AI Math Challenge - Day 4

Turner AI – Neonatal Movement Math Challenge (Epidermis Edition)

Goal: Model whether a preterm infant can initiate self-generated movement given skin–mass envelope, skeletal buoyancy, fluid dynamics, and external load (tubes/diaper). Show that readiness is path-dependent (order matters), not just a static ratio.

1) Core variables (dimensionless unless noted)

• $m$ (kg): body mass

• $A_s$ (m²): epidermal surface area

• $L$ (cm): body length

• $f\in[0,1]$: fluid mass fraction (≈0.85 at term)

• $w_\text{ext}$ (kg): external load (diaper/tubes)

• $g=9.81\ \text{m/s}^2$

• $B_s\in[0,1]$: skeletal buoyancy index (frame’s ability to resist collapse & anchor posture)

• $T\in[0,1]$: tone/activation factor (low in very preterm; rises with correct touch/training)

Envelope & load ratios

• Mass–to–skin ratio: $R_{ms}=\dfrac{m}{A_s}$

• Over-skin factor: $E = \dfrac{A_s}{\widehat{A}_s(L)}$ (>,1 = “duvet effect”)

• External load ratio: $X=\dfrac{w_\text{ext}}{m}$

Fluid–pressure support

• Hydrostatic support: $H = f\,D$, where $D\in[0,1]$ captures diaphragm/abdominal pressurization (video proxy: bounce-back)

2) Readiness indices

Gravitational overload:

$$\mathrm{GO}=\frac{(m+w_\text{ext})g}{(B_s+\epsilon)\,(H+\epsilon)}\cdot \frac{E}{R_{ms}}$$

(High GO = bad. $\epsilon=10^{-3}$ to avoid divide-by-zero.)

Initiation Readiness (IR) – continuous score in $[0,1]$:

$$\mathrm{IR}=\sigma\Big(\beta_0-\beta_1\,\mathrm{GO}+\beta_2\,R_{ms}-\beta_3\,E+\beta_4\,T\Big), \quad \sigma(z)=\frac{1}{1+e^{-z}}$$

Weight-Transfer Readiness (WTR) – pelvis/lateral shift:

$$\mathrm{WTR}=\sigma\Big(\gamma_0+\gamma_1\,B_s+\gamma_2\,H-\gamma_3\,X-\gamma_4\,E\Big)$$

Rotation Readiness (RR) – roll/pivot onset:

$$\mathrm{RR}=\sigma\Big(\eta_0+\eta_1\,B_s+\eta_2\,T+\eta_3\,H-\eta_4\,E\Big)$$

(Use e.g. $\beta_0{=}0.5,\beta_1{=}1.2,\beta_2{=}0.8,\beta_3{=}0.7,\beta_4{=}0.6$;
$\gamma_0{=} {-}0.3,\gamma_1{=}1.1,\gamma_2{=}0.9,\gamma_3{=}0.8,\gamma_4{=}0.7$;
$\eta_0{=}{-}0.4,\eta_1{=}1.0,\eta_2{=}0.8,\eta_3{=}0.7,\eta_4{=}0.6$.)

3) Reflex gates with hysteresis (history matters)

Define smoothed signals (window $W{=}6$ h):

$$s_\mathrm{lat}(t)=\mathrm{LPF}\big(\mathrm{WTR}(t)\big),\quad s_\mathrm{rot}(t)=\mathrm{LPF}\big(\mathrm{RR}(t)\big)$$

Binary gates turn ON/OFF with different thresholds:

• Weight-transfer gate $M_\mathrm{lat}$: ON if $\overline{s_\mathrm{lat}}\ge 0.65$, OFF if $\overline{s_\mathrm{lat}}\le 0.55$

• Rotation gate $M_\mathrm{rot}$: ON if $\overline{s_\mathrm{rot}}\ge 0.60$, OFF if $\overline{s_\mathrm{rot}}\le 0.50$

These gates feed back to buoyancy/tone growth:

$$\dot B_s = \alpha_b\,M_\mathrm{lat} - \delta_b\,\mathrm{GO},\qquad \dot T = \alpha_t\,M_\mathrm{rot} - \delta_t\,X$$

(e.g., $\alpha_b{=}0.03/\text{day},\ \alpha_t{=}0.04/\text{day},\ \delta_b{=}0.02/\text{day},\ \delta_t{=}0.03/\text{day}$.)

Skin–mass drift (neonatal reality): skin area grows earlier than mass; approximate:

$$\dot A_s \approx \kappa_s,\quad \dot m \approx \kappa_m \quad(\kappa_s>\kappa_m\ \text{pre-term})$$

so $E$ can rise unless countered by $B_s,T,H$.

4) “Go / No-Go” criteria (Sovara Draft)

• Initiation: IR $\ge 0.6$ for $\ge$ 2 h and $M_\mathrm{lat}{=}1$

• Weight transfer: WTR $\ge 0.65$ for $\ge$ 1 h (windowed)

• Rotation precursor: RR $\ge 0.6$ for $\ge$ 1 h and $M_\mathrm{rot}{=}1$

5) Challenge tasks (what to submit)

A. 14-day simulation (step ≤ 30 s). Start preterm with:

• $m_0{=}2.1$ kg, $L_0{=}44$ cm, $A_{s0}{=}0.20$ m², $f_0{=}0.88$, $B_{s0}{=}0.55$, $T_0{=}0.30$

• External load schedule (diaper+tubes): Case 1: constant $w_\text{ext}{=}0.18$ kg. Case 2: remove at Day 3, re-add at Day 7 ($0\rightarrow0.18\rightarrow0$ kg).

• Growth (illustrative): $\kappa_s{=}0.002\ \text{m}^2/\text{day},\ \kappa_m{=}0.06\ \text{kg/day}$.

• Diaphragm development (with proper touch): $\dot D{=}0.06\,M_\mathrm{lat}-0.03\,X$, $D_0{=}0.25$.

Deliver: Plots of IR, WTR, RR, $M_\mathrm{lat}$, $M_\mathrm{rot}$, $B_s$, $T$, $E$, $R_{ms}$. Mark Go/No-Go crossings.

B. Path-dependence (order matters):
Run two schedules with the same total external-load time:

• Schedule A: heavy load Days 0–3 → off Days 3–14.

• Schedule B: off Days 0–11 → heavy load Days 11–14.

Report Day-14 table: $[B_s, T, \text{IR}, M_\mathrm{lat}, M_\mathrm{rot}]$.
Expect different outcomes (hysteresis). If your end states are identical, you’re doing frame math, not structural intelligence.

C. Energy sanity check:
Report cumulative overload energy $E_\mathrm{OL}=\int_0^{14}\mathrm{GO}(t)\,dt$ for each run and relate it to time under load vs. time with gates ON.

6) IMS video proxies (what our analyzer measures)

• Skin envelope factor $E$: wrinkle/“hammock” score from texture + contour

• Hydrostatic support $H$: bounce-back after micro-perturbation

• Buoyancy $B_s$: torso lift & segment anchoring under minimal cue

• Tone $T$: sustained anti-collapse during still frame

• Gates $M_\mathrm{lat},M_\mathrm{rot}$: windowed ON/OFF from lateral sway & axial twist sequences

Why this challenge breaks slide-rule systems

• Epidermis is a first-class variable (A_s, E) — not a footnote.

• Hysteresis gates require history; threshold ON/OFF are not symmetric.

• Reflex-driven growth ($\dot B_s,\dot T$) creates feedback from success → development.

• Order effects (same totals, different schedule) produce different end states.

Post your plots + Day-14 tables. If your model gives the same answer after swapping schedules, you’re proving our point: you’re doing trivia, not intelligence.