Tighten the statistics in the design-band section

Re-examining §3.2a's inference rather than its numbers found one claim
that was wrong and one that does not survive correction.

Wrong: "two independent tests agree on that onset." The gap test
(countable - unrestricted) and the vs-1.0 test (countable - 1.0) share
the countable arm and are correlated at rho ~ 0.73. The vs-1.0 test is
not confirmation, it is the same measurement with one noisy arm removed
-- which makes it ~1.4x more sensitive, and is why it flags 4 cells
where the gap test flags 1. What IS informative is the contrast between
arms: unrestricted shows chance-level deviation, countable shows a
shortfall concentrated at every uncle cap at the top of the band.

Does not survive: the whole-band pooled gap, -0.00048 +- 0.00033
(t = 2.8). Counting every test in the section -- 15 per-cell gap, 5
pooled-per-delay, 1 whole-band, 30 vs-1.0 = 51 -- Bonferroni requires
z = 3.30. It is now labelled unresolved. The sign imbalance (11/15
negative, p = 0.12) is labelled likewise.

Holds, and more firmly than claimed: delta_max = 5, pooled t = 3.68.
Added a 20 000-draw permutation test on the same cells, which assumes
no distributional form: p = 0.0003, with every other delay at p >= 0.21.
It clears the 51-test Bonferroni threshold. Replicate distributions are
clean (sd ~0.003, |skew| <= 0.5, no outliers), and the peering graph
re-rolls per replicate -- `replicate` is in config.key() and the graph
is seeded from the full-key spawn hierarchy -- so the replicate SEM
captures graph-to-graph variance rather than pseudo-replicating one
graph.

Added a weighted regression of gap on delay as a single pre-specifiable
test with no choice of which delay to inspect: slope -0.00024 +-
0.00012 per slot (t = -2.05). Consistent with a monotone cost, marginal
on its own, and reported as such.

Added the limitation that actually caps the resolution: the comparison
is unpaired. uncle_model enters the config key (which is what makes
--old bit-reproduce the earlier runs), so the two arms share neither
graph nor lottery draws and each comparison pays the between-run
variance twice -- a +-0.0015 per-cell floor, an order of magnitude
above the effect. A paired design would cancel most of it and is named
as the single change that would most improve the measurement.

The §1 summary is corrected to match: indistinguishable from zero up to
delta_max = 4, resolved only at the top of the band at ~0.14%.

Co-Authored-By: Claude Opus 5 (1M context) <noreply@anthropic.com>

reports/tsi/tsi-report.md

@@ -21,7 +21,7 @@
 
 ---
 
-> **The uncle-reference model analysed here.** Uncle references are *counting-only* — they never affect a block's validity ([§6.7](#s6-7)–[§6.8](#s6-8)) — and are deduplicated by **slot**, not by block ([§8.5](#s8-5)). The reference window is derived from a **window absorption parameter**, `w_u = W_abs/f` with `W_abs = 10` expected block-intervals and bound `W_abs ≤ 0.6·k`, which is the `W = 300 = 10/f` sizing rule of [§3.4](#s3-4). One structural constraint shapes the results: only the **first block of a fork** — one whose parent lies on the referencing chain — is countable, which is the price of verifying every reference from chain data alone. This report calls that the **countable** model, and measures it against an **unrestricted** baseline in which any orphan in the window is referenceable at any fork depth ([§2.1](#s2-1), [§3.2](#s3-2), [§9](#s9)). The result: **in the design regime `ρ < 1` the two are practically identical.** A dedicated 40-replicate sweep of the operating band (`δ_max` 1–5, `ρ ≈ 0.21–0.41`) puts every `U ≥ 1` cell of both models in 0.998–1.001, resolves no difference in any individual cell (widest 95 % CI ±0.15 pp), and bounds the first-fork cost at **≤ 0.2 %** — indistinguishable from zero through `δ_max` ≤ 3 and rising monotonically with delay thereafter ([§3.2a](#s3-2a)). Differences appear only at `ρ ≥ 1`: the restriction trims ~1.4 % at `ρ ≈ 1`, and under overload (`ρ ≈ 1.8`) recovery saturates near 0.95 where the unrestricted model reaches ~1.00. Deep forks are unrecoverable by construction, so past `ρ ≈ 1` the ceiling is set by the fork *depth* profile and **no uncle cap buys it back** — a second, independent reason for the [§8](#s8) "keep `ρ < 1`" rule. The closed-form accuracy with the measured effective utilization, `log(1−f)/log(1−f/q_u)`, predicts every countable cell to within 0.2 %.
+> **The uncle-reference model analysed here.** Uncle references are *counting-only* — they never affect a block's validity ([§6.7](#s6-7)–[§6.8](#s6-8)) — and are deduplicated by **slot**, not by block ([§8.5](#s8-5)). The reference window is derived from a **window absorption parameter**, `w_u = W_abs/f` with `W_abs = 10` expected block-intervals and bound `W_abs ≤ 0.6·k`, which is the `W = 300 = 10/f` sizing rule of [§3.4](#s3-4). One structural constraint shapes the results: only the **first block of a fork** — one whose parent lies on the referencing chain — is countable, which is the price of verifying every reference from chain data alone. This report calls that the **countable** model, and measures it against an **unrestricted** baseline in which any orphan in the window is referenceable at any fork depth ([§2.1](#s2-1), [§3.2](#s3-2), [§9](#s9)). The result: **in the design regime `ρ < 1` the two are practically identical.** A dedicated 40-replicate sweep of the operating band (`δ_max` 1–5, `ρ ≈ 0.21–0.41`) puts every `U ≥ 1` cell of both models in 0.998–1.001, resolves no difference in any individual cell (widest 95 % CI ±0.15 pp), and bounds the first-fork cost at **≤ 0.2 %** — statistically indistinguishable from zero up to `δ_max = 4`, and resolved only at the top of the band, where it is ~0.14 % ([§3.2a](#s3-2a)). Differences appear only at `ρ ≥ 1`: the restriction trims ~1.4 % at `ρ ≈ 1`, and under overload (`ρ ≈ 1.8`) recovery saturates near 0.95 where the unrestricted model reaches ~1.00. Deep forks are unrecoverable by construction, so past `ρ ≈ 1` the ceiling is set by the fork *depth* profile and **no uncle cap buys it back** — a second, independent reason for the [§8](#s8) "keep `ρ < 1`" rule. The closed-form accuracy with the measured effective utilization, `log(1−f)/log(1−f/q_u)`, predicts every countable cell to within 0.2 %.
 
 <a id="s1"></a>
 ## 1. Executive summary
@@ -195,7 +195,7 @@ So: **at the operating loads (`ρ < 1`) no difference between the models is dete
 
 **The question.** [§3.2](#s3-2) samples the mixing delay at 4/8/16/32. That resolves the overload regime, but it leaves the band the parameters are actually chosen in — the low-delay end, where every operating point sits — measured at four-fold spacing and five replicates, which is enough to say "no difference detected" and nothing more. How accurate is TSI across `δ_max` = 1–5, and how large a first-fork cost can be *excluded* there?
 
-**The finding.** One uncle slot holds the estimate at the true stake across the whole band: every `U ≥ 1` cell under both referencing models lands in **0.998–1.001**, flat in delay, while the uncle-free baseline decays 0.810 → 0.640. The two referencing models are **indistinguishable through `δ_max` ≤ 3**; at the top of the band the first-fork restriction becomes measurable and costs **0.1–0.2 %**. Two independent tests — the model-vs-model gap, and each model against the exact target 1.0 — agree on that onset.
+**The finding.** One uncle slot holds the estimate at the true stake across the whole band: every `U ≥ 1` cell under both referencing models lands in **0.998–1.001**, flat in delay, while the uncle-free baseline decays 0.810 → 0.640. The two referencing models are **indistinguishable through `δ_max` ≤ 3**; at the top of the band the first-fork restriction becomes measurable and costs **0.1–0.2 %**. Only the `δ_max = 5` result is statistically resolved; the weaker claims below are labelled as such.
 
 `configs/fine-delay.yaml` spends replicates instead of range — `δ_max` ∈ {1,2,3,4,5}, `U` ∈ {0,1,2,4}, **40 replicates** per cell (8× [§3.2](#s3-2)), N = 1 000, blend, 3 hops — run under both models ([§9](#s9)). The band spans `ρ ≈ 0.21` to `0.41`, entirely inside the design regime.
 
@@ -209,9 +209,11 @@ So: **at the operating loads (`ρ < 1`) no difference between the models is dete
 
 **No individual cell resolves a model difference.** Across the 15 `U ≥ 1` cells the widest 95 % CI half-width is ±0.0015, one cell clears `t = 2` (0.75 are expected to by chance), and its `t = 2.59` does not survive the Bonferroni threshold of 2.94 for 15 tests. Read cell by cell, the honest statement is that any difference is **smaller than ±0.15 pp**.
 
-**Pooled, a small delay-dependent cost appears.** The three uncle caps are independent measurements of the same underlying difference, so inverse-variance pooling across them buys ~√3 in precision. Pooled per delay, `δ_max` 1–4 stay unresolved (|t| ≤ 1.3) while **`δ_max = 5` resolves at −0.0014 ± 0.0007 (t = 3.7)** — surviving correction for the five delays tested. Over the whole band the pooled gap is −0.00048 ± 0.00033 (t = 2.8), and 11 of 15 cells are negative. So the first-fork restriction does cost something, in the direction theory predicts, and the cost is **monotone in delay and below 0.15 % everywhere in this band** — negligible against the ±0.9 % per-epoch sampling noise of [Appendix B](#sB). (The pooling was chosen after inspecting the per-cell results; the per-delay trend, not the whole-band figure, is the defensible claim.)
+**Pooled, a delay-dependent cost appears at the top of the band.** The three uncle caps are separate draws (each `(δ_max, U, replicate)` cell is its own RNG stream, and the peering graph re-rolls with it), so inverse-variance pooling across them buys ~√3 in precision. Pooled per delay, `δ_max` 1–4 stay unresolved (|t| ≤ 1.3) while **`δ_max = 5` resolves at −0.0014 ± 0.0007 (t = 3.7)**. That one result is robust: a 20 000-draw permutation test on the same cells, assuming no distributional form, gives **p = 0.0003** (all other delays: p ≥ 0.21), and it survives Bonferroni correction for **every one of the 51 tests reported in this section** (threshold z = 3.30). Two weaker statements should not be read as resolved: the whole-band pooled gap, −0.00048 ± 0.00033 (t = 2.8), **fails** that same correction, and the sign imbalance (11 of 15 cells negative) is p = 0.12 on its own. A weighted regression of gap on delay — one pre-specifiable test, no choice of which delay to look at — gives a slope of −0.00024 ± 0.00012 per slot (t = −2.05): consistent with a monotone cost, but on its own marginal. The defensible reading is that the first-fork restriction costs nothing measurable up to `δ_max = 4` and about 0.14 % at `δ_max = 5`, which is negligible against the ±0.9 % per-epoch sampling noise of [Appendix B](#sB). (Pooling across `U` was chosen after inspecting the per-cell results; it is reported here because the `δ_max = 5` result survives the harshest correction available, not because the pooling was pre-registered.)
 
-**The absolute test agrees.** Asking the same question without reference to the other model — is each cell's equilibrium exactly 1.0? — reproduces the onset independently. Under the **unrestricted** model 1 of 15 cells sits significantly below 1 (`t` = −2.09, consistent with chance). Under the **countable** model 4 of 15 do, and they are not scattered: `δ_max = 4` at `U = 1` (−0.0012, `t` = −2.6) and **all three uncle caps at `δ_max = 5`** (−0.0012 to −0.0019, `t` = −2.5 to −3.7). A shortfall that appears at every cap simultaneously, only at the top of the band, and only under the restricted model, is the first-fork cost — the same effect the gap test resolves at `δ_max = 5`, seen from the absolute side.
+**The same data, measured against 1.0 rather than against the other arm.** Dropping the unrestricted arm and asking whether each cell's equilibrium is exactly 1.0 removes one noisy measurement, so it is ~1.4× more sensitive. It is **not independent evidence** — it shares the countable arm with the gap test, and the two are correlated at ρ ≈ 0.73 — but the *contrast between the arms* is informative. Under the **unrestricted** model 1 of 15 cells sits below 1 (`t` = −2.09), consistent with chance. Under the **countable** model 4 of 15 do, and they are not scattered: `δ_max = 4` at `U = 1` (−0.0012, `t` = −2.6) and **all three uncle caps at `δ_max = 5`** (−0.0012 to −0.0019, `t` = −2.5 to −3.7; the `U = 2` cell survives the 51-test correction). A shortfall appearing at every cap at once, only at the top of the band, and only under the restricted model, is the signature of the first-fork cost rather than of noise.
+
+**What limits the resolution: the comparison is unpaired.** The two models draw independent RNG streams by construction (`uncle_model` enters the config key, which is what makes `--old` bit-reproduce the earlier runs, [§9](#s9)), so a countable replicate and an unrestricted replicate share neither the peering graph nor the lottery draws. Every comparison therefore pays the full between-run variance twice, and at 40 replicates that floor is ±0.0015 per cell — an order of magnitude above the effect being measured. A **paired** design — same graph, same lottery outcomes, the uncle rule the only difference — would cancel almost all of that shared variance and could resolve the per-cell effect directly instead of only after pooling. It would require seeding the graph and lottery from topology-only entropy, and re-running both arms; it is the single change that would most improve this measurement, and it is not done here.
 
 **The negative control passes.** The `U = 0` arms — identical models by construction — show |gap| ≤ 0.016 with max `t` = 1.26, i.e. no spurious signal, but a 95 % CI of ±0.025: **17× wider than the entire `U ≥ 1` range.** The unrecovered regime is intrinsically noisy, which is precisely why the model comparison has to be made where uncles are active.