Close verification-coverage gaps; harness finds two more bugsHEADmaster

Extends the verification harness from sorts-only to the whole repo, and in
doing so surfaces two further latent bugs (on top of the earlier hash-shift one):

Bugs found and fixed:
- queue/elementarypriority.go: max() seeded at the zero value, so an
  all-negative queue reported a phantom max of 0 and DeleteMax returned/removed
  the wrong element. Caught by the new queue permutation property (testing/quick
  generates negatives; the old test data never did). Seed from a[0] instead.
- sort/sleep.go: result built on NewArrayList(len(a)) -- a slice of that LENGTH
  (len(a) zeros) -- then appended to, yielding double-length output with leading
  zeros. The old .Sorted()-only test passed because zeros-then-ascending is
  sorted. Caught by the new Sleep permutation check. Build from an empty slice.

Coverage added:
- queue/property_test.go: ordering + permutation (completeness) for both queues.
- TestSleepSort now also checks permutation, not just Sorted().
- docs/verification.md: paper proofs for all search/set structures (Elementary,
  Hash, BST, red-black BST invariants, GoMap) and both priority queues.
- formal/tla/ParallelSort.tla: exhaustive fork/join model of ParallelMerge/
  ParallelQuick -- disjoint write-ranges (no data race) + termination. Wired
  into make verify-model.
- formal/selection.go: second Gobra proof (memory safety + sortedness). Wired
  into make verify-formal.
- docs/case-study-bugs-found.md: extensive write-up of all three bugs, how each
  was caught, why the old tests missed it, and the fix (supersedes the earlier
  single-bug case study).

Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>

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+# Case study: three latent bugs the verification harness found
+
+Adding the verification layers (see [`verification.md`](verification.md)) did not
+just re-confirm working code — it surfaced **three real, pre-existing bugs**, all
+invisible to the original test suite. Each is documented below with how it was
+caught, why it is genuinely wrong, why the old tests missed it, and the fix.
+
+A theme runs through all three: the original tests were **too weak in a specific
+dimension**, and the new checks are strong in exactly that dimension.
+
+| # | Location | Bug | Caught by | Old test's blind spot |
+|---|----------|-----|-----------|-----------------------|
+| 1 | `search/hash.go` | shift wider than a narrow key type → term is always 0 | `go vet` (in `make verify`) | only ever used 64-bit `int` keys |
+| 2 | `queue/elementarypriority.go` | `max()` seeded at `0` → wrong for all-negative queues | queue permutation property | test data was never negative |
+| 3 | `sort/sleep.go` | result built on a pre-sized slice → doubled length with leading zeros | sleep permutation property | only checked `.Sorted()`, not completeness |
+
+---
+
+## Bug 1 — a shift wider than the key type (`search/hash.go`)
+
+### The offending code
+
+```go
+func (h *Hash[K,V]) hash(key K) int {
+	i := key + key*2 + key<<10 + key>>2
+	...
+}
+```
+
+`K` is constrained by `ds.Integer`, so it may be **any** width down to `int8`.
+The `key<<10` term is meant to spread low bits into high bits.
+
+### What the tool reported
+
+```
+$ make verify
+go vet ./...
+search/hash.go:29:21: key (may be 8 bits) too small for shift of 10
+```
+
+`go vet`'s shift analyzer is a lightweight formal check: for every shift it
+computes a conservative lower bound on the left operand's bit width and flags any
+shift count `>=` that width. The narrowest `K` can be is `int8` (8 bits), and
+`10 >= 8`.
+
+### Why it is genuinely a bug (Go shift semantics)
+
+The Go spec defines non-constant left shifts operationally: *"Shifts behave as if
+the left operand is shifted n times by 1 … There is no upper limit on the shift
+count."* Shifting an 8-bit value ten times pushes every bit out of the value's
+width, so for `K = int8`/`uint8`:
+
+```
+key<<10  ==  0    // always, for every key
+```
+
+The intended high-bit mixing silently disappears. Note it is width-dependent
+(fine for `int16`+), and it is a *distribution/quality* bug, not a Set-contract
+violation — chaining keeps the table correct, but narrow-key instantiations
+degrade toward `O(n)` per operation. Exactly the kind of silent rot no assertion
+would flag.
+
+### Why the tests missed it
+
+Every test uses `int` keys (`test[int,int](NewHash[int,int](i*2), …)`), where
+`int` is 64 bits and the shift is fine. The bug lives in the **type dimension**,
+not the value dimension — no value-space test or fuzzer over `int` could reach
+it; only a type-aware tool (or an actual `int8` instantiation) can.
+
+### The fix
+
+```go
+func (h *Hash[K,V]) hash(key K) int {
+	// Mix the key in a full-width int64 rather than in K. ...
+	i := int64(key)
+	i = i + i*2 + i<<10 + i>>2
+	...
+}
+```
+
+Widening to `int64` before the shift keeps the result **byte-identical for
+64-bit `int` keys** (so all existing tests still pass unchanged) while making the
+mixing well-defined for every width. It does not *suppress* the warning; it
+removes the condition (`shift >= width`) that made it true.
+
+---
+
+## Bug 2 — a maximum seeded at zero (`queue/elementarypriority.go`)
+
+### The offending code
+
+```go
+func (q *ElementaryPriority[T]) max() (ind int, max T) {
+	for i, a := range q.a {
+		if a > max {          // max starts at the zero value of T, i.e. 0
+			ind, max = i, a
+		}
+	}
+	return ind, max
+}
+```
+
+`max` is a named return, so it starts at `T`'s zero value, `0`.
+
+### How it was caught
+
+The new completeness/permutation property in `queue/property_test.go` drives
+each queue with `testing/quick`, which generates **negative** values too. It
+failed immediately for `ElementaryPriority` (and passed for `HeapPriority`):
+
+```
+ElementaryPriority violated ordered-permutation property:
+  #1: failed on input []int{-1881664299226649700, 1264358012858162353, ...}
+```
+
+### Why it is genuinely a bug
+
+If **every** element in the queue is negative, no element is `> 0`, so the loop
+never updates and `max()` returns `(0, 0)` — reporting a maximum of `0`, a value
+that is not even in the queue. `DeleteMax` then removes the wrong element (index
+0) and returns a phantom `0`. Both the ordering and completeness of a drain
+break.
+
+### Why the tests missed it
+
+The original queue test builds inputs with
+`ds.NewRandomArrayList[int](l, -1)`, whose values come from `rand.Int()` — always
+**non-negative**. And `queue_test.go` only checked that `DeleteMax` was
+non-increasing; it never checked that all inserted elements come back. So a queue
+of non-negative numbers, checked only for ordering, sailed through.
+
+### The fix
+
+```go
+	if len(q.a) == 0 {
+		return 0, 0
+	}
+	ind, max = 0, q.a[0]   // seed from a real element, not the zero value
+	for i, a := range q.a {
+		if a > max { ind, max = i, a }
+	}
+```
+
+Seeding from `q.a[0]` makes the scan correct for any value range.
+(`HeapPriority` was already immune: it compares actual array elements and returns
+`a[1]` as the max.)
+
+---
+
+## Bug 3 — a sort that doubled its output (`sort/sleep.go`)
+
+### The offending code
+
+```go
+func Sleep[V ds.Integer](a ds.ArrayList[V]) ds.ArrayList[V] {
+	sorted := ds.NewArrayList[V](len(a))   // slice of LENGTH len(a): len(a) zeros
+	...
+	for num := range numCh {
+		sorted = append(sorted, num)       // appends AFTER those zeros
+	}
+	return sorted
+}
+```
+
+`ds.NewArrayList(len(a))` is `make(ArrayList, len(a))` — a slice of that
+**length**, pre-filled with `len(a)` zeros. Appending then adds the real values
+*after* them.
+
+### How it was caught
+
+`TestSleepSort`, strengthened to check permutation, failed:
+
+```
+Sleep sort output is not a permutation of input:
+  in =[1 1 2 8 8 7 3 0 6 7]                       (10 elements)
+  out=[0 0 0 0 0 0 0 0 0 0 0 1 1 2 3 6 7 7 8 8]   (21 elements!)
+```
+
+The output is **eleven leading zeros followed by the ten real values** — more
+than double the input length.
+
+### Why the tests missed it
+
+The original `TestSleepSort` only asserted `a.Sorted()`. Zeros followed by an
+ascending sequence **is** sorted, so the wildly-wrong 21-element result passed.
+This is the textbook case for the *permutation* invariant: ordering alone cannot
+detect dropped, duplicated, or (here) invented elements.
+
+### The fix
+
+```go
+	// Start empty with capacity len(a): the received values are appended below.
+	sorted := make(ds.ArrayList[V], 0, len(a))
+```
+
+Length `0`, capacity `len(a)`: the appends now fill it to exactly `len(a)`
+elements with no spurious zeros.
+
+---
+
+## The through-line
+
+Two independent lessons, each reinforced twice:
+
+1. **Ordering is not correctness.** Bugs 2 and 3 both produced *ordered* output
+   that was wrong (missing/extra elements). Only the **permutation / completeness**
+   invariant — added to the sort and queue property tests — catches them. This is
+   the single most valuable check added by this work.
+
+2. **Test data has blind spots the code doesn't.** Bugs 1 and 2 both hid behind
+   the test suite's fixed input distribution — always 64-bit, always
+   non-negative. `go vet` (reasoning over *types*) and `testing/quick` (sampling
+   the *whole* value range, negatives included) each see past a blind spot that
+   hand-picked or `rand.Int()` data does not.
+
+None of these required the heavy layers (TLA+, Gobra). The cheapest checks —
+`go vet` and a stronger property assertion — found all three. Breadth first;
+depth where it earns its keep.