# RUN-007 — Round 03-A: the coefficient frontier, and the minimum anchor measured **Arm:** 數學戰士「墜衡」 / AMRAL Research Lab **Subject:** Neo.K + Aletheia, `Hard_Zeta_Phase_I_Round_03A_Coefficient_Frontier_v0.1.md` + `Hard_Zeta_ROUTE_MAP_v0.3.md` (2026-08-11 14:17) — source item 24 **Tools:** `hz_zeta_measure.rs` (`--tau-records`) · `hz_chart_algebra.py` (Round 03-A layer) · `src09_hardzeta_round03a_recheck.py` · `src09_drill.py` **Logs:** `src09-hardzeta-round03a-recheck.json` · `src09-drill.json` · `hz-tau-records-2p32.json` **Result: 36/36 checks. 17/17 planted defects caught by the check named for each — 9 in Round 03-A's own formulas, 3 in this run's own headline measurement. 2/2 null controls undisturbed.** --- ## What Round 03-A does It takes Round 02's coefficient compartment `C_k` and compresses it until only one unknown is left. * `C_k` is a cylinder union over the **irrational ballot tree** `S_k = { w : 3^{u_j(w)} > 2^j ∀ j ≤ k }`; * it can only shrink at **Beatty event depths** `K_u = ⌈u log₂3⌉`, so `C_k` is a staircase; * its atomic mass is an exact finite Hurwitz-zeta sum, `C_k(s) = 2^{-ks} Σ_{w∈S_k} ζ(s, x_w)`, `x_w = r_w/2^k`; * the **Head–Tail Reduction** `0 ≤ C_k(s) − H_k(s) ≤ ζ(s)·2^{-k(s-1)}` kills every progression tail; * so §29: **`C_k(s) → 0 ⟺ m_k → ∞`**, where `m_k = min C_k`. All of it checks. The survivor DP reproduces the enumerated tree at every depth to 18; the U-child always survives and the D-child survives exactly when the parent clears the next depth; at most one U-count layer can cross at any depth; the Beatty schedule is exactly the bit lengths of powers of three; the exact Hurwitz mass lands inside the brute-force bracket; the duplication transfer, the lift assignment, the anchor-ejection rule, the event-loss operator, the product criterion and both forms of the Head–Tail bound all hold. And §38 keeps an explicit **已證/未證 ledger** whose first unproved item is `m_k → ∞`. That one quantity now carries the entire coefficient conjecture. --- ## The headline: `m_k` measured, and what it bounds `m_k = min{ n ≥ 2 : τ_c(n) > k }` is fixed entirely by the **τ_c record holders**, so one scan of `2, 2^32)` determines it for every `k` up to the largest τ_c there. There are **23 records**, and each was re-derived by exact iteration rather than trusted. Feeding each measured `m_k` into §28's own inequality `C_k(s) ≤ Σ_{n≥m_k} n^{-s} + ζ(s)·2^{-k(s-1)}` turns it into a **rigorous upper bound on the true infinite `C_k(s)`**: | `k` | `m_k` | `C_k(2) ≥` | `C_k(2) ≤` | |---|---|---|---| | 6 | 7 | 2.04e-02 | 1.79e-01 | | 58 | 27 | 1.37e-03 | 3.77e-02 | | 80 | 703 | 2.02e-06 | 1.42e-03 | | 104 | 10,087 | 9.83e-09 | 9.91e-05 | | 134 | 35,655 | 7.87e-10 | 2.80e-05 | | 245 | 8,088,063 | 1.53e-14 | 1.24e-07 | | 375 | 63,728,127 | 2.46e-16 | 1.57e-08 | | 446 | 2,788,008,987 | 1.29e-19 | 3.59e-10 | | **447** | **≥ 2^32** | — | **2.33e-10** | These bound the **real** quantity, not a truncation — §26 already bounds every progression tail and §28 bounds the heads by their minimum, so nothing is left outside. **The measured answer to §38's open question is: the minimum anchor has escaped 23 times, reaching 2,788,008,987, and no further than the scan.** A sequence that has increased 23 times may still be bounded, and this measurement cannot tell the difference — which is exactly why §38 lists it as unproved. It is not evidence for the conjecture. The plateau structure §30 predicts holds exactly: `m_k` stays put until the current holder's own `τ_c`, then jumps to the next record. And the record list ends at `n = 2,788,008,987, τ_c = 447` — the same start and the same value as the maximum σ that [`RUN-004` measured, which is Terras agreeing with itself from a third direction. --- ## A scope note on the Beatty schedule §7 fixes the crossing U-count at `u ≥ 1`, so its event set is `{K_u : u ≥ 1}`. The frontier **also changes at depth 1** — that is the `u = 0` layer, the even numbers, leaving. So §8's "`C_{k+1} = C_k` unless `k+1 = K_u` for some `u`" is exact for `k ≥ 2` and not at `k = 1`. It costs the paper nothing: §11 works at `k ≥ 2` (survivors start `UU`, so the `u = 0` branch is already gone) and §24's product is based at `C_1`, i.e. *after* that event. Admitting `u = 0` — where `K_0 = 1` — accounts for every observed change exactly, and this run checks that it does. --- ## Three more gaps the drill found in my own work **A circular check.** The anchor plateau test derived `m_k` *from* the record list and then validated it against the same list, so an overstated `τ_c` was consistent with it by construction and the planted defect passed silently. The records are now re-derived by exact iteration, and every record below the brute-force range is confirmed to actually be a record. **A tautological check.** The "power form and slope form agree" test computed both sides from the same expression. It now compares `3^u > 2^k` against the floor form `u ≥ ⌊k·log₃2⌋ + 1` — with a companion check that the comparison saw **both** outcomes, since agreement on a one-sided sample proves nothing about a boundary. **`>` versus `>=` was a no-op for the fourth time.** `3^u` and `2^k` are never equal on the range in use, so that mutation tests nothing. Fourth occurrence in this arm; it is a property of the subject, not an accident, and mutations here are now written to change the answer. Also, one defect crashed instead of failing; the recheck now degrades to a named failure. --- ## The drill 17 defects. **Nine** damage Round 03-A's own formulas — the survival condition, the crossing depth, the DP recursion, the tree pruning, the normalized residue, the lift assignment, the loss ratio's scale factor, the cylinder scale, the head representative. **Three** damage this run's own headline: a fake record, an overstated `τ_c`, a reordered list — because a headline nobody drills is a number nobody has checked. One of them is worth naming: damaging `survives()` is caught by the cross-check against `first_crossing_words`, which carries its own independent condition. That is the whole reason for keeping two routes to the same tree. Two null controls disturb nothing. --- ## What this does not establish Nothing about Collatz. Nothing about `m_k` beyond `2^32`. Nothing about whether `C_k(s) → 0`. What it does establish: Round 03-A's compression of the coefficient frontier is correct everywhere it was checked, and the single quantity that compression leaves behind now has 23 measured values and a rigorous numerical upper bound at each of them.