arah2/docs/ALT-DIRECTION-SPLIT-EXPERIMENT.md

# Direction-split ALT: proof gate (research branch only)

**Verdict: NO-GO** for deployment and the 8 ms Worker budget. P-1 and P-2 proved for **pair-restricted domain and independent A* stopping** below; the measured branch-only implementation is 4.35× slower at paired p95 than the original control despite fewer expansions. This is not a proof for the shipped Ikeda sum-of-tops rule with asymmetric families.

## Graph and tables

Let `c(u,v)` be the stored finite nonnegative edge seconds, for directed CSR edge `u→v` **only when bit0 is clear**. Neither search traverses bit0 edges, even in reverse. Let `q>0` be `landmarkQuantumS`, `ĉ(u,v)=floor(c(u,v)/q)` and `D` be shortest path in integer quanta. Row-major `to[ℓ*n+v]=D(v,ℓ)`, `from[ℓ*n+v]=D(ℓ,v)`; `0xffff` means unreachable, **never** the number 65535. All finite stored values are at most 65000. Distances in seconds obey `qD` triangle inequalities against real `c` since `qĉ≤c`.

For this **fixed query** `o→g`, select to-columns `T={ℓ | to[ℓ*n+g] finite}` and from-columns `F={ℓ | from[ℓ*n+o] finite}`. If `to[o]` is sentinel for any ℓ in T, or `from[g]` is sentinel for any ℓ in F, return unreachable: otherwise an `o→g` path would respectively concatenate with `g→ℓ` or `ℓ→o`. Define the search domain `S` as nodes where `to[ℓ*n+v]` is finite for **every** ℓ in T and `from[ℓ*n+v]` finite for **every** ℓ in F. Exclude nodes outside S in **both** traversals. Every vertex of every actual `o→g` path belongs to S: its suffix reaches g then ℓ in T; its prefix is reached from o, which ℓ in F reaches. Thus domain restriction cannot remove any solution. It can remove dead-end branches and can change landmark shortest distances, but the original full-graph labels still satisfy the required triangle inequalities for every remaining edge. When T or F is empty the respective condition is vacuous and haversine remains available. Both endpoints are in S after the early reachability checks.

**P-1, independently:** on S use

`h_f(v)=max(haversine(v,g)/vmax, 0, max_{ℓ∈T} q*(to[ℓ*n+v]-to[ℓ*n+g]))`;

`h_b(v)=max(haversine(v,o)/vmax, 0, max_{ℓ∈F} q*(from[ℓ*n+v]-from[ℓ*n+o]))`.

For each traversable `u→v` inside S, `to[u]≤ĉ(u,v)+to[v]` implies `h_f(u)≤c(u,v)+h_f(v)` for each to-family. Independently `from[v]≤from[u]+ĉ(u,v)` implies `h_b(v)≤c(u,v)+h_b(u)` for each from-family on the reverse search edge `v→u`. Spherical haversine obeys both inequalities if real edge seconds lower-bound its endpoint chord at `vmax`; use the same `maxSpeedKmh` scaling as shipped A*, and require this graph assumption on test fixtures. The max of individually consistent families is consistent. Both heuristics vanish at their targets; by path telescoping they are admissible on S. Family values can be signed; max with zero/haversine keeps h nonnegative. Active sets T and F are query-scoped and unchanged across each edge **within S**. There is no numeric sentinel subtraction or per-node family skip.

Clean-column selection on the actual 0/24-clean binary gives 0 active landmarks globally. This domain construction instead selects endpoint-reachable columns for each query and excludes their holes *before search*: giant-component queries can activate 24 columns despite sentinel island cells; island queries may activate none. It cannot be replaced by evaluating a family only at the finite end of a cross-edge: if `to[u]=100`, `to[v]=0xffff`, and `u→v` costs 1, returning 100 at u and fallback 0 at v violates forward consistency. Likewise `from[u]=0xffff`, `from[v]=100` on `u→v` violates reverse consistency if the family is skipped at u. These edges are excluded when the corresponding column is selected. If they lie on an actual path to g or from o respectively, the endpoint selection/concatenation proves they cannot occur; if the column is not selected it contributes nowhere.

## P-2: independent frontier lower bound, NOT the BH sum rule

Maintain actual seconds distances `df` from o, `db` to g on the reversed induced graph, with keys `Kf(v)=df(v)+h_f(v)` and `Kb(v)=db(v)+h_b(v)`. These keys are not opposite potentials; **do not** add their minima or apply a v-independent offset. There is no v-independent cancellation of `h_f(v)+h_b(v)` for unrelated directed families: for example, a oneway chain `o→a→g` may have `h_f(a)=1,h_b(a)=1`, while a parallel dead-end has either bound 0 or a very large bound. Instead use the conservative stop `min_valid(Kf)≥U` **or** `min_valid(Kb)≥U`, where U is the best *real* discovered complete route (node meet `df(v)+db(v)` or traversable bridge `df(u)+c(u,v)+db(v)`). Strict floating-point tolerance must not weaken the inequality; all emitted seconds come from a traversable reconstructed path, not from subtracting potentials.

For each side, consistency gives reduced edge `c+h(next)-h(current)≥0` on that side, so settling min key fixes its actual distance; duplicate heap entries whose stored g differs from current g or whose vertex is closed are stale and must be discarded *before reading the frontier minimum*. Better g to a closed vertex is impossible under this premise; assert/fail loud if it occurs (or reopen if consistency is abandoned, but then this proof and stop do not apply). For a hypothetical route P of cost `<U`, consider the first vertex on P not yet forward-settled. Its predecessor is settled (or it is o); relaxation inserted a tentative g no larger than P's prefix, and admissibility bounds `Kf≤cost(P)<U`. Its live entry must still be in the forward heap. Therefore a forward minimum ≥U excludes P; the symmetric argument applies to the backward minimum. A bridge meet can only lower U via a genuine chain. If a frontier empties after stale removal, it has exhausted its induced reachable set: if no complete path was found, return null; if U finite the same first-unsettled argument shows U optimal. No assumption of simultaneous frontier growth or of a sum-of-minima bound is needed. Signed raw family differences are harmless since h is nonnegative and keys use real g; each direction has its own independent reduction and no shared potential.

Hand checks: oneway ring `0→1→2→0` with a forbidden bit0 mirror `1→0`, forward to[0] labels (0,2,1) for unit edges and backward from[0] (0,1,2); all allowed edges respect their *respective* oriented inequalities while mirror never relaxes. Bridge `A→B` costing 1 with no return, anchor in A: from[B] finite and to[B] sentinel; anchor in B: to[A] finite and from[A] sentinel. For a query A→B, applicable columns require origin/goal finite and S retains the bridge; B→A fails the endpoint reachability check and returns null. Disconnected anchored island vs unanchored component has sentinel endpoint(s); a route wholly in the unanchored component retains an empty active set and can still succeed, while a cross-component route is null. In all three examples, cross-boundary edges are excluded only if they cannot be on an o→g path.

## Measured experiment and verdict

Branch-only `routing/scripts/experiment-alt-direction-split.mjs` implements these independent A* keys, single-vertex meets, stale-key pruning and a query-domain mask. New suite `routing/test/alt-direction-split-experiment.test.mjs` tests **every pair** across one-way rings with blocked mirrors/spurs and equal costs, both directions of one-way bridges, isolated anchored/unanchored islands, endpoints and interior `0xffff` cells, mixed columns and a finite→sentinel cross-edge per-node skip adversary. Each checks independent O(V²) Dijkstra seconds, null classes and legal reconstructed edges. On real graph, `sweepOracle` uses the *exact* 200 compared snapped cross-metro pairs (seed `0x08f1`, 600 draws, 12 km minimum, 120 rejected snap misses) and independent heap-Dijkstra; **182 routed, 18 unreachable; suboptimal 0, oracle-mismatch 0, class-break 0** at symmetric ±1e-6 s. No synthetic or real pair disproved this conservative stop; the counterexample to naive per-node masking is above, not a measured result of this runner.

Measured 2026-10-26 on `HAFu`, Node v24.18.0, Linux, binary SHA-256 `366e1be2ca270e5e67cb0debc98887b6dcb66f29849747d59f4e0e58739ec5b8`, ARG2 311,361 nodes / 677,126 edges. This checkout's `routing/data/kl-roadgraph.v1.bin` is an **existing read-only symlink** to the other checkout; no binary modifications. Reproduction: `node routing/scripts/experiment-alt-direction-split.mjs`. The experimental instrumented comparison used an automatically removed temporary *source-identical* shipped clone with only two settled counters injected, and also timed original unmodified `routeAstar` on the same snapped endpoints. Clone/original seconds agreed to ±1e-6 for each routed sample. A dedicated warm-up (not included) took experimental 0.935 ms, original 20.005 ms, clone 27.471 ms.

The seed `0x08a1` 100-pair, ≥12 km extended draw produced **41 snap misses, 6 unreachable and 53 routed**; each routed pair was measured twice, reversed run order across rounds (106 routed measurements each). Misses are **outside** routed p95; each pair was compared at identical snapped endpoints. Times are Node `hrtime` wall elapsed for search call on this host, *not Cloudflare Worker CPU instrumentation*, so they cannot establish a Worker CPU PASS.

| Search | n | Median ms | Mean ms | p95 ms | Total settled expansions (106 calls) |
|---|---:|---:|---:|---:|---:|
| Direction-split / domain-mask experimental | 106 | 150.845 | 197.490 | **516.778** | 9,958,938 (forward/backward split NOT MEASURED) |
| Source-identical isolated shipped clone + settled counters | 106 | 73.588 | 72.331 | 129.327 | 14,037,662 (6,764,880 fwd / 7,272,782 bwd) |
| Original shipped haversine-only `routeAstar`, unmodified | 106 | 73.619 | 71.864 | **118.923** | NOT MEASURED directly (clone is expansion proxy) |

Experimental/clone expansion ratio **0.709×**; experimental/original shipped paired p95 **4.35× slower** despite fewer settles. The experiment selected an average of 22.64 to- and 18.87 from-columns per routed call (2400/106 and 2000/106 respectively); shipped has 0/24 clean columns and is the actual haversine-only control. No claim of runtime speedup. Search-scoped domain scanning, per-vertex `Math.max`, and heaps/allocations in this research runner may account for some overhead, but optimization is NOT demonstrated and no performance claim is transferable to an optimized Worker implementation.

Control gates unchanged: `npm run routing:test` **85 passed, 0 failed**, including shipped A1/625; `npm run routing:typecheck` **PASS**. `node routing/scripts/measure-optimality-gate.mjs` D-03 frozen record **PASS** (53 routed equal, 6 unreachable, 41 unsnappable; 0 class breaks). `node routing/scripts/measure-route-cpu.mjs --extended` shipped-control **FAIL**: 53 routed/100, p95 **127.933 ms**, >8 ms; 47 misses printed separately. Early test invocation lacked installed dependencies and failed L2/tsc; `npm ci --ignore-scripts --no-audit --no-fund` installed local dependencies, and both gates then passed; no source changes to those gates.

**Proof status:** P-1 PASS for induced pair domain; P-2 PASS for the conservative independent-frontier stopping bound, **NOT** a proof of asymmetric Ikeda sum stopping. **Oracle status:** 0 violations in synthetic exhaustive and real seeded runs. **Budget status:** local original control FAIL on 8 ms test; candidate local p95 FAIL; representative free-plan Worker 8 ms **NOT MEASURED**. **Recommendation:** NO-GO on integration; retain original A*, baseline and bot/web unchanged. A later, separately approved investigation could evaluate typed-array/precomputed component labels and a separately proved stronger meet bound, then rerun identical oracle and actual Worker-host CPU gates. No merge or Worker-budget retirement is justified.