Appendix A — The Measured Boundary

This appendix carries the technical weight behind Part IV of the essay: exact definitions, the validated regime, the control family and its mechanism, the crossover tables with confidence intervals, the fixed-k caveat, and the stated limits of the claim. Every number below is copied from a committed repository artifact; canonical/RESULTS_CANONICAL.md is the registry that binds each value to its source file and commit, and no statement here is permitted to be stronger than what that registry records.

A.1 The estimator

The blind dimension estimator is E_CAP from Kleindessner & von Luxburg (AISTATS 2015), reconstructed in experiments/harness_valid/B2_2_1/PAPER_EXTRACTION.md directly from the paper’s PDF. Given only a directed kNN graph (an edge i → j iff x_j is among the k nearest samples to x_i):

• B_SP(i, 1) is the radius-1 ball around vertex i in shortest-path distance — the vertex plus its k out-neighbors, so |B_SP(i,1)| = k + 1 always.
• L_CAP(i) = min_{j : i→j} |B_SP(i,1) ∩ B_SP(j,1)| / (k+1) — the smallest normalized overlap between i’s ball and a neighbor’s ball.
• In continuous space, the relative volume of the lens where two unit balls at unit distance overlap is S(d) = I_{3/4}((d+1)/2, 1/2) (regularized incomplete beta function), which depends injectively on d.
• The estimate is E_CAP = S⁻¹(mean L_CAP): measure the overlap in the graph, invert the geometry that would have produced it.

The intuition in one line: in low dimension, near-neighbors share most of their neighborhoods; as dimension grows, that shared overlap dies away at a rate that encodes the dimension.

A.2 What “blind” means, operationally

The learner path receives the kNN graph and nothing else. This is not a promise; it is enforced and recorded:

• The fit path is AST-scanned for forbidden truth-names (coordinates, distances, generator labels, truth_axes, …) — the scan is static analysis of the code, not self-report.
• The evaluation-oracle scan checks that no ground-truth hint enters the evaluation call sites. The B2.2.1 decision records classification_success_depends_on_harness_hint: false (B2_2_1/decision.json).
• The predecessor experiment B2.1 is the honest contrast: there the 3D-control classification did depend on a harness-provided hint (truth_axes=3), the harness flagged it, and the canonical registry forbids citing it as blind discovery. The flag existing — and firing on my own earlier result — is what makes the B2.2.1 “false” meaningful.
• One interface wart, surfaced by an external read of this repository: the world generator’s learner_view also exposes a seed_id alongside the adjacency. The estimator path never reads it — enforced in CI by an AST scan of every estimator function with the harness’s own leakage scanner (scripts/check_learner_view_hygiene.py) — but a stricter interface would not have exposed it at all. The signed run’s code is preserved as it ran; the check, not a retroactive edit, carries the claim.

A.3 The validated regime

B2.2.1 reproduces the paper’s dimension-table behavior from the graph alone, across 20 seeds, with every cell’s expected outcome preregistered — including the expected failure. From B2_2_1/decision.json (key per_cell):

The d=12, n=1000 failure is the estimator faltering exactly where its own literature predicts, preregistered as a failure before the run. The competing estimator E_DP was worse than E_CAP in every cell (e_dp_worse_than_e_cap_all_cells: true). The supported claim is bounded: literature-table rows up to d ≤ 7, with d = 12 as boundary checks.

A.4 The control family, and why it fools the estimator

The preregistered null is a directed random k-out graph: each vertex draws k out-neighbors uniformly at random — no geometry at all. Its mimicry of high-dimensional geometry is not a mystery; it is derived and verified in B2_3/outputs/random_control_mechanism.json:

For an edge i → j, the balls always share at least j itself; with high probability the minimum over out-neighbors is exactly that one forced vertex, giving L_CAP ≈ 1/(k+1) — which E_CAP reads as a spuriously high dimension that depends on k:

That k-dependence is also the tell. Real geometry’s estimate moves only mildly under a k-sweep; the random graph’s estimate tracks k itself. The preregistered discriminator is k-spread — the range of estimates across k ∈ {10, 15, 20} — with a kill threshold of 2.5. The random control’s k-spread is 3.70 (B2_2_1/decision.json, key random_graph_control); the geometric worlds in the validated regime stay well under it (per-cell spreads 0.20–2.03 in A.3).

A.5 The crossover

B2.3 asks where paired discrimination (geometric world vs. random control, same n, same pipeline) collapses as d grows. From B2_3/decision.json and outputs/crossover_results.json:

Selected cells with Wilson 95% intervals (N = 50 paired runs per cell):

Five times the data, and the discrimination boundary of this test moved from d ≈ 130 down to d ≈ 24. That is the essay’s “more data made things worse” — scoped, as it must be, to this diagnostic against this control.

A.6 The mechanism diagnostic

Two candidate explanations were compared at the two crossover points (outputs/local_knn_mechanism_results.json):

The generic story (“distances concentrate in high dimension”) does not unify the two collapse points — its value differs by 2.4× between them. The local story does: the probability that an actual edge of the graph carries shared structure beyond the single forced overlap is nearly identical (≈0.8) at both crossovers. The estimator dies when its edges run out of surplus local overlap — when near stops meaning geometry and starts meaning echo.

A.7 The fixed-k caveat (load-bearing)

An exploratory diagnostic held k = 15 fixed instead of sweeping it (outputs/b2_3_diagnostics.json, key k_confound_fixed_k15): under a fixed-k paired E_CAP gap, n = 1000 showed no crossover through d = 200, while n = 5000 crossed at d = 38.

So the inward shift of the boundary is established for the preregistered k-spread diagnostic; it is not established as stable under fixed k. The canonical registry states this must not be claimed otherwise, and the essay’s Part IV points here. The honest mechanism statement is correspondingly weaker: local edge-conditioned overlap aligns the two observed core crossover points better than the global explanation does — no more than that.

A.8 What this licenses, and what it does not

Verbatim scope warning from B2_3/decision.json:

This result rejects tautology only with respect to the preregistered random k-out graph control. It does not establish that the estimator cannot be fooled by other non-geometric graph families.

Explicitly untested (not_tested_against): small-world graphs, scale-free graphs, other non-Euclidean structured graphs.

And the limits that matter for the essay’s argument:

• No information-theoretic lower bound was proven. Nothing here shows the geometric/random distinction has left the data past the crossover — only that this validated channel stops extracting it. A different statistic might separate these families farther out.
• The boundary is a property of the ensemble — world family, observation model (kNN graph), diagnostic (E_CAP + k-spread), control family, and decision rule together — not a demonstrated property of “contact” in the abstract.
• All of it is one forge. Designer, experimenter, and interpreter are the same author with the same AI partners; the harness makes self-deception harder, not impossible. Independent replication is invited and would outweigh any further internal check.

A.9 External references

• M. Kleindessner, U. von Luxburg. Dimensionality estimation without distances. AISTATS 2015, PMLR vol. 38, pp. 471–479. proceedings.mlr.press/v38/kleindessner15.html
• F. Locatello, S. Bauer, M. Lucic, G. Rätsch, S. Gelly, B. Schölkopf, O. Bachem. Challenging Common Assumptions in the Unsupervised Learning of Disentangled Representations. ICML 2019, PMLR vol. 97, pp. 4114–4124. proceedings.mlr.press/v97/locatello19a.html — the identifiability theorem referenced in Part II.
• M. Yannakakis. The complexity of the partial order dimension problem. SIAM Journal on Algebraic and Discrete Methods, 3(3):351–358, 1982. doi:10.1137/0603036 — the anchor for the essay’s “computational wall”: already deciding whether a partial order has dimension at most three is NP-complete.

A.10 Where every number lives

Every value above appears in canonical/RESULTS_CANONICAL.md with the source file, JSON key, and commit hash recorded next to it. The decisions that carry these numbers are harness-signed; gate_harness.verify_decision accepts all five under experiments/harness_valid/ and rejects every superseded pre-harness decision — which you can re-run yourself from the repository root:

for f in experiments/harness_valid/*/decision.json; do
  PYTHONPATH=. python3 -m gate_harness.verify_decision "$f"
done