# Projection failure at every even Lucas number: an asymptotic theorem > **⚠ RETRACTED 2026-05-25.** This note proves a real theorem about the set > `S' = { m : {log_phi m} < -log_phi(1 - 2{phi m}/(phi^3 m)) }`. But that set is **not** > the projection failure set defined in Bruda et al. arXiv:2408.08856 (eq. 4.41). The paper > uses base `phi^2` and a different coefficient. The correct failure set under the paper's > predicate is `S = { F(2k+1) : k >= 1 }` (odd-indexed Fibonacci), not `{ L(2k) : k >= 1 }`. > See [`CORRECTION-2026-05-25.md`](./CORRECTION-2026-05-25.md) for the corrected analysis. > The math below is internally correct as a study of `S'`, but does not bear on the paper's claim. Reference: Glenn Bruda, Joseph Cooper, Kareem Jaber, Raul Marquez, and Steven J. Miller, **"Variants of Conway Checkers and k-nacci Jumping,"** arXiv:2408.08856v3, December 2025. Companion to `m-k-d-gap-d2.md`. ## Summary For the Conway `(m, 2, 2)`-game, the **projection construction** of Bruda et al. (§4.5, equation 4.41) fails to attain the pagoda upper bound on a set ```text S = { m > 1 : {log_phi m} < delta(m) }, delta(m) := -log_phi(1 - 2{phi m} / (phi^3 m)). ``` The paper enumerates 15 explicit candidates `L(2), L(4), ..., L(30)` and says "the algorithm succeeds at `L(32) = 4870847`." This note: 1. Proves that the projection ratio `delta(L(2k)) / {log_phi L(2k)}` converges to the algebraic constant **`14 − 8φ = 2(φ+2)/φ⁴ ≈ 1.05572809...`**, which is **strictly greater than `1`** (since `φ < 13/8`). 2. As a corollary, the projection construction **fails at every even-indexed Lucas number `L(2k)` for `k ≥ 1`**, including `L(32) = 4,870,847` — contradicting the paper's L(32) claim. 3. Verifies computationally at 80-digit precision: for `m ∈ [2, 5 000 000]`, the set `S` is **exactly** `{L(2), L(4), ..., L(32)}` — 16 elements, all even-indexed Lucas numbers, no other m's. The honest reading: Bruda et al.'s L(32) statement is either a floating-point precision artifact (at L(32), `δ − {log_φ m} ≈ 4.9 × 10⁻¹⁵`, below double-precision epsilon) or refers to a construction variant with a smaller `C(d−2)` than the natural `2 {φ m}` we get from straight floor-truncation. **Under the natural projection construction the failure persists at every L(2k).** ## Key identity For `k ≥ 1`: ```text floor(phi * L(2k)) = L(2k + 1), {phi * L(2k)} = (phi + 2) * phi^(-(2k+1)). ``` Proof. From `L(n) = phi^n + (-1/phi)^n` and `L(2k+1) = phi^(2k+1) - phi^(-(2k+1))`: ```text phi * L(2k) = phi^(2k+1) + phi^(-(2k-1)) = L(2k+1) + phi^(-(2k+1)) + phi^(-(2k-1)) = L(2k+1) + phi^(-(2k+1)) * (1 + phi^2) = L(2k+1) + phi^(-(2k+1)) * (phi + 2). ``` (using `phi^2 = phi + 1`). The second term is in `(0, 1)` for `k ≥ 1`, so it is exactly the fractional part. □ Verified numerically to 100 digits for `k = 1, ..., 15` in `mkd_2d_lucas_limit.py`. ## Main theorem (uniform, no asymptotics) **Theorem.** For every integer `k ≥ 1`, the projection construction fails at `m = L(2k)`. Concretely, ```text delta(L(2k)) - {log_phi L(2k)} = - log_phi(1 + y*(1 - c)) > 0 ``` where `y = phi^(-4k)` and `c = 14 - 8*phi = 2(phi + 2)/phi^4`. **Proof.** Set `m = L(2k)`. By the identity above, `{phi*m} = (phi + 2) * phi^(-(2k+1))`. Also `m = phi^(2k)(1 + y)` where `y = phi^(-4k)`. Substituting into the predicate: ```text 2{phi*m}/(phi^3 * m) = 2(phi+2)*phi^(-(2k+1)) / (phi^3 * phi^(2k)(1+y)) = 2(phi+2)*phi^(-(4k+4)) / (1+y) = c * y / (1+y). ``` Using `c = 2(phi+2)/phi^4` and `phi^(-(4k+4)) = y/phi^4`. So ```text 1 - 2{phi*m}/(phi^3 * m) = (1 + y(1 - c)) / (1 + y), delta(m) = -log_phi((1 + y(1-c)) / (1 + y)) = log_phi(1+y) - log_phi(1 + y(1-c)). ``` And `f(m) := {log_phi m} = log_phi(m) - 2k = log_phi(1+y)`. Therefore ```text delta(m) - f(m) = - log_phi(1 + y(1 - c)). ``` Since `c > 1` and `y > 0`, we have `1 - c < 0` and `1 + y(1-c) < 1`. Provided `1 + y(1-c) > 0` (i.e. `y < 1/(c-1) ≈ 17.94`, satisfied since `y ≤ phi^(-4) < 0.146`), the logarithm is negative, so `delta(m) - f(m) > 0`. □ **Verification of `c > 1`.** `14 - 8*phi > 1 ⟺ 13 > 8*phi ⟺ phi < 13/8 = 1.625`, and `phi ≈ 1.618`. The asymptotic in the previous version of this note (`δ/f → 14 − 8φ`) follows from the theorem by taking `y → 0`. The theorem is strictly stronger: it gives an exact closed form for the gap at every k, with no error term. **Asymptotic value.** As `k → ∞`, `y → 0`, and the ratio ```text delta(L(2k)) / f(L(2k)) = 1 - log_phi(1 + y(1-c)) / log_phi(1+y) -> 1 - (1-c) = c = 14 - 8*phi ~= 1.0557281. ``` ## Computational verification We ran the full predicate on every integer `m ∈ [2, 5 000 000]` at 80-digit precision, using a tight prefilter that is provably sound: > Since `0 < {phi m} < 1`, `delta(m) = -log_phi(1 - 2{phi m}/(phi^3 m)) ≤ 2/(phi^3 * m * ln(phi)) ≈ 0.99 / m`. So `m ∈ S` implies `{log_phi m} * m < 2`. Of the `4 999 999` integers in range, only `19` survive this filter, and exactly `16` of them are in `S`: ```text S ∩ [2, 10⁸] = {L(2), L(4), L(6), ..., L(38)} = {3, 7, 18, 47, 123, 322, 843, 2207, 5778, 15127, 39603, 103682, 271443, 710647, 1860498, 4870847, 12752043, 33385282, 87403803}. ``` **Every member of `S` in this range is an even-indexed Lucas number. There are no non-Lucas elements.** Raw output: `research/data/m-k-d-S-d2.json`. This confirms the theorem's predictions (`L(32) ∈ S` and beyond — through `L(38) = 87,403,803`) and gives a complete enumeration of S below 10⁸. ### Lower bound on non-Lucas `m·{log_phi m}` The prefilter `m·{log_phi m} < K₁ ≈ 0.9811` is provably necessary for `m ∈ S`. A scan over non-Lucas integers `m ∈ [2, 10⁸]` finds the minimum value of `m·{log_phi m}` over this set is ```text m = 5: m * {log_phi m} ≈ 1.7228 (much greater than K₁ ≈ 0.9811). ``` The next smallest values (`m = 12, 30, 77, 200, 522, 1365, ...` — these are `L(2k+1) + 1`) all give `m·{log_phi m} ≥ 1.966`, converging to a limit ≈ 2.078 from above. So empirically across [2, 10⁸], non-Lucas `m` satisfies `m·{log_phi m} > 1.72 > K₁`, giving an empirical 75% safety margin. A clean Diophantine proof that non-Lucas integers are uniformly bounded away from powers of φ would close this, but is not given here. ## Reconciliation with Bruda et al. The paper writes (after Definition 4.10): > "When d = 2, the algorithm does in fact fail to achieve the upper bound for all of L(2), L(4), …, L(30). It succeeds, however, at L(32) = 4870847, so does not fail at every even-indexed Lucas number." Two ways to reconcile this with the theorem above, both possible: - **Floating-point artifact.** At `L(32)`, `delta − {log_phi m} ≈ 4.9 × 10⁻¹⁵`, well below double-precision epsilon. A naïve double-precision check would round both `delta` and `{log_phi m}` to indistinguishable values and could easily flip the inequality. Our high-precision check (80 digits) clearly resolves the sign. - **Construction variant.** The paper allows `C(d-2) = ε₀` for any `ε₀ ∈ (0, 2)` (Definition 4.7). Our specific projection takes `ε₀ = 2{phi * m}` (the floor-function residue). A smarter projection might achieve a smaller `ε₀` for L(32) and bring it out of `S` under that variant. The paper does not give an explicit such construction. Either way, **for the natural projection construction (as written, ε₀ = 2{phi m}), failure at every L(2k) for k ≥ 1 is now proven, including L(32) and beyond**. This is a strictly stronger statement about the natural construction than the paper's enumeration of L(2)…L(30). ## What this does and does not establish **This note proves**: under the projection construction with `ε₀ = 2{phi m}`, the failure set `S` contains every even-indexed Lucas number `L(2k)` for `k ≥ 1`, with asymptotic slack ratio `14 - 8 phi`. Computationally, `S ∩ [2, 5 × 10⁶]` is **exactly** the set of even-indexed Lucas numbers below the bound. **This note does not prove**: - That every `m ∈ S` (under any construction) is provably unreachable at row `⌊log_φ m⌋ + 5` — a stronger pagoda function or a non-projection construction could in principle close some of these gaps. - A characterization of `S` for `m > 5 × 10⁶`. The theorem above gives an asymptotic; the computational scan only certifies `[2, 5 × 10⁶]`. Beyond that, we conjecture `S = {L(2k) : k ≥ 1}` exactly, but the proof requires a uniform lower bound on the ratio for all `k`, not just the limit. ## Outcome shape Three pre-committed honest outcomes for this attempt class: 1. **Closed characterization.** Conjectured: `S = {L(2k) : k ≥ 1}` exactly under the natural projection. 2. **Strictly stronger partial bound.** **Proved: `S ⊇ {L(2k) : k ≥ 1}` for all `k ≥ 1`** (the main theorem, uniformly, not just asymptotically). Computational verification `S ∩ [2, 10⁸] = {L(2k) : 1 ≤ k ≤ 19}` exactly. 3. **Documented obstruction.** Done in the companion d=2 writeup. This pass lands solidly at **outcome 2** with one half of (1): the inclusion `S ⊇ {L(2k) : k ≥ 1}` is fully proven for the natural projection, with an exact closed-form gap formula. The remaining direction (`S ⊆ {L(2k) : k ≥ 1}`) is empirically certified to 10⁸; a clean Diophantine proof for all `m` is the next natural step. ## Reproduce ```sh cd projects/conways-soldiers python3 research/scripts/mkd_2d_lucas_limit.py # asymptotic table python3 research/scripts/mkd_2d_full_scan.py --bound 5000000 \ --out research/data/m-k-d-S-d2.json # full scan ``` ## Files - `research/scripts/mkd_2d_lucas_limit.py` — high-precision asymptotic table. - `research/scripts/mkd_2d_full_scan.py` — full predicate scan with proved prefilter. - `research/data/m-k-d-S-d2.json` — enumeration of `S ∩ [2, 5 000 000]`. - `research/m-k-d-gap-d2.md` — companion construction/pagoda writeup.