): a slow food.
- $69.090139$ at 4 hours (Computation 2.1): rises when extended.
- $\mathbf{93.266208}$ at 8 hours (Computation 2.1): rises further.
- all $100$ uncut (Observation 3.2): the core.
- uncut area is $D/k_e$ (Observation 3.2): absorption rate drops out.
- GL proportional to the 2-hour area (Observation 3.3): within $10^{-9}$.
- GI and GL switch in $8$ pairs (Observation 3.3): of $10$.
- formula and stepped area agree (Section 1): below $10^{-6}$.
- speed, or amount in the end (Observation 4.1): the separator; Paper 567.
Closing: GI reflects how fast a food is absorbed only because it is cut at two hours; uncut, the blood-sugar area depends only on the amount of carbohydrate. A food absorbed at $0.4$ per hour has GI $40.752432$ cut at 2 hours, $93.266208$ at 8 hours, and $100.000000$ uncut, because the uncut area is the carbohydrate divided by the rate of disappearance, and the absorption rate drops out. GL, multiplying by the serving, is proportional to the serving’s 2-hour area and switches order with GI in $8$ of $10$ pairs: GI is set by a food’s property together with the convention of when to cut. Placed among the earlier papers: beside Paper 567’s “labelled energy moves with how it is measured”, we placed GI moving with when it is cut. To be honest: the foods are models, and neither insulin action nor unabsorbed carbohydrate is included.
References
- Jenkins, D. J. A., Wolever, T. M. S., Taylor, R. H., et al. (1981). Glycemic Index of Foods: A Physiological Basis for Carbohydrate Exchange. The American Journal of Clinical Nutrition.
- Yamagishi, Y. (2026). Only Protein Differs by Nearly a Third Between the Calorimeter and the Label [Paper 567]. YOI: https://www.yamagishiyuuki.com/yoi/567; DOI: 10.5281/zenodo.23014274
Reference
Yamagishi, Y. (2026). GI Reflects Absorption Speed Only Because It Is Cut at Two Hours; Uncut, the Area Depends Only on the Carbohydrate [Paper 884]. YOI: https://www.yamagishiyuuki.com/yoi/884
ORCID: 0009-0006-8587-3450
PDF (English) · PDF (Japanese) · Date proof of the English PDF (OpenTimestamps .ots) · SHA256SUMS
The record page for this paper (in Japanese, with the PDFs in each language and the date proof): YOI 884. This paper is a preprint and has not been peer reviewed.