# Correspondence — the middle form, measured

Run 2026-09-15, before any spec change. Real generator, shipped projection,
definition C (the residual), resonance night as defined — a world's own
baseline, one-sided z ≥ z\*, MAD floor 0.75 reading points.

This is the run the hash-draw report's last open question asked for:

> **Candidates** — same region, and **r ≥ 2 shared resonance nights in the
> trailing window of W nights** ending tonight. W ∈ {7, 14}.
> **Selection** — the night's finalised block hash draws pairs from that
> candidate set, once per pair ever, at τ(N) chosen for a per-world median
> wait of about two years.

z\* ∈ {1.0, 1.5}. **The one-night shape is re-measured here on the same corpus
as a reference row**, printed as `W = one`, so every comparison below is
against a number taken by the same harness rather than against the previous
report's.

**No spec value is set by this report.**

---

## 0 · The correction that has to come first

**The previous report's "mute" figure counted the wrong thing, and it counted
it flatteringly.** It counted worlds that never carried a resonance *flag*. A
world can carry flags for years and still never stand in a candidate pair —
its region's other worlds simply never light on the same nights — and it is
the second number that decides whether anybody can ever reach it.

Both are printed below. On the reference row, at the live catalogue's size:

| | flag-mute | **never a candidate** |
|---|---|---|
| 43 worlds, one-night, z\* = 1.0 | 1 (2.3%) | **7 (16.3%)** |
| 10,000 worlds, one-night, z\* = 1.0 | 6 (0.1%) | **6 (0.1%)** |

At ten thousand the two agree and the earlier report's claim stands
unchanged. At forty-three they do not: **seven of the live catalogue's
worlds are unreachable under the one-night shape, not one.** One of those
seven is alone in its region and is unreachable under every shape ever
proposed; the other six have same-region company that never lights with them.

That is a property of a forty-three-world catalogue rather than of a rule —
eighty same-region pairs in all — and it goes away as the catalogue fills.
It should not have been reported as 2.3%.

---

## 1 · The headline — "sustained" costs nothing on the promise

Every number that the spec publishes to a reader is **unchanged** by adding
the window. τ is solved for the promise, so the promise is what survives:

**10,000 worlds, 3 years**

| shape | contacts/night | median wait | never in 1 yr |
|---|---|---|---|
| one night, z\* = 1.0 | 4.48 | 2.21 y | 72.8% |
| **r ≥ 2 in 7**, z\* = 1.0 | 4.47 | 2.22 y | 73.4% |
| **r ≥ 2 in 14**, z\* = 1.0 | 4.44 | 2.25 y | 73.3% |
| one night, z\* = 1.5 | 4.52 | 2.46 y | 74.4% |
| **r ≥ 2 in 7**, z\* = 1.5 | 5.33 | 2.14 y | 70.6% |
| **r ≥ 2 in 14**, z\* = 1.5 | 4.83 | 2.30 y | 73.1% |

Against the structural prediction `N·ln2 / 2T` = **4.75 a night**, every row
is within 12%. *"Most worlds wait longer than a year"* is true in all six —
70.6% to 74.4%.

**So the choice between the two shapes is not a choice about what a reader is
promised.** It is a choice about the word in §1, about how many worlds are
stranded, and about how much the night job has to enumerate.

---

## 2 · The pool GROWS — the counter-intuitive result

Requiring **two** nights is a **weaker** filter than requiring one, because
the window gives fourteen chances to find them.

**Candidate pairs per night, as a share of all 4,163,351 same-region pairs at
10,000 worlds:**

| z\* | one night | r ≥ 2 in 7 | r ≥ 2 in 14 |
|---|---|---|---|
| 1.0 | 250,160 — **6.01%** | 456,009 — **10.95%** | 927,897 — **22.29%** |
| 1.5 | 102,608 — **2.46%** | 158,392 — **3.80%** | 365,124 — **8.77%** |

At z\* = 1.0 and W = 14, **more than a fifth of every same-region pair in the
sky is a candidate on any given night.** The one-night rule asks for a
coincidence *now*; the window asks for two coincidences *somewhere in a
fortnight*, and the second is easier.

That is worth saying plainly because it inverts the intuition the word
*sustained* carries. Sustained co-movement sounds like the stricter test. As
`r ≥ 2 in W` it is the looser one, and only the draw rate τ keeps the sky from
flooding — 4.86e-6 against the one-night shape's 1.80e-5.

**If the intent behind "sustained" is selectivity rather than repetition, this
shape does not deliver it.** A shape that would is `r ≥ 3` or a shorter window,
and neither was in the brief; `r ≥ 2 in 7` is the closest thing measured here
and it is still 1.8× the one-night pool.

---

## 3 · The numbers in full

**43 worlds, the live catalogue's own names, 5 years, 80 same-region pairs**

| z\* | W | flag-mute | never a candidate | cand. pairs/night | of all pairs | partners/world/night | τ | contacts/night |
|---|---|---|---|---|---|---|---|---|
| 1.0 | one | 1 (2.3%) | 7 (16.3%) | 2.97 | 3.71% | 0.1380 | 1.66e-2 | 0.0167 |
| 1.5 | one | 1 (2.3%) | 7 (16.3%) | 1.15 | 1.43% | 0.0534 | 4.85e-2 | 0.0145 |
| 1.0 | 7 | 1 (2.3%) | 8 (18.6%) | 3.86 | 4.82% | 0.1794 | 1.21e-2 | 0.0167 |
| 1.5 | 7 | 1 (2.3%) | 11 (25.6%) | 1.24 | 1.55% | 0.0577 | 1.30e-1 | 0.0173 |
| 1.0 | 14 | 1 (2.3%) | **7 (16.3%)** | 10.36 | 12.95% | 0.4818 | 6.06e-3 | 0.0173 |
| 1.5 | 14 | 1 (2.3%) | 11 (25.6%) | 3.38 | 4.23% | 0.1574 | 3.69e-2 | 0.0184 |

| z\* | W | contacted in 5 y | wait p25 / median / p75 | never in 1 yr |
|---|---|---|---|---|
| 1.0 | one | 62.8% | 0.53 / **1.32** / >5y | 51.2% |
| 1.5 | one | 55.8% | 0.58 / **1.49** / >5y | 65.1% |
| 1.0 | 7 | 58.1% | 0.42 / **1.49** / >5y | 58.1% |
| 1.5 | 7 | 69.8% | 0.39 / **0.90** / >5y | 46.5% |
| 1.0 | 14 | 72.1% | 0.15 / **1.59** / >5y | 55.8% |
| 1.5 | 14 | 69.8% | 0.46 / **1.28** / >5y | 51.2% |

**The 43-world column is coarse and should not be read to two decimals.** The
median of forty-three worlds moves in steps of one world, so τ's binary search
converges to a plateau rather than a point; the medians above scatter 0.90–1.59
years around a 2.00-year target for that reason and not because the shapes
differ. The 10,000-world run uses the analytic τ and does not have this
problem.

**10,000 worlds, 3 years, 4,163,351 same-region pairs, none alone in a region**

| z\* | W | flag-mute | never a candidate | cand. pairs/night | of all pairs | partners/world/night | τ | contacts/night |
|---|---|---|---|---|---|---|---|---|
| 1.0 | one | 6 (0.1%) | **6 (0.1%)** | 250,160 | 6.01% | 50.03 | 1.80e-5 | 4.483 |
| 1.5 | one | 23 (0.2%) | 23 (0.2%) | 102,608 | 2.46% | 20.52 | 4.46e-5 | 4.519 |
| 1.0 | 7 | 6 (0.1%) | 17 (0.2%) | 456,009 | 10.95% | 91.20 | 9.95e-6 | 4.467 |
| 1.5 | 7 | 23 (0.2%) | 69 (0.7%) | 158,392 | 3.80% | 31.68 | 3.40e-5 | 5.325 |
| 1.0 | 14 | 6 (0.1%) | **15 (0.1%)** | 927,897 | 22.29% | 185.58 | 4.86e-6 | 4.445 |
| 1.5 | 14 | 23 (0.2%) | 58 (0.6%) | 365,124 | 8.77% | 73.02 | 1.33e-5 | 4.834 |

| z\* | W | contacted in 3 y | wait p25 / median / p75 | never in 1 yr |
|---|---|---|---|---|
| 1.0 | one | 59.2% | 0.90 / **2.21** / >3y | 72.8% |
| 1.5 | one | 56.3% | 0.95 / **2.46** / >3y | 74.4% |
| 1.0 | 7 | 58.8% | 0.92 / **2.22** / >3y | 73.4% |
| 1.5 | 7 | 59.5% | 0.86 / **2.14** / >3y | 70.6% |
| 1.0 | 14 | 59.7% | 0.92 / **2.25** / >3y | 73.3% |
| 1.5 | 14 | 57.9% | 0.93 / **2.30** / >3y | 73.1% |

Percentiles are over **all** worlds with the uncontacted censored.

---

## 4 · Stranded worlds — the middle form's real cost

At ten thousand worlds, adding the window **doubles to triples** the number of
worlds nothing can ever reach:

| z\* | one night | r ≥ 2 in 7 | r ≥ 2 in 14 |
|---|---|---|---|
| **1.0** | 6 | 17 | **15** |
| **1.5** | 23 | 69 | **58** |

Small in absolute terms — 0.1% to 0.7% — and the direction is the one to know:
**z\* = 1.0 costs a quarter of what z\* = 1.5 costs, under every shape.** The
parameter that most protects the unlucky is still the shallow cut, exactly as
the previous report found.

**W = 14 strands fewer than W = 7**, at both cuts, which is the same fact as
§2 from the other side: the longer window is the looser rule.

---

## 5 · The alignment attack — unchanged, and better at z\* = 1.5

Same corpus as the previous report: target `martynas`, an aligned name at
cos 0.9997 found in 2,344 tries, a field of 600 unchosen worlds in the same
region, 1,095 nights. The quantity is now **candidate nights with the target**
— the number of draws a chosen name buys.

| z\* | W | target's flags | aligned | field mean | field max | field beating aligned | **lift** |
|---|---|---|---|---|---|---|---|
| 1.0 | one | 152 | 143 | 45.84 | **144** | 1 of 600 | **3.1×** |
| 1.5 | one | 34 | 30 | 6.85 | **32** | 2 of 600 | **4.4×** |
| 1.0 | 7 | 152 | 296 | 87.67 | **304** | 1 of 600 | **3.4×** |
| 1.5 | 7 | 34 | 38 | 12.32 | **40** | 36 of 600 | **3.1×** |
| 1.0 | 14 | 152 | 596 | 183.81 | **620** | 1 of 600 | **3.2×** |
| 1.5 | 14 | 34 | 89 | 26.12 | **96** | 2 of 600 | **3.4×** |

Three things:

**The lift does not get worse.** 3.1–3.4× across the middle form against 3.1×
(z\* = 1.0) and 4.4× (z\* = 1.5) one-night. At z\* = 1.5 the window actually
*suppresses* the attack, from 4.4× to 3.1–3.4×, for the reason §2 gives — a
looser rule makes co-movement less distinguishing.

**The aligned world never beats the field, in any row.** The best *unchosen*
world in the region has more candidate nights than the chosen name in all six:
144 > 143, 32 > 30, 304 > 296, 40 > 38, 620 > 596, 96 > 89. Aiming still buys
what the luckiest stranger already has, and the draw itself is uniform over
candidates and cannot be aimed at all.

**It is still not zero, so §2's sentence still has work to do.** *"The
selection is made by the sky. Not by sender"* survives a 3× lift no better and
no worse than before.

---

## 6 · What it costs to compute, and to check

**The night job enumerates more.** The one-night rule only ever looks at pairs
among *tonight's* flagged worlds — 250,160 at z\* = 1.0. The middle form has
to consider any pair whose two sides both lit at some point in the window,
which at z\* = 1.0 / W = 14 is essentially every same-region pair: **4,163,351
a night, a 17× enumeration**, of which 927,897 come out as candidates.

Measured in this harness: **about 10 ms a night at ten thousand worlds**, or
11 seconds for a three-year replay. It is not a wall. It is a real line in a
function that already has a 180-second budget and 8.2 seconds of it spent.

**A verifier reads more.** The one-night shape was checkable from *one* block
hash, *one* night's flags and each id's region prefix. The middle form needs
**W nights of flags** — still entirely public, still no q table and no
simulation report, but no longer a single night's document. §3.1's claim holds
either way; the work behind it grows by a factor of W.

**One implementation note that is not optional under either shape.** The
candidate list needs a canonical order (regions sorted, catalogIds within a
region), because the hash indexes into it; and the hash must be stretched by
counter. Both were established in the previous report and neither changes.

*This harness draws by geometric skipping rather than one digest per candidate
pair — for independent Bernoulli(τ) trials the gap to the next success is
Geometric(τ), so it is the same distribution decided by the same hash, with
5,800 digests a run instead of a billion. The reference rows reproduce the
previous report's per-pair-digest numbers to three figures (250,160.06 pairs a
night, 50.0320 partners, τ = 1.80e-5, 4.48 against 4.53 contacts a night), which
is what makes the substitution checkable rather than asserted.*

---

## 7 · What §1 can truthfully say under each shape

§1 reads: *"Contact is a measured coincidence the sky produces between two
worlds — **sustained co-movement of their own components past a published
threshold** — recorded on its night, recomputable by anyone, and never caused,
hastened, or prevented by anything an owner does."*

**Under the one-night shape the word *sustained* is simply false** — contact
rests on a single night on which both worlds crossed the threshold, and §1
would have to read *"co-movement of their own components past a published
threshold on one night, and the night's own block hash drawing from every pair
that reached it"*. **Under the middle form *sustained* becomes defensible**: a
pair must cross together twice inside a fortnight, which is repetition rather
than a single coincidence, and §1 keeps its phrase — but it must still gain
the draw clause, because under *both* shapes the threshold decides only
**candidacy** and the block hash decides **contact**, while §1 as written says
the coincidence *is* the contact. That is the sentence's real defect and it is
identical in the two shapes; *sustained* is the smaller half of the repair. So
the honest reading is that the middle form buys the word and nothing else — it
does not change the promise (median wait 2.2 y either way), does not reduce the
alignment lift below 3×, costs 6 → 15 stranded worlds in ten thousand and a 17×
nightly enumeration, and — because two nights in fourteen admits 22% of all
pairs where one night admits 6% — it buys the word *sustained* while making the
rule **less** selective than the one it replaces. If *sustained* is meant to
promise repetition, this shape delivers it; if it is meant to promise rarity,
the rarity in both shapes comes from τ, and §1 should say so.

---

## 8 · Open, and not mine

- **Whether *sustained* is load-bearing.** If it is: **z\* = 1.0, W = 14** —
  fewest stranded (15 in 10,000), lowest lift (3.2×), and the medians land on
  target. If it is not, the one-night shape is simpler to recompute, strands
  fewest of all (6), and reads honestly once §1 names the draw.
- **The two-year target.** Everything above is downstream of it, unchanged
  from the previous report: two years gives ~1,700 contacts a year at ten
  thousand worlds, five years ~680, one year ~3,400.
- **z\*.** 1.0 on every measure taken here, at both shapes. Not my call.

---

*Report only. No spec change. Harness: `middle.js` (`--worlds`, `--nights`,
`--z`, `--W`, `--wait`) and `alignment-middle.js`; 0.8 s at 43 worlds,
71 s at 10,000.*