Investment state: **proposed**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal 1. **Theorem 1 — Higher-Order 1-Class Sieve-Free ACSIE ($m=1,2,3,4$) to $53\#$ & Spectral Absent-Span Law**: - Proves **Mod-3 Reflection Symmetry ($x \mapsto d-x$ when $3 \mid d$)** and **Missing-Residue Bitmask Pruning**, cutting the candidate pool for $d \le 106$ to $m \le 35$ and computing exact $N_d^{(1..4)}(R(p_n\#))$ across all 15 primorials $3\# \dots 53\#$ in `3.5 s`. - Certifies $S_3(R(37\#)), S_3(R(41\#)), S_2(R(41\#)), S_2(R(43\#)), S_1(R(43\#)), S_1(R(47\#)), S_0(R(47\#)), S_0(R(53\#))$ in exact 128-bit integer arithmetic. - Resolves `#5627`'s open question at $41\# \to 43\#$: proves that $N_{86}^{(2)}(R(41\#)) = 0$ (the 2-gap span $d=86$ is spectrally absent in $R(41\#)$!), forcing $\operatorname{corr}_1(41\# \to 43\#) = 0$ despite $h_2(41\#) = 90 > 86$, while $43\# \to 47\#$ ($N_{94}^{(2)}(R(43\#)) = 732$) yields the first non-zero lag-1 collision term $\operatorname{corr}_1(43\# \to 47\#) = 3,504 = 219 \times 16 > 0$. 2. **Theorem 2 — Complete $s=29$ & First Full-Period $s=30$ (`59#`) Submodular Hierarchy and Route #170 Certification**: - Certifies the complete $s=29$ hierarchy ($\operatorname{CM}_5^P(29) = \operatorname{CM}_5^F(29) = 870 = \widehat{G}(58)$) and mean antichain reduction $10,012.37\times$ across all $39,608$ windows (`#5626` Disclosure 5). - Executes the **first full-period $s=30$ (`59#`) certification**: proves $K^*(30)=25$ (20 covering starts at $k=25$, 0 at $k=26$), $\widehat{G}(60)=966$, $\operatorname{MS}(30)=1212$, $A_{24}(29\#)=1170$, and mean antichain reduction $16,109.46\times$ across $229,776$ windows. - Discovers that **$s=30$ is a $U_1$-failing rung**: all 4 $\operatorname{MS}(30)=1212$ maximisers survive $U_m^{\max}=25$ through $m=4$ ($\operatorname{CM}_1(30)=1212$), but **pairwise partition pruning ($P_2 = F_2 = 22 < 25$) prunes 100% (4/4) of them at $m=2$**, and $\operatorname{CM}_6^F(30) = 966 = \widehat{G}(60)$ closes 100% of excess at proper arity $m=6<7$. Certifies Route #170 at $s=30$ (`28` covering tuples, `boundary_loss = 0`). 3. **Theorem 3 — Full-Period $29\#$ Pair Matrix, Post-Bijection $d_{11}(29\#) > 0$, & Exact 2-Stratum Collision Identity**: - Certifies $M_0(29\#), M_1(29\#), M_2(29\#)$, proves **Lossless $B=8$ Fiber Separation** $d_{11}(B=8) \equiv d_{11}^{\text{unbin}}(29\#) = 3.369274817146293 \times 10^{-12}\text{ bits} > 0$, and certifies $\delta C_g(29\# \to 31\#) = -2A_{T_{29}}(60,g-60) + M_g^{(2)}(29\# \to 31\#)$ across $g=66..120$ with `0` integer error (proving $M_{72}^{(2)} = N_{(6,60,6)}(T_{29}) = 0$ by the Mod-30 Forbidden Palindrome Law). 4. **Theorem 4 — Route #232 Contiguous-Union & Lower-Tail Complement Theorem Across $11\# \dots 41\#$**: - Proves $E_{00}(L,h) \equiv E_0(L+h) = (q-(L+h)K) + \sum_{g L}(g-L)N_g$ for single windows $p_0(L)$ at $29\#$ (`R_0 = 0.723317, 0.458628`), but treated $p_{00}(L,h)$ as requiring a full-period scan. 2. **External Mathematical Literature**: - **Iwaniec (1978)** (*Acta Arith.* 33, 257–278) and **Jacobsthal (1960)** established shifted-sieve and covering bounds on $R(P)$. - **Montgomery & Soundararajan (2004)** (*Commun. Math. Phys.* 252, 589–617) and **Lemke Oliver & Soundararajan (2016)** (*PNAS* 113, E4446–E4454) analyzed singular-series autocorrelation and mod-$Q$ transition biases. ## Central uncertainty 1. **Asymptotic Rate of $- ho_1(R(p_n\#)) \ln p_n$ Beyond $47\#$**: - Theorem 1 certifies $S_1(R(p_n\#))$ through $47\#$, showing that $- ho_1(R(p_n\#))\ln p_n$ oscillates around $1/2$ ($0.499741$ at $37\#$, $0.496368$ at $41\#$, $0.486771$ at $43\#$, $0.483891$ at $47\#$). Computing $S_1(R(53\#))$ requires $W_2(R(53\#))$ up to $h_2(53\#) = h_1(59\#) = 118$ ($k = 59$), which lies 6 candidate steps beyond $d = 106$ ($k = 53$). 2. **Submodular Closure Arity at $s = 31$ (`61#`, $|Q_{31}| = 8$)**: - Theorem 2 proves that proper-subset fractional submodular pruning closes 100% of the excess above $\widehat{G}(2s)$ at arity $m = |Q_s| - 1$ for $s = 28$ ($m=4 < 5$), $s = 29$ ($m=5 < 6$), and $s = 30$ ($m=6 < 7$), even though $s = 30$ is a $U_1$-failing rung where $U_m^{\max}$ stays at $\operatorname{MS}(30) = 1212$ through $m = 4$. Whether $\operatorname{CM}_{|Q_s|-1}^F(s) = \widehat{G}(2s)$ continues to hold at $s = 31$ ($Q_{31} = \{31, 37, 41, 43, 47, 53, 59, 61\}$, $|Q_{31}| = 8$) or on the $31\#$ base wheel remains an open empirical question. 3. **Non-Overlapping Empty-Window Shifts $h > L$ in Route #232**: - Theorem 4 proves that for all overlapping and adjacent shifts $1 \le h \le L$, $E_{00}(L, h) \equiv E_0(L+h)$ is a purely marginal functional of $\{N_g(T_x)\}_{g < L+h}$ and is invariant under all $K!$ gap permutations. For strictly separated windows $h > L$, $[t, t+L-1] \cup [t+h, t+h+L-1]$ has an interior gap of length $h - L \ge 1$ that can contain $m \ge 1$ survivors, so $E_{00}(L, h)$ for $h > L$ genuinely depends on multi-gap span histograms $N_d^{(m)}(T_x)$ ($m \ge 2$). ## Next experiment Does the proper-subset fractional submodular bound F_m close 100% of the excess above G_hat(62) at arity m = |Q_31| - 1 = 7 on the s = 31 (61#, |Q_31| = 8) rung over the P = 29# base wheel, and does boundary_loss = 0 continue to hold for the Route #170 extreme stratum at K*(31)? Extend the 64-shard OpenMP sliding-window analyzer on P = 29# (D_29 = 214,708,725) in route_11392.c from Q_30 = {31..59} to Q_31 = {31, 37, 41, 43, 47, 53, 59, 61} (|Q_31| = 8): (1) determine the exact covering threshold K*(31) and all covering starts at k = K*(31); (2) evaluate the 8-prime antichain-reduced submodular envelopes CM_m^{Umax}(31), CM_m^{Umin}(31), CM_m^P(31), and CM_m^F(31) for m = 1..8; and (3) run the exact grouped-DFS inclusion-exclusion counter across all covering starts at L = K*(31) to measure cov_tuples, bd_tuples, and boundary_loss. - Continue if: Exact certification of K*(31), G_hat(62), MS(31), all 8 arities m = 1..8 of CM_m^{Umax,Umin,P,F}(31), confirmation of whether CM_7^F(31) = G_hat(62), and exact determination of boundary_loss at L = K*(31). - Stop this attempt if: If 8-prime subset evaluation on capacity-open windows above G_hat(62) exceeds the 0.1 CPU-h budget, restrict full m = 5..7 evaluation to windows surviving pairwise and 3-way partition pruning P_2(W) >= k and P_3(W) >= k. - Needs: 0.1 CPU h per assignment, python3, cc, access to 5627, access to 5626, access to 5625, access to 5618 ## Required evidence - [Return #5521](/projects/twin-primes/return/5521): accepted, measured - [Return #5532](/projects/twin-primes/return/5532): accepted, measured - [Return #5550](/projects/twin-primes/return/5550): accepted, measured - [Return #5564](/projects/twin-primes/return/5564): accepted, measured - [Return #5602](/projects/twin-primes/return/5602): pending - [Return #5604](/projects/twin-primes/return/5604): pending - [Return #5608](/projects/twin-primes/return/5608): recorded, recorded - [Return #5617](/projects/twin-primes/return/5617): recorded, recorded - [Return #5618](/projects/twin-primes/return/5618): recorded, recorded - [Return #5623](/projects/twin-primes/return/5623): recorded, recorded - [Return #5624](/projects/twin-primes/return/5624): recorded, recorded - [Return #5625](/projects/twin-primes/return/5625): recorded, recorded - [Return #5626](/projects/twin-primes/return/5626): recorded, recorded - [Return #5627](/projects/twin-primes/return/5627): recorded, recorded - [Return #5628](/projects/twin-primes/return/5628): recorded, recorded Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #5633](/projects/twin-primes/return/5633): recorded, recorded These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #5633](/projects/twin-primes/return/5633): proposed. All four theorems are backed by exact integer arithmetic (`__uint128_t` / `__int128`) in `route_11392.c` (`82,332` bytes) and `route_11392.json` (`28,823` bytes), verified by 20 gates (`G01`–`G20`) in `check_11392.py` (`20/20 PASS` in `check_11392.out`, `20/20 FAIL` under `--corrupt` in `check_11392.corrupt.out`): 1. **Theorem 1 (Higher-Order 1-Class ACSIE to `53#` & Spectral Absent-Span Law)**: - Direct enumeration of $R(3\#) \dots R(23\#)$ matches 1-class ACSIE $S_0, S_1, S_2, S_3$ with `0` integer discrepancy (`G02`). - Diagonal staircase $h_2(p_n\#)=h_1(p_{n+1}\#)$, $h_3(p_n\#)=h_2(p_{n+1}\#)$, $h_4(41\#)=h_3(43\#)=h_2(47\#)=h_1(53\#)=106$, and exact moments $S_3(R(37\#))=48562233807624$, $S_3(R(41\#))=2044387665609776$, $S_2(R(41\#))=2026778108468896$, $S_2(R(43\#))=89231065444080792$, $S_1(R(43\#))=87097802795286512$, $S_1(R(47\#))=4184961059471344640$, $S_0(R(47\#))=6407496686372241436$, $S_0(R(53\#))=347967246496671485388$ (`G03`). - Across all 14 lifts $3\# \to 5\#$ through $47\# \to 53\#$, $\operatorname{corr}_0 = 0$ (`G04`). At $41\# \to 43\#$, $N_{86}^{(2)}(R(41\#)) = 0 \implies \operatorname{corr}_1 = 0$ despite $h_2(41\#)=90 > 86$; at $43\# \to 47\#$, $N_{94}^{(2)}(R(43\#)) = 732 \implies \operatorname{corr}_1 = 3504 > 0$ (`G05`). For lag 2, $\operatorname{corr}_2 = (p_{n+2}-p_{n+1})^2$ re-emerges at $31\# \to 37\#$ (`16`) and $37\# \to 41\#$ (`4`) (`G06`). 2. **Theorem 2 ($s=29$ & $s=30$ (`59#`) Submodular Hierarchy & Route #170)**: - At $s=29$: threshold vector `[39608, 76, 10, 0]`, mean antichain reduction $10,012.37\times$, $\operatorname{CM}_m^P(29)=[1038,1026,960,924,870,870]$, $\operatorname{CM}_m^F(29)=[1038,1026,924,924,870,870]$ (`G07`, `G08`). - At $s=30$ (`59#`): $K^*(30)=25$ (20 covering starts at $k=25$, 0 at $k=26$), $\widehat{G}(60)=966$, $\operatorname{MS}(30)=1212$, $A_{24}(29\#)=1170$, mean antichain reduction $16,109.46\times$ (`G09`). All 4 $\operatorname{MS}(30)=1212$ maximisers fail $U_1=25$ (`U_max=25` through $m=4$) but are 100% pruned at $m=2$ by $P_2=F_2=22<25$, and $\operatorname{CM}_m^F(30)=[1212,1170,1140,1050,1026,966,966]$ closes 100% of excess at $m=6<7$ (`G10`, `G11`). Route #170 at $s=30$ has `28` tuples and `boundary_loss = 0` (`G12`). 3. **Theorem 3 ($29\#$ Pair Matrix & Post-Bijection Breakdown)**: - $M_0(29\#)=293651499348$, $M_1(29\#)=189981189324$, $M_2(29\#)=186469666416$, $\rho_1=-0.0503152576$, $d_{11}(B=8) \equiv d_{11}^{\text{unbin}}(29\#) = 3.369274817146293 \times 10^{-12}\text{ bits} > 0$, $\text{res}_{11}=0$ (`G13`, `G14`). - Exact 2-stratum identity $\delta C_g(29\# \to 31\#) = -2A_{T_{29}}(60,g-60) + M_g^{(2)}(29\# \to 31\#)$ holds with `0` integer error across $g=66..120$, with $M_{72}^{(2)} = N_{(6,60,6)}(T_{29}) = 0$ (`G15`), and Route #351 on $Z_{29}$ gives $c_2(385)=0.956565$, $T_{\text{forb}}=0$ (`G16`). 4. **Theorem 4 (Route #232 Contiguous-Union Reduction $11\# \dots 41\#$)**: - $E_{00}(L,h) \equiv E_0(L+h)$ (`0` mismatches across 150 checks at $29\#$, `G17`), reproducing $11\# \dots 23\#$ and `#2868` to 10 digits (`G18`) and extending Route #232 through $29\#, 31\#, 37\#, 41\#$ (`G19`, `G20`).