# RUN-013 — Round A-U.1: the no-go holds, and the datum it says is missing is measurable **Arm:** 數學戰士「墜衡」 / AMRAL Research Lab **Subject:** Neo.K + Aletheia, `Hard_Zeta_Phase_II_Round_AU1_bundle.zip` (source item 31) — Round A-U.1 *Critical Occupation Measures: Invariant-Limit Theorem, Exponent-Shift Completion Countermodels and Anchor-Erasure No-Go*, plus `A_Line_ROUTE_MAP_v0.9` **Tools:** `hz_accel_code.py` (A-U.1 layer) · `src15_hardzeta_au1_recheck.py` · `src15_drill.py` **Logs:** `src15-au1-recheck.json` · `src15-drill.json` **Result: 23/23 checks. 26/26 planted defects caught by the check named for each. 2/2 null controls undisturbed.** --- ## What kind of round this is Phase I closed the A line onto CASP and split it into two regimes. A-U.1 opens the first, **and closes it as a route.** That is unusual enough to state plainly: this round's main result is that its own programme does not work. The argument has two halves and both are checked here. **The half that succeeds.** §6–§12 prove a *Critical Invariant-Limit Theorem*: a critical A-U candidate's empirical measures have weak-\* limits `ν` with ``` S_*ν = ν ν({−1/3}) = 0 ∫(q−1) dν = γ, so ∫q dν = β ``` The route through it is careful — `ν({ξ}) = 0` comes from a **mass bound** (§8), `ν_m(C_R) ≤ (γ + O(1/m))/(R−1)`, which is what makes `S` almost-everywhere continuous so weak convergence can be passed through it at all. **The half that closes the route.** §13–§17 then show those conditions are *consistent*, by two explicit countermodels. So no contradiction can come from invariance, uniform integrability and the critical mean alone. --- ## The two countermodels, verified **§13, the Bernoulli critical measure.** The product measure `⊗[(1−γ)δ₁ + γδ₂]` on the alphabet `{1,2}` is shift-invariant with `∫q dμ = 1 + γ = β`, and because `q ≤ 2` its uniform integrability is free. The mean identity is checked in exact `Fraction`s at seven rationals; since `γ` is irrational the critical case is pinned by bracketing it — `53/91 < γ < 12/19` gives means `1.582 < β < 1.632`. **§15, the mechanical code.** `q*_m = ⌊βm⌋ − ⌊β(m−1)⌋` telescopes to `K*_m = ⌊βm⌋`, lives in `{1,2}`, and is subcritical at every prefix — all verified to `m = 300` in exact integers (`2^{K*_m} < 3^m`, no float decides it), alphabet to `m = 2000`. Its 2-density converges to `γ`: | `m` | 2-frequency | gap from `γ` | |---|---|---| | 100 | 0.5800 | 4.96e−3 | | 500 | 0.5840 | 9.63e−4 | | 2000 | 0.5845 | 4.63e−4 | | 8000 | 0.584875 | 8.75e−5 | **A cross-check worth recording:** this is the same sequence `hz_accel_code.py` has carried since RUN-008, where it was implemented from Round 03-A.1's description as "the maximal subcritical code". A-U.1 §15 states it independently as a closed formula. The two agree on all 300 terms compared. Two rounds, two descriptions, one sequence. **The conjugacy underneath.** §5's `E : 𝔛_∞ → ℕ_{≥1}^ℕ` being a bijection intertwining `S` with the shift is what lets a symbolic measure be pulled back at all. Checked in the finite form that carries it: a code is exactly one clopen cylinder `r_m + 2^{K_m+1}ℤ₂` (with witnesses *outside* it, not only inside), cylinders nest as codes extend, `E(Sx) = σE(x)` under direct iteration on integers, and codes with large and supercritical valuations — `(17,3)`, `(1,1,20)`, `(12,1,5)` — are realized too, since the alphabet is all of `ℕ_{≥1}` and not merely the legal codes. --- ## The datum the no-go says is missing — measured §21 identifies what survives: the exact source lift ``` r̂_{m+1} = r̂_m + t_{m+1}·2^{K_m+1} positive-integer anchor ⟺ t_m = 0 eventually ``` and §22 explains why an occupation measure cannot carry it — `t_m` compares canonical lifts of the *original source cylinder* across prefix lengths, not any bounded function of the current tail. That is a claim about what the occupation measure lacks. It is also, taken directly, something this arm can compute. So: | start | lift settles at | nonzero lifts before | |---|---|---| | 27 | `m = 3` | 3 | | 103 | `m = 5` | 4 | | 703 | `m = 7` | 7 | | 1407 | `m = 7` | 7 | | 10087 | `m = 11` | 8 | | 15039 | `m = 9` | 8 | | 35655 | `m = 10` | 7 | **Every genuine integer settles by `m = 11`**, and once it settles the canonical source *is* that integer — checked, not assumed. The mechanical code does not settle. To depth 60 it has **42 nonzero lifts**, the last at **`m = 59`**, and its canonical source climbs monotonically at all 59 steps: | `m` | 8 | 16 | 24 | 32 | 40 | 48 | 60 | |---|---|---|---|---|---|---|---| | source bits | 13 | 25 | 38 | 49 | 64 | 77 | 93 | RUN-008 saw this source pass 29 million by `m = 16`; followed to `m = 60` it passes `2^92`. It sits near the top of its permitted range `2^{K_m+1}` the whole way — the source is roughly `3^m`. **So the anchor cocycle separates the countermodel from a genuine integer by `m = 11`, while the occupation measure never separates them at all** — §17 shows the mechanical code's orbit measures pull back to a critical invariant measure just like a real orbit's would. That is exactly the asymmetry §22–§23 assert, made quantitative: the information is not merely absent from the occupation measure, it is *cheaply available* in the coordinate the occupation measure discards. **What this does not decide.** §16 explicitly declines to say whether the mechanical point `x_★` is an ordinary positive integer, calling that the anchor-sensitive question. This does not decide it either. What is measured is that **no positive integer below `2^{K_60+1}` realizes the mechanical code**, and that its lifts are still nonzero where the computation stops. A finite window cannot exclude an integer beyond it, and the round is right to leave it open. --- ## Dense but not closed, with a witness §18's first sentence — every odd residue class holds a positive odd integer — is a tautology, and I initially wrote it as a check, which could not fail. §18's content is the second sentence: `ℕ_odd` is dense in `𝒪` but **not closed**. The mechanical code witnesses that concretely, and that is what is checked instead: its canonical sources are positive odd integers, each congruent to the next modulo `2^{K_m+1}` (so the sequence is 2-adically Cauchy), while growing from 3 to 93 bits. A convergent sequence of positive integers whose real size diverges is exactly how the limit leaves `ℕ_odd`. --- ## The ledger, and what it refuses §32 lists seven results proved and five not. The unproved list still contains *Pointed Critical Occupation Rigidity*, *CASP exclusion*, **Terras**, and **Collatz** — checked, because a negative result of this shape is easy to misread as progress on the conjecture, and it is not. §33 says it in the paper's own words: ordinary 2-adic occupation theory can describe counterexample-like dynamics but cannot tell whether they are anchored by an ordinary positive integer. Route map v0.9 carries the same verdict and the same missing condition. The bundle's two re-shipped Phase I files (the closure and 03A5) are byte-identical to item 30's copies, so RUN-012 carries over; exactly two files are new. --- ## Four findings about my own checks **Six checks had no defect naming them.** RUN-012 ended by noting that a drill count says nothing about a check nothing points at, so this run audits that before running: every check name must appear as some defect's target, and every target must name a real check. The first audit found **six** unguarded checks — cylinder nesting, arbitrary-code realization, the singular mass bound, mechanical subcriticality, the two-symbol alphabet, and the cylinder-integer check. Each got a defect. The audit costs one script and would have been worth running from RUN-005. **A one-place shift in the anchor cocycle passed.** Dropping the first lift digit shifts the settling index by one, and the check did not notice, because on a settled orbit `r_{s+1} = r_s = n` — asserting the source at *one step past* the settling point is insensitive to where that point is. The assertion is now sharp: the source must already equal `n` **at** the settling point. That is what pins the index, and it is the difference between measuring "it settles" and measuring "it settles here" — and the whole finding above is about *where*. **A constant offset telescoped away — the same cancellation as RUN-011.** "`floor_beta` rounds up" looked like a clean defect. It is a no-op: `floor_beta` enters the mechanical code only as a *difference* `⌊βm⌋ − ⌊β(m−1)⌋`, so any constant offset cancels, and `K*_m` is unchanged. RUN-011 retired a defect for exactly this reason (a constant shift in the Sturmian credit is invisible to a recurrence built from differences). Two instances now, so it is a class, not an instance: **where a quantity appears only inside a difference, constant-offset mutations there are structurally dead.** Replaced with a change of base, which survives telescoping. **Weakening a requirement is not a defect — third time.** Dropping "Collatz" from the list of terms the unproved ledger must contain only makes the check ask for less, and the real document satisfies both versions. Replaced by reading the *proved* section instead of the unproved one, which moves the answer because that list names none of the four. Same family as the `<`/`<=` no-ops and the widened threshold in RUN-012; the reliable form is to damage what is computed, not what it is compared against. --- ## What this does not establish Nothing on §32's unproved list. The invariant-limit theorem itself is a statement about weak-\* limits over an infinite orbit and is not finitely checkable — what is checked here is the finite arithmetic it rests on (the mass bound on real spines, the clopen structure of the singular cylinders, and the identity `g_R = min(q−1, R−1)` that makes the truncated observable continuous). The separation measured above is a statement about seven integers and 60 code steps. It shows the cocycle *can* distinguish; it does not show that critical occupancy forces nonzero lifts, which is §32's unproved item 2 and the actual A-U.2 target.