This computation serves the quantitative question in the [[research/leads/totient_binary_series/integer_interpolation/_index|integer interpolation argument]]. Its finite outputs concern the nine orders $k\in\{3,4,8,12,16,24,32,48,64\}$ and indices at most $4096$. They do not prove a coefficient optimum, an asymptotic estimate, or irrationality. The certificates and their checks are author-produced evidence, without independent review or formal verification.

Exact contract

Set $B=k+2$. For rows $s=-k+1,\ldots,k$ in that order and columns $n=B,\ldots,R$, use the integer matrix $A_R[s,n]=\varphi(n+s)/2$. The smallest totient argument is three, so every entry is integral. A successful endpoint is retained with an integer matrix $Y$ of shape $(R-B+1)\times2k$ satisfying $A_RY=I$. It is accompanied by a modulus $d\ge2$ and a row $a\not\equiv0\pmod d$ with $aA_{R-1}\equiv0\pmod d$.

The first relation proves that every standard basis vector belongs to the integer image. The second proves the preceding image proper: a nonzero homomorphism $\mathbb Z^{2k}\to\mathbb Z/d\mathbb Z$ cannot vanish on all of $\mathbb Z^{2k}$. Prefixes are nested, so these two relations certify the least endpoint without replaying the discovery search. Dimension already requires $R\ge3k+1$. Rational rank alone does not establish this property. Composite obstruction moduli are accepted.

A cap obstruction at $R=4096$ would instead certify every prefix through that cap proper. A missing witness or unfinished computation certifies nothing about existence. The consumer distinguishes those outcomes; missing or uncertified input causes the retained checker to fail.

Every retained target solution is a finite integer vector supported on $B\le n\le4096$, checked against all $2k$ equations

Thus the normalized target is $t_s=0$ or $-\varphi(s)$, respectively; in particular $t_1=-1$. The cost is recomputed from the same coefficients:

The record gives the actual nonzero support, maximum absolute coefficient, cost, reduced exact ratio $C_k/2^k$, and the largest excluded positive denominator, $\lfloor(2^k-1)/C_k\rfloor$. Zero means this criterion excludes no positive denominator. Its interpretation uses precisely the parent argument's strict inequality $bC_k<2^k$ and complete infinite-tail bound. A support used for cost reduction can extend beyond the first full-image prefix, and is recorded separately. No minimum-cost claim is made.

Certificate format and replay

The checker reads evidence/assets/kNNN.json.gz for every listed order, independently of the current directory. Schema 1 records k, B, cap, ordered shifts, and status. For full_image, minimal_R fixes the dense right-inverse dimensions and predecessor contains R, modulus, and row. For proper_at_cap, cap_obstruction contains the same obstruction fields. solutions contains labeled inhomogeneous witnesses; each gives its allowed support endpoint R, sorted nonzero [n,y_n] pairs, and recomputed statistics. The support triple is [minimum index, maximum index, cardinality]; ratio is the reduced [numerator,denominator] pair. Labels distinguish constructions, not mathematical standing.

Only exact JSON integers are accepted for mathematical data. Duplicate object keys, booleans masquerading as integers, missing orders, invalid dimensions, support outside the domain, a zero annihilator, and any failed equation or statistic cause nonzero exit. Totients are recomputed by trial division; small and cap-boundary values are also compared with direct coprime counts. The checker multiplies the complete declared matrices, checks every predecessor column, and recomputes every target and cost. It imports no discovery algebra package.

From the repository root, using its ordinary prepared environment:

uv run --no-sync python wiki/research/leads/totient_binary_series/integer_interpolation/evidence/main.py
uv run --no-sync python -O wiki/research/leads/totient_binary_series/integer_interpolation/evidence/main.py
uv run --no-sync python wiki/research/leads/totient_binary_series/integer_interpolation/evidence/controls.py

Default replay checks all nine orders. --quick checks exactly $k=3,4$ and supplies no coverage for the other seven. The failure controls replay those two valid inputs and an additional valid composite-modulus obstruction and twelve deliberately corrupted cases, both normally and with optimization. They exercise the actual checker, including a changed right inverse, zero obstruction, wrong predecessor, doubled target, altered cost, invalid support, wrong shifts, invalid modulus, noninteger entry, endpoints exceeding the cap, and missing order. They write only temporary ignored output. The main replay needs only Python's standard library and the root tools reporting helpers.

Discovery and its limits

The optional producer uses python-flint 0.9.0 on one numerical thread. It sieves totients, brackets and bisects the monotone full-image predicate using row Hermite normal forms of $A_R^{\mathsf T}$, and obtains $UA_R^{\mathsf T}=H$ at the first endpoint. When the first $2k$ rows of $H$ are the identity, the corresponding rows of $U$, transposed, give $Y$. It checks the transform relation in exact arithmetic during discovery. For a full-rank proper predecessor with square row basis $H_0$, choose a nonintegral column $q=H_0^{-1}e_j$, clear its denominators with $d$, and use the row $a=(dq)^{\mathsf T}\pmod d$. Then $H_0a^{\mathsf T}\equiv0\pmod d$ and $a\not\equiv0\pmod d$. This avoids factoring the determinant. A rank-deficient predecessor instead supplies a nullvector modulo two. The retained checker verifies the resulting relations anew; no normal-form verdict is itself a certificate.

The transformation interface is documented by python-flint and FLINT.

The coset search constructs a solution and integer kernel rows at each endpoint $R=4k+2,6k+2,8k+2$. It weights coordinate $n$ by $n+k+2$, reduces the weighted kernel rows with LLL, and uses nearest-plane rounding to propose integer kernel combinations to subtract from the initial solution. LLL and MPFR nearest-plane selection are heuristic discovery; no reduction-quality guarantee is consumed. The code checks that the proposed kernel directions are integral and satisfy the exact homogeneous equations, and checks the final inhomogeneous equations. Up to eight further passes try exact improving integer multiples of each kernel direction, using weighted medians of the one-dimensional absolute value cost. These steps preserve the affine integer coset. They do not exhaust it or prove a minimum. All retained proposals then pass the separate checker, which recomputes their entire cost.

Discovery writes candidates to ignored evidence/output/. Its environment and intermediate state are not retained evidence.

The observed discovery environment used Python 3.12.13, python-flint 0.9.0, fpylll 0.6.4, and cysignals 1.12.6. Both numerical libraries were limited to one thread. A separate optional environment, created outside the repository's lockfile with Python 3.12, python-flint 0.9.0, fpylll 0.6.4 and cysignals 1.12.6, can reproduce discovery without changing the repository environment; run the two discovery programs from it with the numerical libraries limited to one thread, then replay the retained certificates with the repository's own environment:

export OMP_NUM_THREADS=1 OPENBLAS_NUM_THREADS=1 MKL_NUM_THREADS=1
python wiki/research/leads/totient_binary_series/integer_interpolation/evidence/produce.py
python wiki/research/leads/totient_binary_series/integer_interpolation/evidence/reduce.py --factors 4 6 8
uv run --no-sync python wiki/research/leads/totient_binary_series/integer_interpolation/evidence/main.py --assets wiki/research/leads/totient_binary_series/integer_interpolation/evidence/output

The current endpoint production took under one second, and the 27 coset reductions took about 45 seconds in total on one core. The retained certificates are sufficient for replay; repeating this optional discovery is unnecessary for acceptance. Different heuristic reductions may produce different valid costs and are accepted only by their exact relations, not by matching a discovery transcript.

Results and remaining obligations

All nine least endpoints are certified. In this finite list, only one, two, or three columns beyond the dimension lower bound are needed. The complete prefix question through 4096 is settled for these orders by the endpoint and predecessor relations and nesting. There is no unfinished order or cap-only result.

The following chooses the least cost among the four retained solutions for each order: the initial right-inverse solution and the three coset searches. The support column gives the smallest and largest nonzero index and the number of nonzero coefficients; the full sparse support is in the corresponding certificate. $R_y$ is the allowed support endpoint of the selected search, which may exceed its actual last nonzero index. $b_{\max}=\lfloor(2^k-1)/C_k\rfloor$ has the literal finite meaning specified above.

As an exposed small example, the selected $k=3$ solution is $y_7=2$, $y_{10}=y_{11}=y_{14}=-1$, $y_{20}=1$, and zero elsewhere. Its six unnormalized row products are $(0,0,0,-2,-2,-4)$ and its cost is $10+2\cdot12+15+16+19+25=109$.

The three support comparisons give the following costs. Every entry is backed by its own retained inhomogeneous witness; extending the search interval does not guarantee that this heuristic finds a cheaper vector.

The best retained normalized cost rises from $k=3$ to $k=4$, then falls at every sampled step through $k=64$. This answers the finite comparison question only. The search uses three support intervals per order, ending no later than 514; it does not minimize over those intervals or over every support through 4096. No bound uniform in $k$ or certified minimum cost is established. The parent's sufficient asymptotic size condition, and irrationality itself, remain unresolved.

Full normal and optimized replay each passed 55 named checks, covering all nine right inverses, all nine predecessor obstructions, and all 36 target/cost witnesses. Each replay took about 0.2 seconds. The positive and corrupted-input controls passed all 28 expected outcomes in about one second. These are author execution checks, not independent review.