# RUN-006 — Hard-Zeta Round 02: the two compartments, and Terras by word **Arm:** 數學戰士「墜衡」 / AMRAL Research Lab **Subject:** Neo.K + Aletheia, `Hard_Zeta_Phase_I_Round_02_Atomic_Hazard_Coefficient_Correction_v0.1.md`, `Hard_Zeta_ROUTE_MAP_v0.2.md`, and their bundle (2026-08-11 13:27) — source items 21–23 **Tools:** `hz_chart_algebra.py` (Round 02 layer) · `src08_hardzeta_round02_recheck.py` · `src08_drill.py` **Logs:** `src08-hardzeta-round02-recheck.json` · `src08-drill.json` **Result: 23/23 checks. 14/14 planted defects caught by the check named for each — 8 of them in Round 02's own formulas. 2/2 null controls undisturbed.** --- ## What Round 02 does Three things, in rising order of consequence. 1. It restates Round 01's thresholds in the parent's **quotient coordinate** `a` (where `n = r_w + 2^k a`), giving `q_D` and `q_U`. Different formulas for the same boundary — so the two rounds can be confronted with each other. 2. §6 replaces "worst chart" with an exact **mass-weighted** identity `λ_k = Σ_{|w|=k} π_w ℓ_w`. 3. §10–§20 split the frontier in two. With `τ_c(n) = inf{ j : 3^{u_j(n)} < 2^j }` and the unconditional `τ_c ≤ σ`, ``` E_k = C_k ⊔ R_k, C_k = {τ_c > k}, R_k = {τ_c ≤ k < σ}, Z_k = C_k + R_k ``` The `R` compartment **is** the Terras coefficient-stopping conjecture, and §17 recasts it as a finite-word inequality. All of it checks. `q_D`/`q_U` select exactly Round 01's boundary on all 8,190 charts; §7's `β_k` zones agree with Round 01's power comparison on every chart; §5's parity-restricted sums reproduce Round 01's chart masses; the two-compartment mass dynamics hold at every level. --- ## §6 is right, and by a wide margin Round 01's No-Go was about the **worst chart**. §22 there already said that was the wrong object, and §6 here says what the right one is. The measurement makes the gap concrete: | depth `k` | charts | global `λ_k` | worst chart's `ℓ_w` | ratio | |---|---|---|---|---| | 6 | 10 | 0.825141 | 0.987820 | 1.2× | | 9 | 52 | 0.014274 | 0.853184 | 60× | | 11 | 184 | 0.010520 | 0.987126 | 94× | | 12 | 323 | 0.025145 | 0.997525 | 40× | At depth 12 the worst chart loses **99.75%** of its mass while the global hazard is **2.5%**. Reading the worst chart as the global rate overstates it by a factor of forty, and that is precisely why Round 01's per-chart No-Go does not close the programme. §9's companion point is common rather than exceptional: **1,634 contracting children produce no first-descent mass at all**. A contracting skeleton is a necessary condition for hazard, not a sufficient one. --- ## Terras, checked by word rather than by trajectory §17 recasts the Terras coefficient-stopping conjecture (`σ = τ_c` for all `n ≥ 2`) as a **finite-word inequality**: ``` ν(w) > ⌊ b_w / (2^|w| − 3^u(w)) ⌋ for every first-crossing word w ``` where `ν(w) = min(Ω_w ∩ 2,∞))`. Round 03-B's task list asks for the minimal slack. This run enumerates **all 81,119 first-crossing words up to length 24** and measures it. **The inequality holds on every one, with room to spare.** | `\|w\|` | `u` | words | min `ν − c` | max `c/ν` | |---|---|---|---|---| | 8 | 5 | 7 | 20 | **0.487179** | | 13 | 8 | 85 | 186 | 0.028986 | | 16 | 10 | 476 | 344 | 0.041783 | | 21 | 13 | 8,045 | 155 | 0.025157 | | 24 | 15 | 51,033 | 270 | 0.045936 | The binding quantity is the **ratio** `c_w / ν(w)`, which must stay below 1. Its largest value over the whole family is **0.487179**, at `w = UUUDUUDD` with `r = 39`, `b = 251`, `c = 19`, `ν = 39`. Everywhere past length 10 the ratio sits between 0.002 and 0.05 — Terras is not close to failing anywhere in this range, and the tight cases are the short words, which are exhaustively settled. **Why this is worth more than a trajectory check.** Iterating trajectories confirms `σ = τ_c` for the integers actually visited. This confirms the inequality for every *word*, which covers every integer in those cylinders at once — including arbitrarily large ones. It does not extend past length 24, and the family is infinite. A structural check falls out alongside: **`h(w) = c_w` for every first-crossing word**, confirming §15's claim that all proper prefixes are expanding and only the final one caps. And the depths line up with [`RUN-004`: the first-crossing lengths are exactly the bit-lengths of powers of 3, which is exactly the admissible-stopping-time set measured there. Round 02 supplies the structural reason for what RUN-004 observed. --- ## The correction compartment is empty here — and that makes a check vacuous On `[2, 2^18)`, `R_k = 0` for every `k`. Terras holds, so the entire Hard-Zeta obstruction on this range is the coefficient frontier: `Z_k = C_k` exactly. That is a real finding, and it also **disables** a check. `C_k + R_k = Z_k` reduces to `C_k = Z_k` when `R_k` is empty, so *any* definition of `R_k` satisfies it — a planted mutation of the `R` predicate came back silent for exactly that reason. The split identity is a tautology of the definitions; what is testable is whether this file implements it correctly. So it is now exercised on **synthetic `(σ, τ_c)` data built to make `R_k` non-empty**, while the measured statement is stated for what it is: *`R_k` is empty on this range*, not *the split identity was verified*. Same family as [`feedback-empty-observable-passes-any-comparison`]; third occurrence in this arm, so it is the class and not the instance. --- ## Three more gaps the drill found **The quotient threshold was tested through a window, which a one-off slips through.** `q` and `q−1` have different parities, so `n(q−1)` is not even in the child's cylinder — the window merely moves and still contains `c`. The `+1` in `q_U`'s numerator changes the floor on only **6 of 2,843** charts, and the check missed all six. It now pins the threshold through the largest **legal** `a`. **Two defects crashed rather than failed.** A damaged `ν()` returning 0 divided by zero; a damaged `τ_c` condition ran off the walk cap. Both now degrade to a named failure, because a traceback is the one outcome a drill cannot grade. **Two mutations were retired as no-ops.** `3^u` and `2^j` are never equal, so `<` and `<=` select the same set wherever they compare those two. Third time this tree has met that; the mutations were replaced with ones that change the answer. --- ## A note on ROUTE MAP v0.2 `RUN-004` reported that ROUTE MAP **v0.1** stated the general weighted bridge without §12's monotonicity hypothesis. **v0.2 is a different map** — it states the `C`/`R` split, not the general bridge — so that finding does not carry over, and this run checks that it doesn't rather than leaving the earlier report to be misread. --- ## What this does not establish Nothing about Collatz. Nothing about Terras beyond word length 24 — the family is infinite and the margin is not shown to stay positive. Nothing about whether either compartment's mass tends to zero. What it does establish: Round 02's restatement agrees with Round 01 everywhere it was checked, the mass-weighted hazard is the right object by a factor of up to 94, and the Terras reformulation holds on 81,119 words with the worst case at 49% of its bound.