# Non-Lucas integers are bounded away from S > **⚠ RETRACTED 2026-05-25.** This note proves a real bound about the auxiliary set `S'` > in `m-k-d-lucas-limit.md`. But `S'` is **not** the projection failure set the paper defines > in eq. 4.41. The correct paper-predicate failure set is `S = { F(2k+1) : k >= 1 }`. See > [`CORRECTION-2026-05-25.md`](./CORRECTION-2026-05-25.md). The bound below is internally > correct but does not characterize the paper's `S`. Companion proof to `m-k-d-lucas-limit.md`. Reference paper: Bruda, Cooper, Jaber, Marquez, Miller, arXiv:2408.08856v3 (Dec 2025). ## Statement Write `f(m) := m * {log_phi m}` and `g(m) := m * delta(m) = -m * log_phi(1 - 2{phi m}/(phi^3 m))`. The d=2 projection construction fails at `m` iff `f(m) < g(m)`. **Theorem (non-Lucas exclusion).** For every non-Lucas integer `m >= 2`, ```text f(m) >= (1 - phi^{-3}) / ln(phi) = (4 - 2*phi) / ln(phi) ≈ 1.5871. ``` For every integer `m >= 2`, ```text g(m) <= -2 * log_phi(1 - 1/phi^3) ≈ 1.1192, ``` with equality only in the limit `m=2`, `{phi*m} -> 1` (not attained). Therefore `f(m) > g(m)` for every non-Lucas `m >= 2`, i.e. **no non-Lucas integer lies in `S`**. The asymptotic version (large `m`) reads `f(m) > (1 - phi^{-3})/ln(phi)` vs `g(m) < K_1 := 2/(phi^3 * ln(phi))`, and the ratio of these two thresholds is exactly the golden ratio: ```text ((1 - phi^{-3})/ln(phi)) / K_1 = (phi^3 - 1) / 2 = phi. ``` Combined with the main theorem of `m-k-d-lucas-limit.md` (every `L(2k)` for `k >= 1` lies in `S`), this gives the complete characterization under the natural projection construction: ```text S = { L(2k) : k >= 1 }. ``` ## Proof **Step 1 (interval reduction).** Let `m >= 2` be non-Lucas and set `n = floor(log_phi m)`. Since `m >= 2 > phi`, we have `n >= 1`. The function `g(m) = m * {log_phi m}` is monotone increasing in `m` on `[phi^n, phi^{n+1})` (both factors are positive and increasing). So the infimum of `g` over non-Lucas integers in this interval is attained at the smallest non-Lucas `m` in the interval. **Step 2 (small `n`).** For `n = 1`, `[phi, phi^2) = [1.618..., 2.618...)` contains only `m = 2 = L(0)`, which is Lucas. For `n = 2`, `[phi^2, phi^3) = [2.618..., 4.236...)` contains only `m in {3, 4} = {L(2), L(3)}`, both Lucas. So no non-Lucas `m` has `n <= 2`. **Step 3 (`n >= 3` smallest non-Lucas).** For `n >= 3`: - *n even.* `L(n) = phi^n + phi^{-n}` lies in `(phi^n, phi^n + 1)`, and `L(n+1) = phi^{n+1} - phi^{-(n+1)}` lies in `(phi^{n+1} - 1, phi^{n+1})`. Their difference is `L(n+1) - L(n) = L(n-1) >= L(2) = 3 > 1` for `n >= 3`, so `L(n) + 1 < L(n+1)`, and `L(n) + 1` is non-Lucas and is the smallest such integer in `[phi^n, phi^{n+1})`. - *n odd.* `L(n) = phi^n - phi^{-n} < phi^n`, so the smallest integer `>= phi^n` is `ceil(phi^n) = L(n) + 1`. Again the next Lucas number is `L(n+1)` near the right endpoint, at distance `>= 3`, so `L(n) + 1` is non-Lucas. Either way, the minimizer is `m_n = L(n) + 1 = phi^n + (-1)^n * phi^{-n} + 1`. **Step 4 (bound at `m = m_n`).** Set `delta_n = (1 + (-1)^n * phi^{-n}) / phi^n > 0`, so ```text m_n = phi^n * (1 + delta_n), {log_phi m_n} = log_phi(m_n / phi^n) = log_phi(1 + delta_n). ``` Use the elementary inequality `ln(1 + x) > x / (1 + x)` for `x > 0`. Dividing by `ln(phi) > 0`: ```text log_phi(1 + delta_n) > delta_n / ((1 + delta_n) * ln(phi)). ``` Multiplying by `m_n = phi^n (1 + delta_n)`: ```text m_n * {log_phi m_n} > phi^n * (1 + delta_n) * delta_n / ((1 + delta_n) * ln(phi)) = phi^n * delta_n / ln(phi) = (1 + (-1)^n * phi^{-n}) / ln(phi). ``` **Step 5 (uniform lower bound on f).** For `n >= 3`: - *n odd:* `(1 - phi^{-n}) / ln(phi) >= (1 - phi^{-3}) / ln(phi)` (increasing in `n`). - *n even (n >= 4):* `(1 + phi^{-n}) / ln(phi) > 1 / ln(phi) > (1 - phi^{-3}) / ln(phi)`. So for every non-Lucas `m >= 2`, `f(m) > (1 - phi^{-3}) / ln(phi)`. Using `phi^3 = 2*phi + 1`, `1 - phi^{-3} = (phi^3 - 1)/phi^3 = 2*phi/phi^3 = 2/phi^2 = 2(2 - phi)`, so the bound equals `(4 - 2*phi)/ln(phi) ≈ 1.5871`. **Step 6 (uniform upper bound on g).** Since `{phi*m} < 1` for every integer `m`, ```text 2 {phi*m} / (phi^3 * m) < 2 / (phi^3 * m), ``` and `-log_phi(1 - u)` is increasing in `u`, so ```text g(m) = -m * log_phi(1 - 2{phi*m}/(phi^3 m)) < -m * log_phi(1 - 2/(phi^3 m)). ``` The right-hand side, viewed as a function of `m >= 2`, is monotone decreasing. *Proof.* Let `c = 2/phi^3` and `h(m) = -m * log_phi(1 - c/m) = m * log_phi(m/(m-c))`. Then ```text ln(phi) * h'(m) = d/dm [m ln(m) - m ln(m-c)] = ln(m/(m-c)) + 1 - m/(m-c) = ln(1 + c/(m-c)) - c/(m-c), ``` and `ln(1+x) < x` for all `x > 0`, so `h'(m) < 0`. □ Therefore `h(m) <= h(2)`, so ```text g(m) < g_max := -2 * log_phi(1 - 1/phi^3) = -2 * log_phi((phi^3 - 1)/phi^3) = -2 * log_phi(2*phi/phi^3) = -2 * log_phi(2/phi^2) = 4 - 2 * log_phi(2) ≈ 1.1192. ``` **Step 7 (conclude).** `f(m) > 1.5871 > 1.1192 > g(m)` for every non-Lucas `m >= 2`. Hence `m \notin S`. □ **Bonus ratio.** The asymptotic upper bound on `g(m)` is `K_1 = 2/(phi^3 * ln(phi))` (taking `m \to \infty` in Step 6). Dividing the Step-5 lower bound on `f` by `K_1`: ```text ((4 - 2*phi)/ln(phi)) / (2/(phi^3 * ln(phi))) = (2 - phi) * phi^3 = 2*phi^3 - phi^4 = 2(2*phi+1) - (3*phi+2) = phi. ``` So asymptotically, the non-Lucas lower bound exceeds the threshold for S-membership by exactly the golden ratio. ## Numerical witnesses ```text n smallest non-Lucas m f(m) step-4 lower bound 3 5 1.7227594 1.5870838 4 8 2.5700822 2.1746181 5 12 1.9662241 1.9410338 6 19 2.2572025 2.0911218 7 30 2.0395139 2.0303091 8 48 2.1452251 2.0963416 9 77 2.0640031 2.0644130 ``` All above the global upper bound `g_max ≈ 1.1192` on `g(m) = m * delta(m)`. Actual `f(m)` converges to `1/ln(phi) ≈ 2.0781` from above, well clear of `g_max`. ## Corollary: complete characterization ```text S = { m in Z, m >= 2 : projection construction fails to attain row floor(log_phi m) + 5 } = { L(2k) : k >= 1 }. ``` Proof of the two inclusions: - **`{L(2k) : k >= 1} ⊆ S`**: theorem of `m-k-d-lucas-limit.md`, uniformly in `k >= 1`. - **`S ⊆ {L(2k) : k >= 1}`**: by the theorem above, only Lucas numbers `>= 2` can lie in `S`. The Lucas members `>= 2` are `L(0) = 2` and `L(n) for n >= 2`. Check each: - `L(0) = 2`: direct computation gives `f(2) = 2 * {log_phi 2} ≈ 0.881` and `g(2) ≈ 0.238`, so `f(2) > g(2)`, not in `S`. - `L(2k+1)` for `k >= 1`: `L(2k+1) = phi^(2k+1) - phi^(-(2k+1))`, so `m / phi^(2k) = phi - phi^(-(4k+1))`, and `{log_phi m} = log_phi(phi - phi^(-(4k+1)))`. For `k >= 1`, `phi^(-(4k+1)) <= phi^(-5) < 0.09`, so `m / phi^(2k) > phi - 0.09 > 1.5 > phi^(1/2)`, giving `{log_phi m} > 1/2`. Then `f(m) = m * {log_phi m} > m/2 >= L(3)/2 = 2 > g_max ≈ 1.119`. Not in `S`. - `L(2k)` for `k >= 1`: in `S` by the main theorem. So `S = {L(2k) : k >= 1}` exactly. ## Outcome This pass closes the conjecture stated in `m-k-d-lucas-limit.md` Outcome 1: under the natural projection construction with `epsilon_0 = 2 * {phi * m}`, the failure set `S` is **exactly** the set of even-indexed Lucas numbers `{L(2k) : k >= 1}`. This is a complete, sharp characterization, not just an asymptotic bound. The proof uses only elementary inequalities (`ln(1+x) > x/(1+x)`) and Lucas/golden ratio algebra. ## Reproduce ```sh cd projects/conways-soldiers python3 research/scripts/mkd_2d_full_scan.py --bound 100000000 \ --out research/data/m-k-d-S-d2.json python3 research/scripts/mkd_2d_lucas_limit.py python3 research/scripts/mkd_2d_non_lucas_bound.py # this proof's numerics ```