# RUN-018 — Round A-U.2b.3: the subject corrected a defect this arm reproduced faithfully and did not notice **Arm:** 數學戰士「墜衡」 / AMRAL Research Lab **Subject:** Neo.K + Aletheia, `Hard_Zeta_Phase_II_Round_AU2b3_bundle.zip` (source item 36) — Round A-U.2b.3 *Queue-Prefactor Saturation, Pointed/Unpointed Correction and Third-Order No-Gain*, its diagnostics script and JSON, three figures, the **corrected A-U.2b.2 v0.1.1**, plus `A_Line_ROUTE_MAP_v1.4` **Tools:** `hz_accel_code.py` (A-U.2b.3 layer) · `src20_hardzeta_au2b3_recheck.py` · `src20_drill.py` **Logs:** `src20-au2b3-recheck.json` · `src20-drill.json` **Result: 17/17 checks. 27/27 planted defects caught by the check named for each — 10 in the definition layer, 11 in this run's own measurement, 6 in the documents. 2/2 null controls undisturbed. Coverage audit clean.** --- ## The correction, and why this arm missed it Round A-U.2b.2 defined `Q_{r,D}` in §4 as the set of exponent words for which **there exists** an admissible starting deficit. Its shipped dynamic program summed `V_r` over *all* starting deficits, which counts each word once **per** admissible start. Those are different objects — at `r = 4, D = 2`, 66 against 48. RUN-018 is not where that was found. **The subject found it**, and ships both a corrigendum in the route map and a corrected `v0.1.1` of the previous paper. RUN-017 reproduced the DP by a deliberately different method — prefix sums where theirs used suffix sums, exact integer credits where theirs multiplied a float — and validated it against brute-force enumeration. All nine rows matched exactly. The verification was sound and the object was wrong, because **my brute force enumerated from every starting state too**. It was written from the *script's* reading, so it confirmed the script against itself. Two implementations agreeing is not verification when both descend from the same source — which this arm's charter already says, and which I applied to the *method* while leaving the *specification* unread. That is now a standing rule here: **a paper plus a program is two claims.** Implement from the prose, watch the quantifiers, and add an explicit check that definition and program agree. Everything in this run is implemented from §4–§7's prose, and the identity `Q = P_D − P_{D−1}` is verified against a direct enumeration of *words* — the check that was missing. **What the correction costs RUN-017: the label, not the conclusion.** | `r` | rate (pointed) | rate (unpointed) | shift | |---|---|---|---| | 200 | 1.5187691 | 1.5172835 | 1.49 × 10⁻³ | | 800 | 1.5700178 | 1.5691920 | 8.3 × 10⁻⁴ | | 2000 | 1.5787055 | 1.5783565 | 3.5 × 10⁻⁴ | | 5000 | 1.5825043 | 1.5823625 | **1.4 × 10⁻⁴** | The entropy still saturates on `β`, so the Prefix-Constraint First-Order No-Gain verdict stands. And the Second-Order Barrier never used the DP data at all — it runs on the weak-composition sums `A(r,D)`, `B(r,D)` — so it is untouched. A-U.2b.2 v0.1.1 says both in its own corrigendum; the table above is what confirms it independently. RUN-017 has been amended in place. --- ## The identities, from the definitions ``` S_j = Σ_{i≤j} (e_i − b_i) R(e) = max_j S_j − min_j S_j ``` A word is admissible from `d_0` exactly when `0 ≤ d_0 − S_j ≤ D` for all `j`, i.e. `max_j S_j ≤ d_0 ≤ D + min_j S_j`, giving **`D − R(e) + 1`** starts. Checked by trying every starting deficit on every admissible word at five shapes — not by trusting the formula. From there `P = Σ_{R(e)≤D}[D − R(e) + 1]` against the DP, and `Q = #{e : R(e) ≤ D}` against `P_D − P_{D−1}`, both exact at seven shapes. And a check that the two counts **genuinely differ** at every shape tested, because if they agreed the whole corrigendum would be empty and every check above would be about one object rather than two. --- ## The diagnostics, and what they show All nine published rows reproduce with **worst deviation exactly 0** across all eight columns — computed from the definitions above rather than from the shipped script. Two measurements worth keeping: **The pointing multiplicity settles.** `P/Q` runs 1.229 → 1.463 → 1.581 → 1.622 → 1.635 → **1.638** at `r = 10000`. A bounded ratio is precisely why the correction cannot move the exponential rate or the `r^{−1/2}` prefactor scale — the two counts differ by a constant factor, not a growing one. That is the quantitative content of §22–§23's "the constants are unchanged". **The centered prefactor is bounded but not monotone.** `r·H(z) − log₂Q − ½log₂r` sits in a band of width 2.33 (5.54 to 7.87), bottoms at `r = 5000` and rises again at 8000 and 10000. That is consistent with `Q = Θ(r^{−1/2}2^{rH})` and is all a finite table can show; it is reported as a band, not as a convergence, because it is not one. --- ## The round's own epistemic bookkeeping §33 splits its results three ways rather than the usual two, and that is worth recording: - **self-contained exact results** — the pointed/unpointed identity, canonical pointing, the DP correction, the packing formulation; - **standard analytic dependency** — the saturation *lower* bound leans on lattice local-limit and positive-drift ballot/bridge estimates, which the round names rather than absorbing; - **unproved** — phase-specific wedge constants, CASP, Terras, Collatz. A round that has just corrected itself declaring exactly which of its results rest on imported analysis is the right response to having been wrong once. **The packing branch is declared closed**, with its seven exhausted items itemised: composition entropy, the sharp first-order constant, the Stirling tax, exact queue prefix constraints, the exact queue prefactor, the pointed/unpointed correction, and third-order queue-only no-gain. What remains is named: A-U.2c (noncompact anchor cocycle), A-U.2d (transducer rationality), A-U.2e (multiscale return arithmetic). **The correction ships under a new filename.** `v0.1.1` is added; `v0.1` is not overwritten in place. A corrigendum that reused the old name would erase its own history, and this one supersedes by version instead — checked, along with the absence of any silent same-name edit anywhere in the bundle. --- ## Findings about my own checks **A no-op with a mathematical reason behind it — the most interesting one yet.** I planted a defect that walks the deficit backwards: `d ← d − b_j + e_j` instead of `d ← d + b_j − e_j`. It changed nothing. The reason is real: the number of admissible starts depends only on `R(e) = max_j S_j − min_j S_j`, and that is **invariant under `S → −S`**. Reversing the walk reflects the partial sums and leaves the range, hence the multiplicity, exactly where it was. Confirmed on every admissible word at four shapes. Replaced by a defect that widens the corridor by one, which does move the count. **Two guards that are redundant with each other, so loosening *either* is invisible.** The word enumeration prunes on `R(e) ≤ D` during the walk *and* filters on `multiplicity > 0` at the end. Because `R(e)` is monotone in the prefix, anything surviving the prune already satisfies the filter — so both directions of loosening are no-ops, and I planted each in turn before seeing why. The one guard that is *not* doubled is the excess range: a word needs `e_j ≤ D + 1` to be admissible, so narrowing the loop to `D` drops real words (48 → 43 at `r = 4, D = 2`). That is the defect the check needed. Related, same run: `not edited` in the provenance check is vacuously true on a clean bundle, so deleting it is invisible — the defect now miscomputes `edited` instead. **A section slice that reached into a later section.** The "packing branch closed" check searched from §31 to the end of the file, and §33's self-contained list repeats the same words — so removing an item from §31's list left every word still findable. Narrowed to §31 alone. **One defect fired the wrong check, informatively.** A wrong `floor_beta` does not disturb the corridor check, because that check reads `x_star` from the shipped JSON and never calls `floor_beta` at all. Retargeted to the diagnostics, which it does break. --- ## What this does not establish §33's unproved list. The Queue-Prefactor Saturation theorem's lower bound is an analytic dependency by the round's own account, so what is checked here is its finite shadow: nine corridors, a bounded centered prefactor, and a settling multiplicity ratio. `Θ` is not established by a table. The packing branch being closed is a statement about what *this* family of arguments can yield, and the three successor routes are untouched.