# RUN-015 — Round A-U.2b: a real elimination, and how far its own constant can go **Arm:** 數學戰士「墜衡」 / AMRAL Research Lab **Subject:** Neo.K + Aletheia, `Hard_Zeta_Phase_II_Round_AU2b_bundle.zip` (source item 33) — Round A-U.2b *Sparse Lift Rigidity: Exact Return Separation, Valuation-Language Entropy Gap and Logarithmic Deficit Barrier*, plus `A_Line_ROUTE_MAP_v1.1` **Tools:** `hz_accel_code.py` (A-U.2b layer) · `src17_hardzeta_au2b_recheck.py` · `src17_drill.py` **Logs:** `src17-au2b-recheck.json` · `src17-drill.json` **Result: 25/25 checks. 34/34 planted defects caught by the check named for each — 16 in the A-U.2b layer, 14 in this run's own measurement, 4 in the documents. 2/2 null controls undisturbed. Coverage audit clean.** --- ## The first round of Phase II that eliminates something A-U.1 and A-U.2a both ended in no-gos. This one does not. It proves a positive statement — every positive-integer all-prefix subcritical accelerated orbit has ``` limsup_{N→∞} D_N / log₂N ≥ 0.01 D_N = max_{m≤N} d_m ``` and therefore `sup_m d_m = ∞`. From that it eliminates outright: the mechanical critical code, bounded-discrepancy critical codes, bounded-deficit spines, ultimately periodic subcritical codes, and low-complexity candidates with thin deficit envelopes. **It also settles the question RUN-014 left open.** I measured that the mechanical code — the countermodel that defeated A-U.1 — has positive lift flux, so A-U.2a's Zero-Flux Boundary Theorem already reached it. A-U.2b now proves it is unanchored by a completely different route: `K*_m = ⌊βm⌋` exactly, so `d_m = 0` for every `m`, so `D_N = 0 = o(log N)`, and the barrier applies directly. Verified here to `m = 400`. **Two independent arguments, same verdict** — and RUN-013's direct measurement that its lift digits never stop is a third. --- ## The whole thing turns on 2.8395 < 3 The argument is a squeeze between two exponential scales. **Upward — return separation.** §4's Repeated-Block Congruence: if the same length-`r` exponent block follows positions `a` and `b`, with block valuation `Q`, then `Y_a ≡ Y_b (mod 2^{Q+1})`. On an infinite subcritical spine no state repeats — a repeat would force an accelerated cycle, whose equation `(2^{Q_cyc} − 3^p)Y = B_cyc > 0` demands `Q_cyc > pβ`, contradicting `K_m < mβ`. So `|Y_b − Y_a| ≥ 2^{Q+1}`, and §19 turns that into base 3: `Q ≥ ⌊βr⌋ − D` gives `2^{Q+1} > 3^r/2^D`. **Downward — how many thin-deficit blocks exist.** §15's ledger `E_{i,r} = ⌊γ(i+r)⌋ − ⌊γi⌋ + d_i − d_{i+r}` pins the block excess, §16 confines it to a band of width `2D` around `γr`, and §17 counts the compositions. Stirling gives the growth base ``` Λ_γ = (1+γ)^{1+γ} / γ^γ ``` I computed it at 60 digits: **`Λ_γ = 2.83951373049775259640…`**, matching the paper's `2.8395137304…` on every digit stated, and `Λ_γ < 3` with a gap of `0.1605` — 5.35%. That gap is the theorem. Thin-deficit codes cannot supply enough distinguishable length-`r` blocks to keep up with the separation the integers force, so a repeat must occur too early, and §24's excursion bound `M_N < 2^{D_N+1}(n + N/3)` says the states are too small to be that far apart. Contradiction. **The explicit constants, checked rather than believed.** §21 needs `1/β < c < 1/log₂Λ_γ` to be non-empty — measured, `(0.630930, 0.664168)`, and the paper's `c = 0.645` is inside. §26's two inequalities at `c = 0.645`, `ε = 0.01`: | claim | computed | margin | |---|---|---| | `βc − 1 − 2ε > 0.0022` | `0.0023008129…` | `1.008 × 10⁻⁴` | | `c·log₂Λ_{γ+ε/c} < 0.986` | `0.9854014472…` | `5.986 × 10⁻⁴` | Both clear, and both are tight to the fourth decimal — the constants were chosen to just pass, and they just pass. Each is drilled by a perturbation *smaller than its own margin*, so the check has to be reading the real value and not a rounded one. --- ## The measurement: how far this argument can be pushed §37 names the next round **A-U.2b.1 — Sharp threshold for `D_N ∼ c log N`**. So the useful question is what the *existing* scheme can already give, before any new idea is needed. The two constraints, read off §22–§24, are ``` block count is o(N): c · log₂ Λ_{γ + ε/c} < 1 separation beats the peak: βc − 1 − 2ε > 0 ``` Maximising `ε` over `c`: | | `c` | `ε` | |---|---|---| | published | 0.645 | **0.01** | | this scheme's ceiling | 0.650 | **0.01502** | **The published `0.01` is a safe round number, not the scheme's limit — the same argument supports about `0.015`, a 50% improvement with no new ideas.** Anything past `0.0150` needs a different argument, because at that point the two constraints meet. That is a concrete starting point for A-U.2b.1: it says where the cheap gains stop. --- ## Every eliminated family, checked on an instance - **Mechanical code**: `K*_m = ⌊βm⌋` so `d_m ≡ 0` — verified to `m = 400`. And its factor complexity is exactly `p(r) = r + 1` for `r ≤ 24`, confirming §31's Sturmian claim, which excludes it a second time through the complexity–deficit tradeoff. - **Ultimately periodic tails**: §33's source `y = B_per/(2^Q − 3^p)` is negative for every subcritical period tested, and positive exactly when the period is supercritical — both outcomes present, so the implication is not graded on one side. - **Bounded deficit**: `sup d_m < ∞ ⟹ D_N = o(log N)`, so the barrier applies. Everything upstream holds on real spines too: the repeated-block congruence (on actual repeats, with a guard that repeats occur at all), the return separation, the complexity–peak law `M_{p(r)+1} ≥ 2^{r+1}`, the excursion bound in exact rationals, and the complexity–deficit tradeoff. **One clarification worth recording.** The all-ones family `2^{m+1}−1` is *not* a low-deficit family — it is the opposite extreme. `K_j = j` gives `d_j = ⌊γj⌋`, which grows linearly, the largest deficit possible. The zero-deficit object is the mechanical code. Both are extremes of the same ledger and it is easy to conflate them. --- ## What real starts do The theorem constrains a hypothetical: no positive integer is known to have an infinite subcritical spine. What exists are finite runs, and they clear the barrier by two orders of magnitude: | start | lifetime | `D_N` | `D_N/log₂N` | |---|---|---|---| | 27 | 36 | 6 | 1.16 | | 703 | 50 | 6 | 1.06 | | 10087 | 65 | 6 | 1.00 | | 35655 | 84 | 8 | 1.25 | | `2^17−1` | 62 | 11 | 1.85 | The only object with `D_N = 0` is the mechanical code — which §30 proves is not a positive integer. So the check is stated with both outcomes: every real start clears `0.01`, and the one thing that does not is exactly the countermodel. --- ## Findings about my own checks **Five defects belonged to different checks than I named — and each miss said something true.** Replacing the record deficit's `max` by `min` leaves the complexity–deficit tradeoff intact, because its right-hand side `r − log₂(n + N/3)` is negative at small `r`, so *any* non-negative deficit satisfies it. Shortening the subcritical lifetime produces *shorter* orbits and therefore *fewer* repeats, so "no state repeats" survives it. Shifting the valuation list never reaches the excursion bound, which reads the deficit and the endpoints instead. `gamma_decimal()` is read only by the entropy check, not by the interval check, which computes `β` itself. And flipping the sign inside `log₂Λ` makes the constant *smaller* — `0.6006` instead of `1.5057` — so the admissible interval's upper end moves out to `1.665` and `c = 0.645` is still comfortably inside: the interval **widens** rather than emptying. Each was retargeted and each orphaned check was given a defect of its own — including one that makes `orbit_endpoints` return a constant, which is what "no state repeats" actually needs to be able to fail, and one that computes the interval's lower end from `γ` instead of `β`, which does empty it. **The tenth loosening no-op.** Making the peak law read `max(Y)` instead of `max(Y[:N])` cannot fail: `max(Y) ≥ max(Y[:N])`, so the failure condition `max < 2^{r+1}` becomes *harder*, not easier. Replaced by shrinking the window to `Y[:1]`, which moves the answer. Every constant in this drill is instead perturbed by less than the margin it claims to clear. **A check that is a corollary of another check.** §16's thin-deficit range `⌊γr⌋ − D ≤ E ≤ ⌈γr⌉ + D` follows from §15's ledger together with `d ≤ D`, both already checked — so on a degenerate sample it would hold for free. It now requires the block excess to actually **vary** across the sample (measured spread `0…6` at `r = 3`), which is the part that is not entailed. **A contrast whose sample avoided the interesting case.** "Real spines clear `D_N/log₂N > 0.01`" passed with margin ~100×, and would have passed no matter what, because the only object that fails it was not in the sample. It now includes the mechanical code explicitly and requires **both** outcomes — real starts clearing, the countermodel at exactly zero. That is the theorem's content rather than a one-sided sample. No defect in this drill loosens a comparison. That shape was a no-op nine times in this arm; every constant here is instead perturbed by less than the margin it claims to clear. --- ## What this does not establish §36's own list: `D_N ≍ c log N` is not excluded, nor sparse unbounded deficit spikes, nor high-complexity critical inputs, nor zero-density nonzero lift events, nor CASP, CST, or Collatz. The barrier is a statement about infinite orbits and is not finitely checkable; what is checked here is the exact algebra it rests on, the constants at 60 digits, and the eliminated families on explicit instances. The `0.015` ceiling is a property of *this* proof scheme, measured under the two constraints as I read them from §22–§24 — a different reading of the error terms would move it.