Can depth-crossing dynamics predict kappa?

Created 2026-10-20 21:10 +13:00, base 44ed81f71f4bb1fdccdcdc1cbd586bc1b2eb5613. Status: prospective scaffold; instrument, pilot and confirmation NOT RUN. This records hypotheses before measurement. It is not a frozen confirmation registration: cohort, calibrated estimator and resource manifest remain to be frozen after feasibility. No simulation is authorized by the scaffold itself.

Question

Can a reduced description of excursions between depth surfaces predict boundary occupation on independent starts and previously unseen rule families? If so, does it explain why occupation stabilizes as territory grows?

Read the mechanism ledger, earlier tail-budget results, and family hold-out. The tail-budget joint T1 criterion failed; its T2 extrapolation passed but lost to persistence. Its truncated-power-law estimator failed calibration. This experiment does not relabel those failures or reuse that fit as validated. The new test is composition of depth-crossing kernels, rather than using the target trajectory's measured mean excursion time to reconstruct occupation.

Theory and falsifiable assumptions

Let D be a boundary episode and T the following interior episode, both counted at departure times. In a stationary renewal-reward approximation,

f_boundary = E[D] / (E[D] + E[T])
kappa_inst = L * f_boundary,        L = A/P.

Successive episodes need not be independent for the long-run budget identity. Stable representative means are necessary. A growing finite window is not automatically a stationary ensemble.

Candidate tail, above a separately modelled short-return component:

Pr(T > t | L, rule r) ~ a_r t^(-theta) F(t / (b_r L^z)).
E[T] ~ a_r b_r^(1-theta) L^(z(1-theta)) I_F,  0 < theta < 1.

If E[D] scales as L^delta and long excursions dominate, then

d log(kappa_inst) / d log L = 1 + delta - z(1-theta).

Constant kappa therefore requires z(1-theta) = 1 + delta. The conversational proposal z(1-theta)=1 assumed delta=0; measure that assumption. The candidate (theta,z,delta)=(1/2,2,0) is one possibility, not an ant result. Slowly varying amplitudes, a changing shape F, multiple cutoffs, or a short-return-dominated mean invalidate this simplified exponent prediction.

Physical hypothesis: the written field controls the chance that an ordinary shallow excursion continues into a long journey. A local description should predict these chances from heading, phase and nearby field state. Alternatively, large-scale history may remain indispensable. A uniform-preserving diffusion clock alone cannot change stationary occupation on a fixed domain.

Definitions that the instrument must implement

• Territory includes all visited cells, including the origin. P counts visited cells with an unvisited four-neighbour, including hole boundaries.
• Observe before each departure, after marking its current cell visited. Boundary indicator Z_t uses this evolving territory. Episodes are maximal consecutive runs of Z_t=1 (D) or Z_t=0 (T).
• Depth d_t is shortest four-neighbour distance through visited cells to any boundary cell; boundary depth is zero. Primary depth evolves each step. Record geometry-induced milestone crossings separately: depth can change when the territory changes even without a move across an old contour.
• Milestones: depths 0,1,2,4,8,16,32,64. On hitting one, record first subsequent hit of a different adjacent milestone and elapsed time, retaining revisits to the departure milestone. At depth 64, retain the deep excursion until return to 32; never reflect artificially at 64. If this unresolved deep component dominates the budget, the local closure is uninformative.
• Sensitivity: freeze the depth map at each interior entry, terminate at first contact with that frozen boundary, and record disagreement with evolving episodes. This is a different observable, never silently substituted.
• Sample area bands [A0, A0+A0/16), A0 = 32768, 131072, 524288. Retain entry-conditioned complete lengths AND exact in-band time contributions. Follow entries past band end until return or cap; record left fragments.
• Primary outcome is in-band B/S. Secondary kappa uses exactly the probe's window-mean B/S divided by endpoint P/A. Also record sum_t P_t/A_t so the uniform-window reference is directly computable, without fitting exponents.

Stages, seed roles and cohort

1. Calibration: implement the observation-only accountant against a sparse reference. Use planted semi-Markov chains with known hitting times, an unbiased absorbing interval walk, and a hidden-state chain that defeats a memoryless closure. Include heavy tails and cap censoring. Rotor and random walk validate bookkeeping; neither is assumed to obey the candidate ant law.
2. Feasibility pilot: at most 8 rules, 4 starts each, selected by a seeded shuffle of distinct six-letter families in the committed census rule list. Freeze the exact input hash and eligible rows before reading new outcomes. Choose without ranking kappa, tail length or success at the target bands. Pilot seeds 202610207001..202610207004; calibration seed 202610207000. Board 8192, cap 2^32, at most 2 CPU workers, subject to profiling. These are proposed resources, not measured runtime estimates.
3. Confirmation: target 48 non-pilot rules from at least 12 non-pilot six-letter families, balanced across families, retaining pre-labelled stationary and drifting strata. Selection algorithm, exact list and strata must be committed before simulation. If unavailable, revise publicly before running; do not fill missing cells by selecting favourable outcomes. Per rule: 8 kernel seeds 202610208001..8008 and 8 outcome seeds 202610209001..9008 (full integers: 202610208001 through 202610208008, and 202610209001 through 202610209008). Check all against repository seed manifests before freezing. Folds group complete six-letter families. Split seed roles before jobs launch; retain all attainment/escape outcomes.
4. Intervention: not part of this launch. Only design targeted field edits after an independently validated kernel identifies a concrete state variable. Commit a separate protocol with sham edits and response horizons first.

Models and leakage controls

M0: depth and arrival direction (toward/away from boundary). M1: M0 plus heading and current colour phase. M2: M1 plus a radius-2 local colour/visited patch and bounded recent history (last four milestone labels). Exact encoding, pooling of different rule lengths, regularization and treatment of unseen states require calibration and freezing. Do not assume these finite states are Markov.

Fit transition probabilities Q and mean segment times m from kernel-role starts at the two smaller bands. With boundary absorbing, solve u = m + Q u; nonabsorbing communicating classes or divergent means are explicit failures, not rows to drop. Estimate the entry distribution and E[D] from kernel-role data only. Integrate the predicted return budget into predicted B/S. Report how much E[T] comes from the fitted depth-64 return component: measuring nearly the entire return time there would defeat the reduction.

Two different transfer claims: (a) new-start/new-size prediction may use that rule's small-band kernel data; (b) unseen-family prediction cannot use ANY kernel, entry, dwell or outcome data from the held-out family. All these quantities must be predicted by the training families' frozen state model, given rule and initial-condition inputs. If (b) proves infeasible, report it untested; do not call (a) family transfer.

Compare with training-mean, geometry-only, small-band occupation persistence, and small-band empirical E[T] extrapolation baselines. Geometry supplied from target runs makes kappa a geometry-conditional prediction; label it as such. Primary B/S does not divide by outcome area. Fixed-area sampling is a stopping design, not proof of causal control for size; do not interpret associations conditional on area as direct rule effects.

Prospective predictions and decision rules

These thresholds are design choices, not power calculations. Freeze a synthetic power/bias study and any revisions before confirmation.

• P1, scaling: independently estimated delta, theta and z predict the sign and magnitude of held-out large-band kappa drift: median absolute slope error <=0.15 per log L, and lower error than zero-drift persistence. Fit the tail law only if calibration and identifiable cutoff coverage pass. Failure of tail fitting means P1 uninformative, not evidence against every return theory.
• P2, predictive closure: M2 predicts largest-band rule-mean B/S on outcome seeds with median relative error <=20% AND >=10% smaller absolute error than the strongest frozen baseline. The 95% family-bootstrap interval for the paired error improvement must exclude zero (2000 resamples, seed 202610207099).
• P3, physical transfer: unseen-family M2 has pooled out-of-fold R2 >0 and improves R2 over M0 by >=0.10. Bootstrap whole families, never individual excursions. Both criteria required; report uneven family sizes.
• P4, state sufficiency: on validation data, adding radius-4 context to M2 changes predicted E[T] by <10% for >=80% of rules, and observed conditional crossing probabilities are calibrated within the synthetic-derived error tolerance. This is evidence only for the tested scales, not a proof of locality.

P2 alone would establish conditional predictive value, not causal mechanism. P2 passing and P3 failing suggests rule-specific effective kernels. P4 failing suggests missing memory or collective structure. None of the predictions establishes the infinite-time exponent from three finite area bands.

Readiness and stopping gates

See HANDOFF.md for the implementation sequence and SCHEMA.md for the required outputs. Before confirmation: oracle agreement, calibrated estimator bias and coverage, mid-trajectory checkpoints, byte-identical thread/resume tests, frozen hashes, and a measured runtime/memory budget must pass.

Require >=80% of rules to have >=6 starts per role in all bands; report the full denominator. Deepest relevant milestones need >=200 completed segments per rule across kernel seeds, else pool only by the frozen rule or declare insufficient resolution. Require <=5% unresolved in-band time AND return-mean uncertainty small enough to leave the verdict unchanged. A small censored count is not enough: without a justified tail bound, an unresolved duration has no finite upper bound. Increase follow-up under a predeclared cap schedule or mark uninformative. Cap/board failures stay visible.

Scale separation must be checked: compare entry/exit geometry and frozen/evolving depth estimates; >10% disagreement in mean return estimates blocks a quasi-static interpretation. Report remaining predictions only as explicitly finite-band predictions. No ad hoc reflecting walls.

Scaffold use and present results

From WSL, repository root:

export UV_PROJECT_ENVIRONMENT=$HOME/.venvs/turmite UV_LINK_MODE=copy
uv run --frozen experiments/2026-10-20-kappa-first-passage/scripts/preflight.py

This checks the scaffold and prints its provenance; it launches no simulation. The notebook is a planning/readiness document until committed results exist. No new scientific measurements, calibrated estimator, or instrument exist yet.