A zero-surplus rational construction at N=144

Result: the specified finite feasibility problem is feasible. The vector

\[ 20(c_1,\ldots,c_{12})=(20,-20,-20,0,-20,-13,7,6,0,0,0,13) \]

has harmonic balance, coverage one through q=5, mass 119/20, and no positive surplus at any prime power through 144. Its complete saturated cost is exactly psi(144). The earlier endpoint has strictly positive surplus 4.414404304141713666..., so changing coefficients removes that finite gap.

This is an ordinary finite argument with exact rational executable checks and two independent logarithm enclosure implementations. The independent test implementation is by the same producer, not an independent reviewer. Review status is pending independent challenge; there is no Lean proof, novelty claim, uniform estimate, or claimed asymptotic family. The finite construction is an original output of this pass. An unrun literature search establishes nothing about priority.

Exact class and source boundary

Fix N=144. The class consists of all rational vectors supported on 1<=j<=12 with

\[ \sum_{j=1}^{12}\frac{c_j}{j}=0,\qquad W(q):=\sum_{j=1}^{12}c_j\lfloor q/j\rfloor=1\quad(1\le q\le5), \qquad\sum_{j=1}^{12}|c_j|\le23. \]

There are no additional sign conditions. Coverage already forces c_1=1. Deficits, including negative W, are allowed and their complete bill is paid. Rational refers to the coefficient domain, not a finite denominator grid.

This is a relaxation of the balanced-prefix construction at base fd04f1fac9d48a4d769048817e678acdea3b4bdb, scaling, section 3. Every old prefix with cutoff 6<=h<=12 belongs: its balance is zero, it covers q<h, and its mass is at most 2h-1<=23. The new vector is outside their convex hull: every such prefix has c_7<=0, whereas this vector has c_7=7/20>0. In detail, the old c_7 is zero for h=6, -14/15 for h=7, and -1 for h>=8. All seven inclusions and signs are tested.

The reviewed predecessor is af13799d3034b4d6a07e76c0d6a409dc7421f056. Its old zero repair-difference field is not used as a surplus measurement here. The comparison reconstructs the unchanged endpoint h=12 from its rational coefficients and prices the actual positive part. No other worker's source or output is modified.

Finite proof

Write b_j=20c_j. Harmonic balance follows from the integer identity

\[ 840\sum_j b_j/j =16800-8400-5600-3360-1820+840+630+910=0. \]

The absolute numerators sum to 119, proving the mass bound. For 1<=q<=5, the weights reduce to q-floor(q/2)-floor(q/3)-floor(q/5)=1.

For 2<=d<=144, set W_d=W(floor(144/d)). The nested-floor identity gives W_d=sum_j c_j floor(144/(dj)). If d>=25, its quotient lies between 1 and 5, so coverage gives W_d=1 without knowing whether d is a prime power. The prime powers at most 24 are exactly the following thirteen; primes count as powers of exponent one. Direct integer substitution gives every remaining constraint:

Thus every prime power through 144 has W_d<=1. There are 47 such powers: 13 in this table and 34 covered automatically. Eight have strict deficits; 39 have equality; zero have positive surplus. The negative row at d=23 is essential to retain in the repair cost, not a rejected input.

Full cost and what was improved

Let Lambda(p^k)=log p, Lambda(d)=0 otherwise, and t_+=max(t,0). Define

\[ B=\sum_j c_j\log(\lfloor144/j\rfloor!) =\sum_{d\le144}\Lambda(d)W_d,\quad P_\Lambda=\sum_{d\le144}\Lambda(d)(1-W_d)+,\quad C=B+P\Lambda. \]

The factorial equality is finite rearrangement: the coefficient of log p in log(M!) is sum_{k>=1}floor(M/p^k). No limiting interchange is used. Applying w+(1-w)_+=1+(w-1)_+ row by row proves

\[ C=\psi(144)+S,\qquad S=\sum_{d\le144}\Lambda(d)(W_d-1)_+. \]

The table proves S=0 exactly. Summing its eight deficit rows gives

\[ P_\Lambda=\frac{27}{10}\log2+\frac{21}{20}\log3+\log7 +\frac3{10}\log19+\frac{33}{20}\log23. \]

Consequently B=psi(144)-P_Lambda and C=psi(144) exactly. Every nonzero deficit lies at d<=24; using the exact Mangoldt weight on this range is the saturated-cap assumption. Computing that finite bill is not a uniform analytic estimate. Since C>=psi(144) for all real vectors under this same exact cap, the witness also attains the elementary cost lower bound at this cutoff, without any claim that its coefficient mass is minimal.

The old endpoint is (1,-1,-1,0,-1,1,-1,0,0,1,-1,2/385). Its exact surplus is

\[ S_{old}=\frac2{55}\log2+\frac8{385}\log3 +\frac{387}{385}\log7+\frac{387}{385}\log11>0. \]

This strict sign needs no floating approximation. The complete cost decreases by exactly S_old. The following rounded displays are pinned by rational interval endpoints in the saved output (computations/check-001/results.json):

The repair increases; the factorial term decreases enough to give the full improvement. S is already included in B+P_Lambda and is not added twice.

Controls, reproducibility and remaining uncertainty

The optional discovery pass is one linear feasibility program with an auxiliary coefficient-mass objective. Its floating result is reconstructed over the rationals and every constraint is checked exactly. Neither its status text nor its apparent optimality supplies the proof above. The default script verifies the displayed witness without invoking an optimizer.

The independent test code factors integers and factorials using a different implementation, counts the multiples defining W without calling the producer's floor routine, and verifies the complete cost. The producer separately checks 145 floor-versus-divisor-prefix identities. It evaluates both candidate costs with python-flint and mpmath.iv at 35 and 70 requested decimal digits: 40 expression evaluations, with all four enclosures overlapping for each of 10 expressions, and smaller nonzero widths after precision refinement.

Controls retain the complete hypotheses:

• The old endpoint belongs to the class but violates five surplus constraints, including the higher power d=8.
• Adding 3/5 to c_6 and subtracting 6/5 from c_12 preserves class membership but creates four positive-surplus rows, which the check rejects.
• The zero vector passes all no-surplus constraints but fails all five coverage equations. It cannot serve as a fake solution to this class.
• Separate tests reject domain, support, balance, coverage and mass lesions.

These finite coefficient controls address the actual claim. There is no structural RH explanation or proposed zero-location implication to test against the Davenport-Heilbronn rival.

The complete intended target is still a uniform estimate for the whole paid bound, such as C_N<=N+O(sqrt(N) log(N)^2), with all costs included. The original finite feasibility uncertainty has been resolved affirmatively. The first remaining uncertain step is an explicit scale-dependent admissible construction with a proved complete-cost estimate. Even zero surplus at other cutoffs would only identify C_N=psi(N) there, not supply that estimate. There is no new bound for the complete two-sided prime-pair energy E(N). This witness rules out an infeasibility claim for this declared class at 144; it does not rule out obstructions in other classes or at other cutoffs.

Reproduce from the repository root with its installed environment:

.venv/bin/python hunts/paid_surplus_obstruction/construction.py --output /tmp/paid-surplus-check.json
.venv/bin/python -m pytest -q -n 0 tests/test_paid_surplus_obstruction.py

The run record gives validation and failures. The versioned manifest (computations/check-001/manifest.json) records the optional discovery run, source hashes, software, timing, bounded execution and exact output hash. No larger cutoff search was performed after the witness passed.