Longer Records, Broader Invariance: The Hidden Scaling Problem in Longitudinal Contrastive Learning

Rameen Mahmood; Xuhai "Orson" Xu; Zachary Beattie; Jeffrey Kaye; Danny Yuxing Huang

cs.LGSep 29, 2026

Longer Records, Broader Invariance: The Hidden Scaling Problem in Longitudinal Contrastive Learning

Authors: Rameen Mahmood, Xuhai "Orson" Xu, Zachary Beattie, Jeffrey Kaye, Danny Yuxing Huang

Organizations: New York University · Columbia University · Oregon Health & Science University

Abstract

Longitudinal data are valuable because people change. Yet the objectives used to learn from these data can inadvertently erase that change. In person-level contrastive learning, observations from the same person are treated as positives; as records grow, those positives can span increasingly distant---and increasingly different---behavioral states. More history can therefore produce not only more data, but broader invariance. We show that this distinction is fundamental. We separate \emph{record span}, how much history the learner sees, from \emph{supervision span}, how far across that history positive-pair supervision reaches. Across in-home sensing records spanning up to 2.7 years, broader supervision systematically suppresses recoverable changing-state information, even when the available history is held fixed. At the broadest span, less than 10% of the information recoverable from an untrained encoder remains. Yet keeping positives local is not sufficient: as records grow, even distant states that are never paired become increasingly similar. Explicitly contrasting other observations from the same person reverses this loss without shortening the record, revealing a second route by which longitudinal scale can broaden invariance. Finally, we prospectively reproduce the supervision-span effect in 199 GLOBEM participants. Longitudinal scale therefore presents a choice: more history need not mean more invariance. By controlling what is held invariant as records grow, we can preserve the change that made the longitudinal data valuable in the first place.

Figures & tables

History source	Single-resident	Households
Recent ( $t-1,\dots,t-4$ )	$0.2818\pm 0.0054$	$0.2325\pm 0.0097$
Shifted (disjoint block)	$0.0447\pm 0.0291$	$0.0158\pm 0.0043$
Random (four past windows)	$0.0374\pm 0.0169$	$0.0261\pm 0.0045$
Distant (earliest four windows)	$0.0051\pm 0.0143$	$0.0051\pm 0.0058$

Table 1: Usable predictive information (

\Delta I

, nats/dim) by temporal location of history. Values are mean

\pm

standard deviation across five seeds. The households column reports the 44 deduplicated household streams, consistent with the observational unit defined in Appendix A.3 .

Cohort	History	Ordered	Shuffled	Set	Ordered $-$ Shuffled
Single-resident	20d ( $K=4$ )	0.2877	0.2469	0.2454	+0.0408 [0.0252, 0.0576]
Single-resident	40d ( $K=8$ )	0.2773	0.2312	0.2098	+0.0461 [0.0285, 0.0649]
Multi-resident	20d ( $K=4$ )	0.2480	0.2252	0.2253	+0.0228 [0.0142, 0.0313]
Multi-resident	40d ( $K=8$ )	0.2421	0.2055	0.2064	+0.0365 [0.0249, 0.0485]

Table 2: Effect of temporal order on usable predictive information. All arms receive the same recent windows at matched parameter budget. Brackets show 95% bootstrap confidence intervals for the ordered

-

shuffled difference; all comparisons hold across 5/5 seeds.

Appendix figures & tables52 assets

Supplementary material from the paper’s appendix.

Appendix

Channel	Median correlation	Pairs with variance in both
Motion events	1.000	43/43
Motion areas	1.000	43/43
Steps	0.206	14/43
Bed activity	0.382	8/43
Respiratory rate	0.850	8/43
Medication events	0.011	8/43

Appendix

Table 3: Within-home similarity of co-resident sensing streams. Motion and weight channels are duplicated across all 43 paired homes, and bed-derived channels are near-duplicated, motivating household-level deduplication. Only steps and medication events behave as individual channels.

Analysis	87 records	44 households
Recent $\Delta I$	0.2746	0.2325
Shifted $\Delta I$	0.0306	0.0158
Random $\Delta I$	0.0235	0.0261
Distant $\Delta I$	0.0324	0.0051
Recent $-$ shifted	0.2441	0.2167
Recent $-$ random	0.2511	0.2064

Appendix

Table 4: Sensitivity of headline results to household deduplication. The record-level analysis treats the 87 subject records separately; the household-level analysis retains one stream from each of 44 homes.

Channel	Statistic	Mean	SD	Zero rate
Motion events	$\log(1{+}\mathrm{total})$	7.818	0.787	0.00
Motion areas	$\log(1{+}\mathrm{total})$	4.964	0.430	0.00
Steps	$\log(1{+}\mathrm{total})$	6.323	2.949	0.14
Bed activity	$\log(1{+}\mathrm{total})$	3.673	3.048	0.32
Respiratory rate	$\log(1{+}\mathrm{total})$	2.657	2.222	0.33
Medication events	$\log(1{+}\mathrm{total})$	1.077	0.568	0.11

Appendix

Table 5: Summary of the 48 daily input features, with the 24 hourly-profile bins collapsed to their range. Zero rate denotes the fraction of retained days on which the corresponding feature is zero. Exact feature-level summary statistics are computed from the final preprocessing output.

History arm	Single-resident	Households
Recent	0.1335	0.1171
Shifted	0.0241	0.0425
Random	0.0409	0.0499
Distant	0.0553	0.0559

Appendix

Table 6: History-source results using only the 30 motion-derived features. Values are

\Delta I

in nats per output dimension.

Check	Violations
Training chronology violations	0
Test chronology violations	0
Target contained in prediction input	0
Target contained in an SSL positive block	audited separately †
Mismatched anchor eligibility across arms	0
$B_{i}$ uses post-training data	0

Appendix

Table 7: Adversarial audit of temporal splitting and history construction.

History arm	Single-resident	Households
Recent	$-0.0004$	$-0.0001$
Shifted	$-0.0001$	$+0.0002$
Random	$-0.0001$	$+0.0002$
Distant	$-0.0002$	$+0.0002$

Appendix

Table 8: Permuted-target negative control. All estimates are

\Delta I

in nats per output dimension.

Temporal distance	24-window buffer	8-window buffer
5 days	$+0.117$	$+0.285$
10 days	$+0.082$	$+0.235$
15 days	$+0.044$	$+0.194$
20 days	$+0.006$	$+0.173$
25 days	$-0.016$	$+0.153$
30 days	$-0.033$	$+0.126$

Appendix

Table 9: Sensitivity of the temporal-distance analysis to train–test buffer width. The comparison isolates the effect of anchor eligibility on the estimated distance curve.

Experiment	Cohort	Condition	Mean $\pm$ SD	Seeds
History source	Single-resident	Recent	$0.2818\pm 0.0054$	5/5
History source	Single-resident	Shifted	$0.0447\pm 0.0291$	—
History source	Single-resident	Random	$0.0374\pm 0.0169$	—
History source	Single-resident	Distant	$0.0051\pm 0.0143$	—
History source	Households	Recent	$0.2325\pm 0.0097$	5/5
History source	Households	Shifted	$0.0158\pm 0.0043$	—

Appendix

Table 10: Seed variation for headline learned-representation comparisons. “Seeds” counts how many of five favor the first-named condition.

	Single-resident (chance $0.0185$ )		Households (chance $0.0238$ )
Residualized $z$	Logistic	MLP	Logistic	MLP
Recent	0.019	0.183	0.022	0.157
Shifted	0.036	0.510	0.022	0.541
Distant	0.895	0.895	0.888	0.888

Appendix

Table 11: Participant identification accuracy from the residualized representation used for scoring. Values are accuracy, not AUC.

History arm	Single-resident	Households
Recent	$1.9\times 10^{15}$	$2.5\times 10^{8}$
Shifted	$1.8\times 10^{15}$	$2.4\times 10^{8}$
Random	$1.9\times 10^{15}$	$2.4\times 10^{8}$
Distant	$8.0\times 10^{25}$	$9.7\times 10^{15}$

Appendix

Table 12: Condition numbers of the conditional design matrix

[B_{i},z]

by experimental arm.

Cohort	Arm	LOO	HO	LOO	HO	LOO	HO	$\Delta$	$95\%$ CI
		Penalty		Design cond.		Test MSE		Test MSE, LOO $-$ HO
Single-resident	Uniform, 30 d	$0.71$	$78$	$810$	$13$	$1.1551$	$0.8656$	$+0.2895$	$[+0.0726,+0.5975]$
	Uniform, full	$1.60$	$62$	$95$	$14$	$0.9152$	$0.8813$	$+0.0339$	$[+0.0106,+0.0606]$
	Matched, 30 d	$1.12$	$78$	$112$	$13$	$0.9246$	$0.8686$	$+0.0560$	$[+0.0243,+0.0942]$
	Matched, full	$1.16$	$111$	$129$	$11$	$0.9384$	$0.8698$	$+0.0685$	$[+0.0308,+0.1122]$
Households	Uniform, 30 d	$0.89$	$62$	$109$	$13$	$0.8896$	$0.8062$	$+0.0834$	$[+0.0253,+0.1702]$

Appendix

Table 13: Does the held-out-selected penalty predict better on held-out data? Penalties are selected two ways on the same design, split and seeds: RidgeCV ’s internal efficient leave-one-out on the fitting rows, and explicit validation MSE on the held-out validation rows. Neither sees the test rows. Condition numbers are of the regularized design at the selected penalty, averaged over seeds. Neither rule uses the test rows for fitting or for selection: internal leave-one-out sees only the fitting rows, held-out selection sees only the validation rows, and the test rows enter once, after both penalties are fixed, solely to compare how the two generalize. The final columns give the paired difference in test MSE, LOO minus held-out-selected, so positive favors held-out selection; brackets are

95\%

bootstrap intervals over units, five seeds.

	Single-resident			Households
Arm	Learned	Untrained	Raw block	Learned	Untrained	Raw block
Recent	$+0.2818$	$+0.0705$	$+0.3437$	$+0.2325$	$+0.0476$	$+0.2559$
Shifted	$+0.0447$	$-0.0016$	$+0.0340$	$+0.0158$	$+0.0006$	$+0.0017$
Random	$+0.0374$	$-0.0010$	$+0.0176$	$+0.0261$	$+0.0008$	$+0.0175$
Distant	$+0.0051$	$-0.0032$	$+0.0597$	$+0.0051$	$-0.0030$	$+0.0040$
Recent $-$ shifted	$+0.2371$	$+0.0721$	$+0.3097$	$+0.2167$	$+0.0469$	$+0.2543$

Appendix

Table 14: History-source arms under three representations. The untrained column is the mean over 20 initializations of the same architecture train() uses.

Cohort	Drift definition	Stratification	Low	Middle	High	Low/high disjoint
Single-resident	Local	Pooled	+0.1817 [.131,.234]	+0.2305	+0.3215 [.262,.369]	yes
Single-resident	Local	Within	+0.2367 [.185,.281]	+0.2246	+0.2723 [.227,.316]	no
Single-resident	Baseline	Pooled	+0.1793 [.128,.235]	+0.2390	+0.3151 [.256,.363]	yes
Single-resident	Baseline	Within	+0.2422 [.189,.285]	+0.2253	+0.2661 [.223,.307]	no
Single-resident	Person	Pooled	+0.1739 [.138,.210]	+0.2182	+0.3412 [.288,.405]	yes
Households	Local	Pooled	+0.1164 [.094,.138]	+0.1870	+0.3163 [.259,.361]	yes

Appendix

Table 15: Predictive-information results stratified by behavioral drift. Within-participant stratification is not defined for the person-level drift measure.

Within-unit measure	Single-resident ( $n=50$ )	Households ( $n=41$ )
Displacement from own past	$+0.191$ $[+0.122,+0.257]$	$+0.231$ $[+0.147,+0.313]$
Displacement, lagged one block	$+0.037$ $[-0.034,+0.114]$	$+0.130$ $[+0.067,+0.194]$
Rate of change	$+0.008$ $[-0.060,+0.075]$	$-0.033$ $[-0.105,+0.031]$

Appendix

Table 16: Within-unit Spearman correlation between the recency advantage and three measures of behavioral change. Brackets show bootstrap 95% confidence intervals over units.

	Single-resident		Households
Gap	Canonical	Residualized	Canonical	Residualized
0 days	$+0.2818$ $[+0.2289,+0.3385]$	$+0.2796$ $[+0.2272,+0.3363]$	$+0.2325$ $[+0.1810,+0.2883]$	$+0.2324$ $[+0.1807,+0.2881]$
5 days	$+0.2263$ $[+0.1808,+0.2760]$	$+0.2236$ $[+0.1782,+0.2732]$	$+0.1857$ $[+0.1360,+0.2396]$	$+0.1846$ $[+0.1346,+0.2376]$
10 days	$+0.1877$ $[+0.1465,+0.2352]$	$+0.1846$ $[+0.1434,+0.2321]$	$+0.1526$ $[+0.1061,+0.2047]$	$+0.1514$ $[+0.1046,+0.2026]$
15 days	$+0.1715$ $[+0.1328,+0.2162]$	$+0.1668$ $[+0.1283,+0.2105]$	$+0.1445$ $[+0.0990,+0.1953]$	$+0.1420$ $[+0.0963,+0.1931]$
20 days	$+0.1534$ $[+0.1169,+0.1928]$	$+0.1498$ $[+0.1129,+0.1893]$	$+0.1174$ $[+0.0723,+0.1664]$	$+0.1141$ $[+0.0689,+0.1624]$

Appendix

Table 17: Temporal-distance sweep under the canonical and residualized estimators. A fixed 20-day block is moved progressively further from the anchor with the target held at

t+5

days. Brackets are 95% bootstrap intervals over units; five seeds per cell.

Gap	$\Delta I(d)$	$U(d)$	$U/\Delta I$	$\Delta I(d)$	$U(d)$	$U/\Delta I$
	Single-resident			Households
0 days	$0.2818$ ( $1.000$ )	$0.4829$ ( $1.000$ )	$1.7$	$0.2325$ ( $1.000$ )	$0.4453$ ( $1.000$ )	$1.9$
5 days	$0.2263$ ( $0.803$ )	$0.4244$ ( $0.879$ )	$1.9$	$0.1857$ ( $0.799$ )	$0.3993$ ( $0.897$ )	$2.2$
10 days	$0.1877$ ( $0.666$ )	$0.3893$ ( $0.806$ )	$2.1$	$0.1526$ ( $0.656$ )	$0.3624$ ( $0.814$ )	$2.4$
20 days	$0.1534$ ( $0.544$ )	$0.3566$ ( $0.738$ )	$2.3$	$0.1174$ ( $0.505$ )	$0.3298$ ( $0.741$ )	$2.8$
30 days	$0.1458$ ( $0.517$ )	$0.3427$ ( $0.710$ )	$2.4$	$0.0899$ ( $0.387$ )	$0.3025$ ( $0.679$ )	$3.4$

Appendix

Table 18: Persistence with and without conditioning on

B_{i}

, normalized to the value at

d=0

. Raw values in nats per output dimension.

Cohort	30	120	full	Change	95% CI	Seeds	Identity
	$\Delta I$ at record span $L$ (days)
Single-resident	$0.1410$	$0.0963$	$0.0944$	$-33\%$	$[-0.0639,-0.0320]$	5/5	$0.546\!\to\!0.613$
Households	$0.1025$	$0.0539$	$0.0539$	$-47\%$	$[-0.0641,-0.0359]$	5/5	$0.668\!\to\!0.757$

Appendix

Table 19: Changing-state information and identity as the available record grows under local supervision, for the construction used in Section 4 . Nested histories end at the same cut, positive separation stays capped at the

30

-day reference at every

L

, and each unit contributes the same number of anchors per epoch. Intervals are paired unit bootstraps on the

30

-day to full-record change (Appendix B.3 ); “seeds” counts how many of five decline.

Single-resident
Record span $L$ (days)	30	60	120	240	full
$\Delta I$	$0.141$	$0.136$	$0.096$	$0.091$	$0.094$
identity	$0.546$	$0.546$	$0.571$	$0.585$	$0.613$
between-person	$0.333$	$0.329$	$0.357$	$0.373$	$0.371$
within-person	$0.139$	$0.133$	$0.124$	$0.122$	$0.120$
between/within	$2.397$	$2.478$	$2.876$	$3.069$	$3.097$

Appendix

Table 20: Representation geometry on the fixed evaluation anchors as the record grows under local supervision,

L_{2}

-normalized. Between-person is the dispersion of participant centroids and within-person is the mean spread of a participant’s own embeddings. The last two rows give the mean distance between a participant’s own windows, divided by the mean distance to other participants’ windows, for near and far pairs; the far pairs are never used as positives at any

L

Cohort	Encoder	Supervision	30	60	120	240	full	Change	95% CI	Seeds	Identity
			$\Delta I$ at record span $L$ (days)
Single-resident	$160$ k	local	$0.1419$	$0.1356$	$0.1056$	$0.0946$	$0.0878$	$-38\%$	$[-0.0692,-0.0412]$	5/5	$0.54\!\to\!0.59$
Single-resident	$160$ k	unrestricted	$0.1386$	$0.1063$	$0.0673$	$0.0452$	$0.0346$	$-75\%$	$[-0.1281,-0.0833]$	5/5	$0.53\!\to\!0.72$
Single-resident	$40$ k	local	$0.0863$	$0.0898$	$0.0927$	$0.0913$	$0.0771$	$-11\%$	$[-0.0202,-0.0000]$	4/5	$0.46\!\to\!0.48$
Single-resident	$40$ k	unrestricted	$0.0911$	$0.0765$	$0.0711$	$0.0379$	$0.0196$	$-78\%$	$[-0.0900,-0.0534]$	5/5	$0.47\!\to\!0.58$
Households	$160$ k	local	$0.1081$	$0.0899$	$0.0572$	$0.0621$	$0.0533$	$-51\%$	$[-0.0702,-0.0409]$	5/5	$0.67\!\to\!0.74$

Appendix

Table 21: Variants of the nested-history construction: a fixed set of distinct anchors rather than a fresh draw each epoch, and the base-capacity encoder. Unrestricted pairing on the same build is shown for contrast. Nested histories end at the same cut; positives are capped at the

30

-day reference in every condition, so realized separation stays near

20

days throughout, and each unit contributes the same number of anchors per epoch at every

L

. “Local” caps positive separation; “unrestricted” lets positives come from anywhere in the available record and is shown for contrast. Intervals are paired unit bootstraps on the

30

-day to full-record change; the last column gives identity accuracy at the two endpoints.

Construction	Prediction population	Encoder	Cohort	30 days	Full	Change
Expanding	varies with $L$	$40$ k	Single-resident	$0.0971$	$0.0198$	$-80\%$
Expanding	varies with $L$	$40$ k	Households	$0.0958$	$0.0171$	$-82\%$
Fixed-anchor	held fixed	$40$ k	Single-resident	$0.0664$	$0.0243$	$-63\%$
Fixed-anchor	held fixed	$40$ k	Households	$0.0396$	$0.0138$	$-65\%$
Fixed-anchor	held fixed	$160$ k	Single-resident	$0.0578$	$0.0385$	$-33\%$
Fixed-anchor	held fixed	$160$ k	Households	$0.0475$	$0.0304$	$-36\%$

Appendix

Table 22: The record-span decline under every construction reported in this paper.

\Delta I

at the shortest and longest record. The first three blocks use uniform positive sampling: the expanding block varies the prediction population as well as supervision reach, and the two fixed-anchor blocks vary supervision reach alone, at base and at

4\times

capacity. The last two blocks are the nested-history construction with positives held local, which is the one that varies the history available to the encoder. Absolute levels are not comparable across constructions, since the anchor sets differ; only the within-construction change is.

Cohort	Objective	30d	60d	120d	240d	Full	Canon.	Resid.
Single	Person-contrastive	$.0971/.0965$	$.0697/.0686$	$.0565/.0557$	$.0460/.0456$	$.0198/.0195$	$-80\%$	$-80\%$
	sep.-matched	$.0896/.0887$	$.1014/.1010$	$.0827/.0821$	$.1129/.1121$	$.1004/.0998$	$+12\%$	$+13\%$
	Predictive	$.1983/.1979$	$.2129/.2124$	$.2136/.2128$	$.2237/.2239$	$.2299/.2297$	$+16\%$	$+16\%$
	Masked	$.2040/.2034$	$.2042/.2038$	$.1980/.1975$	$.2030/.2022$	$.2049/.2041$	$+0\%$	$+0\%$
Households	Person-contrastive	$.0958/.0954$	$.0744/.0741$	$.0422/.0423$	$.0222/.0212$	$.0171/.0162$	$-82\%$	$-83\%$
	sep.-matched	$.0876/.0788$	$.0862/.0858$	$.0868/.0867$	$.0755/.0690$	$.0634/.0632$	$-28\%$	$-20\%$

Appendix

Table 23: Record-span sweep, canonical / residualized

\Delta I

. The headline collapse and the direction of every arm are unchanged by the estimator.

Cohort	Representation	History	Canonical	Residualized
Single-resident	Learned	Recent	$0.2818$	$0.2796$
Single-resident	Mean	Recent	$0.2818$	$0.2696$
Single-resident	Learned	Shifted	$0.0447$	$0.0442$
Single-resident	Mean	Shifted	$0.0071$	$-0.0038$
Single-resident	Learned	Random	$0.0374$	$0.0357$
Single-resident	Mean	Random	$0.0050$	$-0.0027$

Appendix

Table 24: Canonical and residualized estimates of

\Delta I

under the overlap-free history construction, the Cholesky solver and the deduplicated household cohort.

Cohort	Baseline	Decoder	30d	60d	120d	240d	Full	30d $\to$ full
Single	Fixed $B_{i}$	Ridge	.0940	.0689	.0525	.0384	.0151	$-84\%$
	Fixed $B_{i}$	MLP	.1687	.1317	.0995	.0872	.0563	$-67\%$
	Trailing 30d	Ridge	.0366	.0278	.0234	.0104	.0143	$-61\%$
	Trailing 30d	MLP	.0362	.0244	.0241	.0181	.0112	$-69\%$
	Trailing 60d	Ridge	.0430	.0292	.0220	.0062	.0074	$-83\%$
	Trailing 60d	MLP	.0504	.0316	.0313	.0261	.0152	$-70\%$

Appendix

Table 25: Record-span sweep for uniform positives under a fixed and two recency-matched baselines, each with both decoder families. Values are

\Delta I

in nats per output dimension, computed on the anchors at which all three baselines are defined. The separation-matched arm on the same anchors is Table 26 .

Cohort	Baseline	Decoder	30d	60d	120d	240d	Full	30d $\to$ full
Single	Fixed $B_{i}$	Ridge	0.0950	0.0893	0.0818	0.1016	0.0982	$+3\%$
	Fixed $B_{i}$	MLP	0.1629	0.1488	0.1426	0.1587	0.1437	$-12\%$
	Trailing 30d	Ridge	0.0392	0.0351	0.0314	0.0385	0.0367	$-6\%$
	Trailing 30d	MLP	0.0360	0.0234	0.0343	0.0370	0.0330	$-9\%$
	Trailing 60d	Ridge	0.0433	0.0393	0.0307	0.0405	0.0406	$-6\%$
	Trailing 60d	MLP	0.0536	0.0303	0.0480	0.0508	0.0416	$-22\%$

Appendix

Table 26: The same sweep for the separation-matched arm, computed on the same anchors, baselines and decoders as Table 25 . Values are

\Delta I

in nats per output dimension.

	Single-resident			Households
Cap	Realized	$\Delta I$	Identity	Realized	$\Delta I$	Identity
10 days	$10.0$	$+0.1250$	$0.593$	$10.0$	$+0.0982$	$0.743$
20 days	$15.0$	$+0.0871$	$0.615$	$15.0$	$+0.0539$	$0.763$
30 days	$20.1$	$+0.0728$	$0.617$	$20.1$	$+0.0408$	$0.762$
60 days	$34.5$	$+0.0628$	$0.634$	$34.7$	$+0.0348$	$0.761$
90 days	$49.4$	$+0.0554$	$0.648$	$48.9$	$+0.0366$	$0.771$

Appendix

Table 27: Changing-state information and identity decodability against the maximum permitted positive-pair separation, at fixed record span. Realized mean separation is reported because the cap is an upper bound. No cap produced an empty pool, so none silently reverted to unrestricted sampling.

	Single-resident			Households
Cap	Realized	$\Delta I$	Identity	Realized	$\Delta I$	Identity
2 days	$2.0$	$+0.1515$	$0.537$	$2.0$	$+0.2027$	$0.690$
4 days	$3.2$	$+0.0909$	$0.529$	$3.2$	$+0.0945$	$0.703$
7 days	$4.7$	$+0.0774$	$0.536$	$4.7$	$+0.0790$	$0.703$
10 days	$6.2$	$+0.0726$	$0.531$	$6.3$	$+0.0689$	$0.702$
20 days	$11.2$	$+0.0670$	$0.538$	$11.2$	$+0.0546$	$0.702$

Appendix

Table 28: Daily-window supervision-span sweep, target-overlapping positives excluded. Five seeds per cell. The untrained reference under this geometry is

0.1058\pm 0.0081

and

0.1524\pm 0.0102

, from 20 initializations.

	Single-resident			Households
$\tau$	Realized	$\Delta I$ ( $L_{2}$ )	$\times$ floor	Realized	$\Delta I$ ( $L_{2}$ )	$\times$ floor
2 days	$2.0$	$+0.1211$	$1.16$	$2.0$	$+0.1235$	$1.07$
4 days	$3.2$	$+0.0444$	$0.43$	$3.2$	$+0.0463$	$0.40$
8 days	$5.2$	$+0.0368$	$0.35$	$5.2$	$+0.0254$	$0.22$
16 days	$9.2$	$+0.0292$	$0.28$	$9.2$	$+0.0153$	$0.13$
32 days	$17.2$	$+0.0272$	$0.26$	$17.2$	$+0.0096$	$0.08$

Appendix

Table 29: Changing-state information against the maximum permitted positive-pair separation, at fixed record span. Realized mean separation is reported because the cap is an upper bound; no cap produced an empty pool.

\times

floor is

\Delta I

as a multiple of a same-architecture untrained encoder: values above

1

mean the objective adds information the architecture did not already carry, values below

1

mean it removes some. Every representation is

L_{2}

-normalized before scoring, so trained and untrained arms are compared at matched embedding scale (Appendix H.6 ).

	Single-resident			Households
$\tau$	Fixed-card. ( $L_{2}$ / raw)	Cap-sampled ( $L_{2}$ )	Gap	Fixed-card. ( $L_{2}$ / raw)	Cap-sampled ( $L_{2}$ )	Gap
2 days	$0.1211$ / $0.1644$	$0.1211$ (2.0)	$+0.0000$	$0.1235$ / $0.2065$	$0.1235$ (2.0)	$+0.0000$
4 days	$0.0757$ / $0.1171$	$0.0444$ (3.2)	$+0.0314$	$0.0898$ / $0.1552$	$0.0463$ (3.2)	$+0.0435$
8 days	$0.0599$ / $0.0849$	$0.0292$ (9.2)	$+0.0307$	$0.0452$ / $0.1165$	$0.0153$ (9.2)	$+0.0298$
16 days	$0.0444$ / $0.0774$	$0.0272$ (17.2)	$+0.0173$	$0.0300$ / $0.0895$	$0.0096$ (17.2)	$+0.0204$
32 days	$0.0283$ / $0.0549$	$0.0130$ (32.9)	$+0.0153$	$0.0265$ / $0.0676$	$0.0054$ (32.9)	$+0.0211$

Appendix

Table 30: Fixed cardinality (one positive per anchor) against cap-sampling, scale-matched and paired on realized separation. Every value is

\Delta I

after

L_{2}

-normalizing the representation (Appendix H.6 ); the fixed-cardinality arm realizes a separation of exactly

\tau

, so each row is paired with the cap-sampled arm whose realized separation is closest, given in parentheses. Raw fixed-cardinality values are shown for continuity with the earlier scoring.

Single-resident , untrained floor $0.1042$
Sampling rule	Realized sep. (d)	$\Delta I$ (raw)	$\Delta I$ ( $L_{2}$ )	$\times$ floor
hard cap $\tau$ = 2 d	$2.0$	$0.1644$	$0.1211$	$1.16$
hard cap $\tau$ = 8 d	$5.2$	$0.0693$	$0.0414$	$0.40$
hard cap $\tau$ = 32 d	$17.3$	$0.0642$	$0.0207$	$0.20$
hard cap $\tau$ = unrestricted	$141.9$	$0.0470$	$0.0083$	$0.08$
exponential decay $\lambda$ = 2 d	$4.2$	$0.0783$	$0.0397$	$0.38$

Appendix

Table 31: Hard caps and distance-weighted sampling on the same build, with positives restricted to the training era in both. A hard cap draws positives uniformly within

\tau

; exponential decay draws them with probability proportional to

\exp(-|\Delta t|/\lambda)

over the same candidates. Realized separation is the draw-weighted mean over training anchors.

\times

floor is

\Delta I

as a multiple of a same-architecture untrained encoder, scored after

L_{2}

normalization.

	Single-resident		Households
Cap	Original	Overlap-free	Original	Overlap-free
30 days	$+0.0664$ $[+0.0463,+0.0893]$	$+0.0697$ $[+0.0496,+0.0919]$	$+0.0396$ $[+0.0232,+0.0608]$	$+0.0381$ $[+0.0229,+0.0578]$
60 days	$+0.0631$ $[+0.0444,+0.0836]$	$+0.0614$ $[+0.0425,+0.0823]$	$+0.0299$ $[+0.0151,+0.0495]$	$+0.0320$ $[+0.0176,+0.0510]$
120 days	$+0.0474$ $[+0.0328,+0.0637]$	$+0.0511$ $[+0.0355,+0.0681]$	$+0.0282$ $[+0.0138,+0.0476]$	$+0.0269$ $[+0.0126,+0.0458]$
240 days	$+0.0405$ $[+0.0270,+0.0550]$	$+0.0413$ $[+0.0273,+0.0570]$	$+0.0256$ $[+0.0107,+0.0459]$	$+0.0211$ $[+0.0084,+0.0385]$
Unrestricted	$+0.0243$ $[+0.0162,+0.0334]$	$+0.0259$ $[+0.0172,+0.0353]$	$+0.0138$ $[+0.0051,+0.0256]$	$+0.0142$ $[+0.0056,+0.0259]$

Appendix

Table 32: Span sweep with and without target-overlapping positive pairs. Brackets are 95% bootstrap intervals over units; five seeds per cell.

Single-resident
	4 days	10 days	30 days	60 days	120 days
$K=1$	$+0.1370$ ( $4.0$ )	$+0.0849$ ( $10.0$ )	$+0.0684$ ( $30.0$ )	$+0.0571$ ( $58.6$ )	$+0.0428$ ( $97.4$ )
$K=3$	$+0.1258$ ( $4.0$ )	$+0.0993$ ( $10.0$ )	$+0.0812$ ( $30.0$ )	$+0.0773$ ( $58.5$ )	$+0.0673$ ( $97.3$ )
$K=10$	$+0.0938$ ( $6.5$ )	$+0.0780$ ( $9.5$ )	$+0.0823$ ( $29.5$ )	$+0.0613$ ( $57.7$ )	$+0.0737$ ( $99.5$ )
$K=30$	$+0.0782$ ( $16.4$ )	$+0.0748$ ( $16.4$ )	$+0.0653$ ( $29.2$ )	$+0.0518$ ( $56.6$ )	$+0.0650$ ( $103.9$ )
Households

Appendix

Table 33: Changing-state information by pool size

K

and realized mean separation (parentheses), daily windows. Comparisons across

K

are interpretable only at nominal 30 and 60 days; elsewhere band widening confounds

K

with separation.

Cohort	Series	normalized value at realized separation (days)
Single-resident	Availability	$1.000$ ( $0$ )	$0.666$ ( $10$ )	$0.517$ ( $30$ )	$0.411$ ( $60$ )	—
	$K=1$	$1.000$ ( $4.0$ )	$0.620$ ( $10.0$ )	$0.499$ ( $30.0$ )	$0.417$ ( $58.6$ )	$0.312$ ( $97.4$ )
	$K=3$	$1.000$ ( $4.0$ )	$0.789$ ( $10.0$ )	$0.645$ ( $30.0$ )	$0.614$ ( $58.5$ )	$0.535$ ( $97.3$ )
	$K=10$	$1.000$ ( $6.5$ )	$0.832$ ( $9.5$ )	$0.878$ ( $29.5$ )	$0.653$ ( $57.7$ )	$0.786$ ( $99.5$ )
	$K=30$	$1.000$ ( $16.4$ )	$0.957$ ( $16.4$ )	$0.835$ ( $29.2$ )	$0.662$ ( $56.6$ )	$0.831$ ( $103.9$ )
Households	Availability	$1.000$ ( $0$ )	$0.656$ ( $10$ )	$0.387$ ( $30$ )	$0.276$ ( $60$ )	—

Appendix

Table 34: Values plotted in Figure 6 . Availability is the Section 3 distance sweep; contrastive values are plotted at their realized separation, given in parentheses. Each series is normalized to its own shortest separation.

Single-resident
Contrastive label	Identities	$\Delta I$	Identity	vs. unsegmented	vs. $L{=}30$
person identity (unsegmented)	$54$	$0.0944$	$0.613$	$+0.0000$ $[+0.0000,+0.0000]$	$-0.0466$ $[-0.0638,-0.0317]$
cross-segment pairs masked	$524$	$0.0944$	$0.613$	$+0.0000$ $[+0.0000,+0.0000]$	$-0.0466$ $[-0.0638,-0.0317]$
segments, 30 d	$987$	$0.1474$	$0.578$	$+0.0529$ $[+0.0410,+0.0661]$	$+0.0063$ $[-0.0078,+0.0191]$
segments, 60 d	$524$	$0.1454$	$0.583$	$+0.0510$ $[+0.0393,+0.0635]$	$+0.0043$ $[-0.0096,+0.0170]$
segments, 120 d	$300$	$0.1453$	$0.587$	$+0.0509$ $[+0.0392,+0.0635]$	$+0.0042$ $[-0.0096,+0.0172]$

Appendix

Table 35: Within-person discrimination at the full record. Only the contrastive label changes: the positive identity is (participant, group) rather than participant, so a participant’s windows from other groups become negatives while an anchor’s own positives are always excluded. Positives, their separation, the anchors, the schedule and the evaluation are those of Appendix D.1 . Contrasts are paired unit bootstraps against the unsegmented full-record run and against the

30

-day record.

Single-resident
Arm	$\Delta I$	Identity	Between/within	Own windows $>180$ d
$L{=}30$ reference	$0.1410$	$0.546$	$2.397$	$0.6366$
full record, unsegmented	$0.0944$	$0.613$	$3.097$	$0.5756$
full record, segments 30 d	$0.1474$	$0.578$	$2.759$	$0.6078$
full record, segments 60 d	$0.1454$	$0.583$	$2.756$	$0.6062$
full record, segments 120 d	$0.1453$	$0.587$	$2.789$	$0.6042$

Appendix

Table 36: Geometry under within-person discrimination,

L_{2}

-normalized, on the fixed evaluation anchors. “Between/within” is the ratio of between-participant centroid dispersion to within-participant spread. The last column is the mean distance between a participant’s own windows more than

180

days apart, divided by the mean distance to other participants’ windows; those pairs are never positives in any arm.

Representation	$\Delta I$	$r^{2}$ (%)	Identity
Raw block ( $92$ -d)	$+0.1666$	$21.57$	—
PCA- $32$	$+0.1502$	$20.31$	—
Masked reconstruction	$+0.1360$	$18.59$	$0.159$
Random projection- $32$	$+0.1213$	$16.86$	—
Predictive	$+0.1222$	$13.36$	—
Untrained encoder- $32$	$+0.0910$	$12.79$	$0.085$

Appendix

Table 37: GLOBEM Wave 2, prospective state-feature set. All representations

L_{2}

-normalized;

r^{2}

is the percentage of baseline test error removed beyond

B_{i}

; identity is

199

-way participant decodability (chance

0.0050

8038

anchors,

199

participants,

2719

test rows.

Cap (realized sep., d)	$\Delta I$	$r^{2}$ (%)	Identity
$3.2$	$+0.0093$	$0.80$	$0.703$
$5.2$	$+0.0066$	$0.42$	—
$8.8$	$+0.0078$	$0.48$	$0.724$
$15.2$	$+0.0045$	$0.33$	—
$24.6$	$+0.0023$	$0.14$	$0.779$

Appendix

Table 38: GLOBEM Wave 2 supervision-span sweep. State information falls and identity rises on the same axis. Separations are realized, not nominal.

Cohort	Objective	Decoder	30d	60d	120d	240d	Full	30d $\to$ full
Single	Person-contrastive	Ridge	.0971	.0697	.0565	.0460	.0198	$-80\%$
	Person-contrastive	MLP	.1433	.1352	.0809	.0619	.0398	$-72\%$
	sep.-matched	Ridge	.0896	.1014	.0827	.1129	.1004	$+12\%$
	sep.-matched	MLP	.1390	.1408	.1181	.1308	.1271	$-9\%$
	Predictive	Ridge	.1983	.2129	.2136	.2237	.2299	$+16\%$
	Predictive	MLP	.1752	.1747	.1821	.1984	.2094	$+20\%$

Appendix

Table 39: Record-span sweep under ridge and MLP decoders. Values are

\Delta I

in nats per output dimension; the final column is the change from a 30-day record to the full record.

	Single-resident (52 units)	Households (42 units)
Recent	$0.3046\pm 0.0037$	$0.2443\pm 0.0034$
Shifted	$0.0313\pm 0.0214$	$0.0145\pm 0.0061$
Random	$0.0362\pm 0.0140$	$0.0418\pm 0.0028$
Distant	$0.0085\pm 0.0140$	$-0.0003\pm 0.0033$
Recent $-$ shifted	$+0.2732$ $[+0.2319,+0.3195]$ , 5/5	$+0.2298$ $[+0.1880,+0.2730]$ , 5/5
Recent $-$ random	$+0.2684$ $[+0.2263,+0.3159]$ , 5/5	$+0.2026$ $[+0.1638,+0.2416]$ , 5/5

Appendix

Table 40: History-source results with a GRU last-state encoder. Values are

\Delta I

in nats per output dimension, mean

\pm

standard deviation over five seeds. Contrasts show the paired difference with a 95% bootstrap interval over units.

Gap from present	Single-resident	Households
0 days	$0.3071$ $[0.2562,0.3625]$	$0.2429$ $[0.1936,0.2935]$
5 days	$0.2473$ $[0.2025,0.2949]$	$0.1946$ $[0.1511,0.2391]$
10 days	$0.1970$ $[0.1552,0.2427]$	$0.1549$ $[0.1146,0.2008]$
15 days	$0.1764$ $[0.1356,0.2198]$	$0.1477$ $[0.1053,0.1940]$
20 days	$0.1595$ $[0.1237,0.1983]$	$0.1206$ $[0.0764,0.1678]$
30 days	$0.1414$ $[0.1049,0.1790]$	$0.1044$ $[0.0703,0.1419]$

Appendix

Table 41: Temporal-distance sweep with a GRU last-state encoder. A fixed 20-day block is moved progressively further from the anchor with the target held at

t+5

days. Brackets are 95% bootstrap intervals over units.

Cohort	Context	Ordered	Shuffled	Set	Ordered $-$ Shuffled	Main-text 5d
Single-resident	$K{=}4$ (4d)	0.3016	0.2381	0.2439	+0.0635 [0.0545, 0.0721]	+0.0408
Single-resident	$K{=}20$ (20d)	0.2602	0.1605	0.1569	+0.0997 [0.0824, 0.1183]	+0.0408
Households	$K{=}4$ (4d)	0.3167	0.2481	0.2539	+0.0686 [0.0486, 0.0882]	+0.0228
Households	$K{=}20$ (20d)	0.2598	0.1364	0.1423	+0.1234 [0.0975, 0.1519]	+0.0228

Appendix

Table 42: Temporal order at daily resolution. Values are

\Delta I

in nats per output dimension; brackets give participant- or household-level bootstrap 95% confidence intervals for the ordered

-

shuffled difference.

K{=}4

is matched on window count to the main-text five-day row and

K{=}20

on calendar span. The final column repeats the main-text five-day value for reference.

Experiment	Arm	Representation dim.	Encoder width	Parameters
History source	All four arms	32	95	33,791
Temporal order	Ordered / shuffled	32	95	33,791
Temporal order	Set (deep sets)	32	88	33,908
Temporal order, $K{=}8$	Ordered / shuffled	32	71	34,111
Sequence encoders, $N{=}1/2/4/8$	MLP	48	151 / 134 / 108 / 74	39,999 / 39,920 / 40,200 / 39,992
Sequence encoders, $N{=}1/2/4/8$	GRU (1 layer)	48	84	40,200

Appendix

Table 43: Realized model capacity by experimental arm. For the sequence encoders of Appendix H.3 , width is the recurrent or model dimension and is matched to the same

40

k budget at each context length

N

; the Transformer is a single encoder layer with

4

heads, learned positional embeddings, feed-forward width

2d_{\text{model}}

, no dropout, and mean pooling over positions.

		30-day record		Full record
Cohort	Dim.	$\Delta I$	Identity	$\Delta I$	Identity
Single	$16$	$0.0639$	$0.608$	$0.0214$	$0.842$
	$32$	$0.0664$	$0.619$	$0.0243$	$0.844$
	$64$	$0.0731$	$0.633$	$0.0300$	$0.841$
	$128$	$0.0966$	$0.613$	$0.0376$	$0.844$
Households	$16$	$0.0304$	$0.757$	$0.0112$	$0.861$

Appendix

Table 44: Record-span sweep at four representation dimensions, uniform positives, parameter count matched to within

1\%

\Delta I

is in nats per output dimension; identity is multinomial decoding accuracy.

	Single-resident		Households
Arm	$\Delta I$	identity	$\Delta I$	identity	vs. unrestricted
1 Unrestricted person-contrastive	$+0.0243$	$0.844$	$+0.0138$	$0.863$	—
2 Hand-matched 30-day cap	$+0.0830$	$0.625$	$+0.0458$	$0.757$	$+.0587$ / $+.0320$
3 TNC	$+0.1752$	$0.257$	$+0.1612$	$0.378$	$+.1510$ / $+.1474$
4 InfoNCE $+$ ADF neighborhood	$+0.1109$	$0.106$	$+0.1005$	$0.133$	$+.0867$ / $+.0867$
PCA, 32-d (reference)	$+0.2674$	$0.549$	$+0.2233$	$0.705$

Appendix

Table 45: Raw-score TNC comparison on the five-day-window build, shown for completeness. Because representation norms differ across methods, these values are not used for the final cross-method interpretation; the scale-matched

L_{2}

-normalized analysis follows below. The last column is a paired unit bootstrap against arm 1. Table 46 repeats the raw-score comparison on the daily-window build used in Figure 4 .

Arm	Single-resident	Households
Unrestricted person-contrastive	$+0.0492$	$+0.0223$
TNC	$+0.1716$	$+0.1711$
InfoNCE $+$ ADF neighborhood	$+0.1366$	$+0.1825$
Raw block, 96-d (reference)	$+0.2738$	$+0.3177$
PCA, 32-d (reference)	$+0.2187$	$+0.2648$
Random projection, 32-d	$+0.1536$	$+0.2170$

Appendix

Table 46: Raw-score comparison on the daily-window build, matching Figure 4 . Because representation norms differ across methods, final cross-method conclusions use the

L_{2}

-normalized analysis below. The hand-matched arm and identity probes were not rerun under this geometry, so those cells are unavailable.

	Single-resident		Households
Representation	$\Delta I$	identity	$\Delta I$	identity
Raw block, 96-d	$+0.3013$	$0.614$	$+0.2308$	$0.738$
PCA, 32-d (training-only)	$+0.2674$	$0.549$	$+0.2233$	$0.705$
Random projection, 32-d	$+0.2034$	$0.462$	$+0.1947$	$0.616$
Untrained encoder, 32-d	$+0.0995$	$0.342$	$+0.1087$	$0.445$
Learned, short span	$+0.0664$	$0.619$	$+0.0396$	$0.759$

Appendix

Table 47: Capacity-matched baselines, ridge decoder. PCA is fit on training anchors only and the training mean and components are applied unchanged to test.

Cohort	Negatives	Span	InfoNCE loss	Top-1	$\Delta I$	Change
Single-resident	Easy (uniform)	30d	$2.2061$	$0.595$	$+0.0664$ $[+0.0463,+0.0893]$
Single-resident	Easy (uniform)	Full	$2.7368$	$0.353$	$+0.0243$ $[+0.0162,+0.0334]$	$-63\%$ , 5/5
Single-resident	Hard (mined)	30d	$2.8729$	$0.262$	$+0.0392$ $[+0.0228,+0.0578]$
Single-resident	Hard (mined)	Full	$3.5039$	$0.197$	$+0.0234$ $[+0.0147,+0.0337]$	$-40\%$ , 5/5
Households	Easy (uniform)	30d	$2.1341$	$0.625$	$+0.0396$ $[+0.0232,+0.0608]$
Households	Easy (uniform)	Full	$2.4543$	$0.443$	$+0.0138$ $[+0.0051,+0.0256]$	$-65\%$ , 5/5

Appendix

Table 48: Contrastive difficulty crossed with longitudinal span. Pretext metrics are computed on held-out validation anchors. Brackets are 95% bootstrap intervals over units; “seeds” counts how many of five show a decline from 30 days to the full record.

Cohort	Negatives	Span	Pool size	InfoNCE loss	Top-1	$\Delta I$ [95% CI]
Single-resident	Fixed-8	30d	$8.0$	$2.1453$	$0.622$	$+0.0664$ $[+0.0463,+0.0893]$
Single-resident	Fixed-8	Full	$8.0$	$2.6355$	$0.407$	$+0.0243$ $[+0.0162,+0.0334]$
Single-resident	In-batch	30d	$63.3$	$2.4125$	$0.582$	$+0.0334$ $[+0.0238,+0.0440]$
Single-resident	In-batch	Full	$63.3$	$3.1163$	$0.430$	$+0.0146$ $[+0.0100,+0.0199]$
Households	Fixed-8	30d	$8.0$	$2.0827$	$0.653$	$+0.0396$ $[+0.0232,+0.0608]$
Households	Fixed-8	Full	$8.0$	$2.3975$	$0.471$	$+0.0138$ $[+0.0051,+0.0256]$

Appendix

Table 49: Larger, dynamically resampled negative pool. The fixed-8 rows repeat the standard condition as a paired reference within the same run. Pretext metrics are computed on held-out validation anchors in batches of

128

and averaged across batches, which is the batching the in-batch pool requires; Table 48 scores the whole validation set in a single pass. The fixed-8 checkpoints are the same in both tables, which is why

\Delta I

matches exactly while the pretext columns differ slightly. Brackets are 95% bootstrap intervals over units.

	Single-resident		Households
History	CKA / ceiling	Information	CKA / ceiling	Information
Recent	1.000	100%	1.000	100%
Shifted	0.536	9%	0.592	7%
Random	0.571	9%	0.610	10%
Distant	0.571	5%	0.597	2%

Appendix

Table 50: Normalized linear CKA and predictive information retained relative to the recent-history encoder. CKA is normalized by the similarity between two recent-history encoders trained with different random seeds (raw ceilings

0.876

and

0.874

). Both cohorts use the Cholesky estimator and the deduplicated household panel.

Cohort	Objective	30-day $\Delta I$	Full-record $\Delta I$	Change	Identity (30d $\to$ full)
Single-resident	Person-contrastive	$+0.2030$ $[+0.1494,+0.2718]$	$-0.0115$ $[-0.0335,+0.0136]$	$-106\%$	$0.467\to 0.738$
Single-resident	separation-matched	$+0.2406$ $[+0.1772,+0.3225]$	$+0.3208$ $[+0.2474,+0.4010]$	$+33\%$	$0.471\to 0.560$
Single-resident	Predictive	$+0.5652$ $[+0.4644,+0.6693]$	$+0.6381$ $[+0.5081,+0.7900]$	$+13\%$	$0.500\to 0.500$
Single-resident	Masked reconstruction	$+0.6829$ $[+0.5578,+0.8281]$	$+0.6715$ $[+0.5422,+0.8227]$	$-2\%$	$0.433\to 0.433$
Households	Person-contrastive	$+0.2177$ $[+0.1736,+0.2677]$	$-0.0018$ $[-0.0095,+0.0049]$	$-101\%$	$0.621\to 0.827$
Households	separation-matched	$+0.2039$ $[+0.1627,+0.2493]$	$+0.1850$ $[+0.1343,+0.2416]$	$-9\%$	$0.626\to 0.740$

Appendix

Table 51: Held-out channel evaluation.

\Delta I

is measured on the time-in-bed statistics, which are excluded from all encoder inputs, from the self-supervised targets, and from

B_{i}

. Brackets are 95% bootstrap intervals over units; identity accuracy is reported on the same frozen representations.

		Uniform positives		Separation-matched
Cohort	$N$	$\Delta(\Delta I)$	$\Delta$ identity	$\Delta(\Delta I)$	$\Delta$ identity
Single-resident	15	$-0.0452\pm 0.0273$	$+0.125\pm 0.031$	$-0.0051\pm 0.0054$	$-0.001\pm 0.010$
Single-resident	30	$-0.0404\pm 0.0214$	$+0.171\pm 0.027$	$+0.0001\pm 0.0056$	$+0.003\pm 0.006$
Single-resident	54 (full)	$-0.0428$	$+0.231$	$+0.0136$	$+0.004$
Households	15	$-0.0394\pm 0.0160$	$+0.070\pm 0.020$	$+0.0058\pm 0.0122$	$-0.003\pm 0.005$
Households	30	$-0.0185\pm 0.0046$	$+0.081\pm 0.009$	$-0.0020\pm 0.0029$	$+0.005\pm 0.005$

Appendix

Table 52: Effect of cohort size on the span experiment. Entries are the paired change from 30-day to full-record training,

\Delta I_{\mathrm{full}}-\Delta I_{30\mathrm{d}}

and the corresponding change in identity accuracy, mean

\pm

standard deviation across participant subsets. The full level admits only one subset, so no across-subset deviation is available. Identity is multiclass accuracy, so chance is

1/N

(

0.067

0.033

and

0.019

0.024

at the full level). No cell is near the ceiling: 30-day accuracies run

0.744

–

0.614

and

0.801

–

0.755

. Chance-adjusting the change,

(\mathrm{acc}-1/N)/(1-1/N)

, leaves it essentially unchanged (

+0.134

+0.176

+0.235

and

+0.075

+0.084

+0.111

for uniform positives), so its growth with cohort size is not a chance-level or ceiling artifact.

$\Delta t$ bin	Initial loss	Final loss	Learned weight
2	6.0422	0.0703	0.884
3	6.1339	0.0704	0.732
4–5	6.2152	0.0711	0.578
6–8	6.2789	0.0712	0.424
9–13	6.3430	0.0708	0.272
14–20	6.3631	0.0704	0.169

Appendix

Table 53: InfoNCE loss by temporal-separation bin before and after training, for the seven effective bins (Appendix I.3 ). Loss is averaged over all pairs falling in each bin; learned weights are normalized to the nearest bin.

	Single-resident				Households
Arm	$\Delta I$	identity	InfoNCE	top-1	$\Delta I$	identity	InfoNCE	top-1
Untrained (strong architecture)	$+0.0686$	$0.315$	—	—	$+0.0793$	$0.408$	—	—
Strong, short span (30 d)	$+0.0578$	$0.634$	$7.34$	$0.347$	$+0.0475$	$0.754$	$6.38$	$0.461$
Strong, unrestricted	$+0.0385$	$0.859$	$8.33$	$0.214$	$+0.0304$	$0.864$	$7.99$	$0.280$
Weak ( $40$ k), short span	$+0.0664$	$0.619$	—	—	$+0.0396$	$0.759$	—	—
Weak ( $40$ k), unrestricted	$+0.0243$	$0.844$	—	—	$+0.0138$	$0.863$	—	—

Appendix

Table 54: Strong-contrastive stress test. Pretext metrics are on held-out validation anchors under in-batch negatives. The weak rows repeat the

40

k encoder on the same anchors.

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