FiftyStrong · quiz weight projection · 9 September 2026

The cadence of losing weight, and what our charts should say

A paywall promised a woman 0.2 lb in four weeks and 2.4 lb in twelve, then 50 lb by week 91. The curve was drawn backwards. This page reproduces the bug, sets out what the physiology and the trials say about how weight actually comes off after fifty, proposes a model, and redraws each of the three screens that show it.

−0.2 lb
Week 4, as shown on the paywall (replay 01a08198, 8 Sep). The curve eases in, so a long plan starts flat.
−3.2 lb
Week 4 under the proposed model at the same "steady" pace, front-loaded by the glycogen and water she sheds first.
0 ± 0.5 kg
What 45 minutes a week of exercise does to the scale in 12 weeks without eating changes. "Plan alone" cannot carry a falling line.
≤ 9%
Twelve-month mean in the best lifestyle trials. A 24% gap has no honest date, at any pace.
Sources: 38 physiology and trial references, 40 competitor and regulatory references, in the research folder.Code: web/src/lib/belly-math.ts, ios/FiftyStrong/Quiz/BellyMath.swift

1 · What she saw

One session, three screens, one curve

The session came from a Meta ad on 8 September. She answered the belly branch: shape "fuller all over", change "in the last year or two", BMI well above the healthy range, a goal 24% below her weight, pace "Steady — about ¼ kg a week, the plan alone", no event. From those choices the page computed a 91-week plan. She reached the paywall, opened checkout, and left.

Her weight is not stored in the events, but the gap, the pace and the week count pin it: about 208 lb to 158 lb (94.3 → 71.7 kg, a 22.6 kg gap). Re-running the shipped module with those inputs reproduces the paywall to the decimal.

MilestoneShown to herA straight line at her paceProposed model
Week 4−0.2 lb−2.2 lb−3.2 lb
Week 12−2.4 lb−6.6 lb−6.7 lb
Week 26−9.9 lb−14.2 lb−11.3 lb
Week 52−30.2 lb−28.5 lb−16.3 lb
"End of plan"−49.8 lb at week 91week 91not within 104 weeks

Two problems, not one. The shape is inverted (the first weeks show nothing). And the horizon is a fiction: the last dot is labelled "End of plan" twenty-one months out, for a product that is a twelve-week programme rebuilt every four weeks.

2 · Root cause

An ease-in-ease-out curve applied to a 91-week line

The module keeps two models. trajectory() is a straight line at the chosen pace, and it sets the goal date. But every number that appears on a screen, and every drawn curve, goes through easedShare(t) = t²(3 − 2t): the share of the gap covered at fraction t of the timeline. That is smoothstep. Its slope at the start is zero, so the longer the plan, the flatter the opening: at week 4 of 91 weeks, t is 0.044 and the share is 0.6%. The loss then accelerates through the middle of the plan, which is the opposite of how bodies behave.

Paywall milestones as shippedEased
Her inputs, the shipped renderer's geometry: the curve is a bezier that starts flat, and the dots sit on smoothstep. Same code in the iOS port.
Same inputs, proposed modelFront-loaded
Water and glycogen go first, fat loss slows gently. Week 12 is the emphasised point because it is the product. The last point is one year, not a 91-week "end of plan".

3 · What the physiology says

Fast, then slower, then a plateau nobody's metabolism caused

  1. Weeks 0–2: the beginner's bonus is mostly water. The body holds about 500 g of glycogen, each gram bound to 2.7–4 g of water. Cut intake and it drains: roughly 1 kg on a moderate deficit, 0.5 kg on a gentle one, with a half-life under a week, finished by week 2–3. Only about a third of first-phase loss is fat (Heymsfield 2011; Hall 2018). The bonus is real, honest to show, and must never be extrapolated.
  2. Week 2 onward: fat loss that slows on its own. Every kilogram lost lowers daily expenditure by 22–24 kcal, so a fixed intake buys a shrinking deficit. Hall's model (the NIH Body Weight Planner) has weight approach a new plateau exponentially: with perfect adherence, half the eventual change arrives at about a year, 95% by three. Appetite makes it faster: intake drifts up about 100 kcal/day per kilogram lost, several times the expenditure effect, which is why trials plateau at about six months. The plateau is behaviour, not a broken metabolism (Thomas 2014; Hall & Kahan 2018).
  3. The trials draw the same curve. Look AHEAD, the most intensive lifestyle programme ever run at scale: 8.2% at six months, 8.6% at twelve (women 8.1%; ages 65–74, 9.4%). DPP: 6.5 kg at 24 weeks. CALERIE: 11.5% at a year. Budget 60–70% of the six-month loss by week 12.
  4. Exercise alone barely moves the scale after fifty. DREW, six months of supervised aerobic work in postmenopausal women: −1.4 to −2.1 kg at 72–194 minutes a week. A 12-week, three-day resistance programme: +0.3 kg on the scale, −2.4 kg of fat, +1.2 kg of muscle. Villareal 2011, adults over 65: exercise alone, no loss. Our dose is 45 minutes a week. Expect 0 ± 0.5 kg in 12 weeks, with about a kilogram of fat gone and 1.5 cm off the waist. That is the story the proposal already tells ("the middle is the story; the scale is measured"); the weight chart has to stop contradicting it.
  5. Safe pace and the "without" drift. NHS and NIDDK: 0.5–1 kg a week; for older adults a 500–750 kcal/day deficit with 1.0–1.2 g/kg protein and resistance training to keep lean mass. Our 0.25–0.5 kg/week sits under every cap. Mid-life drift is 0.25–0.7 kg a year (SWAN), driven by age and activity; menopause changes composition more than weight.
Model curves against trial meansKilograms lost, 52 weeks, 94 kg start
Moderate, plan + walking + eating to itSteady, plan + eating guideMost-women band, 60–130%Trial means (Look AHEAD, DPP, DREW)
Trial dots are population means including non-adherers, scaled to a 94 kg start; the lines are for a woman who keeps to the plan. The moderate line is optimistic past six months; its band's lower edge is where trial medians land. DREW is exercise only, and sits near zero.

4 · The model

Two exponentials and four constants

A water term that empties in days, and a fat term whose rate decays slowly. No calories on screen; the pace she chose sets the initial rate, and the average over the first twelve weeks still equals the pace label she read.

L(t) = Wg · (1 − e−t/τw) + r0 · τ · (1 − e−t/τ)    t in weeks, L in kg lost
ConstantMeaningSteadyModerate / fastGrounding
r0Initial fat-loss rate, kg/week0.250.50The pace copy; ≈275 and ≈550 kcal/day deficits, under the NHS cap
WgGlycogen and water bonus, kg0.51.0~500 g glycogen × 2.7–4 g water; a moderate mixed-diet deficit drains about half
τwBonus time constant, weeks0.70.7Heymsfield: half-life under a week, phase over by week 4–6
τSlowing time constant, weeks3636Between Hall's perfect-adherence ~70 weeks and the ~24 weeks a coached plan's adherence implies. The one knob to recalibrate from our own weigh-ins

Consequences worth knowing. The rate at week 12 is 72% of the starting rate, at week 26 about half, at week 52 a quarter. Each pace has a ceiling, Wg + r0τ: 9.5 kg on steady, 19 kg on moderate, which bounds what a single pace can promise. Trial standard deviations are about the size of the mean, so the chart carries a band from 60% to 130% of the line and every number reads "about". The goal is dated only when the curve reaches it inside 52 weeks; otherwise the screen shows the one-year point and says the goal is further.

Try it with her numbers, or yours

Trajectory screen (5.5)Proposed
Paywall milestones (P1)Proposed
WeekShipped (eased)Straight lineModelBand

5 · Screen by screen

What each interstitial and the paywall should say

5.1 Pace (s4-w4). The "Steady" option reads "the plan alone". Nothing in the evidence lets a 45-minute week draw a falling weight line.
  • Rename: "Steady — about ¼ kg a week. The plan, plus the eating guide."
  • Moderate stays: "the plan, plus walking and eating to it."
  • If she declines any eating change, the promise is waist and fat, and the copy should say the scale may barely move in twelve weeks.
4.4 With and without (s4-w3). Illustrative, no numbers, and today an S-curve for "with" and a rising line for "without".
  • "With" becomes a fast-start drop that flattens.
  • "Without" goes flat. The 0.5 kg/year drift is real (SWAN) but invisible over twelve weeks, and a rising line is a comparative claim regulators read literally.
4.4 as shippedS-curve, rising "without"
4.4 proposedFast start, flat "without"
5.5 Trajectory (s4-w8). "You'll reach your goal of X by DATE" with the S-curve under it, and the date from the straight line. The curve and the date disagree today.
  • Draw L(t) with the band. Date the goal only when it lands inside 52 weeks.
  • Otherwise the end bubble is one year: "Your first year takes you to about X. Your goal is further; we re-plan every four weeks."
  • Event marker on the curve at L(weeks to event); never bend the line to land early.
  • Adjacent to the chart, same visual weight: "Estimate. Assumes three sessions a week and the eating guide. Results vary. Check with your doctor before starting."
P1 Milestones (paywall). Week 4, Week 12, event, "End of plan", all on smoothstep. This is where −0.2 lb appeared.
  • Week 4 and Week 12 always; Week 12 emphasised, it is the thing being sold.
  • The last point is "End of plan" only when the goal is dated inside 52 weeks; else "1 year".
  • Numbers as "about −3 lb", band shaded under the curve.
  • Week 12 must never read smaller than a woman would lose by accident.

In the app, later. "A stall around month four to six is normal, and mostly appetite quietly closing the gap; the four-week rebuild is where we reset it." Not in the quiz. And once fortnightly weigh-ins exist, τ gets fitted to them and the trajectory redrawn against actuals, which the proposal already lists as product debt.

6 · Rules

The exposure is precision, not the rate

RegimeRuleWhat it means for the three screens
UK CAP 13.9, 13.10No claims of precise amounts within a stated period; over 2 lb a week is not good practice.Milestones as "about" with a band. Our 0.25–0.5 kg/week is far under the line.
FTC Health Products Guidance, Example 41; 16 CFR 255.2"Results not typical" is not enough; disclose generally expected results next to the claim, clearly and conspicuously.Assumption and range adjacent to the chart, not in a footer. A projection is an implied efficacy claim for the programme as sold.
Meta health and wellnessNo promises of specific outcomes in a set timeframe without qualifiers; no "you" weight framing.The chart is fine on the page; lifting a dated chart into creative needs a visible qualifier.
Apple 1.4.1Disclose methodology for health accuracy claims; remind users to check with a doctor.Be able to state the model in a sentence. Add the doctor line under the trajectory.
CAP 13.2 (BMI over 30)Obesity treatment not advertised to the public without supervision.No "obesity" language, doctor signpost. Already the case in the belly branch.

The market's practice: Noom shows one dated point ("we predict you'll be 57 kg by 15 Oct"), a near-linear line with a shallow curve at the start, an event marker, and hedges with "we predict", "conservatively", and "results not guaranteed". Nobody verified shows weekly milestones as exact figures. Yazio is the only flat "no change" comparison line found; Hers is the only cohort-footnoted milestone chart. BetterMe lost an ASA ruling in 2024 over a 28-day result claim.

7 · Decisions

Four calls, with recommendations

Cap dated goals at 52 weeksRecommendedThe date is the legal exposure and the credibility exposure. Her answer to "what got in the way before" was "I've started before and it didn't stick"; a 21-month line is the thing she has learned not to believe. The alternative is to keep dating every goal, Noom-style, with "we predict, conservatively". Run it as a test and resolve on purchase rate, not leads.
Fold the eating guide into "steady"RecommendedEvery falling line assumes the eating guide. The alternative keeps "plan alone" and drops the weight chart for that branch in favour of waist and fat, which is more honest but loses the number the ad promised.
τ = 36 weeks for display, then fit itRecommendedRecalibrate from our own check-ins once there are about 200 weigh-ins. Until then it is the middle of the evidence, and the band covers the spread.
Flat "without" lineRecommendedThe alternative is +0.5 kg/year with the SWAN citation adjacent. Over twelve weeks that is 0.12 kg, invisible on the chart, and it turns a motivational picture into a comparative claim.

8 · Next

Implementation scope

Product tier. One module and its two copies, three renderers, the Swift port and its vectors, the tests, and the zh-TW cycle for the copy changes: belly-math.ts gains lossAt, a searched weeksToGoal and the band, and retires easedShare; the page copy of the module and the three renderers in the static quiz follow; belly-math-vectors.ts regenerates the fixture that pins BellyMath.swift, and the schedule, paywall and trajectory views take the new curve. Legal review of the three screens before spend, from the proposal's guardrails, still stands.

9 · References

The ones the numbers lean on

  1. Hall KD et al. Quantification of the effect of energy imbalance on bodyweight. Lancet 2011;378:826–37. NIH Body Weight Planner.
  2. Hall KD. Dynamic energy balance and obesity prevention. J Obes Metab Syndr 2018;27:203–12.
  3. Heymsfield SB et al. Voluntary weight loss: systematic review of early phase body composition changes. Obes Rev 2011;12:e348–61.
  4. Kreitzman SN et al. Glycogen storage: illusions of easy weight loss. Am J Clin Nutr 1992;56:292S–3S.
  5. Thomas DM et al. Effect of dietary adherence on the body weight plateau. Am J Clin Nutr 2014;100:787–95.
  6. Hall KD, Kahan S. Maintenance of lost weight and long-term management of obesity. Med Clin North Am 2018;102:183–97.
  7. Polidori D et al. How strongly does appetite counter weight loss? Obesity 2016;24:2289–95.
  8. Wadden TA et al. One-year weight losses in the Look AHEAD study. Obesity 2009;17:713–22.
  9. Wing RR et al. Achieving weight and activity goals among DPP lifestyle participants. Obes Res 2004;12:1426–34.
  10. Franz MJ et al. Weight-loss outcomes: meta-analysis of trials with ≥1-year follow-up. J Am Diet Assoc 2007;107:1755–67.
  11. Church TS et al. Changes in weight, waist and compensatory responses with different doses of exercise among sedentary, overweight postmenopausal women (DREW). PLoS One 2009;4:e4515.
  12. Friedenreich CM et al. Adiposity changes after a 1-year aerobic exercise intervention among postmenopausal women (ALPHA). Int J Obes 2011;35:427–35.
  13. Villareal DT et al. Weight loss, exercise, or both and physical function in obese older adults. N Engl J Med 2011;364:1218–29.
  14. Free-weight resistance training and a high-protein diet in postmenopausal women, 12-week RCT. Sci Sports 2024.
  15. Exercise training and body composition in postmenopausal women: meta-analysis of 101 trials. Front Endocrinol 2023;14:1183765.
  16. Donnelly JE et al. ACSM position stand on physical activity for weight loss. Med Sci Sports Exerc 2009;41:459–71.
  17. Greendale GA et al. Changes in body composition and weight during the menopause transition (SWAN). JCI Insight 2019;4:e124865.
  18. Davis SR et al. Understanding weight gain at menopause. Climacteric 2012;15:419–29.
  19. FTC Health Products Compliance Guidance, December 2022, Example 41; 16 CFR 255.2(b), 2023.
  20. CAP Code section 13, Weight control and slimming, rules 13.2, 13.9, 13.10, 13.11.
  21. Meta Advertising Standards, Health and wellness; Apple App Store Review Guidelines 1.4.1.

Full lists with links: .claude/weight-loss-model/research/physiology-and-trials.md and competitors-and-rules.md in the repo.