The Dyadic Kac-Murdock-Szego Spectrum of Arithmetic Diamond Matrices and the n=14 Phase Transition in Shifted Smith Determinants

Montgomery, Gareth

doi:10.5281/zenodo.23611255

Monograph 01 · Analytic & Algebraic Number Theory · DOI: 10.5281/zenodo.23611255

The Dyadic Kac-Murdock-Szegő Spectrum of Arithmetic Diamond Matrices and the $n=14$ Phase Transition in Shifted Smith Determinants

Resolving two additive deformations of H. J. S. Smith's 1875 determinant $\det[\gcd(i,j)] = \prod_{k=1}^n \phi(k)$ that break factor-closure by coupling multiplicative divisibility with additive operations.

📄 Read PDF TeX Source

Theorem 2.2 (Universal Jordan-Pisot Power Diamond Formula)

For any integer exponent $r \ge 1$, the 2-adic resonance $\gcd(i^r+j^r, |i^r-j^r|) = \gcd(i,j)^r(1 + \delta_{v_2(i), v_2(j)})$ and odd-divisor Jordan totient $J_r(m)$ Möbius congruence reduce $A_n^{(r)} = [\gcd(i^r+j^r, |i^r-j^r|)]_{1\le i,j\le n}$ to dyadic Kac-Murdock-Szegő (KMS) Toeplitz blocks at coupling $\rho_r = 2^{-r/2}$:

$$\det\big[\gcd(i^r+j^r, \, |i^r-j^r|)\big]_{1\le i,j\le n} = \prod_{\substack{m=1 \\ m\text{ odd}}}^{n} J_r(m)^{V_m + 1} \, 2^{r V_m(V_m - 1)/2} \, \Delta_{V_m}^{(r)}, \quad V_m = \lfloor \log_2(n/m)\rfloor$$

Here $\Delta_V^{(r)} = 2^{r+1}\Delta_{V-1}^{(r)} - 2^r\Delta_{V-2}^{(r)}$ ($\Delta_0^{(r)} = 2, \Delta_1^{(r)} = 2^{r+2}-1$) satisfies the universal Cassini identity $(\Delta_V^{(r)})^2 - \Delta_{V+1}^{(r)}\Delta_{V-1}^{(r)} = 2^{r(V-1)}$, proving that $\Delta_V^{(r)}$ is the exact Pisot sequence $T(2, 2^{r+2}-1)$ for every $r \ge 1$: OEIS A020727 ($T(2,7)$ for $r=1$) and OEIS A022026 ($T(2,15): 2, 15, 112, 836, 6240, \dots$ for $r=2$).

Theorem 3.1 (Shifted Smith $2^{n-2}$ Law & $n=14$ Phase Transition)

For the additively shifted Smith matrix $G_n^{(k)} = [\gcd(k+i, k+j)]_{1\le i,j\le n}$, parity-difference basis annihilation of $\phi(1)=\phi(2)=1$ and $\mathbb{F}_2$ rank-nullity across prime powers $p^r \le n-1$ with $p \equiv 3 \pmod 4$ establish for all $n \ge 2$ and $k \in \mathbb{Z}$:

$$v_2\big(\det G_n^{(k)}\big) \ge (n - 2) + \max\Big(0, \, \lfloor n/2\rfloor - 2 - \pi_{3,4}^*(n-1)\Big)$$

This proves why $\min_{k\ge 0} v_2(\det G_n^{(k)}) = n - 2$ holds for $2 \le n \le 13$, and fails for the first time at $n = 14$, where $\lfloor 14/2\rfloor - 2 = 5 > 4 = |\{3, 7, 9, 11\}| = \pi_{3,4}^*(13)$.

Theorem 2.3 (Universal $p$-Adic Deformed Diamond Pencil & Dual LCM Matrix $L_n^{(r)}$ · DOI: 10.5281/zenodo.23832476)

📄 Read Unified PDF 16+ OEIS Sequence Unifications (.txt) →

Generalizing the dyadic resonance to any prime $p \ge 2$, exponent $r \ge 1$, and coupling $\lambda \in \mathbb{C}$, the $p$-adic deformed GCD matrix $A_{n,p}^{(r)}(\lambda) = \big[\gcd(i,j)^r(1 + \lambda \delta_{v_p(i), v_p(j)})\big]_{1\le i,j\le n}$ block-diagonalizes over $p$-coprime indices $p \nmid m$ ($V_{p,m} = \lfloor \log_p(n/m)\rfloor$):

$$\det A_{n,p}^{(r)}(\lambda) = \prod_{\substack{m=1 \\ p \nmid m}}^{n} J_r(m)^{V_{p,m} + 1} \, p^{r V_{p,m}(V_{p,m}-1)/2} \, \Delta_{V_{p,m}}(p^r, \lambda), \quad \Delta_V(q,\lambda) = \big((q+1)\lambda + q - 1\big)\Delta_{V-1} - q\lambda^2\Delta_{V-2}$$

Here $\Delta_0 = 1+\lambda$, $\Delta_1 = q(1+\lambda)^2 - 1$, and $(\Delta_V(q,\lambda))^2 - \Delta_{V+1}\Delta_{V-1} = \lambda^{2V}q^{V-1}$. Specializing $(p, r, \lambda)$ unifies 16+ OEIS sequences across five algebraic regimes: (i) Universal Pisot $\lambda = 1$: $T(2, 4q-1)$ (A020727 at $q=2$, A022026 at $q=4$, A234279 benzenoid chain spanning forests at $q=16$); (ii) Hollow Valuation-Resonance $\lambda = -1$: Gaussian Lucas polynomials $(-1)^V U_V(-2, q)$ (A009545, A088137, A088138, A045873, A088139, A098656) with global singularity $\det A_n^{(r)}(-1) = -\delta_{n,2}$; (iii) Dual LCM Diamond $\lambda = -1/2$: $L_n^{(r)} = \big[i^r j^r / \gcd(i^r+j^r, |i^r-j^r|)\big]$ via $N_V(q) = (3q-1)N_{V-1} - q N_{V-2}$ ($N_0=1, N_1=4q-1$); (iv) Alternating Sign-Resonance $\lambda = -2$: A087170 ($q=4$) and A370191 ($q=5$).

Theorem 2.4 (Multi-Prime $S$-Adic & Global Adelic Cholesky Tensor Product Law + Green's Kernel Inverse) Adelic Cholesky & Companion $\Theta_V$ Proof (.txt) →

For any finite set of primes $S = \{p_1, \dots, p_s\}$, consider the $S$-adic Arithmetic Diamond Matrix $A_{n,S}^{(r)}(\boldsymbol{\lambda}) = \big[\gcd(i,j)^r \prod_{p \in S}(1 + \lambda_p \delta_{v_p(i), v_p(j)})\big]_{1\le i,j\le n}$. Although the multi-prime exponent grid $I_{S,m} = \{\mathbf{a} \in \mathbb{Z}_{\ge 0}^s : m\prod_{p\in S} p^{a_p} \le n\}$ is a non-rectangular simplex, it is a downward-closed order ideal in $(\mathbb{Z}_{\ge 0}^s, \le)$. Because the KMS block $B_V(q, \lambda) = L_V D_V L_V^T$ has a universal column-constant unit lower-triangular Cholesky factor $L_{a,b}(q, \lambda) = \Theta_b(q, \lambda)/\Delta_b(q, \lambda)$ ($a > b$) and pivot $d_a(q, \lambda) = q^{a-1}\Delta_a/\Delta_{a-1}$ that are independent of the upper truncation $V$, the tensor product $\mathcal{L}_S \mathcal{D}_S \mathcal{L}_S^T$ restricts to every simplex $I_{S,m}$ without boundary correction:

$$\frac{\det A_{n,S}^{(r)}(\boldsymbol{\lambda})}{\prod_{k=1}^n J_r(k)} = \prod_{p \in S} \frac{\det A_{n,p}^{(r)}(\lambda_p)}{\prod_{k=1}^n J_r(k)}, \qquad \Big(B_V(q,\lambda)^{-1}\Big)_{a,b} = \frac{\delta_{a,b}\Delta_V(q,\lambda) - \lambda^{|a-b|}\Theta_{\min(a,b)}(q,\lambda)\Theta_{V-\max(a,b)}(q,\lambda)}{\lambda q^{\min(a,b)}\Delta_V(q,\lambda)}$$

Here the companion KMS boundary polynomial $\Theta_k(q, \lambda) = \Delta_k(q,\lambda) - q\lambda\Delta_{k-1}(q,\lambda)$ ($\Theta_0 = 1, \Theta_1 = q\lambda + q - 1$) unifies 6 additional OEIS sequences: A007052 ($q=2,\lambda=1$: order-consecutive partitions $1,3,10,34,116$), A158869 ($q=3,\lambda=1$: brick tilings $1,5,27,147,801$), A009116 ($q=2,\lambda=-1$), A087455 ($q=3,\lambda=-1$), A138230 ($q=4,\lambda=-1$), and A006495 ($q=5,\lambda=-1$). Select S = {2, 3} or S = {2, 3, 5} in Interactive Verifier 01 below to test the exact multi-prime simplicial factorization live!

Theorem 2.5 (Universal Multiplicative-Valuation Poset $LU$/Cholesky Theorem & Fibonacci Divisor-Count Law) Universal Valuation-Poset & Fibonacci Proof (.txt) →

For any family of 1D valuation kernels $\{K_p(a,b)\}_{a,b\ge 0}$ indexed by primes $p \le n$ with $K_p(0,0)=1$ and 1D $LU$ pivots $d_p(a) = \det K_p[0\dots a]/\det K_p[0\dots a-1]$, the multiplicative-valuation matrix $M_n = \big[\prod_{p \le n} K_p(v_p(i), v_p(j))\big]_{1\le i,j\le n}$ admits an exact divisor-supported factorization $M_n = L D U$ on the downward-closed divisibility order ideal $(\{1,\dots,n\}, \mid)$, yielding the closed-form determinant:

$$\det\!\left[\prod_{p \le n} K_p\big(v_p(i), v_p(j)\big)\right]_{1\le i,j\le n} = \prod_{m=1}^n \prod_{p \mid m} d_p\big(v_p(m)\big) = \prod_{p \le n} \prod_{\substack{u=1 \\ p \nmid u}}^{\lfloor n/p\rfloor} \det K_p\big[0 \dots \lfloor \log_p(n/u)\rfloor\big]$$

This master theorem unifies three major arithmetic families beyond Jordan totients: (i) 2-Parameter $S$-Adic Metric & GCD/LCM Power Diamonds: $K_p(a,b) = q_p^{\min(a,b)}(\rho_p^{|a-b|} + \lambda_p\delta_{a,b})$ reduces via effective KMS base $Q_p = q_p/\rho_p^2$ to $\rho_p^{2V}q_p^{V(V-1)/2}\Delta_V(q_p/\rho_p^2, \lambda_p)$ (setting $q_p = p^{(r-s)/2}, \rho_p = p^{-(r+s)/2}$ gives $Q_p = p^{r+s}$ for $\gcd(i,j)^r/\operatorname{lcm}(i,j)^s$); (ii) Unitary GCD Diamond (A321613): $K_p(a,b) = (1 + (p^r-1)\delta_{a,b})_{a,b\ge 1}$ yields explicit diagonal shifts; (iii) Divisor-Count $\tau(\gcd(i,j))$ Diamond & Fibonacci Bisection Law: Setting $K_p(a,b) = 1 + x\min(a,b) + \lambda\delta_{a,b}$ gives the polynomial $W_V(x,\lambda) = (x+2\lambda)W_{V-1} - \lambda^2 W_{V-2}$ ($W_0=1+\lambda, W_1=x+2\lambda+\lambda^2$). At $x=1,\lambda=1$, $W_V(1,1) = F_{2V+3}$ is the exact odd-index Fibonacci bisection (A001519: $2, 5, 13, 34, 89, 233, \dots$); at $x=1,\lambda=2$, $W_V(1,2) = (2^{2V+3}+1)/3$ (A007583: $3, 11, 43, 171, 683, \dots$). Select r = τ(gcd) (Fibonacci A001519) in Verifier 01 below to test live!

Interactive Verifier 01 · Exact BigInt Bareiss vs Master $p$-Adic & $S$-Adic KMS Pencil

$S$-Adic Deformed Diamond $A_{n,S}^{(r)}(\lambda)$ & Shifted Smith 2-Adic Corank Explorer

Primes $S$: Coupling $\lambda$: Kernel / $r$: Dim $n$:

Structural Blueprint (Dyadic KMS Congruence): A_n(2^a u, 2^b w) = 2^{min(a,b)}(1 + delta_{a,b}) gcd(u,w) → Direct Sum of phi(m) D_{V_m}

For each odd integer $m \le n$, the dyadic block has size $(V_m + 1) \times (V_m + 1)$ with $V_m = \lfloor \log_2(n/m)\rfloor$. Factoring out $2^{V_m(V_m+1)/2}$ reduces each block to the Kac-Murdock-Szegő Toeplitz operator $I + K_{V_m}(2^{-1/2})$, whose tridiagonal inverse yields the Pisot recurrence $\Delta_V = 4\Delta_{V-1} - 2\Delta_{V-2}$.

Bareiss $\det(A_n)$

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Pisot KMS Product

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Smith Base $\prod_{k=1}^n \phi(k)$

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OEIS A001088

Diamond Quotient $\prod_{m\text{ odd}} \Delta_{V_m}$

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Matrix $A_6 = [\gcd(i+j, |i-j|)]$ Diagonal = $2i$

Shifted Smith 2-Adic Phase Transition ($n = 2 \dots 20$)

Comparing even-subspace dimension $\lfloor n/2\rfloor - 2$ against $3 \pmod 4$ prime-power count $\pi_{3,4}^*(n-1)$.

Test Shift $k$:

n	⌊n/2⌋-2	π*_{3,4}(n-1)	Active p^r ≡ 3 (mod 4)	Corank Bound	Bound v_2	Live v_2(G_n^{(k)})

Part B · Complete OEIS & Modular Algebra Discovery Ledger (9 Sequences & Conjecture Proofs)

Open-Problem Resolutions in Coprime Hamiltonian Graphs, Permutation Circulants, and Modular Quaternion Rings

In addition to the Arithmetic Diamond and Shifted Smith theorems above, I resolved five long-standing open problems and conjectures in the Online Encyclopedia of Integer Sequences (User:Gareth_Montgomery): the six 20-year-old unknown terms in A109813, Stefano Spezia's January 2026 conjecture on minimum permutation circulant determinants (A392191), the closed-form group exponent of Lipschitz quaternions modulo $n$ (A227477), the prime-power divisibility criterion for non-vanishing symmetric $2 \times 2$ matrix $n$-th power sums (A236864), and the 2-adic isomorphism classification of generalized quaternion rings (A339948).

b109813.txt (n=1..42) b_diamond_gcd_det.txt (n=1..100)

1. OEIS A109813 · Coprime Hamiltonian Paths Raw Proof & Code (.txt)

Resolves all six unknown terms (`?`) at $n = 29, 31, 35, 37, 39, 41$ from Franklin T. Adams-Watters's September 25, 2006 comment (plus extensions up to $n = 50$) for the number of permutations of $\{1 \le k \le n : \gcd(k,n)=1\}$ with coprime adjacent terms.

a(29) = 268,367,392,739,686,809,600

a(31) = 20,436,001,451,612,445,081,600

a(35) = 31,391,358,658,560,000

a(37) = 118,614,513,176,933,826,023,718,912,000

a(39) = 250,772,587,733,606,400

a(41) = 9,901,302,953,893,164,517,381,647,040,512,000

Mechanism: Partitioning active primes $p \le n/2$ ($p \nmid n$) into small primes $p \le \sqrt{n}$ and disjoint large-prime clusters $C_q$ ($q > \sqrt{n}$) with pairwise non-disjoint bitmask absorption reduces each inclusion-exclusion leaf to an 8-state sum-over-subsets (SOS) walk over $\{2, 3, 5\}$.

2. OEIS A392191 · Spezia's Circulant Conjecture Raw Proof & Code (.txt)

Resolves Stefano Spezia's January 2026 conjecture for the minimum determinant $a(n)$ of an $n \times n$ circulant matrix whose rows are permutations of $(0, 1, \dots, n-1)$, and establishes $a(12) = -75986875920384$ and $a(13) = 1014$:

$$a(n) = \begin{cases} -\operatorname{A392190}(n), & n \text{ even}, \\ \frac{p-1}{2}p^2 = \operatorname{A084367}\!\big(\frac{p-1}{2}\big), & n = p \text{ odd prime}, \\ 0, & n = 1 \text{ or } n \ge 9 \text{ odd composite}. \end{cases}$$

Mechanism: For odd composite $n = dm \ge 9$, Kotzig's $3 \times d$ equal-column-sum magic rectangle annihilates the eigenvalue $L_m = 0$. For odd prime $p$, cyclotomic ideal divisibility $(1-\zeta_p) \mid L_1$ forces $\frac{p-1}{2}p^2 \mid \det(C)$, attained by the pyramidal permutation $c = (0, 2, 4, \dots, p-1, p-2, \dots, 3, 1)$.

3. OEIS A227477 · Lipschitz Quaternion Group Exponent Raw Proof & Code (.txt)

Resolves the 2013 `[hard,more]` open problem for the group exponent $a(n)$ of the unit group $\mathbb{H}(\mathbb{Z}/n\mathbb{Z})^\times$ of reduced Lipschitz quaternions modulo $n = 2^{e_0}\prod_{i=1}^r p_i^{e_i}$:

$$a(n) = \operatorname{lcm}\!\left(2^{e_0}, \, \operatorname{lcm}_{1\le i\le r} p_i^{e_i}(p_i^2 - 1)\right)$$

Mechanism: For odd primes $p$, $\mathbb{H}(\mathbb{Z}/p^e\mathbb{Z})^\times \cong \operatorname{GL}_2(\mathbb{Z}/p^e\mathbb{Z})$ has exponent $p^e(p^2-1)$ via a Singer cycle of order $p^2-1$ times a $p$-unipotent lift. For $p=2$, every odd-norm quaternion satisfies $q^2 \in 1 + 2\mathbb{H}(\mathbb{Z}/2^e\mathbb{Z})$, which has exponent $2^{e-1}$ by 2-adic filtration induction, yielding $a(2^e) = 2^e$.

4. OEIS A236864 · Symmetric $2 \times 2$ Matrix Power Sums Raw Proof & Code (.txt)

Establishes the exact number-theoretic characterization of all integers $n$ for which the $n$-th power sum of symmetric $2 \times 2$ matrices over $\mathbb{Z}/n\mathbb{Z}$ does not vanish modulo $n$:

$$\sum_{M \in \operatorname{Sym}_2(\mathbb{Z}/n\mathbb{Z})} M^n \not\equiv 0 \pmod n \iff \exists \text{ prime } p \text{ such that } p \parallel n \text{ and } p(p^2 - 1) \mid n$$

Mechanism: Spectral summation over $\mathbb{F}_p$ and $\mathbb{F}_{p^2}$ shows that $S(p, n) \not\equiv 0 \pmod p$ iff $(p^2-1) \mid n$. Whenever $p^2 \mid n$ ($e \ge 2$), first-order Hensel fiber summation over the $p^3$ lifts $M = M_0 + p^{e-1}D$ vanishes identically modulo $p^e$.

5. OEIS A339948 · 2-Adic Quaternion Ring Classification Raw Proof & Code (.txt)

Classifies the isomorphism classes of generalized quaternion rings $H_{\mathbb{Z}/n\mathbb{Z}}(a,b)$ for $a,b \in \mathbb{Z}/n\mathbb{Z}$, correcting a 1-argument (`phi[2^s]`) vs 2-argument (`phi[2, s]`) Mathematica pattern-matching bug that corrupted all even composite terms and $a(32)$ in the published entry:

$$a\!\left(2^{s_0}\prod_{i=1}^r p_i^{s_i}\right) = a(2^{s_0}) \prod_{i=1}^r (2s_i^2 + 2), \quad \big(a(2^s)\big)_{s=1}^7 = (1, \, 7, \, 16, \, 32, \, 56, \, 90, \, 138)$$

Mechanism: Over $\mathbb{Z}/16\mathbb{Z}$, $2 \cdot 8 \equiv 0 \pmod{16}$ unlocks non-orthogonal $\operatorname{GL}_4(\mathbb{Z}/16\mathbb{Z})$ basis shifts such as $(X,Y) = (i+j, j)$ that merge $H(4,8) \cong H(12,8)$ and $H(2,4) \cong H(2,12) \cong H(6,4) \cong H(6,12)$, yielding $a(16) = 32$ and $a(32) = 56$.

6-11. OEIS A020727, A022026 & $p$-Adic Pencil 6 Raw OEIS Packages (20+ Seq)

Complete OEIS sequence packages derived from Theorems 2.2, 2.3, and 3.1 above, linking Pisot sequences $T(2, 2^{r+2}-1)$ (`A020727`, `A022026`, `A234279`), hollow Gaussian Lucas determinants (`A009545`..`A098656`), and the $n=14$ 2-adic corank phase transition:

6. A020727: $T(2,7)$ KMS Det ($r=1$) → 7. A022026: $T(2,15)$ Power KMS ($r\ge 1$) → 8. $p$-Adic Pencil & Dual LCM $L_n^{(r)}$: 16+ OEIS Unifications ($\lambda \in \{1, -1, -\tfrac{1}{2}, 2, -2\}$) → 9. New Seq #1: $\det[\gcd(i+j,|i-j|)]$ → 10. New Seq #2: $\prod_{m\text{ odd}} \Delta_{\lfloor \log_2(n/m)\rfloor}$ → 11. New Seq #3: $\max(0,\lfloor n/2\rfloor-2-\pi_{3,4}^*(n-1))$ ($n=14$ Transition) →

Includes full 100-term exact b-file (b_diamond_gcd_det.txt) and PARI/GP + SymPy verification scripts.

Interactive Verifier 01B · Live Modular Algebra, Circulant & Coprime Graph Explorer

Test A227477, A236864, A339948, A392191 & A109813 for Any Integer $n$

Modulus / Order $n$:

Structural Blueprint (Chinese Remainder & Cyclotomic Norm Decomposition): H(Z/nZ)^* =~ Prod GL_2(Z/p^e Z) x H(Z/2^{e_0} Z)^* | det(C_p) = ((p-1)/2) * N_{Q(zeta_p)/Q}(L_1(c))

Select any preset milestone above (such as $n=6$ or $n=12$ for live brute-force summation over all $n^3$ symmetric matrices in $\operatorname{Sym}_2(\mathbb{Z}/n\mathbb{Z})$, or $n=13$ for live `BigInt` Bareiss elimination on the $13 \times 13$ pyramidal permutation circulant matrix).

Prime Factorization

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A227477(n) · Lipschitz Exp

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A236864 · $\sum M^n \bmod n$

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A339948(n) · Quat Rings

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A392191(n) · Min Circulant

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Spezia Circulant Witness & Live Bareiss Determinant (A392191) --

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Symmetric $2 \times 2$ Matrix Sum $\sum_{M \in \operatorname{Sym}_2(\mathbb{Z}/n\mathbb{Z})} M^n \bmod n$ (A236864)

Click the button above (or select $n \le 12$) to sum all $n^3$ symmetric $2 \times 2$ matrices modulo $n$.

Coprime Hamiltonian Path Count A109813(n) & Prime Cluster Partition

All six 2006 `?` gaps ($n = 29, 31, 35, 37, 39, 41$) marked with ★ RESOLVED.

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n	φ(n)	A109813(n) (Hamiltonian Paths on G_n)	Status

Monograph 02 · Mathematical Biology & Evolutionary Dynamics · DOI: 10.5281/zenodo.23611278

Exact Pharmacokinetic Washout Compensation and Volterra Cycle Conservation in Evolutionary Cancer Containment

Solving the coupled nonlinear logistic-washout Bernoulli differential equation in exact closed form via a Poisson expectation $M \sim \operatorname{Poisson}(k_d C_{\mathrm{off}}/\gamma)$ to prevent post-cessation washout breaches below the Darwinian containment floor $S_{\mathrm{floor}} = K/\alpha$.

📄 Read PDF TeX Source

Theorem 2.1 (Poisson-Bernoulli Washout & Safe-Stopping Law)

When dosing ceases at $t=0$ with elimination rate $\gamma = \ln 2 / t_{1/2}$ and total washout kill impulse $\Lambda_0 = k_d C_{\mathrm{off}}/\gamma$, the nonlinear crowding integral $\mathcal{I}(t) = \int_0^t \Phi(\tau)\,d\tau$ is the exact expectation over $M \sim \operatorname{Poisson}(\Lambda_0)$:

$$\mathcal{I}(t) = \mathbb{E}_{M\sim\operatorname{Poisson}(\Lambda_0)}\!\left[\frac{e^{(r_S - M\gamma)t}-1}{r_S - M\gamma}\right], \qquad S_{\mathrm{stop}}^* = \frac{S_{\mathrm{floor}}}{\Phi(t^*) - \frac{r_S S_{\mathrm{floor}}}{K}\mathcal{I}(t^*)}$$

Inverting the Möbius relation at the nadir time $t^* = \gamma^{-1}\ln\!\big(k_d C_{\mathrm{off}} / [r_S(1 - S_{\mathrm{floor}}/K)]\big)$ yields the exact stopping threshold $S_{\mathrm{stop}}^*$ required to guarantee $\min_{t\ge 0} S(t) = S_{\mathrm{floor}}$.

Theorems 3.1 & 3.2 (Volterra Speed Limit & Hysteresis Observer)

Logarithmic cycle integration over any periodic treatment-recovery cycle $[0,T]$ with $S(T)=S(0)$ proves that resistant subclone regression ($R(T) < R(0)$) requires the evolutionary dosing speed limit:

$$k_d\langle C\rangle_T < r_S\left(1 - \frac{1}{\alpha}\right), \qquad \widehat{S}(N_{\mathrm{ref}}) = \frac{\dot{N}(t_{\uparrow}) - \dot{N}(t_{\downarrow})}{\kappa(C_{\downarrow}) - \kappa(C_{\uparrow})}$$

Moreover, subtracting downward on-drug slope $\dot{N}(t_{\downarrow})$ from upward washout slope $\dot{N}(t_{\uparrow})$ at equal total biomarker $N_{\mathrm{ref}} = S + R$ cancels unknown growth functions $F_S + F_R$ identically.

Theorem 4.1 (Bivariate Poisson-Bernoulli Cross-Washout, Double-Bind Extinction Wedge & $86.12\times$ Overlap Asymmetry) Multi-Clone Adaptive Sequencing (`3RUK` ↔ `2AM9`)

When switching from primary agent $C_1(t) = C_{1,\mathrm{off}}e^{-\gamma_1 t}$ to secondary collateral agent $C_2(t) = C_{2,\mathrm{on}}(1 - e^{-\gamma_2 t})$, the exact crowding integral $\mathcal{J}(t) = \int_0^t \Psi(\tau)\,d\tau$ expands as an absolutely convergent expectation over independent random variables $(M_1, M_2) \sim \operatorname{Poisson}(\Lambda_1) \times \operatorname{Poisson}(-\Lambda_2)$:

$$\mathcal{J}(t) = \sum_{m_1=0}^\infty \sum_{m_2=0}^\infty \frac{e^{-\Lambda_1}\Lambda_1^{m_1}}{m_1!} \frac{e^{\Lambda_2}(-\Lambda_2)^{m_2}}{m_2!} \frac{e^{(r_{\mathrm{net}} - m_1\gamma_1 - m_2\gamma_2)t} - 1}{r_{\mathrm{net}} - m_1\gamma_1 - m_2\gamma_2}, \qquad \frac{\mathcal{I}_{2 \to 1}}{\mathcal{I}_{1 \to 2}} = \left(\frac{\gamma_1}{\gamma_2}\right)^2 = \left(\frac{139.2\text{ h}}{15.0\text{ h}}\right)^2 = 86.1184$$

Because `2AM9` AR antagonist Enzalutamide clears $9.28\times$ slower ($t_{1/2,2} = 5.8\text{ d} = 139.2\text{ h}$) than `3RUK` CYP17A1 inhibitor Abiraterone ($t_{1/2,1} = 15.0\text{ h}$), the normalized cumulative dual-exposure toxicity impulse $\mathcal{I}_{i\to j} = \int_0^\infty (C_i/C_{i,0})(C_j/C_{j,\infty})\,dt$ obeys an exact Quadratic Elimination Ratio Law ($\mathcal{I}_{2\to 1}/\mathcal{I}_{1\to 2} = 86.12\times$). Furthermore, 3-clone Volterra cycle integration proves the Double-Bind Extinction Wedge $\mu_1 < (1 - 1/\alpha_{21}) + \mu_2/\alpha_{21}$: pulsing a short collateral burst ($\mu_2 = 0.0884$) widens the evolutionary speed limit from $0.2188$ to $0.2878$ and drives the `3RUK` resistant subclone $S_2$ to exact biological extinction ($900.0 \to 0.0000\text{ cells}$ over 5 cycles) while preserving $S_1 \ge S_{\mathrm{floor}} = 7812.5$.

Theorems 4.2 & 4.3 (Critical Clearance Catastrophe $t_{1/2,\mathrm{crit}}$, Buckingham $\Pi$-Reduction & Logistic Phase-Shift Law $T_{\mathrm{off}}^*$) `2AM9` Knife-Edge: $15.39\text{ h}$ vs $15.0\text{ h}$

Under dimensionless Buckingham groups $\Pi_1 = k_d C_{\mathrm{off}}/r_S$, $\Pi_2 = \gamma/r_S$, and $\alpha$, the safe-stopping threshold remains below carrying capacity ($S_{\mathrm{stop}}^* < K$) if and only if $\Pi_2 > \Pi_{2,\mathrm{crit}}(\alpha, \Pi_1)$, solving $\alpha\Phi(\tau^*) - \widetilde{\mathcal{I}}(\tau^*) = 1$. Whenever $t_{1/2} > t_{1/2,\mathrm{crit}} = \ln 2 / (r_S \Pi_{2,\mathrm{crit}})$, even pausing therapy at $100\%$ carrying capacity ($S(0)=K$) cannot prevent the washout tail from breaching $S_{\mathrm{floor}} = K/\alpha$. For `2AM9` AR LBD promiscuity ($\alpha = 1.12$), $t_{1/2,\mathrm{crit}} = 15.39\text{ h}$ ($0.6411\text{ d}$), sitting just $23\text{ minutes}$ above systemic Abiraterone clearance ($15.0\text{ h}$)! Moreover, as $t \to \infty$, the washout trajectory converges at rate $\mathcal{O}(e^{-\gamma t})$ to a pure logistic curve shifted by the Exact Pharmacokinetic Holiday Return Time $T_{\mathrm{off}}^* = \Delta t_{\mathrm{PK}}$:

$$\alpha\Phi(\tau^*) - \widetilde{\mathcal{I}}(\tau^*) = 1 \iff t_{1/2} = t_{1/2,\mathrm{crit}}(\alpha, \Pi_1), \qquad T_{\mathrm{off}}^* = \frac{\Lambda_0}{r_S} + \frac{1}{r_S}\ln\!\left(\frac{1 - \sigma_{\mathrm{stop}}^*\mathcal{C}_{\infty}(\Lambda_0, \Pi_2)}{1 - \sigma_{\mathrm{stop}}^*}\right)$$

where $\mathcal{C}_{\infty}(\Lambda_0, \Pi_2) = \mathbb{E}_{M\sim\operatorname{Poisson}(\Lambda_0)}[(1 - M\Pi_2)^{-1}]$ is the Poisson-Moment Laplace constant, matching the exact numerical return time $S(T_{\mathrm{off}}^*) = S_{\mathrm{stop}}^*$ to within $0.0007\%$ across all clinical regimes.

Theorem 4.4 (Multi-Cycle Kendall-Volterra Stochastic Extinction Law & Bubble-Kill Horizon $P_{\mathrm{ext}}(kT)$) Bubble Compression: $\Omega_{\mathrm{bubble}}: 15.79 \to 3.31$ · $99\%$ Eradication in $17.45\text{ Cycles}$

When the resistant subclone $S_2$ approaches low cell numbers, demographic birth-death stochasticity during periodic Volterra cycles of period $T$ is governed by Kendall's inhomogeneous master equation with cycle contraction multiplier $\eta = e^{\gamma_{\mathrm{net}}T} > 1$. Over $k$ cycles, the cumulative hazard integral factors geometrically as $\mathcal{J}(kT) = \mathcal{J}_1(\eta^k - 1)/(\eta - 1)$ with exact Cycle Birth-Fluctuation Identity $\mathcal{J}_1 - \mathcal{B}_1 = \eta - 1$ and Stochastic Lineage Bubble Factor $\Omega_{\mathrm{bubble}} = \mathcal{J}_1/(\eta - 1) = 1 + \mathcal{B}_1/(\eta - 1) > 1$:

$$P_{\mathrm{ext}}(kT) = \left[1 + \frac{1}{\Omega_{\mathrm{bubble}}\big(e^{k\gamma_{\mathrm{net}}T} - 1\big)}\right]^{-n_2(0)} \simeq \exp\!\left(-\frac{N_{\mathrm{scale}}S_2(kT)}{\Omega_{\mathrm{bubble}}}\right), \qquad k_p = \frac{1}{\gamma_{\mathrm{net}}T}\ln\!\left(1 + \frac{\eta - 1}{\mathcal{J}_1\big(p^{-1/n_2(0)} - 1\big)}\right)$$

Because intracycle birth fluctuations inflate the surviving lineage size to $\mathbb{E}[N_2(kT) \mid N_2 > 0] \to \Omega_{\mathrm{bubble}}$ cells, single-drug PK containment ($\Omega_{\mathrm{bubble}} = 15.79$) requires $94.0\text{ cycles}$ ($2819\text{ d}$) for $99\%$ eradication of $n_2(0)=800\text{ cells}$. Activating the Bivariate Double-Bind Trap compresses $\Omega_{\mathrm{bubble}}$ by $4.77\times$ to $3.31$ and slashes the $99\%$ stochastic eradication horizon by $5.4\times$ to $k_{0.99} = 17.45\text{ cycles}$ ($523.4\text{ d}$) for `3RUK` ($S_2$) and $12.94\text{ cycles}$ ($388.3\text{ d}$) for `AR-V7` ($S_3$), verified to $0.06\%$ across $20,000$ exact Gillespie trajectories.

Interactive Verifier 02 · Real-Time Pharmacokinetic Trajectory Solver

Poisson-Bernoulli Washout & Volterra Containment Calculator

Structural Biophysics Grounding of $\alpha$ (PDB 2AM9 & 3RUK / AlphaFold P10275 & P05093)

In crystal coordinates of human CYP17A1 bound to abiraterone (3RUK, $2.60\text{ \AA}$), the 20 active-site residues within $4.5\text{ \AA}$ of the ligand sit at extreme core burial ($10.69 \pm 4.39\text{ \AA}$ vs $22.19 \pm 6.78\text{ \AA}$, Mann-Whitney $p = 9.95 \times 10^{-11}$), making active-site point mutations catalytically lethal and forcing resistance into energetic metabolic amplification ($\alpha \approx 1.28$), a 6-residue steric filter in the AR ligand-binding domain (2AM9, $p = 0.0045$, $\alpha \approx 1.12$), or splice deletion of the folded LBD in AR-V7 ($\Delta 629\text{--}920$, $\alpha \approx 1.65$).

Elimination Half-Life $t_{1/2}$ (days): 4.5 d

Controls washout decay rate $\gamma = \ln(2)/t_{1/2}$.

Competition Coefficient $\alpha$: 1.28

Sets default Darwinian floor $S_{\mathrm{floor}} = K/\alpha$.

Target Floor $S_{\mathrm{floor}}$ ($K=10000$): 7812.5

Minimum sensitive burden needed to suppress $R(t)$.

Exact Safe Stop $S_{\mathrm{stop}}^*$

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Poisson Integral $\mathcal{I}(t^*)$

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Uncompensated Nadir

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Volterra Speed Limit

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Max cycle $k_d\langle C\rangle_T$ for $R\downarrow$

Critical Half-Life $t_{1/2,\mathrm{crit}}$

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Holiday Return $T_{\mathrm{off}}^*$

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Kendall-Volterra $99\%$ Eradication Horizon $k_{0.99}$

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Post-Cessation Washout Trajectory $S(t)$ Over 60 Days ($r_S = 0.035\text{ d}^{-1}, k_d C_{\mathrm{off}} = 0.085\text{ d}^{-1}$)

Exact PK-Compensated Stop ($S(0) = S_{\mathrm{stop}}^*$) Uncompensated Floor Stop ($S(0) = S_{\mathrm{floor}}$) Containment Floor $S_{\mathrm{floor}}$

5-Cycle Multi-Clone Double-Bind Verification ($S_1(0)=8950, S_2(0)=900, S_3(0)=150, K=10000$) Overlap Asymmetry: $\mathcal{I}_{2\to 1}/\mathcal{I}_{1\to 2} = (\gamma_1/\gamma_2)^2 = 86.1184\times$

Dosing Protocol Arm	Stop Threshold	Min $S_1$ (Floor 7812.5)	Final $S_2$ (3RUK Res)	Final $S_3$ (AR-V7)	Evolutionary Outcome
1. Continuous MTD (No Holiday)	None	0.0	2,148.2	7,481.9	Competitive Release Relapse
2. Naive Adaptive Stop ($S_1 = S_{\mathrm{floor}}$)	7,812.5	5,969.4	501.3	0.0	Washout Floor Breach (-1,843.1)
3. PK-Compensated Single-Drug Stop	8,951.1	7,812.5	98.6	0.0	Zero Breach; Slow $S_2$ Decay
4. Uncompensated Double-Bind Switch	7,812.5	5,436.2	0.0	0.0	Severe Cross-Washout Over-Kill
5. Bivariate PK-Compensated Double-Bind ★	9,264.1	7,812.5	0.0000	0.0000	Exact Floor + Total $S_2, S_3$ Extinction ✓

Research & Non-Clinical Disclaimer: This monograph and interactive calculator present theoretical mathematical models and in silico analytical results for scholarly research in mathematical biology. They do not constitute clinical advice, a treatment protocol, or Software as a Medical Device (SaMD) under UK MHRA, EU MDR, or US FDA regulations, and must not be used to guide patient dosing decisions outside an approved clinical trial.

Monograph 03 · Quantitative Corpus Linguistics & Psychometrics · DOI: 10.5281/zenodo.23832151

The Split-Mark Vector $(m_L, m_R)$ Poset Identifiability Theorem & Macro/Micro Decoupling

Proving why recording the ordered split-mark tuple $(m_L, m_R) \in \{0,1\}^2$ on Cambridge Part 4 transformations resolves all 10 hierarchical CEFR grammatical mastery states in $N_{\min} = 3$ static items ($2.30$ adaptive items), alongside my longitudinal corpus findings on Macro/Micro Decoupling and Closed-Class Syntactic Cadence (SCQS).

📄 Read PDF 🔬 Open C1/C2 Lab & Scrubber ⚡ 3-Question Adaptive Test TeX Source

1. Poset Identifiability ($N_{\min}=3$)

Each Cambridge Part 4 item carries a two-channel Q-matrix $(q_L, q_R)$ over 5 syntactic skills. While dichotomous scoring (`0/1`) conveys at most $1\text{ bit/item}$ (requiring $\ge 4$ items for 10 states) and scalar sum scoring (`0/1/2`) conflates $(1,0)$ with $(0,1)$, the ordered vector $(m_L, m_R)$ yields $4^3 = 64$ response patterns and resolves all 10 Poset states into exact singletons ($\log_2 10 \approx 3.3219\text{ bits}$) in 3 items.

2. Macro/Micro Decoupling ($p = 0.7258$)

Across 67 annotated learner sentences ($N=47$ post-toolkit), candidates execute C1/C2 macro-frames (Wh-clefts, negative inversions, passive reporting) at 96.9% structural accuracy, yet L1 micro-slips (articles, uncountable plurals, prepositions) distribute orthogonally across C1 and non-C1 sentences ($\chi^2 = 0.3113$, Fisher $p = 0.7258$).

3. Syntactic Cadence Score ($\mathrm{AUC} = 1.000$)

My closed-class Syntactic Cohesion & Restraint Score (`SCQS`) penalises memorised template clichés and measures opener diversity alongside restrained C1 density (25% to 65% of sentences), separating authentic C1/C2 prose from formulaic template stuffing with $\mathrm{AUC} = 1.000$ without relying on rare-word lookup tables.

4. Diachronic Clause-Bitrate Law (1800 to 2026)

Across 15 paired 19th-century literary passages and 2026 C1/C2 transfers, per-word syntactic coding efficiency $\eta_{\mathrm{syn}} = \mathrm{PPS}/\bar{L}$ surges by $+53.14\%$ ($0.1101 \to 0.1687\text{ prop/word}$, $t(14) = +6.844, p = 8.0 \times 10^{-6}$) and propositional bitrate rises from $17.46 \to 26.73\text{ bits/100w}$ while conserving sentence-level load ($\mathrm{PPS}: 2.70 \to 2.92, p = 0.4778$).

Theorem 5.1 (Noisy Split-Mark Channel Capacity & Antipodal KL Blind-Spot Law) Stochastic DINA Psychometrics ($s, g \in [0, 0.25]$)

Under any stochastic DINA response channel with slip probability $s = P(M_{j,c}=0 \mid \eta_{j,c}=1)$ and guess probability $g = P(M_{j,c}=1 \mid \eta_{j,c}=0)$, antipodal half-credit states $\eta = (1,0)$ and $\eta' = (0,1)$ induce identical scalar-sum distributions $P(S_j \mid \eta) \equiv P(S_j \mid \eta')$, forcing zero Kullback-Leibler divergence $D_{\mathrm{KL}}(S_j) \equiv 0$ across all $(s,g) \in [0,1]^2$ (infinite sample complexity under scalar 0/1/2 marking). By contrast, the ordered Split-Mark vector $V_j = (M_{j,L}, M_{j,R})$ achieves exact symmetric Jeffreys divergence:

$$D_{\mathrm{KL}}\!\Big(P(S_j \mid 1,0) \,\Big\|\, P(S_j \mid 0,1)\Big) \equiv 0, \qquad J(V_j) = 2(1 - s - g)\log_2\!\left(\frac{(1-s)(1-g)}{sg}\right)\text{ bits/item}$$

At standard C1 testing noise ($s=0.05, g=0.05$), each split-mark item delivers $J(V_j) = 15.29\text{ bits}$ of antipodal discrimination ($D_{\mathrm{KL}} = 7.64\text{ bits}$ directional) and $I(\Theta; Y^{(3)}) = 2.4193\text{ bits}$ ($85.6\%$ 3-item Bayes accuracy vs $70.9\%$ for scalar sum and $61.7\%$ for dichotomous scoring).

Theorems 5.2 & 5.3 (Sequential Bayes Antipodal Trapping Law & $99.73\%$ Macro/Micro Shannon Orthogonality) $\mathbb{E}[N_{\mathrm{stop}}] = 2.30$ Items · $I(M;E) = 0.0047\text{ bits}$

In sequential Wald/Bayes active learning maximizing Expected Posterior Mutual Information $I(\Theta; Y_j \mid \mathcal{D}_{t-1})$, C1 states $\mathrm{S08} = (1,1,1,0,1)$ and $\mathrm{S09} = (1,1,1,1,0)$ have antipodal responses on $J_4$ ($\textsc{account}$) and identical responses on $\{J_2, J_3\}$, forcing the scalar-sum posterior odds ratio to remain identically $\pi_t(\mathrm{S08})/\pi_t(\mathrm{S09}) \equiv 1$ ($\pi_t \le 1/2$) along every trajectory $\mathcal{D}_t$ of any length $t \ge 1$ and permanently trapping scalar-sum adaptive tests below any $1-\delta > 50\%$ confidence threshold. Conversely, Split-Mark Vector scoring resolves $\{\mathrm{S08}, \mathrm{S09}\}$ in $\mathbb{E}[N_{\mathrm{stop}}] = 2.00$ noiseless items ($3.37$ items at $s=g=0.05$ with $99.73\%$ confidence completion). Across our empirical C1/C2 learner corpus ($N=47$ post-toolkit sentences), the Shannon mutual information between macro-syntactic frame deployment $M$ and L1 micro-syntactic slips $E$ is $I(M;E) = 0.004715\text{ bits}$ ($\mathrm{NMI}_{\mathrm{joint}} = 0.002709$, $99.73\%$ information-theoretic orthogonality), yielding the 2D Orthogonal Remediation Law:

$$\frac{\pi_t(\mathrm{S08})}{\pi_t(\mathrm{S09})}\Bigg|_{\text{Scalar Sum}} \equiv 1 \implies \pi_t \le \frac{1}{2} \;\; \forall t \ge 1, \qquad \mathbb{E}[\mathrm{EFSR}] \approx \hat{\theta}_{\mathrm{micro}}\big(1 - \rho_M(1 - \hat{\theta}_{\mathrm{macro}})\big), \qquad \frac{\partial\,\mathbb{E}[\mathrm{EFSR}]}{\partial\hat{\theta}_{\mathrm{micro}}} \approx 1.0$$

Theorem 5.4 (Fano-Le Cam Minimax Sample Complexity & Pairwise Chernoff Exponent Law) $P_{e,\max}^{\mathrm{sum}}(N) \ge 50\%\;\forall N$ vs $C_{\mathrm{Chernoff}}^*(\mathcal{B}_3) = 0.8304\text{ nats}$

By Fano's inequality, any $N$-item diagnostic test on the 10-state Poset $\mathcal{S}$ with Bayes error $P_e \le \varepsilon$ requires $N \ge N_{\min}^{\mathrm{Fano}}(m;\varepsilon) = (\log_2 10 - H_2(\varepsilon) - \varepsilon\log_2 9)/C_m(s,g)$, where $C_{\mathrm{vec}}^{\mathrm{Shan}}(s,g) = 2C_{\mathrm{BAC}}(s,g)$ doubles the dichotomous capacity. More stringently, Le Cam's two-point lemma proves that because $\mathrm{TV}^{\mathrm{sum}}(P_{\mathrm{S08}}^{(N)}, P_{\mathrm{S09}}^{(N)}) \equiv 0$ on $\mathcal{B}_3 = \{J_2, J_3, J_4\}$, the minimax worst-case error under Scalar Sum scoring is bounded below by $P_{e,\max}^{\mathrm{sum}}(N) \ge 1/2 = 50\%$ for all $N \to \infty$ ($N_{\min}^{\mathrm{LeCam}}(\mathrm{sum}, \varepsilon) = +\infty$). Conversely, across all $\binom{10}{2} = 45$ state pairs on $\mathcal{B}_3$, the Split-Mark Vector achieves the exact positive single-channel Chernoff exponent:

$$P_{e,\max}^{\mathrm{sum}}(N) \ge \frac{1}{2}\Big(1 - \mathrm{TV}^{\mathrm{sum}}(\mathrm{S08},\mathrm{S09})\Big) \equiv \frac{1}{2} \;\;\forall N \ge 1, \qquad C_{\mathrm{Chernoff}}^*(\mathcal{B}_3) = D_{\mathrm{KL}}\!\Big(\mathrm{Bern}(p^*) \,\Big\|\, \mathrm{Bern}(1-s)\Big) = -\ln\!\left((1-s)^{1-p^*}g^{p^*} + s^{1-p^*}(1-g)^{p^*}\right)$$

where $p^* = \ln[(1-s)/s] / \ln[(1-s)(1-g)/(sg)]$ ($p^* = 1/2$ when $s=g$). At $s=g=0.05$, $C_{\mathrm{Chernoff}}^*(\mathcal{B}_3) = -\ln(2\sqrt{0.0475}) = 0.8304\text{ nats}$ ($1.198\text{ bits}$), guaranteeing exponential error decay $P_e^{\mathrm{vec}}(N) \le 9\exp(-N \bar{C}_{\mathrm{Chernoff}}^*)$. Even on the full 6-item calibrated bank $\mathcal{B}_6$, Dichotomous `0/1` scoring suffers a $54.6\times$ Chernoff exponent degradation on $\mathrm{S02}$ vs $\mathrm{S03}$ ($0.0152\text{ nats}$ vs $0.8304\text{ nats}$), requiring $196$ item repetitions versus $3.6$ for Split-Mark Vector scoring.

Interactive Verifier 03 · Partition Entropy & Singleton Resolution

10-State Prerequisite Poset & $(m_L, m_R)$ Mutual Information Explorer

Select Active Diagnostic Items (Toggle to compare 1-, 2-, 3-, or 4-item batteries):

1. Dichotomous Scoring (0 / 1)

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2. Scalar Sum Scoring (0 / 1 / 2)

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3. Split-Mark Vector $(m_L, m_R)$

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4. Antipodal Jeffreys $J(V_j)$

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DINA Slip Probability $s = P(M=0 \mid \eta=1)$: 0.05

Models careless execution slips on mastered C1 syntactic channels.

DINA Guess Probability $g = P(M=1 \mid \eta=0)$: 0.05

Scalar Sum antipodal $D_{\mathrm{KL}}(S_j) \equiv 0.0000\text{ bits}$ at every $(s,g)$.

Exact Response Signatures Across the 10 Hierarchical Poset States ($\log_2 10 \approx 3.3219\text{ bits}$ Total Prior Entropy) Skills: [a1: Clause, a2: Unreal Past, a3: Passive, a4: Inversion, a5: Nominalisation+Prep]

Poset State	Skill Vector	CEFR Profile	Dichotomous (0/1)	Scalar Sum (0/1/2)	Split-Mark Vector (m_L, m_R)	Vector Status

Working Papers & Live Mathematical Verifiers

The Dyadic Kac-Murdock-Szegő Spectrum of Arithmetic Diamond Matrices and the \(n=14\) Phase Transition in Shifted Smith Determinants

\(S\)-Adic Deformed Diamond \(A_{n,S}^{(r)}(\lambda)\) & Shifted Smith 2-Adic Corank Explorer

Shifted Smith 2-Adic Phase Transition (\(n = 2 \dots 20\))

Open-Problem Resolutions in Coprime Hamiltonian Graphs, Permutation Circulants, and Modular Quaternion Rings

Test A227477, A236864, A339948, A392191 & A109813 for Any Integer \(n\)

Coprime Hamiltonian Path Count A109813(n) & Prime Cluster Partition

Exact Pharmacokinetic Washout Compensation and Volterra Cycle Conservation in Evolutionary Cancer Containment

Poisson-Bernoulli Washout & Volterra Containment Calculator

The Split-Mark Vector \((m_L, m_R)\) Poset Identifiability Theorem & Macro/Micro Decoupling

10-State Prerequisite Poset & \((m_L, m_R)\) Mutual Information Explorer