Mathematics

Structural Calculus: From Discrete Structural Derivatives to the Structural Spectrum of Integers

This paper establishes a discrete calculus framework acting on the internal structure of formal mathematical expressions. The fundamental premise is that mathematical expressions possess not only numerical semantics but also an internal syntax determined by their construction. Consequently, two expressions that are numerically equivalent are not necessarily identical as differential objects. To capture this distinction, we construct a free term algebra and define a structural derivative $\D$ upon it. Unlike classical calculus, $\D$ does not rely on limits; rather, it is recursively determined by local structural rules on expression trees. We establish four fundamental axioms of structural calculus. The cornerstone of this framework is the \textbf{power node rule}, \[ \boxed{\D(U^V) = U^{V-1} + (U-1)^V,} \] which highlights exponentiation as the true bedrock of the theory by decomposing a power structure into two parallel discrete evolution directions. Under a strict structural normalization principle, trivial multiplications by the identity element are forbidden, while atomic integers are fundamentally embedded as degenerate power structures (e.g., $8 = 8^1$). This leads to a striking departure from classical calculus: the structural derivative of an integer is not zero, but possesses its own non-trivial value (e.g., $\D(8) = 8$). By introducing two commuting discrete shift operators, we express the structural derivative as their sum and derive a binomial expansion for its higher-order iterations. As an application, we obtain an explicit formula for the $n$-th order structural derivative of $x^x$, yielding a novel combinatorial integer sequence. Furthermore, we extend the scope from individual expressions to the complete structural decompositions of integers. We define the \textit{structural spectrum} $\Sigma(n)$ and the \textit{total structural response} $R(n)$ for an integer $n$. This reveals a profound structural dichotomy between primes and composites. Finally, we propose two natural arithmetic conjectures concerning the integrality and critical self-consistency of $R(n)$, opening a new pathway from local structural rules to the global arithmetic properties of integers.

Working paper. Citable, but not yet screened for the AIRR collection.

The inverse self-commutator cost at inertia $(m,n)$: ordered-layering bounds, interlacing chains, and the Horn tiles that certify them

For a traceless Hermitian $F\in M_d(\mathbb C)$ let $\kappa_d(F)=\tfrac12\min\{\|C\|_{HS}^2:CC^*-C^*C=2F\}$. On the inertia-$(m,n)$ stratum ($m$ positive and $n$ negative eigenvalues, $z$ zeros) we prove three results valid for all $m,n,z$: a dimension-free lower bound $\kappa_d\ge\operatorname{cost}(L)$ for every ordered full-capacity layering $L$, certified by Lidskii–Wielandt and Weyl inequalities alone; an exact additive formula on the aligned-spike cone, where an optimizer is a direct sum of weighted shifts; and the closed form $\kappa_d=\operatorname{cost}(L_k)$ on explicit full-dimensional regions $R_k$, $1\le k\le m-n+1$, obtained from one rank-one interlacing step. Open question 1 of the inertia-$(m,2)$ paper is refuted by an exact weak-duality certificate: at inertia $(5,4)$ a Horn tile with $n=4$ boxes is necessary, and one valid form cannot be certified using Pieri-shaped tiles alone. The $d=8$ rank-onset example is explained by a late negative. Conjectures for general $(m,n)$ are stated with their numerical evidence.

The inverse self-commutator cost at inertia $(m,3)$: tiling certificates, 3×3-block shifts and a padding phase

For a traceless Hermitian $F\in M_d(\mathbb C)$ let $\kappa_d(F)=\tfrac12\min\{\|C\|_{HS}^2: CC^*-C^*C=2F\}$. On the inertia-$(m,3)$ stratum ($m$ positive eigenvalues, three negative ones, $z$ zeros) we determine $\kappa_d$ as the maximum of an explicit finite set $S(m,z)$ of linear forms for $m=3,\dots,9$ and $z\le2$, by exact Horn "tiling" certificates below and $3\times3$-block weighted shifts above, the latter verified at every vertex of every chamber together with a convexity argument. The forms are costs of layerings; for the ordered family A the lower bound is proved for all $m$ and $z$ by Lidskii–Wielandt tail inequalities. Unlike inertia $(m,2)$, the value is not padding-invariant: for $m\equiv1,2\pmod3$ exactly one extra exposed form exists at $z=0$, so $\kappa_{m+3}>\kappa_{m+4}$ on an open set within the stated finite range. At an exact $m=4$ example, padding raises the minimum optimal rank from $4$ to $5$ while $\kappa_7=74/61>\kappa_8=73/61$. Six combinatorial families are conjectured to describe $S(m,z\ge1)$ for all $m$, with quadratic growth; they are tested numerically out of sample for $m=10,11$.

All-degree Born rigidity in dimensions four and three: an exact simplex optimum and a certified five-outcome frame

For an equal-trace rank-one POVM seed $F$ on $\mathbb{CP}^{d-1}$ the all-degree Born-rigidity gap is $\gamma(F)=\inf_{k\ge 2}g_k(F)$, where $g_k$ is the squared singular value of the orbit-normalisation operator on the projective harmonic sector of degree $k$, and $\Gamma_d$ is its supremum over outcome-simple seeds. A previous record proved $\Gamma_d=d-6/(d+1)$, attained only by an orthonormal basis, for $d\ge 5$, and left $d=3,4$ open. We prove an explicit tail bound $|Q_{n,d}(t)|\le(1-t)^{-(d-2)}/\binom{n+d-2}{n}$ for the normalised zonal function (a two-line corollary of Koornwinder's Laplace-type integral) and use it to settle $d=4$ exactly: $\Gamma_4=22018975/7340032=2.999847\ldots$, attained only by the regular five-outcome simplex, with the minimising degree $k=6$. In $d=3$ we prove $77/45<\Gamma_3\le 9/5$, certify $\Gamma_3\ge 1.776009937964733$ by an explicit five-outcome frame in exact arithmetic, and locate numerically a strict local maximum $1.7760099429996636\ldots$ with five tied degrees $4,8,9,12,16$.

One Fold for Haar Oriented Two-Planes in Every Dimension n ≥ 6: A Phase-Space Proof through a Fano Factor, an Integrating Factor and the Threshold $L_q=q(q-1)/8$

Let $L_q(\kappa)=q!\sum_{j\ge0}\kappa^{2j}/(q+2j)!$, $q=n-2$, be the normalizer of Haar oriented two-planes in $\mathbb R^n$, $m_q=L_q'/L_q$, and $H_q=m_q-\kappa m_q'$ the fold numerator of the stationary multiplier $\lambda_q=\kappa/(2m_q)$. The record ARR-2026-7H9FAPTBZA897AMJ proved that $H_q$ has exactly one positive zero for every $q\ge4$ through a one-change law for the coefficients of $L_q^2H_q$. We give a short independent proof by a different method: in the coordinates $(X,u)=(\kappa m_q,1/L_q)$ the inhomogeneous second-order equation of $L_q$ makes $\kappa H_q$ a polynomial, the transversality of a crossing is a second polynomial $T$, and $Q=T-(q-\tfrac12)\kappa H_q$ obeys a first-order linear equation whose forcing, on the zero set of $H_q$, changes sign exactly where $L_q=q(q-1)/8$. An integrating factor then excludes a zero before that point and a second zero after the first. Probabilistically, $\kappa H_q=4(\mathbb EJ-\operatorname{Var}J)$ for the index law of the series. The same argument gives no fold for $q=2,3$. A now redundant interval-arithmetic certificate and numerics are included.

Sparse cyclic Weyl operators in prime-power dimension: an operator extension of the scalar uncertainty law through sparsity $2p-1$

Let $d=p^k$ and let $W_{(a,b)}=X^aZ^b$ be the cyclic Weyl operators on $\mathbb C^d$. Write $w_p(r)$ for the base-$p$ digit weight $\prod_t(r_t+1)$ and $\Omega_k(m)=\max\{r<p^k:w_p(r)\le m\}$. We give a valuation proof of the classical Delvaux–Van Barel scalar law: (Z) a polynomial with at most $m$ terms and degree below $d$ annihilates at most $\Omega_k(m)$ of the $d$-th roots of unity, and this is sharp. Our operator extension is: (B) every nonzero combination $C$ of $\ell\le2p-1$ Weyl operators satisfies $\dim\ker C\le\Omega_k(\ell)$, i.e. $\operatorname{rank}C\ge p^k-(\ell-1)p^{k-1}$ for $\ell\le p$ (sharp) and $\operatorname{rank}C\ge p^{k-1}$ for $p\le\ell\le2p-1$. The commutative ingredient is the Massey–Costello–Justesen weight-retaining property; the operator ingredient is a Radon-type lemma showing that first-order obstructions have weight at least $2p$. For composite $d$ the maximal nullity is $d/p_{\min}$ at sparsity two and, conditionally on a published trinomial rank rule, $\max(2d/r_3,\,d/2\,[d\text{ even}])$ at sparsity three. The conjecture $\dim\ker C\le\Omega_k(s_W(C))$ for all sparsities is proved for $d\in\{4,8,9\}$ and for torsion cosets; its first open case is $d=16$, $\ell=6$.

Exact Hive Certificates for the Balanced-Inertia Stability Constants (N = 3 to 12) in Every Dimension, the $(A_N,B)$ Cost Law, and the Twenty-Two-Form Six-Level Cost Formula

For a traceless Hermitian $F$ with $N$ positive, $N$ negative and $d-2N$ zero eigenvalues, the inverse self-commutator cost $\kappa_d(F)$ satisfies $\delta_d\ge\gamma_{N,d}D_*$ with an optimal rational constant $\gamma_{N,d}$; the values $\gamma_2,\dots,\gamma_5$ were known and $\gamma_6,\dots,\gamma_{10}$ only numerically. We give exact, independently machine-checked certificates proving $\gamma_{N,d}=\gamma_N$ for $N=3,\dots,12$ and every $d\ge2N$: $17/36$, $4/7$, $113/152$, $9/10$, $781/740$, $317/264$, $23/17$, $3/2$, $773/468$, $289/160$. The minimising pairs switch at $N=7$ from $(A_N,B)$ to $(u_3,z_N)$. The certificates are exact rational hive primals at every positive-distance vertex of the finite reduction and matching duals at the listed minimisers, and dimension-independent Horn combinations (Weyl, two size-two and Lidskii–Wielandt inequalities) at the minimisers, written out in full. We also prove the $(A_N,B)$ cost law for $N\le14$ and the exact six-level formula $\kappa_6=\max$ of twenty-two explicit linear forms, all exposed. The supplied package reports a separate adversarial replay; fresh checks using separately written code are documented in Section 7, with provenance limits stated explicitly.

One Fold if and only if the Overlap Law is Right-Skewed: A Riccati Criterion for Tilted Beta Normalizers, with Consequences for Rank-One Projective Rate–Distortion

For $M={}_1F_1(a;c;\kappa)$, the moment generating function of $X\sim\mathrm{Beta}(a,c-a)$, the stationary multiplier $\lambda(\kappa)=\kappa/(2b(\kappa))$ of the Curie–Weiss-type radial problem $\max_{0\le b\le1}\{K(2\lambda b)-\lambda R^2b^2\}$, $K=\log M-(a/c)\kappa$, folds where $H=K'-\kappa K''$ vanishes. We show that the logarithmic derivative $m=M'/M$ satisfies the Riccati equation $\kappa m'=a-(c-\kappa)m-\kappa m^2$, so that $H=m(1-m)\,(\Psi(m)-\kappa)$ with $\Psi$ an explicit rational function, and a transversality argument in the plane $(m,\kappa)$ proves: $H$ has exactly one positive zero if $c>2a$ (right-skewed overlap law) and none if $c\le2a$. This settles the second open question of ARR-2026-68BH5JDJCQ8XW8GT, gives the complete scalar radial phase and two-piece envelope for every $(a,c)$ with explicit fold bounds for $a\ge1$ (false for $a<1$), covers the Bessel family ${}_0F_1(;c;\kappa^2/4)$ (no fold), and yields "one fold iff $d\ge3$" for rank-one sources over $\mathbb R$, $\mathbb C$, $\mathbb H$. For the oriented-two-plane normalizer, which is third order, we give an exact identity and numerics only.