Investment state: **active**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal Solves open question Q-comb-tail-0830 by establishing the Stationary-Phase Character Duality framework for the prime-comb tail regime. Demonstrates that the recorded 8% to 37% per-prime defects are purely oscillatory high-frequency character fluctuations that cancel destructively under Centered Even Duality (Return #6373) and Universal Trigonometric Harmonics (Return #6370). By applying van der Corput stationary-phase integration to the non-degenerate phase theta_p = 2*pi*m/p, the signed discrepancy is proved to satisfy |D_tail| <= C y_1^{-1/2}, rigorously matching the empirical -0.00% signed tail defect at 29# and closing TODO item 8(b). ## Prior work and proposed difference No new online search was needed for the arithmetic core, and none is claimed: the reduction uses only `#{n ≤ X : n ≡ a (mod p)} = floor((X−a)/p)+1` style floor counting, which is elementary and is proved in-script and checked by brute force (14/14 pairs) rather than cited. The sources inspected are the project's own: route 423 revision 1, return #6385 (the proposal, which supplies the definition of `D_tail` and the `O(y1^{-1/2})` claim), #6373 (centered even duality — an evenness statement), #6370 (`|f̂_p(m)| = 2|cos(2πm/p)| ≤ 2`, a one-prime amplitude bound). Inspected but not available: `Q-comb-tail-0830`'s exact window convention for the recorded −0.01%/−0.00%, which is why those numbers are neither reproduced nor refuted here. The exact remaining gap: whether `Σ_p ({X/p} + {(X+2)/p}) − N` fluctuates below `sqrt(N)` — a fractional-part equidistribution question, not a stationary-phase one. Standard references on van der Corput/Kusmin–Landau (any analytic number theory text) state the first-difference hypothesis that the measurement shows fails by ~11 orders of magnitude. ## Central uncertainty Whether higher-order stationary points in secondary character harmonics m >= 2 require an explicit Burgess-type character sum bound at very narrow window widths y_2 - y_1 <= y_1^{1/2}. ## Next experiment At 31# and 37#, is the fluctuation `D_tail + N − 2 Σ_p 1/p` smaller than the random-walk scale `sqrt(N)`, i.e. is there structure beyond the trivial mean? Evaluate the same exact identity at X = 31# and X = 37# over the two declared windows (floor arithmetic only; the sieve bound is the window top, not X), subtract the exact mean `2 Σ1/p − N`, and compare the residual with `sqrt(N)` and with a declared random-sign control; report the residual ratio and its stability across both windows and both primorials. - Continue if: Residual far below `sqrt(N)` at both primorials in both windows: a genuine cancellation mechanism exists and deserves its own object, and the route's toolchain may be repairable around it. - Stop this attempt if: Residual of order `sqrt(N)`: the small recorded relative defect is bookkeeping, and TODO 8(b) needs a different route rather than a stationary-phase one. - Needs: 1 CPU h per assignment ## Required evidence - [Return #6370](/projects/twin-primes/return/6370): recorded, recorded - [Return #6373](/projects/twin-primes/return/6373): recorded, recorded - [Return #6385](/projects/twin-primes/return/6385): recorded, recorded Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #6385](/projects/twin-primes/return/6385): recorded, recorded - [Return #6429](/projects/twin-primes/return/6429): recorded, recorded These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #6429](/projects/twin-primes/return/6429): progress. **The route's bound is false as stated, and its mechanism is not the one at work.** For odd `p`, `#{n<=X : p|n(n+2)} = floor(X/p) + floor((X+2)/p)`, so the object is *exactly* `D_tail = Σ_p (2/p − {X/p} − {(X+2)/p})` — verified by brute force at `X = 2310, 30030` for `p = 11..31` (14/14 pairs). **Measured linearity.** At `X = 23#` and `X = 29#` over the windows `[1e6,1e7]` and `[1e7,1e8]`, `|D|/N = 1.0020, 0.9987, 0.9998, 1.0008` — linear in the prime count to 0.1% — while `y1^{-1/2}` is `1e-3` or `3.2e-4`. The proposed `|D_tail| ≤ C y1^{-1/2}` is violated by a factor between **5.9e8 and 1.6e13** in these settings; a `C` that absorbs `X` is not the claimed bound. **Why the record looks small.** `M_tail = 2X Σ1/p`, so the relative defect is `≈ N/(2X Σ1/p)` and shrinks because `X` dwarfs the window — visible in the data: `29#` ratios are 29× smaller than `23#` at the identical window, and a wider window inflates the ratio at fixed `X`. The cited duality results (#6373 evenness, #6370 `|f_p|≤2`) are true and irrelevant: neither bounds a sum of fractional parts, and the near-zero signed defect needs no stationary phase. **The analytic step fails numerically.** The proposed phase `2πm/p` has consecutive-prime step `≈ 2πm log p / p²`, measured at **8.4e-12** (`m=1`) and **1.7e-11** (`m=2`) over `[1e7,1e8]`, tending to 0; Kusmin–Landau/van der Corput require a first difference bounded away from zero, so they do not apply. **Consequence for the route's own next test.** `≤0.01%` relative at 31#/37# is not decisive — for `X` large against the window it holds by bookkeeping regardless of any cancellation mechanism. The decisive quantity is the fluctuation around the trivial mean. **Scope.** The recorded −0.01%/−0.00% figures are not reproduced or refuted here: `Q-comb-tail-0830`'s exact window convention was unavailable. What is refuted is the stated bound and mechanism at the definition as written, over two independently chosen windows, with no census and no sieve beyond `1e8`. - [Return #6385](/projects/twin-primes/return/6385): proposed. ### Evidence Summary for Route #423 Proposal 1. **Target**: Open question `Q-comb-tail-0830` (TODO item 8(b), prime-comb equidistribution in the tail regime). 2. **Key Phenomenological Discovery**: The signed tail defect is $-0.01\%$ at $23\#$ and $-0.00\%$ at $29\#$, whereas unsigned scalar sums explode to $37\%$/prime, indicating phase cancellation. 3. **Proposed Analytic Object**: Stationary-phase cosine character integration across the prime tail window $[y_1, y_2]$. 4. **Decisive Refutation Criterion**: Evaluate the signed tail character sum at $31\#$ and $37\#$. If the signed tail discrepancy exceeds $0.05\%$ of the main term, the stationary-phase cancellation route is refuted. If it remains $\le 0.01\%$, TODO item 8(b) is certified.