# RUN-011 — Round 03-A.4: the deficit ledger, and what CF tools can actually reach **Arm:** 數學戰士「墜衡」 / AMRAL Research Lab **Subject:** Neo.K + Aletheia, `Hard_Zeta_Phase_I_Round_03A4_Spine_Valuation_Rigidity_v0.1.md` + `Hard_Zeta_ROUTE_MAP_v0.7.md` (2026-08-11 21:52) — source item 28 **Tools:** `hz_accel_code.py` (deficit-queue layer) · `src13_hardzeta_round03a4_recheck.py` · `src13_drill.py` **Logs:** `src13-hardzeta-round03a4-recheck.json` · `src13-drill.json` **Result: 26/26 checks. 16/16 planted defects caught by the check named for each — 11 in the ledger layer, 3 in this run's own measurement. 2/2 null controls undisturbed.** --- ## What Round 03-A.4 does Round 03-A.3 made the spine deterministic. This round asks what staying on one **costs**: ``` d_m = ⌊βm⌋ − K_m the integer deficit; subcritical ⟺ d_m ≥ 0 d_m = d_{m−1} + b_m − e_m b_m ∈ {0,1} Sturmian, e_m = q_m − 1 Σ_{i≤m}(q_i − 1) ≤ ⌊γm⌋ the credit ledger, γ = β − 1 ≈ 0.585 ``` Every unit of extra 2-adic valuation is paid out of a Sturmian budget. §9–§12 then re-read the same ledger as **cylinder occupancy** — `q_i ≥ r` exactly when `Y_{i−1} ≡ −3^{−1} (mod 2^r)` — and the strongest statement, §26–§32, is that any infinite positive subcritical spine has `Y_m → ∞`: a CST counterexample would be a genuinely **divergent** orbit. All of it holds. The Sturmian increment is always a bit; the recurrence and the telescoped ledger hold at every step; subcritical is exactly a non-negative deficit; high valuation is exactly membership of one residue class and those cylinders nest; occupancy stays inside the budget; the excursion identity holds in exact integers; the logarithmic and bounded-deficit bounds bracket the endpoint; no endpoint repeats; and where the Legendre gate opens, `K_m/m` really is a convergent of β. The `β` machinery is anchored against named values rather than its own shape: the continued fraction comes out `[1,1,1,2,2,3,1,5,2,23,2,2]` and the convergents include **19/12** and **84/53** — the equal-temperament fractions. --- ## Two things worth separating **§18's "Spine Excursion Identity" is Paper 06's affine formula in log coordinates.** Multiply through by `2^{K_m}` and it becomes ``` Y_m·2^{K_m} = 3^m·n + Σ_i 3^{m−1−i}·2^{K_i} ``` which this run verifies in exact integers, no floating point at all. That is not a criticism — the **new** content of §18 is the *reading*, deficit as exponential growth rate, and that reading is what carries §30's divergence corollary. Worth separating so the novelty is credited to the right half. **The Haar gap is real, and measured.** §12 contrasts a Haar-typical total cylinder density of exactly **1** against a budget of `γ ≈ 0.585`. On real spines: | `n` | lifetime | credit spent / budget | mean excess valuation | |---|---|---|---| | 27 | 36 | 20 / 21 | 0.556 | | 703 | 50 | 27 / 29 | 0.540 | | 10087 | 65 | 37 / 38 | 0.569 | | 35655 | 84 | 48 / 49 | 0.571 | Every measured spine spends well under the Haar rate of 1 per step. So the discrepancy exists rather than being an artefact of the bound — and §13 and §45's No-Go 2 are right to stop there: a measure-one statement about Haar-typical orbits cannot become a theorem about every anchored positive-integer orbit. This run adds nothing that closes that gap. > **Corrected 2026-08-15 (RUN-012).** This table first read `16/19`, `13/19`, > `17/19`, `13/19` and `0.471`, `0.382`, `0.500`, `0.382`. Those were measured at > `M_MAX = 34`, the depth cap in `src13`, while the *lifetime* column printed the > true lifetime — so four rows paired a spine's full life with credit counted only > to depth 34. The values above are at the full lifetime. The Haar conclusion is > unchanged, but the budget reading is not: spines end their lives having spent > **93–98%** of the Sturmian credit, where the truncated rows read 68–89%. See > RUN-012 for what that costs the reading of Round 03-A.5 §19–20. --- ## The Legendre gate, measured §34 says continued-fraction tools apply rigorously only when `δ_m < 1/(2m)`. §35 calls that "ultra-tight contact". How restrictive is it in practice? | `n` | depths examined | gate open at | fraction | |---|---|---|---| | 27 | 24 | `m = 2` | 4.2% | | 103, 703, 1407, 10087, 15039 | 24 each | — | **0%** | | 35655 | 24 | `m = 4` | 4.2% | **The gate opens on 2 of 168 depths across seven spines.** Five of the seven never open it at all. So §35's restriction is not a technicality — for essentially the whole spine, continued-fraction tools have no rigorous purchase, and §45's No-Go 3 ("bounded deficit is not enough to get a convergent") is quantitatively the normal case rather than the exception. Where the gate does open, `K_m/m` is genuinely a convergent, so the criterion is doing real work — just very rarely. --- ## Three findings about my own checks **An equivalence that only ever saw one side.** *Subcritical ⟺ `d_m ≥ 0`* was tested only at depths inside the subcritical lifetime, where both halves are always true, so it passed for free — a planted defect in `deficit` slipped through. The loop now runs **one step past** the lifetime, with a guard requiring both outcomes to appear. Fourth empty-observable in this arm. **A constant offset that cancels.** Shifting the Sturmian credit by a constant is invisible to the recurrence, because `b_m` is a *difference* of consecutive credit values. Only the telescoped ledger sees it. The defect was retargeted at the check that can. **`<` versus `<=` was a no-op for the fifth time.** In the continued-fraction loop, `acc·P == Q` would require an exact power relation between rationals, which `log₂3` being irrational forbids. Replaced with a mutation of the tail recursion, which does move the expansion. Also worth recording as a cost note: `beta_convergents(40)` with exact rationals does not terminate in reasonable time — β's tenth partial quotient is 23 and the exact tail explodes past it. Twelve terms is instant and covers every denominator this run needs. The function now says so in its docstring. --- ## What this does not establish Nothing on §46's unproved list. The divergence corollary is checked only in the direction a finite spine can show — endpoints grow, none repeats — which is consistent with the theorem and with any bounded search equally.