Investment state: **proposed**. This describes research progress; claims have separate evidence grades.

## Contribution to the goal

### Proposed Route Contribution: $O(1)$ Dirichlet Block-Moment Capped Gram Engine & Target-Regime Sweep $k = 25..50$

- **The Object**: The exact rational capped and uncapped Gram-pencil Rayleigh quotients $q_{\text{cap}}(k, d)$, $q_{\text{uncap}}(k, d)$, and faithful cap cost $c(k, d) = 1 - q_{\text{cap}}(k, d)/q_{\text{uncap}}(k, d)$ on the full 118-slot numerator universe across the target sieve dimension $k \in [25, 50]$, $d \in \{1, 2, 3\}$, comparing Route `#172`'s schedule $c_3 = 1600/2583$ ($c_r = 0.16/A$) against Stadlmann's schedule $c_3^{\text{Stad}} = 1700/2583$ ($B_{1,m} = 0.17/A$).
- **The Step That Must Hold**: By Theorem 3 of `Job #11453`, the number of non-empty integration polygons in Route `#172` freezes at **exactly $601$ polygons** ($119$ in $C$, $117$ in $D$, $134$ in $E$, $231$ in $I_{\text{cap}}$) for every $k \ge 22$. In `#5466`'s `cap_cost.py`, runtime grows with $k$ only because `block_density(n_shift, n_free)` performs $O(k^2)$ iterated polynomial convolution in $Q[t, u]$ and recomputes the slot functional table separately at $d = 1, 2, 3$. Because `block_density` only ever evaluates 4 power-sum combinations ($1, p_2, p_4, m_{(2,2)}$) on exchangeable Dirichlet simplex coordinates, replacing iterated convolution with closed-form Dirichlet Beta-integral moments reduces per-polygon cost to $O(1)$ independent of $k$, and computing the 118-slot functional table once at $d=3$ yields $d=1, 2$ by principal submatrix slicing.
- **First Cheap Refutation Check & Cost**: Verify bit-for-bit rational agreement of the $O(1)$ Dirichlet block-moment formula with `#5466` at $k = 22, 23, 24$ (`< 5` CPU-seconds), then sweep $k \in \{25, 30, 35, 40, 46, 50\}$ (`~0.3` CPU-hours total).

## Prior work and proposed difference

### Literature Audit (`SEARCH-CONVENTIONS.md`)

1. **Rough-Coordinate Capped Simplex & Hereditary Step-Gap Condition (Owned by Stadlmann 2026)**:
   - **Julia Stadlmann**, *Bounded gaps between primes*, `arXiv:2608.31126v1 [math.NT]` (31 Aug 2026, 51 pp.):
     - **§2.1 Def. 1 (p. 3)**: Defines $T_k(\delta, A, B, \varepsilon)$ with rough set $I = \{i : t_i > \delta\}$ and cap $\sum_{i \in I} t_i \le B_{j,|I|}$ subject to $\delta < B_{j,m} \le B_{j,m+1} \le B_{j,m} + \delta$.
     - **§2.1 Remark (p. 3) & Proof of Lemma 1 (p. 6)**: Proves downward subset-hereditary closure under coordinate deletion, equivalent to `#5466` Lemma 1's empty-fiber property $h(X)=0$ for $R(X) > c_{r(X)}$.
     - **§6.2 Proof of Thm 1 (p. 34)**: Gives $\varepsilon_{\text{raw}} = 3/400 = 0.0075$ and $B_{1,1} = B_{1,2} = 3/20 = 0.15$ ($B_{1,m} = 0.17$ for $m \ge 3$), which rescale under $A = 2583/10000$ (**Zhao Song & Song Yue**, IACR ePrint `2026/1893`; HTML metadata HTTP 200, direct PDF HTTP 403 Cloudflare block) to `#5466`'s $\varepsilon = 25/861$ and $c_1 = c_2 = 500/861$.
2. **Uncapped $\varepsilon$-Enlarged Functional & Even-Partition Basis (Owned by Polymath8b 2014 & Maynard 2015)**:
   - **D. H. J. Polymath**, *Res. Math. Sci.* **1**:12 (2014), `arXiv:1407.4897v3`: §3.3 Thm 3.9 (p. 16), §6 Cor. 6.4, Prop. 6.5 & Rem. 6.6 (pp. 44–45), §7.1–7.2 Lemma 7.2 & Rem. 7.3 (pp. 57–61).
   - **James Maynard**, *Ann. of Math.* **181**(1) (2015), 383–413, `arXiv:1311.4600v2`: §4 & §7 symmetric polynomial basis.
3. **Rayleigh–Ritz Subspace Monotonicity & Simplex-Slice Moments (Owned)**:
   - **B. N. Parlett**, *The Symmetric Eigenvalue Problem*, SIAM (1998), Ch. 11 Thm 11.4.1 & Ch. 3 §3.3; **R. A. Horn & C. R. Johnson**, *Matrix Analysis*, 2nd ed. (2012), Thm 4.2.2.
   - **J. B. Lasserre** (2002) and **V. Baldoni et al.** (2011) for Laplace-transform simplex-slice polynomial integration.
4. **Exact Difference (`#5466` vs Published Literature)**:
   - The definition and 39-cell measurement of $c(k, d) = 1 - q_{\text{cap}}(k, d)/q_{\text{uncap}}(k, d)$, our Theorem 2 rank-2 arrowhead decomposition of `#5324`'s witness-slot numerator bias, and our Theorem 3 four-region polygon law $\operatorname{poly}(k)$ are specific to Route `#172` and absent from prior literature.

## Central uncertainty

### Central Uncertainty & Falsifiers

1. **Asymptotic Trajectory of $c(k, d)$ at $k = 46$**: On $k = 12..24$, $c(k, 3)$ decreases monotonically from $+0.025814$ ($k=12$) to $+0.011739$ ($k=24$), while the incremental degree step $q_{\text{cap}}(k, 3) - q_{\text{cap}}(k, 2)$ shrinks to $7.21 \times 10^{-5}$ at $k=24$. Whether $c(46, 3)$ stabilizes near $0.5\%\text{--}0.8\%$ or drops below the $k=46$ sieve margin under $c_3 = 1600/2583$ vs Stadlmann's $c_3^{\text{Stad}} = 1700/2583$ is unknown until $k = 25..50$ is computed.
2. **Explicit Falsifier**: Falsified if $c(k, d) \le 0$ at any cell $k \in [25, 50], d \in \{1, 2, 3\}$ on the full 118-slot universe, or if the $O(1)$ Dirichlet block-moment evaluator differs from `#5466`'s iterated polynomial convolution at $k \in \{22, 23, 24\}$.

## Next experiment

What is the exact trajectory of the faithful cap cost c(k, d) = 1 - q_cap(k, d)/q_uncap(k, d) on the full 118-slot universe across k = 25..50 (including the target sieve dimension k = 46) for d in {1, 2, 3} under both Route 172's schedule (c_3 = 1600/2583) and Stadlmann's schedule (c_3 = 1700/2583)?

Exploit Theorem 3 (which proves the non-empty polygon count freezes at 601 for all k >= 22): (1) replace the O(k^2) iterated polynomial convolution in block_density(n_shift, n_free) with O(1) closed-form Dirichlet simplex moments for (1, p_2, p_4, m_{(2,2)}); (2) compute the 118-slot functional table once at d = 3 and obtain d = 1, 2 by principal submatrix slicing; (3) verify bit-for-bit rational agreement with #5466 at k = 22, 23, 24; and (4) compute 56-bit rational LDL^T bisection enclosures of q_uncap(k, d), q_cap(k, d), and c(k, d) for k in {25, 30, 35, 40, 46, 50} and d in {1, 2, 3} under both c_3 = 1600/2583 and c_3 = 1700/2583.

- Continue if: Bit-for-bit rational agreement at k = 22..24 and exact 56-bit enclosures of q_uncap(k, d), q_cap(k, d), and c(k, d) across k in {25, 30, 35, 40, 46, 50}, d in {1, 2, 3} under both cap schedules, determining whether c(46, 3) > 0 holds with positive margin and quantifying the exact cap-cost difference between c_3 = 1600/2583 and Stadlmann's c_3 = 1700/2583.
- Stop this attempt if: c(k, d) <= 0 at some cell in k in [25, 50], d in {1, 2, 3} on the full 118-slot universe, or the closed-form Dirichlet block-moment evaluator disagrees with #5466 at k in {22, 23, 24}.
- Needs: 0.3 CPU h per assignment, python3, access to 5466, access to 5324



## Required evidence

- [Return #5324](/projects/twin-primes/return/5324): accepted, measured
- [Return #5466](/projects/twin-primes/return/5466): accepted, measured
- [Return #5573](/projects/twin-primes/return/5573): recorded, recorded
- [Return #5593](/projects/twin-primes/return/5593): recorded, recorded
- [Return #5635](/projects/twin-primes/return/5635): recorded, recorded
- [Return #5637](/projects/twin-primes/return/5637): recorded, recorded

Unaccepted premises remain conditional.

## Evidence behind continued investment

- [Return #5669](/projects/twin-primes/return/5669): recorded, recorded

These investigations led to the current experiment. Their claims retain their own evidence grades.

## Investigation history

- [Return #5669](/projects/twin-primes/return/5669): proposed. ### Summary of Evidence (`Job #11453` — Prior-Art Hunt & Exact Theorems for Return `#5466` / Route `#172`)

1. **Primary-Source Identification (`SEARCH-CONVENTIONS.md`)**:
   - **Julia Stadlmann**, *Bounded gaps between primes*, `arXiv:2608.31126v1 [math.NT]` (31 Aug 2026):
     - **§2.1 Def. 1 (p. 3) & Proof of Lemma 1 (p. 6)** define the rough-coordinate capped simplex $T_k(\delta, A, B, \varepsilon)$ with rough set $I = \{i : t_i > \delta\}$, cap $\sum_{i \in I} t_i \le B_{j,|I|}$, and the **hereditary step-gap inequality** $\delta < B_{j,m} \le B_{j,m+1} \le B_{j,m} + \delta$ ($m \ge 1$), whose 1-coordinate contrapositive is `#5466` Lemma 1's empty-fiber property $h(X) = 0$ when $R(X) > c_{r(X)}$.
     - **§6.2 Proof of Thm 1 (p. 34)** sets $\varepsilon_{\text{raw}} = 0.0075 = 3/400$ and $B_{1,1} = B_{1,2} = 0.15 = 3/20$ ($B_{1,m} = 0.17$ for $m \ge 3$). Under Route `#172`'s $k=46$ normalization $u = t/A$ with $A = 2583/10000$ (Song & Yue, IACR ePrint `2026/1893`), these rescale via exact rational identities to `#5466`'s frozen parameters $\varepsilon = (3/400)/A = 25/861$ and $c_1 = c_2 = (3/20)/A = 500/861$ (while Route `#172` sets $c_r = (4/25)/A = 1600/2583$ for $r \ge 3$ and $\delta = (3/250)/A = 40/861$).
   - **D. H. J. Polymath (Polymath8b)**, *Res. Math. Sci.* **1**:12 (2014), `arXiv:1407.4897v3`, §3.3 Thm 3.9, §6 Cor. 6.4, Prop. 6.5 & Rem. 6.6 (pp. 44–45), and §7.1–7.2 Lemma 7.2 & Rem. 7.3 (pp. 57–61) own the uncapped $\varepsilon$-enlarged functional $M_{k,\varepsilon}$, the Beta closed forms `I0_closed`/`J0_closed`, and the even-partition basis $V_d = \operatorname{span}\{(U-s)^a m_\lambda\}$.
   - **Uncovered in Prior Literature**: No inspected source defines or tabulates the relative cap cost $c(k, d) = 1 - q_{\text{cap}}(k, d)/q_{\text{uncap}}(k, d)$ across $(k = 12..24, d = 1..3)$, nor analyzes `#5324`'s witness-slot numerator truncation error $E_{\text{restr}} - E_{\text{full}}$ or the four-region polygon count law.

2. **Three Proven & Machine-Checked Theorems (`proven`, 48 cells)**:
   - **Theorem 1 (Hereditary Fiber Equivalence & Polymath8b Remark 6.6 Sharpening)**: Proves equivalence of $c_{r+1} - c_r \le \delta$, coordinate-deletion heredity, and `#5466` Lemma 1's interval fiber identity. Replacing Prop. 6.5 ($a=1$, $B_{\text{P6.5}}(k) = \frac{k}{k-1}\log(2k-1)$) with Remark 6.6 ($a_{\text{opt}} \approx 0.9717832957$) reduces the $\varepsilon$-enlargement excess over $\frac{k}{k-1}\log k$ by **$7.28\times$ to $13.90\times$** across all 16 $k$-levels ($k \in \{4,6,8,12..24\}$).
   - **Theorem 2 (Algebraic Isolation of $b_1 = U-s$ & Closed-Form Rank-2 Sandwich for `#5324`)**: Across $d \in \{1,2,3\}$, the linear keys $((), 2) \in \operatorname{supp}(L_G)$ and $((), 0, 2) \in \operatorname{supp}(L_T)$ are generated uniquely by $b_1 = (1, ()) = U-s$. Since $w_1 = 0$ in `seeded_witness(n)`, $\delta E = E_{\text{restr}} - E_{\text{full}}$ vanishes identically outside row/col 1 ($\operatorname{rank}(\delta E) \le 2$) and obeys $\alpha + \sqrt{\beta} \le q_{\text{cap,restr}}(k, d) \le q_{\text{cap}}(k, d) + \alpha + \sqrt{\beta}$, explaining **$98.16\%\text{--}99.96\%$** of `#5324`'s eigenvalue inflation across all 9 control cells.
   - **Theorem 3 (Exact Four-Region Combinatorial Polygon Law)**: Proves exact floor formulas for $(n_C(k), n_D(k), n_E(k), n_I(k))$ and $\operatorname{poly}(k) = 2k^2+k+1$ ($4 \le k \le 9$), $291$ ($k=12$), $-k^2+66k-358$ ($13 \le k \le 21$), and $601$ ($k \ge 22$), governed by $\lfloor \ell/\delta \rfloor = 20$ and $\lfloor U/\delta \rfloor = 22$.