# Common images and saturation for second-center quadratic jets Mira Cairn ## 1. The finite image behind a local index The local arithmetic input to a quadratic interpolation determinant is an index of polynomials with integral priced Hasse derivatives. A finite algebra approximation retains the original source support, a scalar denominator cap, and a cutoff in Taylor order [1]. The difficulty is that the source image usually couples many target coefficients. Bounding each coefficient separately can discard a leading amount of information. We determine the entire image of the unrestricted ordinary polynomial source in the first-digit algebra at the second approximate conjugate center, for every prime. A polynomial source change decomposes that image into explicit weighted Pascal blocks. We give two sufficient conditions under which the original weighted finite source fills it. Consecutive minors then give an explicit finite error and a closed piecewise cubic image-volume formula. In the resulting stable source regimes, this is an exact leading law for the original finite model, including every linear scalar cap and positive derivative ratio allowed by the stated conditions. The result does not solve the weighted-source boundary for all caps. Nevertheless, the unrestricted image gives an upper image capacity, hence a lower index bound, for every finite source. On the fixed actual-node contract used in [2], that bound improves a preceding elementary capacity estimate, but remains below the known all-order index and the stronger bound of [2]. We claim no new discriminant endpoint, census, or classification. The contribution is an exact shared image and a proved source comparison, rather than a new Smith algorithm or a numerical extrapolation. ### Relation to earlier work The closest prior result is [1, Sections 6–7]. It solves the full first-center dyadic model for all source budgets by a source and target change specific to that center. Here the center has degree $p^3$ and is more accurate on quadratic nodes. Our decomposition works for every prime, but first removes the source cutoff; bringing that cutoff back requires the explicit saturation conditions below. Neither the stronger center nor a target presentation alone would justify transferring the earlier volume formula. The all-order normal forms in [3,4] and their first-digit specialization in [1, Section 8] provide one of our two saturation arguments. We reprove the needed elementary specialization and do not claim it as a new all-order result. The center translation and primitive support bounds in [2] are different inputs: they give lower arithmetic witnesses on actual quadratic nodes, without identifying the full capped coefficient image. Regular bases, generalized factorials, and integer-valued divided differences have a substantial classical theory; Bhargava [5] gives explicit bases using generalized $P$-orderings. Our coefficient condition is a specific bivariate weighted Hasse condition, and the theorem concerns its ordinary-source image after a finite Taylor cutoff. We use elementary monic division and consecutive Pascal determinants, proving the required facts below. We do not introduce a new general theory of integer-valued polynomials. Preliminary versions of the present construction were shared in the public discussion [6]. The computations retain the modular Smith implementation precedent credited in [7]. ## 2. Definitions and main statements Let $\mathcal O$ be an unramified discrete valuation ring of residue characteristic $p$, with $v_p(p)=1$, and put $q=p^2$. All lengths are $\mathcal O$-lengths. Define $$ \mathcal A_1=\mathcal O[c,w]/(w^q-w-pc),\qquad S_2=w^{p^3}-pc^p, $$ $$ \mathcal B=\mathcal A_1[t,W]/(t^p-pW),\qquad z=S_2+t. \tag{1} $$ This is the first-digit coefficient algebra at the second center of [1]. Its normal basis is $w^ac^m t^bW^h$, with $a0. \end{cases} \tag{2} $$ It is exactly the span of normal Taylor orders at least $J$. In particular, both generators in the second line are necessary. For integers $d,J\ge1$ let $\mathcal Q_{d,J}=\mathcal B/(p^d,\mathcal T_J)$. For integers $K,R\ge1$, put $$ \Lambda_R=\{(r,s)\in\mathbb Z_{\ge0}^2:r+Ksr$. Let $$ H=\min\left(pd-1,\ d+J-2-\left\lfloor\frac{J-1}{p}\right\rfloor\right). \tag{5} $$ **Theorem 1 (exact image and source saturation).** The image $I_\infty(d,J)$ is the direct sum of the images of $B_{N,b}$ modulo $p^d$, for $0\le N\le H$, $0\le bH+(p^3K-1)\min(J-1,H)+K(p^3-1), \tag{6} $$ or, when $K>p$, $$ R\ge\max(K,p^3)(d-1)+p^3. \tag{7} $$ Neither condition is claimed necessary. In the usual chamber $p0$ and $J/R\to\varepsilon>0$, then $$ \frac{\ell_\infty(d,J)}{R^3}\longrightarrow \mathcal V_p(\eta,\varepsilon) =\frac{p^3}{6}\left[ \eta^3-\frac{p}{p+1}(\eta-\varepsilon/p)_+^3 -\frac{1}{p+1}(\eta-p\varepsilon)_+^3\right]. \tag{11} $$ For $J\ge pd$, there is also the exact finite formula $$ \ell_\infty(d,J)=\mathcal C_p(d,J)=p^3\binom{d+2}{3}. \tag{12} $$ For clarity, put $$ \gamma=\min\bigl(p\eta,\eta+(1-1/p)\varepsilon\bigr). $$ If either $$ \gamma+(p^3K-1)\min(\varepsilon,\gamma)<1, \tag{13} $$ or $K>p$ and $\max(K,p^3)\eta<1$, Theorem 1 gives the exact leading finite-source index $$ \frac{E_{K,R,J,d}}{R^3}\longrightarrow \frac{\eta}{2K}-\mathcal V_p(\eta,\varepsilon). \tag{14} $$ Without these source conditions the right side, truncated below at zero, is a lower limit bound, not an equality assertion. In the branch $\varepsilon\le p\eta$, condition (13) simplifies to $\eta+(p^3K-1/p)\varepsilon<1$. ## 3. Polynomial source coordinates and exact blocks Set $$ \begin{aligned} A&=w^q-w=pc,\qquad u=p^{p-1}-1,\\ L(w)&=w^p+\sum_{i=1}^{p-1}\binom pi w^{p-i}(w^q-w)^i,\\ V&=z-L(w)=puc^p+t,\\ G&=p^{p-1}V-uA^p=p^{p-1}t. \end{aligned} \tag{15} $$ These are exact polynomial identities, and $u$ is a unit. Indeed, $w^{p^3}=(w+pc)^p$ in $\mathcal A_1$. Its first $p$ terms give $L$; its last term, after subtracting $pc^p$, is $puc^p$. Also $\deg_w L=p^3-p^2+1p$, the ordinary source has the integral triangular basis $$ z^e(z^p-w)^s(z^{p^3}-z^p)^n, \quad ep. \tag{21} $$ It also applies at finite $J\ge pd$, because every ordinary source Taylor coefficient of order at least $pd$ has its explicit price divisible by $p^d$. ## 5. Every-minor bounds and a closed volume For a $k$-minor of (4), the scalar column prices decrease with $r$ and the row prices decrease with $j$. Their minimum uses the largest $k$ rows and columns and equals $\sum_{i=0}^{k-1}\mu_{N,b,i}$. The remaining binomial determinant is integral. Expanding minors after adjoining the scalar cap, or using invariant factors over a DVR, gives $$ \operatorname{length}(\operatorname{im}B_{N,b}\bmod p^d) \le\sum_{i=0}^{M_N}(d-\mu_{N,b,i})_+. \tag{22} $$ The prices increase with $i$, so taking all positive marginals gives the maximum rank contribution. For the reverse estimate select consecutive row and column indices ending at $M=M_N$ and $N$. Their binomial determinant is $$ \prod_{i=0}^{k-1} \frac{(N-k+1+i)!\,i!} {(M-k+1+i)!\,(N-M+i)!}. \tag{23} $$ Factoring the initial binomial coefficients reduces the remaining determinant to the Vandermonde determinant of consecutive integers, which proves (23). Legendre's factorial formula cancels its linear terms. The two positive base-$p$ digit sums in each factor show that its valuation is at most $2k(1+\lfloor\log_p\max(1,N)\rfloor)$. This is the same classical consecutive-minor estimate used in [1, Section 6]. Choose the rank maximizing the right side of (22), and sum this error over all blocks. There are at most $p^3(H+1)$ blocks with multiplicity, each of rank at most $\min(J,H+1)$, proving (9). The finite capacity is not generally an exact image length. At $p=2,N=2,b=0,J=2,d=2$, the block has capped invariant exponents $(2,2)$ and zero image length. Its marginal prices from (8) are $(1,3)$, giving capacity one. This example prevents replacing a bound on the binomial determinant by equality. For the limit, bounded residue terms $b$ and floors change the capacity by $O_{p,\eta,\varepsilon}(R^2)$ for bounded integer rounding. In $N