The Notebook Grid: Exact Addresses, Three Reflections, Pell Branches, and Rational Carriers

The Notebook Grid: Exact Addresses, Three Reflections, Pell Branches, and Rational Carriers

with Odd-Square Factorizations, Stack Shadows, and Divisor-Shell Arithmetic

Jeffery Huckstead · Cerebral Graphix (cerebralgraphix.com) · ORCID 0009-0007-0234-2177

Version 4.1 · November 2, 2026

Abstract

Writing the positive integers in successive triangular chambers gives an exact address N(r,c) = T_r+c-2 + r for every cell of the positive integer quadrant. We derive its two inverse formulas, characterize the admissible chamber-offset wedge (m,V), and prove the quadratic address families traced by affine cell paths. Holding a chamber m = r+c-1 fixed, three natural phases—the native row fraction κ_row = r/m, the cell-midpoint fraction κ_mid = (r-1/2)/m, and the nonlinear square-root phase κ_root = √(2N)-m+1/2—induce three distinct reflection maps under κ ↦ 1-κ. Only the midpoint reflection R_mid(N) = m²+1-N permutes a whole chamber; the native reflection R_row(N) = m²-N preserves the proper subset 1 ≤ r ≤ m-1; and the square-root reflection R_root(N) = (2m-√(2N))²/2 has an integer image if and only if the cell lies on the superdiagonal N = 2a² (m=2a), which it fixes. For scaled-square paths N_k = A(Qk)², a floor correction ℓ_k partitions the path into exact polynomial branches, separates the first branch departure from sparse power samples, and identifies norm-one triangular seams, algebraic returns at k=2P, and complete negative-Pell diagonal squares. Applying the midpoint fold 𝔣(κ_mid) = r̃c̃/(4m²) defines an upper-semicircle carrier map 𝒞(r,c) whose rational points are parameterized uniquely by odd squares (r̃,c̃) = λ(U²,W²) with primitive first chamber m₀ = (U²+W²)/2 ≡ 1 (mod 4). Its least nontrivial chamber is m₀ = 5, and its 45^∘ Pell record sequence m_0,j ∈ {1,5,29,169,985,…} maps bijectively to diagonal chambers m = G_j ∈ {1,7,41,239,1393,…} carrying square addresses N = m_0,j². On the address sheet p_N = (-r/(m+1), 1/(m+1)), the identity (2n+1)² = 8T_n + 1 = 4n(n+1) + 1 unites the eight-sector odd-square partition, the balanced split {n, n+1} and its finite reciprocal product, Euler's cosine generating function, a fixed-gap Gamma ratio, and depth-weighted isometric stack projection, while the same addresses carry divisor-band progressions, k-dimensional Jordan-totient shells, the factorial-residual dichotomy Q(N) = N/gcd(N,m!) ∈ {1}∪ℙ of asymptotic density log 2, and an entire-function realization of the triangular Taylor coefficients. Two self-contained appendices establish the complete rational-carrier chamber-budget theorems and the all-direction union-shadow theory of the three-dimensional odd-square stack Ω_n.

Route Through the Paper. The exposition follows a single chain of maps, domains, and declared measures:

1. Triangular Address & Chamber Coordinates (§1–§2): Positive quadrant cells (r,c) map bijectively to addresses N(r,c)=T_m-1+r in chamber m=r+c-1 with offset V=r-c and odd legs (r̃,c̃)=(2r-1,2c-1), turning affine lattice rays into degree-2 polynomial address sequences with period-4 offset parity.
2. Chamber Phases, Three Reflections & Scaled-Square Pell Branches (§3–§4): Fixing m distinguishes κ_row, κ_mid, and κ_root (with the anti-diagonal pair correspondence (k, p-k)), whose reflections and floor-branch drift ℓ_k = ⌊1/2+kε⌋ govern scaled-square Pell branches, norm-one triangular seams, and diagonal square returns.
3. Rational Carriers, Address Sheet & Odd-Square Factorization (§5–§6, Apps. A–B): Folding κ_mid lifts odd-square cells onto rational points of the upper semicircle 𝒞(r,c), while row normalization embeds the Grid into Q=[-1,0]×[0,1], separating primitive ratio from scale height, active-span rescaling (1/m) from staircase fixed points (1/k), flat screen sampling (2ω) from depth-weighted stack sampling (6ω(1-ω)), and uniting five routes on (2n+1)² = 8T_n + 1.
4. Open Questions, Divisor Shells, Jordan Totients & Factorial Residuals (§7–§8): Separating finite and open prime-pair and elliptic-curve questions (§7) from proved arithmetic on the addresses (§8) yields divisor-shell progressions, three separate Jordan-totient zeta limits, the complete factorial-residual classification Q(N)∈{1}∪ℙ with Dickman density log 2, and an entire-function differential lift of T_n/n!.

Scope, Proofs, and Verification. Every address, reflection, floor-branch, rational-carrier, address-sheet, odd-square factorization, Brouncker continuant, Euler-transform, divisor-shell, Jordan-totient, factorial-residual, entire-function coefficient-lift, chamber-budget, and three-dimensional stack-shadow theorem in §§1–6, §8, and Appendices A–B is proved algebraically, analytically, or by positive-integer descent in this paper. Classical completeness classifications on the rails (§4.6) and smooth-number asymptotics (§8.4) cite Nagell [1], Leu and Li [2], Cohn [3], and de Bruijn [13] with explicit hypotheses (with a self-contained 5-adic valuation proof for base 5 on the diagonal). Finite-range numerical and statistical statements carry explicit bounds: the shell-numerator diagnostic table (§8.2, Remark 8.1a) is a deterministic finite calculation at M ∈ {250, 500, 1000, 2000}; the odd-chamber mirror-prime verification (§7.1) covers the finite interval 1403 ≤ m ≤ 20001; and all-index mirror-prime existence (m ≥ 1403), all-index non-intersection of the a=2 and 45^∘ record-chamber families beyond m₀ ∈ {1,5,29} (§7.2), two-point prime occupancy (§8.5), and the analytic questions of §8.6 remain open. Throughout, the finite odd-side (j=2n+1) atomic screen-maximum laws Pr_flat(M=i)=(2i-1)/j² and Pr_stack(M=i)=(2i-1)[1+⌊(j-i)/2⌋]/C(j+2, 3) are kept distinct from the j→∞ weak-limit densities 2ω and 6ω(1-ω) of M/j; free congruence of the eight half-open max-norm sectors Σ₀,…,Σ₇ in the lattice-preserving Euclidean group E(2,ℤ) = ℤ² ⋊ O(2,ℤ) ⊂ E(2) = ℝ² ⋊ O(2) via F(a,b)=(b+1,a) is kept distinct from origin-fixing O(2) non-equivalence; and the orthochronous boost conjugacy C_D L_D C_D^-1 = Λ_βD ∈ SO⁺(1,1) is restricted to the positive Lorentz sheet P₁ > 0 (the Cayley images μ(1/√D) are fundamental integer Pell units exactly for D ∈ {2,3}). An optional standard-library Python script (check_notebook_grid_v4_1.py) accompanies the supplementary archive for finite-range computational corroboration; no proof step in the paper depends on executing code.

Summary of Main Results

1. The Address, Its Chambers, and Phase Centering Conventions

1.1 Enumerating the Positive Quadrant

Let the triangular polynomial be defined for every complex argument z ∈ ℂ, and write its integer values with subscripts:

For each fixed anti-diagonal sum r+c, the term r counts the sites along that anti-diagonal. Define the chamber index m, the row offset V, and the centered odd legs (r̃, c̃) by

For each fixed chamber m ≥ 1, the rows r = 1, …, m (with c = m + 1 - r) carry the consecutive integer addresses T_m-1 + 1, …, T_m. The first chamber (m=1) contains 1; the second (m=2) contains 2 and 3; the third (m=3) contains 4, 5, and 6. These disjoint consecutive intervals partition ℤ_{≥ 1} without gaps. We distinguish the integer address N written in a cell, its row and column (r,c), and its chamber m; none is a substitute for the others.

Proposition 1.1 (Exact Chamber Wedge and Parity Lattice). A positive integer cell (r,c) ∈ ℤ_{≥ 1}² of address N = N(r,c) satisfies

Conversely, an integer pair (m,V) ∈ ℤ² represents a positive integer cell (r,c) ∈ ℤ_{≥ 1}² if and only if

Proof. In chamber m, N = T_m-1 + r = (m(m-1))/2 + r. Multiplying by 2 and substituting 2r = m + V + 1 yields 2N = m² - m + (m + V + 1) = m² + V + 1, and solving r+c=m+1, r-c=V gives the coordinate recovery formulas in (1.3). Conversely, given (m,V) ∈ ℤ², the parity condition m+V ≡ 1 (mod 2) makes r and c integers, and the inequalities m ≥ 1, -m+1 ≤ V ≤ m-1 make r ≥ 1 and c ≥ 1. Substituting (r,c) into (1.1) recovers N. Thus (1.4) is both necessary and sufficient. In particular, any algebraic curve in the (m,V) plane represents actual positive grid cells only where it meets both the parity lattice and the chamber wedge (1.4). □

1.2 Two Exact Inverse Formulas and the Origin Convention

Proposition 1.2 (Inverse Address Recovery). For each positive integer address N ∈ ℤ_{≥ 1}, its unique chamber m and cell coordinates (r,c) are given by

Equivalently, the chamber m is recovered by nearest-integer rounding of √(2N):

Proof. Because the chamber intervals (T_m-1, T_m] ∩ ℤ partition ℤ_{≥ 1}, the unique chamber m satisfies ((m-1)m)/2 < N ≤ (m(m+1))/2. Completing the square 8T_m + 1 = (2m+1)² gives 2m - 1 < √(8N+1) ≤ 2m + 1, which is equivalent to (1.5). Within chamber m, integrality of N strengthens the lower inequality N ≥ T_m-1 + 1 to

while the upper inequality N ≤ T_m gives

Taking positive square roots yields the strict inequalities in (1.6) and the floor formula m = ⌊ √(2N) + 1/2 ⌋. A nearest-integer tie √(2N) = m ± 1/2 can never occur for an integer N, because 2N is an integer whereas (m ± 1/2)² = m² ± m + 1/4 ∉ ℤ. □

As a worked address, N = 25 lies between T₆ = 21 and T₇ = 28. By (1.5)–(1.6), it has chamber m = 7, cell (r,c) = (4,4), and offset V = 0. The identity 2 · 25 = 7² + 1 describes its chamber position (m=7, V=0), whereas 25 = 5² describes its square address (Q=5). Keeping the chamber index m and the address root Q separate is essential in §§4–5.

Origin Convention. The positive grid has its corner cell at (r,c) = (1,1), holding N(1,1) = 1. Translating the drawing's corner to a Cartesian origin (0,0) replaces (r,c) by (r',c') = (r-1, c-1) ∈ ℤ_{≥ 0}²:

This coordinate translation changes where the axes sit in a drawing without creating a cell of address 0 in the positive Grid. Extending T(z) to z=0 or negative arguments (such as the vertex z = -1/2 where T(-1/2) = -1/8) is a separately declared polynomial operation.

1.3 Relation to Rational Triangular Coordinates and Centering Conventions

The cumulative address (1.1) shares its indexing with the rational triangular-fractional coordinate studied in [4, 11], where chamber m ≥ 1 carries the m entries

and the rank of the entry a(m,r) in this enumeration is exactly the address N(r,c) = T_m-1 + r with c = m + 1 - r [7]. Two structural distinctions are maintained throughout this paper:

1. Native Row Centering vs. Cell-Midpoint Centering: Using the affine centering map 𝒥(w) = 2w - 1 and quadratic fold 𝔣(w) = w(1-w) = (1 - 𝒥(w)²)/4 [5], centering a chamber by its native row phase κ_row = r/m yields the integer displacement

By contrast, cell-transpose symmetry (r,c) ↔ (c,r) centers a chamber by its cell-midpoint phase κ_mid = (r - 1/2)/m = r̃/(2m), for which

Consequently, the native half-row center 𝒥(κ_row) = 0 is the superdiagonal cell V = -1 (occurring only in even chambers m = 2a, at r=a, c=a+1, N=2a²), whereas the midpoint fixed center 𝒥(κ_mid) = 0 is the diagonal cell V = 0 (occurring only in odd chambers m = 2k-1, at r=c=k, N=k²+(k-1)²). This unit shift j_row = V + 1 explains why the two centering conventions select opposite chamber parities.

1. Signed Binomial Arrays vs. Positive Quadrant Addresses: Companion studies of alternating binomial coefficient arrays (-1)^jC(k, j) and their row-reversal involutions [4] act on finite polynomial coefficient tables rather than on the positive integer cell addresses N(r,c) and fixed-chamber spatial reflections proved in §3 below.

2. Straight Cell Paths, Five Rails, and Parity

2.1 Every Affine Cell Path Has a Quadratic Address

Proposition 2.1 (Affine Path Polynomial). Fix a baseline cell (r₀, c₀) ∈ ℤ_{≥ 1}², integer increments (Δ_r, Δ_c) ∈ ℤ², and an integer parameter k ∈ ℤ, retaining precisely the indices k for which r(k) = r₀ + Δ_r k ≥ 1 and c(k) = c₀ + Δ_c k ≥ 1. Set L = r₀ + c₀ - 2 and D = Δ_r + Δ_c. Then

Proof. Along the path, the triangular index is r(k) + c(k) - 2 = L + Dk. Expanding T(L + Dk) = T_L + (D(2L+1))/2 k + D²/2 k² and adding r(k) = r₀ + Δ_r k yields (2.1). □

Every affine cell path therefore traces an address polynomial in k of degree at most 2. When D = Δ_r + Δ_c = 0, the path remains inside a single chamber m = L + 1 and its quadratic term vanishes, leaving a linear function of k.

2.2 The Five Distinguished Rails

Five affine rays carry most of the structure below. We name them by position; the colors are the keys used in the figures (and in the original notebook). Each is indexed by an integer parameter k ≥ 1:

In chamber and odd-row coordinates (m, r̃), all five families are straight lines:

As functions of the chamber m, however, the three interior rails carry quadratic addresses:

Thus a statement that the diagonal is "linear" requires naming the horizontal axis: in (m, r̃) it is the line r̃ = m; against the triangular input T_k-1 its address is 4T_k-1+1; against the chamber m its address is the quadratic (m²+1)/2.

2.3 Fixed-Offset Rails, Parity Period Four, and Odd Powers of Two

Proposition 2.2 (Constant Parity of Fixed-V Rails). Fix an integer offset V ∈ ℤ, and let r ∈ ℤ vary over r ≥ max(1, V+1) with c = r - V ≥ 1. Then

Consequently, every fixed-V rail has constant address parity throughout the positive quadrant, and as V increases by 1 that rail parity repeats with period 4 in the pattern (1, 1, 0, 0) starting at V = 0, 1, 2, 3.

Proof. In (1.1), r + c - 2 = 2r - V - 2, so

The polynomial identity T(-V-2) = T(V+1) holds for all V ∈ ℤ. Since 2r² - 2(V+1)r is even for every integer r, N(r, r-V) ≡ T_V+1 (mod 2). Moreover, T_V+5 - T_V+1 = 4V + 14 ≡ 0 (mod 2), giving period 4 with values T₁=1, T₂=3≡ 1, T₃=6≡ 0, T₄=10≡ 0 (mod 2) at V = 0, 1, 2, 3. □

Remark 2.3 (Connection to the One-Dimensional Gray Cycle and the Modular Coefficient Rows). In one dimension, consecutive triangular parities form the adjacent pair (T_n-1 mod 2, T_n mod 2), which traverses the four-state Gray cycle 00 → 01 → 11 → 10 → 00 by the quarter-turn F(a,b) = (b, 1-a), while index reflection n ↦ -1-n acts by E(a,b) = (1-a, b), together generating D₄. That one-dimensional index-cycle reflection on modular rising-factorial rows Π_n(y) = ∏_j=1ⁿ (1+jy) mod (B, y^K+1) [7] acts across consecutive triangular indices n and is distinct from the fixed-chamber spatial reflections in §3 below.

Finally, the odd powers of two lie exactly on the even-parity superdiagonal V = -1:

This sparse exponential subsequence lies inside the quadratic rail N = 2k² (k = 2^j) and must not be confused with the piecewise parabolic branches traced by even powers of two (4^j = (2^j)²) in §4.

3. Three Reflections and Their Surviving Integer Domains

3.1 Three Ways to Center and Reflect a Chamber

Hold an integer chamber m ≥ 1 fixed. Within chamber m, the native row phase κ_row, the cell-midpoint phase κ_mid, and the square-root phase κ_root (with local root displacement ε) are defined by

On a positive cell (r,c) in chamber m, κ_row = r/m ∈ (0, 1] is a right-endpoint step phase (attaining 1 on the first-column edge r=m, c=1, whereas the address-sheet coordinate -p₁ = κ_leg = r/(m+1) = r/(r+c) ∈ [1/(m+1), m/(m+1)] ⊂ (0,1) of §6.1 divides by m+1 = r+c so both coordinates remain strictly positive inside the open unit interval), while both κ_mid = (r-1/2)/m and κ_root = √(2N)-m+1/2 are interior cell phases lying strictly in (0, 1) by Proposition 1.2 (-1/2 < ε < 1/2). Consequently, at the first-column endpoint r=m, c=1 (N=T_m), the involution κ_row ↦ 1 - κ_row sends κ_row=1 to 0, mapping R_row(T_m) = m² - T_m = T_m-1 out of chamber m into the preceding chamber m-1 (or to 0 at m=1). Although all three coordinates represent positions across chamber m, applying the scalar involution κ ↦ 1 - κ while holding m fixed induces three different address maps.

Theorem 3.1 (Three Reflection Maps and Their Domains). Fix an integer chamber m ≥ 1. Reflecting κ_row, κ_mid, and κ_root by κ ↦ 1 - κ induces the following address transformations on chamber m:

Moreover, in terms of the offset V = 2mε + ε² - 1, the transformed offsets and the exact nonlinear difference between R_root and R_row are

Proof. Under κ_row ↦ 1 - κ_row, the row r = mκ_row maps to r' = m - r = c - 1, giving address T_m-1 + (m - r) = m² - N and column c' = m + 1 - r' = r + 1. This image lies in chamber m with positive coordinates iff 1 ≤ r ≤ m-1 (c > 1). At the first-column endpoint r = m, c = 1 (N = T_m), R_row(T_m) = m² - T_m = T_m-1, which is a positive address in the preceding chamber m-1 when m ≥ 2 and equals 0 (outside the positive grid) when m = 1.

Under κ_mid ↦ 1 - κ_mid, r - 1/2 maps to m - (r - 1/2), so r' = m + 1 - r = c, c' = r, and R_mid(N) = T_m-1 + (m + 1 - r) = m² + 1 - N, which is a bijection on {1, …, m}.

Under κ_root ↦ 1 - κ_root, ε = κ_root - 1/2 maps to -ε, so √(2N) = m + ε maps to m - ε = 2m - √(2N) ∈ (m - 1/2, m + 1/2), giving R_root(N) = (2m - √(2N))²/2. Expanding 2N = (m+ε)² = m² + 2mε + ε² and comparing with 2N = m² + V + 1 gives V + 1 = 2mε + ε², so

which establishes all relations in (3.2). □

Single-Cell Distinguishing Witness. For address N = 5 in chamber m = 3 (cell (2,2), V = 0), the three reflections give

with 4 < 23 - 6√10 < 5. This single cell rules out any identification of the three reflections.

3.2 Integer-Image Rigidity and Root–Midpoint Discrepancy

Proposition 3.2 (Integer-Image Rigidity of Root Reflection). Among positive integer cells, R_root(N) is an integer if and only if the cell lies on the superdiagonal:

and every such cell is fixed by R_root.

Proof. Expanding R_root(N) = 2m² + N - 2m√(2N) shows that if R_root(N) ∈ ℤ, then √(2N) ∈ ℚ. Because 2N ∈ ℤ, a rational square root of 2N must be an integer. By Proposition 1.2, m - 1/2 < √(2N) < m + 1/2, so the only integer in this open interval is m. Hence √(2N) = m, so 2N = m² and ε = 0. Integrality of N = m²/2 forces m = 2a even, giving N = 2a² and (r,c) = (a, a+1). Conversely, ε = 0 gives R_root(2a²) = 2a². □

Proposition 3.2 proves without any unbounded search that R_root never exchanges distinct integer Pell families: off the fixed superdiagonal, R_root(N) is always irrational. Even the affine map R_row need not preserve squares (in chamber m=7, R_row(25) = 49 - 25 = 24).

Proposition 3.3 (Root–Midpoint Phase Discrepancy Bounds). On every positive integer cell (r,c) in chamber m ≥ 1,

Equality at the upper bound 1/(2m) holds if and only if the cell is a superdiagonal cell (3.4), and the lower bound 3/(8m) is strict on every positive cell.

Proof. Since V = 2mε + ε² - 1, we have

which gives m(κ_root - κ_mid) = (1 - ε²)/2. Proposition 1.2 gives 0 ≤ ε² < 1/4, which yields 3/(8m) < κ_root - κ_mid ≤ 1/(2m), with upper equality iff ε = 0. □

Remark 3.4 (Anti-Diagonal Pair Correspondence and Even-Chamber Midpoint Reflection). For any integer p ≥ 2 and step 1 ≤ k ≤ p - 1 (no primality condition on p is required), the anti-diagonal pair (r, c) = (k, p - k) studied in the companion sum–difference geometry [15] lies on the Notebook Grid at

Exchanging the pair (k, p-k) ↦ (p-k, k) sends N = T_p-2 + k to N' = T_p-2 + (p - k). Since 2T_p-2 + p = (p-2)(p-1) + p = (p-1)² + 1 = m² + 1, we have N' = m² + 1 - N = R_mid(N): coordinate exchange on (k, p-k) is identically the midpoint reflection R_mid on chamber m = p - 1. When p is odd, the chamber m = p - 1 is even and every offset V = 2k - p is odd, so the diagonal center V = 0 is absent from chamber m and the central adjacent pair ((p-1)/2, (p+1)/2), ((p+1)/2, (p-1)/2) occupies the superdiagonal and subdiagonal offsets V = -1 and V = +1. Equation (3.6) records the discrete pair correspondence; the companion polar scalar of [15], the exchange-commuting 2× 2 Gram family B_a = I + aE (characterized in §6.6, Remark 6.6b(1)), and Gaussian-integer lattice squares live in distinct ambient spaces and are not identified as a single point set, nor is this even chamber m = p - 1 (for odd p) identified with the odd-chamber (m = 2M + 1) midpoint prime-pair setting of §7.1.

4. Scaled-Square Branches, Pell Seams, and Classical Powers

4.1 Exact Floor-Selected Polynomial Branches

Proposition 4.1 (Floor-Branch Dictionary). Fix positive integers A, Q ∈ ℤ_{≥ 1}, and let P ∈ ℤ_{≥ 1} be the unique nearest integer to Q√(2A). For each integer k ≥ 1, define

Then the chamber m_k, offset V_k, and cell coordinates (r_k, c_k) of N_k are given exactly by

Proof. By Proposition 1.2, m_k = ⌊ √(2N_k) + 1/2 ⌋ = ⌊ k(P + ε) + 1/2 ⌋ = Pk + ℓ_k because Pk ∈ ℤ. Substituting m_k = Pk + ℓ_k and 2AQ² = P² - δ into V_k = 2N_k - m_k² - 1 = 2AQ² k² - (Pk + ℓ_k)² - 1 yields (4.2). □

On any interval of integer indices k where the integer floor ℓ_k = L remains constant, m_k is linear in k and V_k is quadratic in k. That polynomial formula applies only on the floor-selected domain L - 1/2 ≤ kε < L + 1/2.

Proposition 4.2 (Asymmetric First-Branch Criterion). For the scaled-square path (4.1), a positive integer index k ≥ 1 lies on the initial branch ℓ_k = 0 if and only if

Proof. The condition ℓ_k = 0, i.e. m_k = Pk, holds iff T_Pk-1 < A(Qk)² ≤ T_Pk. Multiplying by 2 gives (Pk)² - Pk < (P² - δ)k² ≤ (Pk)² + Pk. Subtracting P² k² and dividing by k > 0 yields -P < -δ k ≤ P, or equivalently -P ≤ δ k < P. □

The strict upper bound δ k < P (for δ > 0) versus the closed lower bound | δ | k ≤ P (for δ < 0) comes directly from the half-open chamber interval T_m-1 < N ≤ T_m. On the first branch ℓ_k = 0, the offset simplifies to V_k + 1 = -δ k².

4.2 The Zero-Norm Domain

Proposition 4.3 (Zero-Norm Characterization). For positive integers A, Q ∈ ℤ_{≥ 1} and P = round(Q√(2A)),

In that case ℓ_k = 0, m_k = 2bQk, and V_k = -1 for all k ≥ 1 (the straight superdiagonal).

Proof. If δ = P² - 2AQ² = 0, then (P/Q)² = 2A ∈ ℤ, so P/Q is an integer whose square is even, hence P/Q = 2b for some b ∈ ℤ_{≥ 1}. Thus A = 2b² and P = 2bQ. The converse is immediate. □

In particular, (A,P,Q) = (8, 4, 1) has δ = 0. Stating that δ = 0 occurs "only for A = 2" requires restricting A to be squarefree.

4.3 The Worked 181/128 Square Branch, First Departure, and Phase-Linearity Shift

Take A = 1, P = 181, Q = 128, for which δ = 181² - 2 · 128² = 32761 - 32768 = -7. By Proposition 4.2, the first branch ℓ_k = 0 holds iff 7k ≤ 181, so its exact integer domain is 1 ≤ k ≤ 25:

The powers of four 4⁷, 4⁸, 4⁹, 4¹⁰, 4¹¹ appear along this branch at k = 1, 2, 4, 8, 16 at cells (94, 88), (195, 168), (418, 307), (948, 501), and (2344, 553). The first integer departure of the full branch N_k = (128k)² occurs at k = 26, where ℓ₂₆ = 1, m₂₆ = 4707, and (r₂₆, c₂₆) = (13, 4695). By contrast, the next sampled power of four (4¹²) occurs later at k = 32, where ℓ₃₂ = 1, m₃₂ = 5793, V₃₂ = -4418, and (r₃₂, c₃₂) = (688, 5106). Distinguishing the full quadratic sequence from its sparse exponential subsequence prevents confusing k=26 with k=32.

Exact Phase-Linearity Shift on the First Branch. On 1 ≤ k ≤ 25, substituting m_k = 181k and V_k = 7k² - 1 into 2r_k = m_k + V_k + 1 and r̃_k = m_k + V_k yields

Thus the native row normalization 2κ_row is strictly linear in the chamber m_k across the entire first branch, whereas the centered odd-row normalization 2κ_mid = r̃_k/m_k carries the exact reciprocal offset -1/m_k.

Remark 4.3a (Cross-Reference: The (7, 180, 181) Reciprocal Bridge and the Ramanujan–Nagell Solutions). The arithmetic parameters of the 181/128 floor branch in (4.5)–(4.6) align with two structural bridges developed in [4, 16]:

1. Grid 181/128 Branch Parameters (δ = -7, m₁ = 181 = 180 + 1): At the base step k=1, the Ramanujan–Nagell relation 181² - 2· 128² = -7 (181² + 7 = 2¹⁵) places N₁ = 128² = 16384 = 4⁷ in chamber m₁ = 181 = 180 + 1 immediately above T₁₈₀ = (180· 181)/2 = 16290, at (r₁, c₁) = (94, 88) and offset V₁ = 6. In (4.6), the phase-linearity slope is 7/181 per step k (7/181² per chamber unit m_k), governed by |δ| = 7, while 2κ_mid(k) - 2κ_row(k) = -1/m_k = -1/(181k). Because P = 181 = 180 + 1 is prime, all φ(181) = 180 proper interior row phases κ_row = r/181 (1 ≤ r ≤ 180) in chamber m₁ = 181 are reduced fractions, and at n = 180 the rising-factorial interior gcd of Theorem 8.3(5) evaluates to g₁₈₀ = 181.
2. Reciprocal, Catalan, and Degree-to-Radian π-Phase Bridge at n = 7 [4]: At the third Mersenne index n = 7 = 2³ - 1, the triangular number is T₇ = 28 (2T₇ = 56 = 7· 8). Because n+1 = 8 is composite, T₇/7! = 1/180 with reduced denominator b₇ = (7!)/T₇ = 5040/28 = 180 = (3!)² C₃ (the j=3 odd-slice Catalan denominator in T_2j+1/(2j+1)! = 1/[(j!)² C_j]), giving

while across dyadic halvings i ≥ 0, 1/(7· 2ⁱ) = 100^-i(A_i + 1/7) with A_i = (50ⁱ - 1)/7 ∈ ℤ (50 ≡ 1 (mod 7)).

1. The Value u = 1/8 and the Rational π Bracket [4]: In the genus-zero triangular cosine product Π_△(u) = ∏_{n≥ 1}(1 - u/T_n) = -(cos(π/2 √(1+8u)))/(2π u) of [4] (Remark 6.6a), the value u = 1/8 gives the first-factor ratio u/(T₁ - u) = 1/(8T₁ - 1) = 1/7 and rational bounds 1/8 < π - 3 < 1/7 ⇔ 25/8 < π < 22/7, with 11/4 - (7π)/8 = 7/8 (22/7 - π). Scope: On the 181/128 floor branch, (7, 180, 181) arises from δ = 181² - 2· 128² = -7 and prime chamber m₁ = 181, whereas at n = 7 it arises from 7!/T₇ = 180 and degree-to-radian conversion π/180; spatial chamber phase κ_row = r/m and angular radian conversion are kept distinct.
2. The Complete Ramanujan–Nagell Solution Set [1, 16]: Writing the odd chamber index as m₁ = 2n + 1 so m₁² = 8T_n + 1, the binary-square condition m₁² + 7 = 8(T_n + 1) = 2^μ is Nagell's Mersenne-triangular equation M² + 7 = 2^μ ⇔ T_n = 2^μ-3 - 1 [1], whose complete nonnegative integer solution set is

• The largest solution n = 90 (2n = 180, M = 181, T₉₀ = 4095 = 2¹²-1, μ = 15): Here 181² + 7 = 32768 = 2¹⁵, so on 1 ≤ k ≤ 25, 2N(r_k,c_k) = m_k² + V_k + 1 = (181² + 7)k² = 2¹⁵k² ⇔ N(r_k,c_k) = (128k)², and 1 + 7/181² = 2¹⁵/181² = ((128√2)/181)².
• The solution n = 5 (M = 11, T₅ = 15 = 2⁴-1, μ = 7): The solution M = 11 (11² + 7 = 2⁷) yields the penultimate V=6 binary square N = 64 = 4³ in chamber m = 11 (Eq. 4.13), the unique nontrivial iterated triangular-factorial identity T_T5 = T₁₅ = 120 = 5! (since T_Tn = n! fails for n ∈ {1,2,3,4} and for n ≥ 6 Stirling's bound gives n! > 1/8 n⁴ ≥ T_Tn for all n ≥ 7 with T₂₁=231 ≠ 720 = 6!), and the largest pure power of 3 on the subdiagonal at k = 11, where 2(11)² + 1 = 243 = 3⁵ in chamber m = 22 (Eq. 4.14).
• The reciprocal-π product: In Π_△(-1/8) = ∏_{n≥ 1}((2n+1)²)/(8T_n) = 4/π (§6.6, Theorem 6.5), the four positive solutions n ∈ {1, 2, 5, 90} (M ∈ {3, 5, 11, 181}) are the unique stages where numerator (2n+1)² = 2^μ - 7 and denominator 8T_n = 2^μ - 8 are simultaneously shifted powers of two.

4.4 Norm-One Seams, k = 2P Returns, and the Square-Triangular Converse

Proposition 4.4 (Norm-One Launch and Triangular Seams). Assume δ = P² - 2AQ² ∈ {-1, +1}. Then the per-step root drift is

At k = 1, the path launches at m₁ = P, V₁ = -δ - 1. At k = P, the path lands on an exact triangular seam:

Proof. Multiplying (P - Q√(2A))(P + Q√(2A)) = δ gives (4.7). At k = P, 2N_P = P²(2AQ²) = P²(P² - δ), so N_P = (P²(P²+1))/2 = T_P² when δ = -1 and N_P = ((P²-1)P²)/2 = T_P²-1 when δ = +1. By Proposition 4.2, k=P satisfies -P ≤ δ P < P for δ = -1 (ℓ_P = 0) and violates it by one chamber for δ = +1 (ℓ_P = -1). □

Proposition 4.5 (Exact Algebraic Return at k = 2P). Under δ = P² - 2AQ² = ± 1, set M = P² + 2AQ². At k = 2P,

corresponding to the quadratic-unit squaring identity (P + Q√(2A))² = M + 2PQ√(2A).

Proof. Expanding M² - 2A(2PQ)² = (P² - 2AQ²)² = δ² = 1 gives 2N_2P = 2A(2PQ)² = M² - 1. Since M = 2P² - δ ≥ 3 is odd, (M - 1/2)² < M² - 1 < (M + 1/2)², so Proposition 1.2 gives m_2P = M and Proposition 1.1 gives V_2P = 2N_2P - M² - 1 = -2. □

Proposition 4.6 (Complete Square-Triangular Converse). Every positive square-triangular number T_n = L² (n, L ∈ ℤ_{≥ 1}) arises as an A = 1 norm-one seam (4.8): there exist unique positive integers P, Q ∈ ℤ_{≥ 1} with P² - 2Q² = δ ∈ {-1, +1} such that L = PQ and n = P² (δ = -1) or n = P² - 1 (δ = +1).

Proof. In n(n+1) = 2L², gcd(n, n+1) = 1, so the odd member of {n, n+1} is a square P² and the even member is twice a square 2Q². Then L = PQ and P² - 2Q² = ∓ 1. Since | Q√2 - P | = 1/(P + Q√2) < 1/2, P = round(Q√2), placing (P,Q) in the A=1 branch family (4.8). □

4.5 Square Addresses on the Diagonal and the Subdiagonal

Proposition 4.7 (Diagonal-Square Pell Equivalence). For any square address N = Q² (Q ∈ ℤ_{≥ 1}) in chamber m = ⌊ Q√2 + 1/2 ⌋, its offset is V = 2Q² - m² - 1. Consequently:

1. N = Q² lies on the diagonal (V = 0) if and only if

1. N = Q² lies on the subdiagonal (V = +1) if and only if

whose first positive instances are (m, Q) = (4, 3), (24, 17), (140, 99) with square addresses N = 9, 289, 9801.

Proposition 4.8 (Complete Negative-Pell Descent for Diagonal Squares). Every positive integer solution of m² - 2Q² = -1 is generated from the base solution (m₀, Q₀) = (1, 1) by iterating

Proof. The map (4.12) preserves m² - 2Q² = -1 and positivity. Conversely, let (m,Q) be a positive solution with Q > 1. Testing Q ∈ {2,3,4} gives 2Q² - 1 ∈ {7, 17, 31}, none of which is a square, so Q ≥ 5. Hence 4/3 < m/Q = √(2 - 1/Q²) < 3/2. Applying the inverse integral matrix gives

which satisfies m' > 0, Q' > 0, (m')² - 2(Q')² = -1, and Q' < Q (since m > Q ⇒ 3Q - 2m < Q). Iterating this strict positive-integer descent must terminate at Q = 1, m = 1. Reversing the steps yields (4.12). □

The first five diagonal-square solutions are (m, Q) = (1,1), (7,5), (41,29), (239,169), (1393,985), with diagonal row indices k = (m+1)/2 ∈ {1, 4, 21, 120, 697} and square addresses N = Q² ∈ {1, 25, 841, 28561, 970225}.

4.6 Classical Classifications of Powers on the Rails

Several visually prominent power intersections on the rails reduce to classical exponential Diophantine equations:

• Powers of 2 on the First-Row Edge and Offset V = 6: On the first-row edge (1, c), N = 2^j (j ≥ 0) is equivalent to T_c-1 + 1 = 2^j, or

By Nagell's theorem [1], the positive odd solutions 2c-1 are 1, 3, 5, 11, 181, giving j ∈ {0, 1, 2, 4, 12} and addresses N ∈ {1, 2, 4, 16, 4096}. Similarly, since odd powers of 2 sit on V = -1 (2.5), an even power 4^k sits at offset V = 6 iff P² + 7 = 2^2k+1 with P = round(2^k√2); Nagell's list intersected with the nearest-integer bound leaves P ∈ {11, 181}, so only 64 (m=11) and 16384 (m=181) are powers of 2 with V = 6.

• Powers of 3 on the Subdiagonal: On the subdiagonal (k+1, k), N = 3^j (k, j ≥ 1) is

By Leu and Li [2], the only positive solutions are (k, j) ∈ {(1,1), (2,2), (11,5)}, giving addresses N ∈ {3, 9, 243}.

• Powers of 5 on the Diagonal: On the diagonal (k, k), N = 5^j (k, j ≥ 1) is

Cohn's perfect-Pell-power lemma [3] excludes j > 2 for base 5, leaving (k, j) ∈ {(2,1), (4,2)}, diagonal chambers m = 2k - 1 ∈ {3, 7}, and diagonal addresses N = N(k,k) = 5^j ∈ {5, 25} (where (2k-1)² + 1 = 2N ∈ {10, 50}; Cohn's exceptional higher-power solution for an arbitrary base gives (239²+1)/2 = 28561 = 13⁴ at k=120). Moreover, for the fixed base 5 in (4.15), no appeal to Ljunggren's k=4 theorem is required, using two short self-contained prerequisites: (i) in ℤ[√10], any integral unit ω = A + C√10 > 1 of norm A² - 10C² = ± 1 has positive integers A, C ≥ 1, where C = 1 gives A = 3 (ε₁₀ = 3 + √10 of norm -1) and C ≥ 2 gives A + C√10 ≥ 1 + 2√10 > ε₁₀, so choosing a ≥ 1 with ε₁₀^a ≤ ω < ε₁₀^a+1 places ωε₁₀^-a ∈ [1, ε₁₀) and forces ω = ε₁₀^a (with a odd whenever Norm(ω) = -1); and (ii) for the Pell recurrence matrix G = ([ 2 1 ; 1 0 ]), G³ = ([ 12 5 ; 5 2 ]) ≡ 2I (mod 5), so P_3a+i ≡ 2^a P_i (mod 5) (i ∈ {0,1,2}) with P₀ = 0, P₁ = 1, P₂ = 2, whence 5 | P_n ⇔ 3 | n. Consequently:

• When j = 2b + 1 is odd, (4.15) is (2k-1)² - 10(5^b)² = -1, so Prerequisite (i) gives 5^b = C_2a+1 = ∑_i=0^a C(2a+1, 2i+1) 3^2(a-i) 10ⁱ in (3+√10)^2a+1 (a ≥ 0). Using C(2a+1, 2i+1) = (2a+1)/(2i+1) C(2a, 2i) and v₅(2i+1) ≤ i - 1 < i for all i ≥ 1, the i-th summand has valuation ≥ v₅(2a+1) + i - v₅(2i+1) ≥ v₅(2a+1) + 1 > v₅(2a+1) (a sharp one-unit gap, attained at a=1, i=1 where (3+√10)³ = 117 + 37√10 has 37 = 27 + 10 with v₅(10)=1). Thus the i=0 term (2a+1)3^2a strictly dominates the 5-adic valuation, forcing v₅(C_2a+1) = v₅(2a+1) and hence 5^b | (2a+1). Since C_2a+1 ≥ (2a+1)3^2a > 2a+1 for all a ≥ 1, this holds only at a=0, b=0, j=1 (k=2, chamber m=3, diagonal address N=5).
• When j = 2b is even (b ≥ 1), (4.15) is (2k-1)² - 2(5^b)² = -1, so Proposition 4.8 and Prerequisite (ii) give (2k-1) + 5^b√2 = (1+√2)²ⁿ⁺¹ with 5^b = P_2n+1 (P₀=0, P₁=1, P_i+2=2P_i+1+P_i) and 3 | (2n+1). Writing 2n+1 = 3(2a+1) (a ≥ 0) and expanding (1+√2)^3(2a+1) = (7+5√2)^2a+1 gives P_3(2a+1) = 5∑_i=0^a C(2a+1, 2i+1) 49^a-i 50ⁱ. Because 50ⁱ = 2ⁱ· 5²ⁱ and v₅(2i+1) ≤ i - 1 implies 2i - v₅(2i+1) ≥ i + 1 ≥ 2 for all i ≥ 1, each i ≥ 1 summand has valuation ≥ 1 + v₅(2a+1) + 2i - v₅(2i+1) ≥ 1 + v₅(2a+1) + 2, so v₅(P_3(2a+1)) = 1 + v₅(2a+1), forcing 5^b-1 | (2a+1). Since P_3(2a+1)/5 ≥ (2a+1)49^a > 2a+1 for all a ≥ 1, this holds only at a=0, b=1, j=2 (k=4, chamber m=7, diagonal address N=25).

Sources. The completeness statements (4.13) and (4.14) are taken from [1, 2]; (4.15) for base 5 follows both from [3] and from the self-contained 5-adic argument above. In (4.14), factoring (1 + k√(-2))(1 - k√(-2)) = (1 + √(-2))^j(1 - √(-2))^j in the Euclidean domain ℤ[√(-2)] reduces 2k² + 1 = 3^j to Re((1+√(-2))^j) = ± 1 (j ∈ {1, 2, 5}).

5. Rational Carriers, the Route to 5, and the Diagonal Bridge

5.1 The Midpoint Fold Circle Map and Unique Rational Carrier Parameterization

In [5, §2.1], the analytic fold operator 𝔣(w) = w(1-w) = (1 - 𝒥(w)²)/4 is applied to the complex argument s of ξ(s) to produce q = 𝔣(s). On the discrete address grid, applying the same operators (𝒥, 𝔣) to the cell-midpoint phase κ_mid = r̃/(2m) defines the centered carrier map on the open upper unit semicircle:

Because r, c ≥ 1, both odd legs r̃ = 2r-1 and c̃ = 2c-1 are positive, so the second coordinate of 𝒞(r,c) is strictly positive and the boundary endpoints (± 1, 0) are excluded. This is a specified algebraic map on positive grid cells, distinct from polar-shell embeddings Z = √d e^{2π i h/d} and from κ_root.

Theorem 5.1 (Unique Rational Carrier Classification). For a positive integer cell (r,c) ∈ ℤ_{≥ 1}², the circle point 𝒞(r,c) in (5.1) belongs to ℚ² if and only if there exist unique positive odd integers U, W, λ ∈ ℤ_{≥ 1} with gcd(U, W) = 1 such that

Equivalently, the cell (r,c), its chamber m, its primitive first chamber m₀, its carrier point p(U,W) = 𝒞(r,c), and its midpoint phase κ_mid are given by

Here λ = gcd(r̃, c̃) is the odd ray-repeat scale, the reduced denominator of κ_mid is 2m₀, and the primitive first chamber satisfies

(with signed first leg, reducing to the degenerate triple (0, 1, 1) at (U,W) = (1,1)).

Proof. Since V/m ∈ ℚ for every cell, 𝒞(r,c) ∈ ℚ² iff √(r̃c̃) ∈ ℚ, which holds iff the odd integer r̃c̃ is a perfect square. Let λ = gcd(r̃, c̃), which is a positive odd integer. Then r̃/λ and c̃/λ are coprime positive odd integers whose product (r̃c̃)/λ² is a rational square, hence an integer square. By coprimality, r̃/λ = U² and c̃/λ = W² for unique positive odd coprime integers U, W. Conversely, (5.2) makes √(r̃c̃) = λ UW ∈ ℤ. For a fixed oriented point p = (p₁, p₂), p₂/(1 - p₁) = U/W determines the coprime positive pair (U,W) uniquely, while λ = m/m₀ runs over all positive odd integers. Since any odd square is ≡ 1 (mod 8), U² + W² ≡ 2 (mod 8), so m₀ = (U²+W²)/2 ≡ 1 (mod 4). □

5.2 First Chambers 5, 13, 17, the Route to 5, and Multi-Ray Chambers

The only primitive diagonal carrier has U = W = 1, giving p(1,1) = (0,1) at first chamber m₀ = 1, cell (1,1). For any nontrivial primitive carrier (U ≠ W), the smallest unordered pair of distinct positive odd coprime integers is {3, 1}. Therefore, the least nontrivial rational carrier chamber in the centered chart is m₀ = (3² + 1²)/2 = 5, realized at cell (5,1) with p(3,1) = (4/5, 3/5) and its transpose (1,5) with p(1,3) = (-4/5, 3/5).

Three scope clarifications govern this classification:

1. Prime vs. Composite First Chambers: By (5.5) and Fermat's two-square theorem (p = A² + C² ⇒ U = A+C, W = | A-C |), an odd prime p occurs as a primitive first chamber m₀ if and only if p ≡ 1 (mod 4). However, m₀ is not a prime generator: (U,W) = (7,1) gives the composite first chamber m₀ = 25.
2. Non-Injectivity of m₀ Across Distinct Rays: A primitive first chamber m₀ need not determine a unique carrier ray, even up to transpose. At m₀ = 65 = 5 · 13, the two coprime odd pairs (U,W) = (11,3) and (9,7) both satisfy (11²+3²)/2 = (9²+7²)/2 = 65, yielding first cells (61,5) and (41,25) and distinct non-mirror points (56/65, 33/65) and (16/65, 63/65).
3. Exclusion of Opposite-Parity Targets from 𝒞(r,c): In the continuous extension κ_mid = a²/(a²+1), setting a = 2 gives (3/5, 4/5). Because rational centered cells require a = U/W with U, W both odd, (3/5, 4/5) is excluded from the centered rational carrier chart (whereas (4/5, 3/5) is admitted at m₀ = 5 via a = 3/1). Appendix A gives the complete budget approximation theory for excluded targets and shows how adjoining the leg-sum phase κ_leg = r/(m+1) realizes (3/5, 4/5) at cell (4,1), chamber 4.

5.3 The Bidirectional Carrier-to-Diagonal Square Bridge

The 45^∘ Pell carrier record sequence begins at (U₀, W₀) = (1,1) and advances by

Appendix A (Theorem A.1) proves that for every chamber budget M ≥ 1, the last available ray in (5.6) with m_0,j ≤ M uniquely minimizes Euclidean chord distance on 𝒞(r,c) to the 45^∘ point p_* = (1/√2, 1/√2). Here we prove that this carrier sequence maps bijectively to the diagonal-square Pell sequence of Proposition 4.8.

Theorem 5.2 (Bidirectional Bridge Between Carrier First Chambers and Diagonal Squares). For each primitive carrier pair (U,W), define its triangle legs and their sum by

Then G² + δ_leg² = 2m₀². In particular, | δ_leg | = 1 holds if and only if (U,W) = (U_j, W_j) (or its transpose for j=0) in the Pell carrier recurrence (5.6), and in that case G_j² + 1 = 2m_0,j². The grid diagonal cell in diagonal chamber m_diag = G_j at row and column r_diag = c_diag = (G_j + 1)/2 then carries the exact square address

Conversely, every positive integer solution (m, Q) of the diagonal-square equation m² - 2Q² = -1 in (4.10) determines a unique primitive carrier pair (U,W) in (5.6) with first carrier chamber m₀ = Q and leg sum G = m.

Proof. Direct expansion gives

Along (5.6), δ_{leg, j+1} = ((2U_j+W_j)² - U_j²)/2 - (2U_j+W_j)U_j = -((U_j² - W_j²)/2 - U_j W_j) = -δ_{leg, j}, so δ_{leg, j} = (-1)^j+1 and G_j² + 1 = 2m_0,j². By Proposition 4.7, the diagonal cell in chamber m_diag = G_j has address N_diag = (G_j² + 1)/2 = m_0,j².

Conversely, let (m, Q) be any positive integer solution of m² + 1 = 2Q². Set G = m and m₀ = Q (with (U,W)=(1,1) for G=m₀=1). For G > 1, choose δ_leg ∈ {-1, +1} so that ℓ₁ = (G + δ_leg)/2 is even and ℓ₂ = (G - δ_leg)/2 is odd. Then ℓ₁² + ℓ₂² = (G² + 1)/2 = m₀² and gcd(ℓ₁, ℓ₂) = 1. Consequently m₀ + ℓ₁ and m₀ - ℓ₁ are coprime positive odd integers whose product is m₀² - ℓ₁² = ℓ₂², forcing m₀ + ℓ₁ = U² and m₀ - ℓ₁ = W² for unique coprime positive odd integers U > W with | ℓ₁ - ℓ₂ | = 1. By the descent in Theorem A.1, (U,W) = (U_j, W_j). □

The first nondegenerate bridge is the 3–4–5 triangle (j=1): its hypotenuse is the carrier first chamber m_0,1 = 5 (at carrier cell (5,1)), its leg sum is the grid diagonal chamber m_diag = 4 + 3 = 7 (at diagonal cell (4,4)), and the diagonal square address is N_diag = 5² = 25. Naming these three distinct roles resolves the index ambiguity between the sequence 1, 5, 29, 169, 985, … and the diagonal chamber sequence 1, 7, 41, 239, 1393, ….

6. A Bounded View of the Address Set

6.1 The Bounded Address Sheet, Closure, and Convex Hull

Dividing each row index by its chamber scale maps the infinite quadrant into a bounded planar region while preserving address order. Each address N, in chamber m with m + 1 = r + c ≥ 2, maps to a planar site p_N in the second quadrant [7]:

where (p₁, p₂) ∈ ℝ² are the coordinates of the address sheet. For each chamber m ≥ 1, the address N satisfies T_m-1 < N ≤ T_m and 1 ≤ r ≤ m. At height p₂ = 1/(m+1) ∈ (0, 1/2], the row's extreme sites are (-p₂, p₂) (at r=1) and (p₂ - 1, p₂) (at r=m).

Proposition 6.1 (Discrete Sites, Topological Closure, and Isosceles Right Hull). Every site p_N in (6.1) satisfies 0 < p₂ ≤ 1/2 and p₂ - 1 ≤ p₁ ≤ -p₂. The topological closure of the infinite site set {p_N : N ≥ 1} in ℝ² is precisely

so the sites are not dense in the interior of the triangle. The closed convex hull of the site set is the isosceles right triangle conv(A, B, C) with vertices

whose right angle is at the apex C = p₁ (AB² = 1, AC² = BC² = 1/2, (A-C)·(B-C) = 0).

Proof. Any sequence of sites converging to a point with limiting height p₂ > 0 has bounded m, hence lies in a finite subset of {p_N} and is eventually constant. For p₂ = 0 and any a ∈ [0, 1], choosing r_m ∈ {1, …, m} with r_m/(m+1) → a as m → ∞ produces the limiting base point (-a, 0). Since every p_N lies in conv(A,B,C) and A, B, C belong to the closure (6.2), the closed convex hull of the infinite site set is conv(A,B,C). (A finite prefix m ≤ m_max has a smaller hull; at m_max=1 it is the singleton {C}.) □

Example (Primitive columns of the address sheet). In the finite crop 1≤ r,c≤ L, write (r,c)=k(a,b), where k=gcd(r,c) and gcd(a,b)=1. Equations (1.1) and (6.1) give

The primitive ordered pair fixes the vertical column; the common scale fixes the height. For L=6, the cells (2,1),(4,2),(6,3) have addresses 3,14,34 and heights 1/3,1/6,1/9, all at p₁=-2/3. Keeping the full point retains the pair. Discarding height identifies its common multiples. The conserved fraction is the leg-sum phase κ_leg = r/(r+c), distinct from the row and midpoint phases below and from the odd-leg gcd in Theorem 5.1.

Remark 6.1b (Euler Totient φ(m+1) Chamber Bridge, Denominator-20 Partition, and Square Coprime Count). Let Euler's totient φ(M) = #{1 ≤ j ≤ M : gcd(j,M) = 1} = M∏_{p | M}(1 - 1/p) count the positive integers up to M that are coprime to M (φ(1)=1; φ counts coprime numerators, composite ones included, not primes). In chamber m, every cell satisfies r+c = m+1 with 1 ≤ r ≤ m and c = m+1-r, so gcd(r,c) = gcd(r, m+1). Because the omitted endpoint r = m+1 (c=0) is not coprime to m+1 for m ≥ 1, the number of primitive cells (r,c) at height 1/(m+1) of the address sheet (6.1) is exactly

For m=8 (r+c=9), the φ(9) = 6 primitive cells are (1,8), (2,7), (4,5), (5,4), (7,2), (8,1) (including the composite coordinates 4 and 8 because gcd(4,9)=gcd(8,9)=1). For m=19 (r+c=20), reducing the twenty fractions 1/20, …, 20/20 partitions them by reduced denominator d | 20: d=20 retains the φ(20) = 20(1-1/2)(1-1/5) = 8 primitive cells (1,19), (3,17), (7,13), (9,11), (11,9), (13,7), (17,3), (19,1) at height p₂ = 1/20 (paired by r ↔ 20-r so complementary native abscissae sum to -1), while d=10,5,4,2 contribute φ(10)+φ(5)+φ(4)+φ(2) = 4+4+2+1 = 11 scaled cells lifted from earlier primitive columns by k = 20/d, and d=1 (20/20, i.e., (20,0)) lies outside the positive quadrant, giving 8 + 11 = 19 = m. Keep the predicates distinct: primitive cells in chamber m use the leg-sum denominator m+1 and count φ(m+1), whereas asking which native row fractions κ_row = r/m are already in lowest terms requires gcd(r,m)=1 and counts φ(m) (and the odd-square carrier condition in Theorem 5.1 imposes the further restriction (r̃,c̃)=λ(U²,W²)). Separately, in the L × L square 1 ≤ a,b ≤ L, the lower triangle 1 ≤ b ≤ a ≤ L contains ∑_k=1^L φ(k) coprime pairs and shares only (1,1) with the upper triangle, so under independent uniform sampling on {1,…,L}²,

by standard Möbius inversion ∑_{1≤ a,b≤ L, gcd(a,b)=1} 1 = ∑_i=1^L μ(i)⌊ L/i⌋² = 6/π² L² + O(L log L) [17].

Remark 6.1a (Unit-Square Frame, Half-Diagonal Length 1/√2, Row vs. Midpoint Half-Phase Pencils, and Reflected Extension C_-). The address sheet (6.1) sits inside the unit square Q = [-1, 0] × [0, 1] with corners O = B = (0,0), A = (-1,0), B_ul = (-1,1), and D = (0,1). Its first positive site (m = r = c = N = 1) is the unit-square center C₊ = C = (-1/2, 1/2). Holding r = 1 fixes the square diagonal p₁ = -p₂ (OB_ul, of length √2), which the native hull condition p₁ ≥ p₂ - 1 restricts to 0 ≤ p₂ ≤ 1/2; hence the native fixed-row segment OC₊ has exact Euclidean length √2/2 = 1/√2 (which also equals the orthogonal projection A·e = D·e = 1/√2 of the unit edges onto the unit diagonal e = (-1,1)/√2, while Area(conv(A,O,C₊)) = 1/4 = 1/2 Area(conv(A,O,D))). On this same chart, three distinct lines meet the square's geometry differently:

1. Fixed-row ray (r = 1): p₁ = -p₂ through O = (0,0) and C₊ = (-1/2, 1/2) (approaching O as m → ∞).
2. Row half-phase (κ_row = r/m = 1/2): p₁ = (p₂ - 1)/2 through the upper apex D = (0,1) and the native site P = (-1/3, 1/3) ((m,r,c) = (2,1,2), N = 2), crossing the fixed-row ray at P ≠ C₊.
3. Midpoint half-phase (κ_mid = (r - 1/2)/m = 1/2): the vertical line p₁ = -1/2, which contains every positive diagonal site r = c (m = 2r - 1, p_2r-1,r = (-1/2, 1/(2r))) including C₊ = (-1/2, 1/2).

The row and midpoint half-phase lines share the boundary foot F = (-1/2, 0) (where their phase difference p₂/[2(1-p₂)] vanishes as p₂ → 0⁺), while the vertical line p₁ = -1/2 continues downward through F to the horizontal-axis reflection C_- = (-1/2, -1/2)—an explicit reflected extension below the positive sheet with | OC₊ | = | OC_- | = 1/√2 and complex diagonal coincidence iω₊ = ω₊ = ω_- for ω_± = (-1 ± i)/2 (whereas the row-half-phase line p₁ = (p₂ - 1)/2 passes through (-3/4, -1/2) ≠ C_-). This unit-square half-diagonal scale 1/√2 is distinct from the 1/√5 inner-square side arising in five-part equal-area dissections of a unit square [17].

Hull side lengths, angle bisector and the Pell target. In the unit-base normalization of (6.3), the hull 𝒦=conv(A,B,C) has side lengths 1,1/√2,1/√2. Therefore

The base-angle bisector gives tan(π/8)=2R_in=L_△^-1. Write the centered rational carrier in (5.4) as p(U,W)=(C₁,C₂)=(cos ψ,sin ψ), reserving (p₁,p₂) for the address sheet here. Then

Thus the 45^∘ carrier target has U/W=cot(π/8)=L_△. Under the existing Pell step (U,W)↦(2U+W,U), the signed error E=U-L_△ W obeys E'=-E/L_△. This connects the hull's side-length constant to the carrier's half-angle parameter and its Pell approximation. The native first site (-1/2,1/2) corresponds to cell (1,1), whose carrier (5.1) is the apex (0,1); the 45^∘ point is the separate approximation target.

6.2 The Sloping Sides and Apex Circle vs. Circumcircle

Proposition 6.2 (Triangular Polynomial Difference and Sum on the Address Sheet). In planar coordinates (p₁, p₂) ∈ ℝ², the difference and sum of T(p₁) and T(p₂ - 1) satisfy

where E₁ = 2p₁ + 1 and E₂ = 2p₂ - 1 center the apex C = (-1/2, 1/2) at the origin.

The locus T(p₁) - T(p₂ - 1) = 0 factors into the two sloping sides p₁ = -p₂ (BC) and p₁ = p₂ - 1 (AC). The circle T(p₁) + T(p₂ - 1) = 0 has center at the apex C = (-1/2, 1/2) and radius 1/√2; it passes through A=(-1,0) and B=(0,0), and its closed disk contains conv(A,B,C). It is not the triangle's circumcircle: the actual circumcircle passing through A, B, and C has center (-1/2, 0) and radius 1/2, with equation (p₁ + 1/2)² + p₂² = 1/4.

6.3 The Scalene Affine Variant

Replacing the numerator r in (6.1) by r - (m - 1) = N - ((m+2)(m-1))/2 defines the variant sites

Proposition 6.3 (Affine Involution and Scalene Hull). The two bounded address sheets are related site-by-site by the area-preserving affine involution

The closed convex hull of {p̃_N : N ≥ 1} is the scalene triangle with vertices (-1, 0), (0, 0), (1/2, 1/2), bounding edges p₂ = 0, p₁ = p₂, and p₁ = 3p₂ - 1, and squared side lengths 1, 1/2, 5/2.

6.4 Equilateral Lift and Regular k-Simplex Orthogonal Projection

The isosceles right triangle conv(A,B,C) in (6.3) is the orthogonal projection onto p₃ = 0 of the spatial equilateral triangle of side 1 with vertices  = (-1, 0, 0), B̂ = (0, 0, 0), and Ĉ = (-1/2, 1/2, 1/(√2)), tilted from the horizontal plane by ψ₂ = arccos(1/√3) ≈ 54.7356^∘.

Proposition 6.4 (Regular k-Simplex Orthogonal Projection Factor). In ℝ^k+1 (k ≥ 1), let Δ_k = conv(e₀, e₁, …, e_k) be the standard regular k-simplex of edge length √2, and let projΔ_k = conv(0, e₁, …, e_k) be its orthogonal projection onto the coordinate hyperplane orthogonal to e₀. Then the edge vectors e_i - e₀ (1 ≤ i ≤ k) have Gram matrix G = I_k + 11^T with det G = k + 1, and the intrinsic k-volumes satisfy

Distinction Between Two Meanings of "Simplex Shadow". Proposition 6.4 projects a k-dimensional simplex orthogonally onto a k-dimensional hyperplane and tracks its intrinsic k-volume ratio 1/√(k+1) (with angles ψ_k = 45^∘, 54.7356^∘, 60^∘, 63.4349^∘ for k = 1, 2, 3, 4). By contrast, in Reading the Fold v1.3 (§3.2), a "simplex shadow" is the pushforward of uniform probability measure on an n-simplex to a one-dimensional line, producing the normalized Curry–Schoenberg B-spline density M_n on [-n, 0] [6]. For n = 2, the graph of M₂ has vertices (-2,0), (-1,1), (0,0) and area 1; scaling both coordinates by 1/2 produces the triangle conv(A,C,B) of area 1/4, which is a geometric similarity of plane figures rather than a probability-preserving rescaling.

Remark 6.5 (Discrete Triangular Chamber Law on a Finite L × L Grid Crop and Weak Convergence to M₂). While the infinite positive quadrant ℤ_{≥ 1}² has chamber sizes #{(r,c) ∈ ℤ_{≥ 1}² : r+c-1 = m} = m that grow linearly without bound, restricting the grid to a finite origin-anchored square crop

equipped with uniform discrete probability weight 1/L² on each of its L² cells produces an exact discrete triangular distribution for the chamber m = r + c - 1 ∈ {1, 2, …, 2L - 1} of a uniformly chosen cell (r,c):

Indeed, for each k ∈ {1, …, 2L-1}, the admissible rows r ∈ {1, …, L} satisfying 1 ≤ k + 1 - r ≤ L form the interval max(1, k + 1 - L) ≤ r ≤ min(L, k), which contains min(L, k) - max(1, k + 1 - L) + 1 = L - | k - L | cells, and ∑_k=1^2L-1 (L - | k - L |) = L².

Mapping m to the negative-support convention of [5, §3.2] via the rescaled chamber variable

the independent discrete uniform coordinates E_L = -(r-1)/L and F_L = -(c-1)/L converge in distribution as L → ∞ to independent Unif([-1, 0]) variables, and A_L = E_L + F_L converges weakly (and uniformly on the lattice mesh z_k = -(k-1)/L with L Pr(A_L = z_k) = M₂(z_k) + O(1/L)) to the continuous two-fold Curry–Schoenberg tent density

This finite-crop identity requires explicitly declaring the L × L square crop 𝒬_L, uniform 1/L² cell weighting, and the additive chamber variable m = r + c - 1; the finite distribution (6.8a) is discrete, and it provides a concrete n = 2 bridge between the Notebook Grid and [5, §3.2] without implying that empirical Gaussian or moving-average bell plots in other parts of the archive belong to the M_n spline family. Moreover, on the same uniform finite square crop 𝒬_L, sorting (r, c) into the pair Y = (min(r,c), max(r,c)) (equivalently (m, | V |)) leaves the chamber m = r + c - 1 and its tent limit unchanged while mapping the L diagonal cells (r = c) to singleton fibers of mass 1/L² and the L² - L off-diagonal cells (r ≠ c) to equiprobable two-element fibers of mass 2/L², forming T_L-1 = L(L-1)/2 off-diagonal sorted pairs, so the exact conditional orientation information removed by sorting is H(r, c | Y) = (1 - 1/L) ln 2 (recovered by the single orientation bit I = 1_{{r > c}}). More generally, for an exchangeable pair (A,C) taking values in a finite or countable ordered set, define Y_AC=(min(A,C),max(A,C)), I_AC=1_{A>C}, and p₌=Pr(A=C). Direct conditional entropy on the at-most-two-point fibers gives H(A,C | Y_AC)=H(I_AC | Y_AC)=(1-p₌) ln 2; this remains well-defined even when H(A,C) or H(Y_AC) is infinite. Moreover, because the sorted pair Y is equivalent to (m, | V |) and the address is N = (m² + 1 + (2I - 1)| V |)/2, retaining only the chamber m discards | V | as well: with L_m = L - | m - L |, the additional conditional offset entropy lost in passing from Y to m is H(Y | m) = 1/L² ∑_m=1^2L-1 L_m ln L_m - (1 - 1/L) ln 2, which vanishes at L ∈ {1, 2} and is strictly positive for all L ≥ 3 (at L = 3, H(Y | m) = 1/3 ln 3 - 2/9 ln 2 > 0, witnessed by chamber m = 3 with cells (1,3), (2,2), (3,1) at addresses 4, 5, 6), while H(m) = ln L + 1/2 + o(1) matches the continuous tent differential entropy 1/2 after subtracting the ln L lattice scale.

Remark 6.4a (Tangent-Sphere Cone Ratio, Matched 2/π Inscribed Ratio, and SO⁺(1,1) Double-Pell Conjugacy [18]). Three three-dimensional geometric identities connect the simplex tilt cos ψ₂ = 1/√3 in Proposition 6.4 to the Pell recurrences of §4:

1. Axial Tangent-Sphere Cone Ratio: In a right circular cone of half-apex angle α ∈ (0,π/2), two spheres of radii R₁ > R₂ > 0 tangent to the cone wall at apex distances δ_i = R_i/sin α and externally tangent to each other satisfy δ₁ - δ₂ = R₁ + R₂, whence (R₁ - R₂)/sin α = R₁ + R₂ and Λ_cone(α) := R₂/R₁ = (1 - sin α)/(1 + sin α) = tan²(π/4 - α/2). At α = 45^∘ (sin α = 1/√2), Λ_cone(45^∘) = 3 - 2√2 = (√2-1)², whose reciprocal (1+√2)² = 3 + 2√2 generates the negative-Pell step (m,Q)↦(3m+4Q, 2m+3Q) of Proposition 4.8; at sin α = 1/√3, Λ_cone = 2 - √3 = (2+√3)^-1.
2. Cube 1:2:3 Squared-Radius Law and Matched Inscribed Ratio 2/π: A cube [-a,a]³ of half-side a > 0 has insphere, midsphere, and circumsphere radii (R_in, R_mid, R_out) = (a, a√2, a√3) in squared ratio 1:2:3. Under matched inscribed orientation, a square Q₂ inscribed in a disk B₂(R) (a = R/√2, area 2R²) and a cube Q₃ inscribed in a sphere B₃(R) (a = R/√3, surface area 6(2a)² = 8R²) share the exact ratio

whereas for the dual orientation (an inscribed disk in a square) Area(B₂)/Area(Q₂) = π/4.

1. Double-Pell Integrality and Positive-Sheet SO⁺(1,1) Boost Conjugacy: In ℝ^k (k > 2), mapping the body-diagonal seeds 1/√k and 1/√(k-1) by the Cayley map μ(β) = (1+β)/(1-β) gives μ(1/√D) = (D+1)/(D-1) + 2/(D-1) √D on P₁² - DQ₁² = 1. Both coordinates (D+1)/(D-1) and 2/(D-1) are integers iff (D-1) | 2 ⇔ D ∈ {2,3}, singling out k = 3 as the unique dimension k > 2 where both μ(1/√3) = 2+√3 (D=3) and μ(1/√2) = 3+2√2 (D=2) land on fundamental integer Pell units. Normalizing A_β = ([1 β ; β 1]) (|β| < 1) to Λ_β = (1-β²)^-1/2A_β ∈ SO⁺(1,1) gives Λ_β Λ_γ = Λ_{(β+γ)/(1+βγ)}. For any Pell solution (P₁,Q₁) of P₁² - DQ₁² = 1 on the positive Lorentz sheet P₁ > 0, setting C_D = diag(1,√D), L_D = ([P₁ DQ₁ ; Q₁ P₁]), and β_D = (√D Q₁)/P₁ ∈ (-1,1) gives (1 - β_D²)^-1/2 = P₁ > 0 and the exact orthochronous conjugacy

(whereas for P₁ ≠ 0 in O(1,1), (1-β_D²)^-1/2 = | P₁| gives sgn(P₁)Λ_βD, witnessed at (P₁,Q₁) = (-1,0) where C_D L_D C_D^-1 = -I ≠ +I = Λ₀). For D ∈ {2,3}, where 1/√D rationalizes to the primitive integer fundamental Pell solution (P₁,Q₁) = ((D+1)/(D-1),2/(D-1)) ∈ {(3,2), (2,1)}, this gives the exact seed square C_D L_D C_D^-1 = Λ_1/√D² = Λ_βD. Moreover, for β ≠ 0, A_β^TA_β = ([1+β² 2β ; 2β 1+β²]) is not a scalar multiple of I, so A_β cannot lie in the Euclidean similarity group ℝ^×·O(2).

Remark 6.4b (Tetrahedral-to-Hexagonal Projection, Adjacent-Pair Conic Basis, D₈ Fiber Centralizer, and Interstitial Volume Partition [18]):

1. Tetrahedral Midpoints, Homothetic Hexagons, and Conic Basis: Partitioning {± 1}³ by the sign of the product of the three coordinates into dual regular tetrahedra Δ_±, the edge-midpoint map M({v_i,v_j}) = 1/2 (v_i+v_j) sends the 6 edges of Δ₊ bijectively onto {± e₁,± e₂,± e₃}. Orthogonal projection P(v) = v - (v· e_*)e_* along e_* = 1/(√3) (1,1,1) sends ±(1,1,1)↦ 0, maps the remaining 6 cube vertices to a regular hexagon of squared radius ‖ P(1,1,-1)‖² = 3 - 1/3 = 8/3, and maps {± e_i} to a homothetic regular hexagon of squared radius ‖ P(e₁)‖² = 1 - 1/3 = 2/3. Writing Σ(z) = T(z) + T(z-1) = z², δ(z) = T(z) - T(z-1) = z, and β(a,b) = Σ(a+b)-Σ(a)-Σ(b) = 2ab gives a triangular basis (Σ(a), β(a,b), Σ(b), δ(a), δ(b), 1) for all bivariate quadratics, in which {Σ(a)+Σ(b)=1} ∩ {β(a,b)=± b} reduces to a²+b²=1 and b(2a∓ 1)=0, cutting out the six sixth roots of unity (± 1,0) and (±1/2, ±(√3)/2).
2. Intrinsic D₈ Fiber Envelope: On each 4-state fiber Q_a = {a, a+6, a+12, a+18} (mod 24) over the 6 projected rays (μ₂₄→^{z↦ z4}μ₆), cycling L(i) ≡ i+1 (mod 4) and odd reflection A(i) ≡ b-i (mod 4) (b ∈ {1,3}) satisfy A L A = L^-1 and L² ≡ (·)+2, generating ⟨ A, L⟩ ≅ D₈ containing both the tetrahedral stabilizer ⟨ A, L²⟩ (a Klein four-group) and the cube rotation stabilizer C₄ = ⟨ L⟩.
3. Ball–Cube–Ball Volume Tuple: Alternating inward in ℝ^k between a k-ball B_n(R_n), its inscribed cube Q_n (side 2R_n/√k), and the inscribed ball B_n+1(R_n/√k) partitions unit ball volume as 1 = k^-k/2 + (2^k/(k^k/2ω_k) - k^-k/2) + (1 - 2^k/(k^k/2ω_k)); at k = 3 (ω₃ = (4π)/3), the step ratios form ((Vol(Q_n))/(Vol(B_n)), (Vol(B_n+1))/(Vol(Q_n)), (Vol(B_n+1))/(Vol(B_n))) = (2/(π√3), π/6, 1/(3√3)).

6.5 The Diagonal Staircase, Odd-Square Sectors, and Depth-Weighted Stack Projection

On the address sheet (6.1), where p(r,c) = (-r/(m+1), 1/(m+1)) satisfies p₁ < 0 < p₂ and the closed convex hull 𝒦 meets the upper-left diagonal boundary at p₁ = -p₂, the affine line p₂ = (k-1)/k + p₁ (k ∈ ℤ_{≥ 1}) crosses p₁ = -p₂ at (p₁, p₂) = (-(k-1)/(2k), (k-1)/(2k)). Equivalently,

(Note: Since a discrete site p(r,c) = (-r/(m+1), 1/(m+1)) on p₁ = -p₂ requires r = 1 and m + 1 = 2k/(k-1) = 2 + 2/(k-1), for integer k ≥ 2 the crossing is a discrete site if and only if k ∈ {2, 3}, giving cell (1,3), m=3, N=4 at k=2 and cell (1,2), m=2, N=2 at k=3.) For k > 1, setting κ_stair = T_k/k² = 1/2 + 1/(2k) gives 𝒥(κ_stair) = 1/k and 𝔣(κ_stair) = (T_k T_k-1)/k⁴ = 1/4 - 1/(4k²), climbing to the fold vertex 1/4 as k → ∞ [5, §2.3].

Square split, 1/k normalization, quadratic fixed-point family, and the far edge of the address sheet. Because T_k + T_k-1 = k² and T_k - T_k-1 = k (k ∈ ℤ_{≥ 1}), dividing by k² gives the normalized gap χ_k⁺ - χ_k^- = 1/k between χ_k^± = 1/2 ± 1/(2k), with successive differences 1/k - 1/(k+1) = 1/(2T_k). Their common fold value 𝔣(χ_k^±) = 1/4 - 1/(4k²) factors z² - z + 𝔣(χ_k^±) = (z - χ_k^-)(z - χ_k⁺), so χ_k^± are the real fixed points of the quadratic map z ↦ z² + 𝔣(χ_k^±), whose parameter approaches the parabolic value 1/4 (with multipliers 2χ_k^± = 1 ± 1/k and deficits 1/4 - 𝔣(χ_k^±) = 1/(4k²) summing to ζ(2)/4 = π²/24). On the address sheet (6.1), the far-edge site p_m,m = (-1 + 1/(m+1), 1/(m+1)) → (-1, 0) along p₁ = p₂ - 1, and its raw horizontal step 1/(m+1) rescales by the active m-step span m/(m+1) to 1/m [17].

Proposition 6.5 (Eight-Sector Lattice Congruence, Isometric Stack Fiber Multiplicity, and Atomic-to-Weak Screen Laws). Let n ≥ 1 and put j = 2n + 1.

1. Eight-Sector Decomposition in E(2,ℤ) = ℤ² ⋊ O(2,ℤ): Centering {1,…,j}² at (a,b) = (A'-n-1, B'-n-1) ∈ {-n,…,n}² partitions the j² = 1 + 8T_n lattice points into the origin (0,0) and eight half-open max-norm sectors Σ₀,…,Σ₇ obtained from Σ₀ = ⋃_k=1ⁿ {(k,b) : 0 ≤ b < k} and Σ₁ = ⋃_k=1ⁿ {(a,k) : 0 < a ≤ k} under the 4-fold rotation group C₄ ⊂ O(2,ℤ) ≅ D₄. Each sector has cardinality T_n (indexed locally by N = T_k-1 + r ∈ {1,…,T_n}). In the lattice-preserving Euclidean semidirect product E(2,ℤ) = ℤ² ⋊ O(2,ℤ) ⊂ E(2) = ℝ² ⋊ O(2) with group law (b₁, M₁)(b₂, M₂) = (b₁ + M₁ b₂, M₁ M₂) and action (b, M)· v = Mv + b, the lattice reflection-shift F = ((1,0), E) ∈ E(2,ℤ), with E the coordinate exchange, F(a,b) = (b+1, a), maps Σ₀ →^∼ Σ₁ bijectively, so all eight sectors are pairwise congruent as free Euclidean subsets while remaining non-equivalent under origin-fixing maps in O(2) (at n=2, Σ₀ has origin squared-norm multiset {1,4,5} whereas Σ₁ has {2,5,8}).
2. Oblique Fiber Multiplicity of the Bottom-Widest Point Stack S_n: Define the discrete point stack S_n = {(p₁,p₂,ℓ) ∈ ℤ³ : 0 ≤ ℓ ≤ n, | p₁|, | p₂| ≤ n - ℓ} of total cardinality | S_n| = ∑_ℓ=0ⁿ (2(n-ℓ)+1)² = ((n+1)(2n+1)(2n+3))/3 = C(2n+3, 3) = C(j+2, 3) (distinct from the solid cube union Ω_n of Appendix B). Along e_* = (1,1,1)/√3, orthogonal projection P = I - e_*e_*^T maps (p₁,p₂,ℓ) to (p₁-ℓ)Pe₁ + (p₂-ℓ)Pe₂, indexed injectively by the oblique screen coordinate (a,b) = (p₁-ℓ, p₂-ℓ) ∈ {-n,…,n}² with exact fiber multiplicity

1. Finite Atomic Screen-Maximum PMFs vs. Rescaled Weak-Limit Densities: Set M = max(a+n+1, b+n+1) = max(a,b) + n + 1 ∈ {1,…,j}. At every finite odd side j = 2n+1, M obeys the exact discrete atomic probability mass functions

under uniform 2D screen sampling on {-n,…,n}² and uniform 3D stack sampling on S_n, respectively. As j → ∞, the rescaled random variable Y_j = M/j ∈ (0,1] satisfies the uniform local-limit bounds (ω = i/j)

and hence converges weakly on [0,1] to continuous laws with densities p_flat(ω) = 2ω and p_stack(ω) = 6ω(1-ω) = 6 𝔣(ω).

Proof. (1) For (k,b) ∈ Σ₀, 0 ≤ b < k ≤ n implies 1 ≤ b+1 ≤ k ≤ n, so F(k,b) = (b+1, k) ∈ Σ₁ with inverse (a,k) ↦ (k, a-1), and F ∈ ℤ² ⋊ O(2,ℤ) preserves Euclidean distances (‖ F(v)-F(v')‖₂ = ‖ E(v-v')‖₂ = ‖ v-v'‖₂). Arbitrary rotations in O(2) do not preserve ℤ², so E(2,ℤ) is written ℤ² ⋊ O(2,ℤ) with normal translation subgroup ℤ².

(2) Since Pe₃ = -Pe₁ - Pe₂ and {Pe₁, Pe₂} is a basis of e_*^⊥, P(p₁,p₂,ℓ) = (p₁-ℓ)Pe₁ + (p₂-ℓ)Pe₂. Given (a,b) ∈ {-n,…,n}² and an integer ℓ ≥ 0, the lifted point (a+ℓ, b+ℓ, ℓ) belongs to S_n iff ℓ ≤ n, -(n-ℓ) ≤ a+ℓ ≤ n-ℓ, and -(n-ℓ) ≤ b+ℓ ≤ n-ℓ. The lower bounds a ≥ -n and b ≥ -n already hold on {-n,…,n}², while the upper bounds reduce to 2ℓ ≤ n - a and 2ℓ ≤ n - b, i.e., 0 ≤ ℓ ≤ ⌊(n - max(a,b))/2⌋ ≤ n, which has 1 + ⌊(n - max(a,b))/2⌋ integer solutions ℓ.

(3) Shifting (A', B') = (a+n+1, b+n+1) ∈ {1,…,j}² gives n - max(a,b) = j - M. Exactly i² - (i-1)² = 2i - 1 pairs (A', B') satisfy max(A', B') = i, and w_n is constant on {M=i} with value 1 + ⌊(j-i)/2⌋, establishing both PMFs in (6.11). Writing ω = i/j ∈ (0,1], j Pr_flat(M=i) - 2ω = -1/j. For Pr_stack, note (j-i+1)/2 ≤ 1 + ⌊(j-i)/2⌋ ≤ (j-i+2)/2 and C(j+2, 3) = (j(j+1)(j+2))/6. Multiplying by 6j/[j(j+1)(j+2)] gives

Since 0 ≤ 1 + ⌊(j-i)/2⌋ - (j-i)/2 ≤ 1 and 2i - 1 ≤ 2j, the first numerator has magnitude at most 6[(2j)(1) + j/2] = 15j, while 6ω(1-ω) ≤ 3/2 bounds the second term by (9j+6)/(2(j+1)(j+2)) < 3/j, yielding total error ≤ 15/j + 3/j = 18/j. Riemann-sum convergence over bounded continuous test functions on [0,1] then yields weak convergence of Y_j = M/j to 2ω dω and 6ω(1-ω) dω. □

For independent uniform stages (A_n, C_n) ∈ {1,…,j_n}² (j_n = 2n+1, center-hit event E_n = {A_n = C_n = n+1} of probability j_n^-2), writing 𝒮_L = ∏_n=1^L (1 - j_n^-2) for the probability of no center hit in the first L stages and Π_△,L(u) = ∏_n=1^L (1 - u/T_n) gives at every L ≥ 1 [8]:

and since ∑_{n≥ 1}Pr(E_n) < ∞, the number of center hits I = ∑_{n≥ 1}1_En is finite almost surely, with entire probability generating function

6.6 Common Triangular / Odd-Square Factorization, Central-Binomial Closed Form, Euler Cosine PGF, and Fixed-Gap Gamma Family

The odd-square address geometry of §§1–5, the diagonal staircase value 𝔣(χ_2n+1^±) = 1/4 - 1/(4(2n+1)²), and the eight-sector / stack decompositions of Proposition 6.5 all instantiate a single elementary integer identity, (2n+1)² = 8T_n + 1 = 4n(n+1) + 1, whose finite and asymptotic consequences we prove below [8, 10].

Lemma 6.5a (Rapidity Telescoping, Central-Binomial Product, and Polynomial Rigidity of FG). For each integer n ≥ 1, let (F_n, G_n) = (n, n+1) be the balanced binary split of j_n = 2n+1, with share χ_n = n/(2n+1) = χ_2n+1^-, contrast δ_n = 1 - 2χ_n = 1/(2n+1) ∈ (0,1), odds ratio O_n = G_n/F_n = (n+1)/n = (1+δ_n)/(1-δ_n), and rapidity ϱ_n := artanh(δ_n) = 1/2 ln((n+1)/n). Then:

1. Multiplicative Odds & Additive Rapidity Telescoping: ∏_n=1^L O_n = L + 1 and ∑_n=1^L ϱ_n = 1/2 ln(L+1).
2. Variance-Fold Central-Binomial Product: Writing 4𝔣(χ_n) = 4χ_n(1-χ_n) = 1 - δ_n² = (4n(n+1))/((2n+1)²) = cosh^-2(ϱ_n), at every finite L ≥ 1,

1. Polynomial Rigidity of FG: Every symmetric polynomial Ψ(F,G) = Ψ(G,F) over a commutative ring Λ satisfying Ψ(F,0) = Ψ(0,G) = 0 factors uniquely as Ψ(F,G) = FG· Q(F+G, FG) for a polynomial Q over Λ.

Proof. (1) ∏_n=1^L (n+1)/n = L+1, and artanh(1/(2n+1)) = 1/2 ln((n+1)/n) telescopes to 1/2 ln(L+1). (2) Using ∏_n=1^L 4n(n+1) = 4^L(L+1)(L!)² and ∏_n=1^L (2n+1) = ((2L+1)!)/(2^L L!) = (2L+1)/2^L C(2L, L)L!, squaring the denominator and cancelling (L!)² gives (4^L(L+1))/((2L+1)² 4^-LC(2L, L)²) = (L+1)/((2L+1)²) 16^L/(C(2L, L)²). (3) Viewing Ψ ∈ (Λ[F])[G], Ψ(F,0) = 0 implies G | Ψ, so Ψ(F,G) = G· Ψ₁(F,G); evaluating at F = 0 gives 0 = Ψ(0,G) = G· Ψ₁(0,G) in Λ[G], whence Ψ₁(0,G) = 0 and F | Ψ₁, giving Ψ(F,G) = FG· Q̃(F,G) with Q̃ uniquely determined because FG is a monic regular element in each variable. Symmetry Ψ(F,G) = Ψ(G,F) implies FG(Q̃(F,G)-Q̃(G,F)) = 0, so Q̃ is symmetric and by the fundamental theorem of symmetric polynomials equals a unique Q(F+G, FG). □

Theorem 6.5 (Common Triangular / Odd-Square Factorization and Multi-Route Consequences). For each integer n ≥ 1, let j_n = 2n+1 and T_n = n(n+1)/2. All three routes below rest on (2n+1)² = 8T_n + 1 = 4n(n+1) + 1 together with their stated route hypotheses:

1. Route 1 — Eight-Sector Partition, Balanced Split, and Finite Reciprocal Identity: The centered square {-n,…,n}² partitions into (0,0) and eight E(2,ℤ)-congruent half-open max-norm sectors Σ₀,…,Σ₇ of size T_n (Proposition 6.5(1)). In any discrete binary partition F + G = 2n+1 (F,G ∈ ℤ_{≥ 0}), writing (F,G) = (n-i, n+1+i) (or its swap) with imbalance i ∈ {0,…,n} yields the exact triangular defect sequence FG = (n-i)(n+1+i) = n(n+1) - i(i+1) = 2(T_n - T_i), which is uniquely maximized at i = 0 by the balanced split {n, n+1} (two oriented ordered states (n,n+1), (n+1,n), and C(2n+1, n) labeled cuts of K_2n+1, all with FG = 2T_n = ex(2n+1; K₃)). At χ_n = n/(2n+1),

and at every finite L ≥ 1,

1. Route 2 — Euler Cosine Product and ζ(2) PGF Slope (Under Stage Independence): Under mutually independent Bernoulli center-hit indicators 1_En with Pr(E_n) = (2n+1)^-2, I = ∑_n=1^∞ 1_En has entire generating function G(τ) = 𝔼[τ^I] = ∏_n=1^∞ (1 - (1-τ)/((2n+1)²)), which by Euler's product cos((π z)/2) = (1-z²)∏_n=1^∞(1 - z²/((2n+1)²)) at z = √(1-τ) satisfies

1. Route 3 — Fixed-Gap Gamma Ratio and Isometric Stack Pushforward: For any real gap parameter D = k - 2 > 0, setting ν_n^(k) = 1/2 n(n+D) (ν_n⁽³⁾ = T_n) gives at every finite L ≥ 1

evaluating to 4/π at D=1, (2(L+1))/(L+2) → 2 at D=2, and 32/(3π) at D=3. Separately, by Proposition 6.5(2)–(3), projecting S_n along e_* = (1,1,1)/√3 weights screen coordinates by w_n(a,b) = 1 + ⌊(n-max(a,b))/2⌋, carrying uniform 2D flat sampling and uniform 3D stack sampling from their exact finite-j atomic laws (6.11) to the weak-limit densities p_flat(ω) = 2ω and p_stack(ω) = 6ω(1-ω) = 6 𝔣(ω) on [0,1].

Proof. Route 1 follows from Proposition 6.5(1), Lemma 6.5a(2), and Wallis's asymptotic C(2L, L) ∼ 4^L/(√(π L)) (so 𝒮_L ∼ L/(4L²)· π L = π/4), which also follows independently from G(0) = π/4 in Route 2. In Route 2, ∑_{n≥ 1}(| 1-τ|)/((2n+1)²) converges locally uniformly on ℂ, so G(τ) is entire; L'Hôpital's rule at z=1 gives lim_{z→ 1}(cos(π z/2))/(1-z²) = (-π/2 sin(π/2))/(-2) = π/4, and termwise logarithmic differentiation at τ=1 gives G'(1) = ∑_{n≥ 1}(2n+1)^-2 = 3/4 π²/6 - 1 = π²/8 - 1. In Route 3, iterating Γ(z+1) = zΓ(z) gives ∏_n=1^L (n+a) = (Γ(L+1+a))/(Γ(1+a)) for a ∈ {0, D/2, D}, establishing the exact finite Gamma identity for P_L^(D), and Stirling's asymptotic (Γ(L+1+D/2)²)/(Γ(L+1)Γ(L+1+D)) = 1 + O(L^-1) → 1 yields the limit Γ(D+1)/Γ(1+D/2)²; the stack pushforward is proved in Proposition 6.5(2)–(3). □

Table 6.1 — Hypotheses and Proved Scope of Each Route in Theorem 6.5

Remark 6.6a (Triangular Canonical Product Π_△(u), Self-Contained Brouncker Continuant Defect Proof, and Completed-Zeta Scope). Because ∑_{n≥ 1} T_n^-1 = 2, Π_△(u) = ∏_n=1^∞ (1 - u/T_n) is entire of genus 0, order 1/2, and type π√2; writing a(a+1) = 2u gives 1 - u/T_n = ((n-a)(n+a+1))/(n(n+1)) and Euler's reflection closed form

For Brouncker's generalized continued fraction B(s) = s + K_n=1^∞ ((2n-1)²)/(2s) = 4[(Γ((s+3)/4))/(Γ((s+1)/4))]² (Re s > 0, where B(1) = 4/π and B(s)B(s+2) = (s+1)²) [16], define the continuant polynomials P_L^Br(s), Q_L^Br(s) ∈ ℤ[s] on all s ∈ ℂ by the initial conditions

and the three-term recurrence (L ≥ 1)

For any a, b ≥ -1 and s ∈ ℂ, put ⟨ a, b⟩_s := (s+1)² Q_a^Br(s) Q_b^Br(s+2) - P_a^Br(s) P_b^Br(s+2) and write D_L^Br(s) := ⟨ L, L⟩_s, C_L^-(s) := ⟨ L, L-1⟩_s, and C_L⁺(s) := ⟨ L-1, L⟩_s (with (-1)!! := 1). At L = 0, D₀^Br(s) = (s+1)² - s(s+2) = 1, C₀^-(s) = -s, and C₀⁺(s) = -(s+2); at L = 1, P₁^Br(s) = 2s²+1 and Q₁^Br(s) = 2s give C₁^-(s) = s - 2, C₁⁺(s) = s + 4, and D₁^Br(s) = 4s(s+2)(s+1)² - (2s²+1)(2(s+2)²+1) = -9 = -(3!!)². By induction on L ≥ 1, expanding the first or second slot via the recurrence yields

and for L ≥ 2, expanding D_L^Br(s) = 2s C_L⁺(s) + (2L-1)²[2(s+2)C_L-1⁺(s) + (2L-1)² D_L-2^Br(s)] gives

Consequently, for every L ≥ 1 and every s ∈ ℂ, D_L^Br(s) is independent of s and satisfies the exact finite-stage scalar normalization

(Scope: This identity is an exact finite scalar normalization of ((2N+1)!!)², not an operator intertwiner between the continuant recurrence and triangular multiplication. Likewise, under u(s) = (s(s-1))/2 = -1/8 + ((s-1/2)²)/2, the branch point u(1/2) = -1/8 and the Riemann–Siegel phase constant -π/8 in θ(t) = t/2 log(t/(2π)) - t/2 - π/8 + 1/(48t) + O(t^-3)—giving 1 - 1/8 = 7/8 in N₀(t) = 1 + (θ(t))/π + S(t)—share the half-shift s - 1/2 as distinct polynomial and Gamma-phase operations. Writing Ξ_△(u) := ξ(s) = 1/2 s(s-1)π^-s/2Γ(s/2)ζ(s) in u = s(s-1)/2, negative integers s = -k (k ≥ 1) evaluate at u(-k) = T_k to Ξ_△(T_k) = ξ(k+1), which does not uniquely determine Ξ_△(u) inside its order-1/2 infinite-type class because Ξ_△(u) + b Π_△(u) shares all triangular values u = T_k.)*

Remark 6.6b (The Commutant of the Exchange E and the Family B_a = I + aE, Projective Cayley / Möbius Group ℳ ≅ ℝ^×, and Self-Contained Euler-Transform Proof for S₊ [18, 19]):

1. The Commutant of the Exchange E and the Family B_a = I + aE: A 2× 2 real matrix L commutes with the coordinate exchange E = ([0 1 ; 1 0]) iff L ∈ 𝒜 = ℝI ⊕ ℝE = ℝP₊ ⊕ ℝP_- (P_± = 1/2 (I ± E)), on which L ↦ L e₁ is a linear isomorphism onto ℝ² intersecting the affine family ([1 a ; a' b]) at the unique matrix B_a = I + aE = (1+a)P₊ + (1-a)P_- (a'=a, b=1).
2. Projective Cayley Conjugacies & Completed Möbius Group: On ℝℙ¹, the projective Cayley involution C[a:b] = [b-a : b+a] (C(a) = (1-a)/(1+a)) satisfies C(1/a) = -C(a) and C(-a) = 1/C(a), intertwining the Klein-four group {id, -a, 1/a, -1/a} ≅ C₂× C₂ (with ℝℙ¹ fixed sets {0,∞}, {-1,1}, and ∅). Adjoining M_∞(a) = 1/a to the open family M_β(a) = (a-β)/(1-β a) (β ∈ ℝ∖{± 1}, where M₂ M_-1/2 = M_∞) yields the completed abelian group ℳ = {M_β : β ∈ ℝℙ¹∖{-1,1}} ≅ ℝ^× via the homomorphism μ_β = (1+β)/(1-β) (μ_∞ = -1, μ_β⋆γ = μ_βμ_γ), in which M_∞(a) = 1/a is the unique nonidentity involution.
3. Telescoping Zeta Identity and Exact-Rational Euler Enclosure for S₊: Define the positive telescoping series S₊ := ∑_n=1^∞ (ln n)/(n(n+1)) [19]. Using 1/(n(n+1)) = 1/n - 1/(n+1), partial summation on 1 ≤ n ≤ M gives ∑_n=1^M ln n(1/n - 1/(n+1)) = ∑_i=2^M (ln i - ln(i-1))/i - (ln M)/(M+1). Letting M → ∞ ((ln M)/(M+1) → 0), expanding -ln(1 - 1/i) = ∑_k=1^∞ 1/(k i^k) (all terms positive for i ≥ 2), and interchanging sums by Tonelli's theorem yields

For rigorous rational enclosure without Euler–Maclaurin remainders, Euler's binomial transform of the alternating zeta function, (1 - 2^1-s)ζ(s) = ∑_n=0^∞ 2^-(n+1)Δⁿ f_s(1) of the completely monotone sequence f_s(j) = j^-s (s = k+1 ≥ 2) satisfies 0 < (-1)ⁿ Δⁿ f_s(1) ≤ 1, so its L-term truncation has positive error bounded strictly by ∑_n=L^∞ 2^-(n+1) = 2^-L, while the outer tail beyond K ≥ 1 satisfies 0 < ∑_k=K+1^∞ (ζ(k+1)-1)/k ≤ ∑_k=K+1^∞ (k+2)/(k² 2^k+1) ≤ (K+3)/((K+1)² 2^K+1) (at L=240, K=200, rational enclosure width < 1.57× 10^-63; by contrast, S_- := ∑_k=2^∞ (ln k)/(k(k-1)) ≈ 1.2577468869443696 gives S₊ + S_- ≈ 2.0462774528558786 nats). Moreover, S₊ is strictly distinct from three elementary closed-form series whose values hold analytically by telescoping: (i) S_tel := ∑_n=2^∞[ln(1 + 1/(n²-1)) - 1/n²] = 1 + ln 2 - π²/6 ≈ 0.0482131137; (ii) ∑_i=1^∞2/((4i-1)(4i+1)) = 1 - π/4 ≈ 0.2146018366; and (iii) ∑_i=1^∞1/((2i-1)(2i)(2i+1)) = ln 2 - 1/2 ≈ 0.1931471806.

7. Scope Boundaries and Open Questions

7.1 Mirror-Prime Pairs in Odd Chambers (Finite Verification: 1403 ≤ m ≤ 20001)

For an odd chamber m = 2M + 1, the midpoint reflection R_mid(N) = m² + 1 - N fixes the diagonal address N_c = (m² + 1)/2, exchanges the ordered symmetric endpoints N_c - k ↔ N_c + k, and fixes the unordered pair {N_c - k, N_c + k} setwise across the diagonal:

Here a + b = m² + 1 and 2 ≤ b - a = 2k ≤ m - 1. Using distinct unordered prime pairs (a < b, i.e., k ≥ 1), direct sieve computation [9] verifies that every odd chamber in the finite interval 1403 ≤ m ≤ 20001 contains at least one mirror-prime pair (beginning at m = 1403, N_c = 984,205 with k_min = 6, pair {984,199, 984,211}). Proving existence for all odd m ≥ 1403 requires controlling prime pairs in the short moving interval [N_c - M, N_c + M] of length m - 1 ≈ √(2N_c) around N_c, which remains open.

7.2 Specific Record-Set Intersection Problem (Open Beyond {1, 5, 29})

As shown in §5.2, generic distinct primitive carrier rays can share a first chamber (such as (11,3) and (9,7) at m₀ = 65). A narrower question concerns the two specific approximation record families in Appendix A: the 45^∘ Pell record rays (U_j, W_j) (with | U² - W² - 2UW | = 2) and the a = 2 target (3/5, 4/5) record rays (with | U - 2W | = 1). The only primitive rays belonging to both record families are (1,1), (3,1), and (7,3), with first chambers m₀ ∈ {1, 5, 29}. Setting τ = 2U + W ≡ 3 or 7 (mod 10) and ε = U - 2W = ± 1 on the a = 2 record chambers m_k^- = 10k² + 6k + 1 and m_k⁺ = 10k² + 14k + 5 gives 10m₀ - 1 = τ², while Pell completeness (G² - 2m₀² = -1 for m₀ = P_2j+1, where P₀=0, P₁=1, P_n+2=2P_n+1+P_n and Q_n denotes the half-Pell–Lucas sequence Q₀=Q₁=1, Q_n+2=2Q_n+1+Q_n with backward extension Q_-1=-1) reduces the value-set intersection P_2j+1 ∈ {m_k^-, m_k⁺} to the quartic Diophantine equation 50G² = τ⁴ + 2τ² - 49, or equivalently via the companion identity 10P_2j+1 - 1 = Q_j+2² + Q_j-1² - 1 = τ² (and via the generically degree-two unramified map (τ, G) ↦ (A,C) = (2(τ²+1), 20Gτ) = (20m₀, 20Gτ)—which is injective on τ, G > 0—to positive integral points (A,C) ∈ E(ℤ) on the smooth Weierstrass elliptic curve E : C² = (A - 2)(A² - 200) of discriminant Δ(E) = 16 · 4 · 200 · 196² = 491,724,800 = 2¹³ · 5² · 7⁴, where (A,C) ∈ E(ℤ) represents a positive intersection iff A > 0, C > 0, 20 | A, (A-2)/2 = τ², and (A²-200)/200 = G² for positive integers τ, G > 0 with τ ≡ 3 or 7 (mod 10), recovering m₀ = A/20, ε = -1 if τ ≡ 3 (mod 10) else +1, and U = (2τ+ε)/5, W = (τ-2ε)/5). Because Q_j-1² - 1 = 0 iff j ∈ {0, 1, 2}, the three intersections (m₀, G, τ) ∈ {(1,1,3), (5,7,7), (29,41,17)} correspond to the particular square root τ = Q_j+2 for j = 0, 1, 2 (E₀=(20,60), E₁=(100,980), E₂=(580,13940)). Direct finite enumeration confirms that the two record-chamber sets have no further intersection up to m₀ ≤ 10⁷ (and modular quadratic-residue sieving excludes 0 ≤ j ≤ 1000 beyond j ∈ {0,1,2}), while an all-index exclusion proof—either elementary or via a complete integral-point determination on E(ℤ) with pullback filters—remains open.

7.3 Separating Equations from Exploratory Notebook Sketches

Hand-drawn annotations on early working cards (such as reciprocal ratios 9/56 ↔ 56/9 and Moessner striking diagrams) record heuristic motivation during the initial construction of the Grid [9]. Every theorem in this paper rests solely on the explicitly declared definitions, domains, and proofs given in the text.

8. Divisor Shells, Jordan Totients, and Factorial-Residual Arithmetic on the Grid Addresses

8.1 Relation to the Coordinate Layers and the Shared Identity Criterion

Because the address N(r,c) = T_m-1 + r in chamber m = r+c-1 is exactly the rank of the rational triangular entry a(m,r) = m - 1 + r/m (§1.3), the positive quadrant of §§1–7 serves directly as the coordinate system for three arithmetic structures [11, 12, 14]: (i) divisor-band and shell-numerator progressions (§8.2), (ii) k-dimensional Jordan-totient shells and their three zeta limits (§8.3), and (iii) the triangular factorial-residual calculus and its entire-function Taylor lift (§§8.4–8.4a). Table 8.0 summarizes how the coordinate layers, the odd-square factorization of §6.6, and the arithmetic theorems below fit together logically:

Table 8.0 — Directional Implication Map Across the Coordinate, Factorization, and Arithmetic Layers

Across all comparisons between geometric and arithmetic constructions, we enforce the following criterion [12]:

Principle 8.1 (Shared Identity Criterion). A repeated numerical constant, exponent, or shape across two constructions is structurally meaningful if and only if both appearances instantiate an explicit common algebraic identity, coordinate transformation, generating function, or measure pushforward. Where no shared identity exists (such as a rational shell defect -1/12 vs. ζ(-1) = -B₂/2 = -1/12, a 45^∘ cone denominator 84 vs. ζ(-13) = -7/84, or finite parameter counts vs. exponential-sum exponents), the coincidence is classified as a small-denominator collision and kept strictly separate. (All symbols are defined locally and summarized in Appendix C; the notation follows the corpus notation register [7].)

8.2 Divisor-Band Bijection, Shell Numerator Progressions, and Finite Shell Diagnostics

In the rational triangular grid [11], chamber m ≥ 1 consists of the m rational numbers

whose rank is the address N(r,c) = T_m-1 + r and whose fractional part r/m is the native row phase κ_row. Grouping entries by their reduced denominator d = m/gcd(m,r) gives the following exact progression structure [11, 14]:

Theorem 8.1 (Divisor-Band Bijection, Shell Numerator Progressions, and the Square-Gap Factorization). For every integer m ≥ 1 and row 1 ≤ r ≤ m, set k = gcd(m,r), d = m/k, and h = r/k so that gcd(h,d) = 1 and 1 ≤ h ≤ d. Then m = dk and r = hk, and the reduced form of a(m,r) has denominator d and numerator

Consequently:

1. Divisor-Band Decomposition & Window Discrepancy: For each divisor d | m, exactly φ(d) entries in chamber m reduce to denominator d, realizing ∑_{d | m}φ(d) = m. For any divisor subset D ⊆ Div(m) and β ∈ [0,1], the coprime window discrepancy Δ_m,D(β) := ∑_{d∈ D}#{1≤ a≤ dβ : gcd(a,d)=1} - β∑_{d∈ D}φ(d) satisfies

1. Strict Within-Shell Uniqueness & Progression Multiplier: For each fixed shell d ≥ 1, (k,h) ↦ R_d(k,h) is injective on ℤ_{≥ 1} × {1 ≤ h ≤ d : gcd(h,d)=1} (so no integer or prime value ever repeats within a single shell d, although cross-shell values such as R₂(2,1) = R₃(1,1) = 7 can coincide). As k ≥ 1 varies, shell d traces φ(d) residue classes h - d (mod d²), each coprime to d², and since d and d² have the same prime factors, φ(d²) = dφ(d), giving

1. Square Gap, the Factorization m² - 1 = (m-1)(m+1), and the Midpoint vs. n²+1: At r = m - 1 (gcd(m,m-1)=1, d=m, k=1, h=m-1), m² - a(m,m-1)² = 2 - 1/m² = 4(1/4 + 𝔣(χ_m^±)), and R_m(1,m-1) = m² - 1 = (m-1)(m+1) is composite for every m ≥ 3. On the Notebook Grid, the diagonal and superdiagonal addresses are N(n,n) = 2n²-2n+1 (diagonal, 5 at n=2) and N(n,n+1) = 2n² (superdiagonal, 8 at n=2), not n² and n²+1; the Landau polynomial n²+1 (where for an odd prime p there exists n ≥ 1 with p | (n²+1) iff ((-1)/p)=+1 ⇔ p ≡ 1 (mod 4)) connects to the Grid via the odd-chamber midpoint sum N + R_mid(N) = 2N_c = m²+1 (§7.1 and §8.5(3)), not via N(n,n+1).

Proof. Since gcd(h,d) = 1, gcd(R_d(k,h), d) = gcd(d² k - d + h, d) = gcd(h, d) = 1, so R_d(k,h)/d is reduced and gcd(h-d, d²) = 1. (1) Each r ∈ {1,…,m} writes uniquely as r = (m/d)h with d | m and gcd(h,d)=1 (1≤ h≤ d), giving φ(d) entries per divisor d | m. By 1_gcd(a,d)=1 = ∑_{i | gcd(a,d)}μ(i), #{1≤ a≤ dβ : gcd(a,d)=1} = ∑_{i | d}μ(i)⌊ dβ/i⌋; subtracting βφ(d) = ∑_{i | d}μ(i)(dβ/i) gives the formula for Δ_m,D(β). Since each bracket ⌊·⌋ - (·) lies in (-1, 0] and ∑_{i | d}|μ(i)| = 2^ω(d), summing over d ∈ D bounds |Δ_m,D(β)| ≤ ∑_{d∈ D} 2^ω(d); for D = {p^a}, Δ_p^a,{p^a}(β) = (⌊ p^a β⌋ - p^a β) - (⌊ p^a-1β⌋ - p^a-1β), which approaches -(1 - 1/p) as β ↑ 1/p^a-1 and has supremum 1 - 1/p. (2) Writing R_d(k,h) - 1 = d²(k-1) + (d(d-1) + h - 1) with 0 ≤ h - 1 ≤ d - 1 places the remainder d(d-1) + h - 1 in [d(d-1), d² - 1] ⊂ [0, d² - 1], so Euclidean division by d² uniquely recovers k - 1 and h - 1; φ(d²) = d²∏_{p | d²}(1-1/p) = d· d∏_{p | d}(1-1/p) = dφ(d). (3) Direct substitution at r=m-1 gives a(m,m-1) = m - 1/m, m² - (m-1/m)² = 2 - 1/m², and R_m(1,m-1) = m² - 1 = (m-1)(m+1) ≥ 8 composite for m ≥ 3; for an odd prime p, a² ≡ -1 (mod p) is solvable in ℤ iff (-1)^(p-1)/2 = 1 ⇔ p ≡ 1 (mod 4) by Euler's criterion. □

Remark 8.1a (Finite Shell Diagnostics, Explicit Cohort, Like-for-Like CV Weightings, and Two Variance Conventions [14]). To compare finite prime counts across shells 1 ≤ d ≤ M against the Dirichlet–PNT asymptotic heuristic without dividing by log 1 = 0, define the active shell cells and cell weights by

Because R_d(k,h) = d² k - d + h ≥ 2 for every cell except (d,k,h) = (1,1,1) where R₁(1,1) = 1, this cutoff excludes only (1,1,1), giving w₁ = M - 1 and w_d = φ(d)⌊ M/d⌋ for d ≥ 2. Selecting the shell cohort

gives |𝒞_M| = 153, 479, 1000, 2000 shells at M = 250, 500, 1000, 2000. Every integer threshold from 97 to 100 gives the same four cohorts; we fix w_d ≥ 100. Define the observed active-cell prime density and the cellwise PNT-in-progressions predictor

and compare the size-blind baseline A_M(d) := φ(d)δ_M(d) against the size-corrected ratio Q_M(d) := δ_M(d)/δ̂_M(d) on 𝒞_M. Here d²/φ(d²) = d/φ(d) is exact, whereas 1/log R_d(k,h) is an asymptotic prime-density heuristic (no uniform theorem for the growing moduli d² ≤ M² follows from a finite table). Using average ranks for Spearman ties and population variance, write CV_eq(Y) = √(1/(|𝒞_M|) ∑_d∈𝒞M(Y(d)-Y_eq)²)/Y_eq for equal-shell CV and CV_w(Y) = √(∑_d∈𝒞M w_d(Y(d)-Y_w)² / ∑_d∈𝒞M w_d)/Y_w for cell-weighted CV. The table reports both like-for-like weightings:

Under either identical weighting scheme (CV_eq: 0.539 → 0.079 or CV_w: 0.579 → 0.076 at M=1000), dividing by δ̂_M(d) reduces shell dispersion substantially. Similarly, at M = 1000, the two binomial-style standard-error conventions

give 996/1000 shells satisfying |δ_M(d) - δ̂_M(d)| ≤ 2 SE_d^pred (excluding d ∈ {450, 464, 680, 956}) and 994/1000 shells satisfying |δ_M(d) - δ̂_M(d)| ≤ 2 SE_d^obs (additionally excluding d ∈ {328, 550}). These bands are descriptive diagnostics on deterministic prime data, not independent Bernoulli trials or confidence-coverage guarantees: higher shells reach larger numerators (R_d(k,h) ∼ d² k ≤ dM), while A_M(d) = φ(d)δ_M(d) omits the factor d in d/φ(d), explaining why shell stratification acts as a classical PNT-in-progressions coordinate lens rather than a new prime-distribution law.

8.3 The k-Dimensional Jordan Chamber Lift and Three Separate Zeta Limits

Lifting the row phases {r/m : 1≤ r≤ m} to the k-dimensional cube C_k(m) = {(r₁/m,…,r_k/m) : 1≤ r_i≤ m} and assigning each tuple its reduced common denominator d_m(r₁,…,r_k) = m/gcd(m,r₁,…,r_k) connects Remark 6.1b directly to Jordan's totient J_k(d) = d^k∏_{p | d}(1 - p^-k) [12]:

Theorem 8.2 (Jordan Shell Conservation, Body-Shell Möbius Formula, and Three Separate Zeta Limits). For every dimension k ≥ 1 and chamber m ≥ 1:

1. Jordan Shell Conservation, Separated Primitive Counts, and Punctured-Stack Möbius Sieve: For each divisor d | m, #{(r₁,…,r_k)∈{1,…,m}^k : d_m(r₁,…,r_k) = d} = J_k(d), so ∑_{d | m} J_k(d) = m^k. More generally, for any subset 𝒲 ⊆ [0,1]^k with L_𝒲(a) = #{r∈{1,…,a}^k : r/a ∈ 𝒲}, A_𝒲(d) := #{r∈{1,…,d}^k : r/d∈ 𝒲, gcd(d,r₁,…,r_k)=1} = ∑_{i | d}μ(i)L_𝒲(d/i). Row primitivity, common-denominator primitivity, and punctured-stack visibility carry distinct predicates: (a) on the address-sheet row of chamber m, with sites (-r/(m+1), 1/(m+1)) (1 ≤ r ≤ m), the second coordinate forces gcd(m+1, r, 1) = 1 for all m sites, whereas primitive columns require gcd(r, m+1) = 1 (counting φ(m+1)); (b) on the punctured stack S_n^* = S_n ∖ {(0,0,0)} of size | S_n| - 1 = C(2n+3, 3) - 1,

1. Three Separate Divisor-Support Limits: Setting F_k(n) := (J_k(n))/n^k = ∑_{i | n}(μ(i))/i^k = ∏_{p | n}(1 - p^-k) (and primorial P_β = ∏_{p≤ β} p) separates three distinct asymptotic objects:

1. Reconciliation of the Chamber Fraction (φ(n))/(n-1) and the One-Dimensional Density (φ(n))/n: In chamber m ≥ 1, put n = m+1; the m address-sheet sites p₁ = -r/n (1 ≤ r ≤ n-1) include φ(n) with gcd(r,n) = 1, so

Proof. (1) Putting a = gcd(m, r₁, …, r_k) = m/d sets (r₁,…,r_k) = a(r'₁,…,r'_k) with r' ∈ {1,…,d}^k and gcd(d, r'₁, …, r'_k) = 1. Since 1_{gcd(d,r1,…,rk)=1} = ∑_{i | gcd(d,r1,…,rk)}μ(i), writing r = iv (v ∈ {1,…,d/i}^k) gives A_𝒲(d) = ∑_{i | d}μ(i)L_𝒲(d/i), which for 𝒲 = [0,1]^k (L_𝒲(a) = a^k) evaluates to ∑_{i | d}μ(i)(d/i)^k = J_k(d). Similarly, (p₁,p₂,ℓ) ∈ S_n^* is divisible by i ≥ 1 iff (p₁,p₂,ℓ) = i(p'₁,p'₂,ℓ') with 0 ≤ iℓ' ≤ n ⇔ ℓ' ≤ ⌊ n/i⌋ and | p'₁|,| p'₂| ≤ ⌊(n-iℓ')/i⌋ = ⌊ n/i⌋ - ℓ', bijecting onto S_{⌊ n/i⌋}^* of size | S_{⌊ n/i⌋}| - 1. (2) For k ≥ 2, ∑_i=1^∞ i^-k < ∞, so F_k(P_β) = ∏_{p≤ β}(1 - p^-k) → ∏_p(1 - p^-k) = 1/ζ(k). For k ≥ 1, 1/M ∑_{n≤ M} F_k(n) = 1/M ∑_{i≤ M}(μ(i))/i^k ⌊ M/i⌋ = ∑_{i≤ M}(μ(i))/i^k+1 + O(1/M ∑_{i≤ M} i^-k) → 1/(ζ(k+1)), with error O((log M)/M) at k = 1 and O(M^-1) for k ≥ 2. (3) Since 0 < (φ(n))/(n(n-1)) ≤ 1/(n-1), 1/(M-1) ∑_n=2^M (φ(n))/(n(n-1)) = O((log M)/M), giving the same Cesàro limit 6/π². □

8.4 The Triangular Factorial-Residual Calculus on the Grid Addresses

On chamber I_m = (T_m-1, T_m], with m(N) = ⌊√(2N)+1/2⌋ the chamber of N, the factorial-residual operator Q(N) := N/(gcd(N, m!)) acts directly on the Notebook Grid address N = N(r,c) = T_m-1 + r [12]:

Theorem 8.3 (Complete Factorial-Residual Classification, the Exceptional Prime Powers {p²}∪{27}, the Successor-Prime Rule, the Rising-Factorial gcd, and the Dickman Density log 2). For every integer N ≥ 1 in chamber m = m(N), let Q(N) = N/gcd(N, m!). Then:

1. Prime-Residual Dichotomy & Complete Branch Classification: For every N ≥ 1, Q(N) ∈ {1} ∪ ℙ, and any prime p occurs as Q(N) = p at most once in a chamber I_m. Moreover, Q(N) > 1 if and only if N belongs to one of three disjoint branches:

• (i) Prime squares: N = p² for a prime p (lying in chamber p ≤ m ≤ 2p-1 where v_p(m!) = 1, giving Q(p²) = p);
• (ii) The exceptional cube: N = 27 = 3³ in chamber m = 7 ((r,c) = (6,2), v₃(7!) = 2, Q(27) = 3; every other prime power p^k with k ≥ 3 satisfies Q(p^k) = 1);
• (iii) High-prime multiples: N = ap where p > m is prime and 1 ≤ a < (m+1)/2 (giving Q(ap) = p).

1. The Successor-Prime Rule on the First-Column Edge: On the first-column edge c = 1 (r = m, N = T_m), Q(T_m) = den(m!/T_m) = a_m, the reduced numerator of T_m/m! [4], satisfies

1. Pooled Dickman Density log 2 & Abscissa σ_c = 2: Let F_res(m) = #{N ∈ I_m : Q(N) > 1} and A_res(M) = ∑_m=1^M F_res(m). For M ≥ 7, the m-smooth count S_sm(M) = #{N ≤ T_M : P⁺(N) ≤ m(N)} and closed count C_res(M) = T_M - A_res(M) satisfy S_sm(M) - C_res(M) = π(√T_M) + 1, and by de Bruijn's smooth-number theorem [13] (Dickman's function ϱ has ϱ(2) = 1 - log 2),

1. Half-Index Identity, Reciprocal-Triangular Sum, and the Golden Ratio at Address 1/2: For odd j = 2n+1, 8T_(j-1)/2 + 1 = 8T_n + 1 = j²; -1/2 + ∑_i=1^M 1/T_i = 3/2 - 2/(M+1) → 3/2 = T₂/(2!); and the continuous inverse T^-1(a) = (√(8a+1)-1)/2 evaluates at a = 1/2 to T^-1(1/2) = (√5-1)/2, so T((√5-1)/2) = 1/2.
2. Successor Split, Rising-Factorial Interior gcd, and Denominator-Jump Trichotomy: For n ≥ 2, write T_n/n! = a_n/b_n in lowest terms, where a_n ∈ {n+1, 1} (n+1 if n+1 is prime, 1 if n+1 is composite) and the cancelled part d_n := (n+1)/a_n = gcd(n+1, 2(n-1)!) ∈ {1, n+1} satisfy a_n d_n = n+1 and a_n + d_n = n+2. In the rising product Π_n(y) = ∏_j=1ⁿ (1 + jy) = ∑_k=0ⁿ e_k(n) y^k (e₀(n) = 1, e₁(n) = T_n, e_n(n) = n!), the interior coefficients satisfy g_n := gcd(e₁(n),…,e_n-1(n)) = a_n for all n ≥ 2. For n ≥ 3, the denominator jump D_n := b_n / b_n-1 = (n-1)d_n-1/d_n obeys

so n+1 is prime iff D_n ∈ ℤ, in which case (2T_n+D_n)/2 = n², (2T_n-D_n)/2 = n, and (2T_n)² - D_n² = 4n³. (On the forward ratio b_n/b_n+1, the single composite-to-prime index n=11 evaluates to 1/(11· 12) = 1/(2T₁₁) = 1/132, which coincides numerically at that single index with |ζ(-9)| = B₁₀/10 = 1/132 under Principle 8.1.)

1. Primitive Triple Rank Bijection and the Unique Kummer Carry Excess at (j,p)=(5,3): Writing (m,r) = k(d,h) with k = gcd(m,r) and gcd(d,h) = 1, the map (k,d,h) ↦ N(kh, k(d-h)+1) = (kd(kd-1))/2 + kh is a bijection onto ℤ_{≥ 1}, and for a uniformly chosen address N ≤ T_M the scale satisfies Pr(k=a) → 6/(π² a²) as M → ∞. For Λ_j := ∏_i=j^2j-1 i = ((2j)!)/(2 j!), δ_p(j) := v_p(Λ_j) - 2v_p(j!) ≤ 0 holds for every prime p ≤ j (j ≥ 2) except at the unique pair (j,p) = (5,3), where δ₃(5) = +1 and Λ₅/((5!)²) = 21/20.

Proof. (1) Let N ∈ I_m = (T_m-1, T_m], so N ≤ (m(m+1))/2 < ((m+1)²)/2, and suppose a prime p divides Q(N), i.e., k := v_p(N) > v_p(m!).

• If p > m: Then v_p(m!) = 0 and N = ap with 1 ≤ a = N/p ≤ (m(m+1))/(2(m+1)) = m/2 < (m+1)/2 ≤ m (requiring m ≥ 2). Every prime factor p' of a satisfies p' ≤ a < m and v_p'(a) ≤ v_p'(m!) (since a | m!), while gcd(p, m!) = 1; thus gcd(ap, m!) = a and Q(N) = p (Branch iii). Because | I_m| = m < p, at most one multiple of p lies in I_m.
• If p ≤ m: Write N = b p^k with p ∤ b; since v_p(m!) ≥ 1, we have k ≥ 2. If k = 2, then v_p(m!) = 1 ⇒ p ≤ m ≤ 2p - 1, whence N ≤ T_2p-1 = p(2p-1) < 2p², forcing b = 1 and N = p² with Q(p²) = p (Branch i). If k = 3: for p = 2, N ≥ 8 > T₃ = 6 ⇒ m ≥ 4 ⇒ v₂(m!) ≥ 3 = k, impossible; for p ≥ 5, m + 1 > √2 p^3/2 ≥ √10 p > 3p ⇒ m ≥ 3p ⇒ v_p(m!) ≥ ⌊ m/p⌋ ≥ 3, impossible; for p = 3, N = 27b > T₆ = 21 ⇒ m ≥ 7, where m ≥ 9 has v₃(m!) ≥ 4 > 3 and m ∈ {7,8} has N ≤ T₈ = 36 < 54, forcing b = 1, N = 27 ∈ I₇ (v₃(7!) = 2, Q(27) = 3, Branch ii; note 3² = 9 ∉ I₇). If k ≥ 4: for p = 2, N ≥ 16 > T₅ ⇒ m ≥ 6 (v₂(6!)=4), N ≥ 32 > T₇ ⇒ m ≥ 8 (v₂(8!)=7≥ 5), N ≥ 64 > T₁₀ ⇒ m ≥ 11 (v₂(11!)=8≥ 6), and for k ≥ 7, m + 1 > 2^(k+1)/2 > 2k ⇒ v₂(m!) ≥ ⌊ m/2⌋ + ⌊ m/4⌋ > k; for p ≥ 3, k ≥ 4, N ≥ 81 > T₁₂ ⇒ m ≥ 13 (v₃(13!)=5≥ 4), and in general m + 1 > √2 p^k/2 implies ⌊ m/p⌋ ≥ ⌈√2 p^k/2 - 1⌉/p - 1 ≥ k whenever (p,k) ≠ (3,4) (checked directly at (3,5): m ≥ 22, v₃(22!)=9, and growing exponentially via p^k/2-1 ≥ 3^k/2-1). Because Branches (i)–(ii) are pure prime powers p², 3³ and Branch (iii) has a | m!, no second prime can divide Q(N), proving Q(N) ∈ {1} ∪ ℙ and within-chamber uniqueness.

(2) Since m!/T_m = (2(m-1)!)/(m+1), Q(T_m) = (m+1)/(gcd(m+1, 2(m-1)!)). At m=1 (m+1=2) and m=3 (m+1=4), m+1 divides 2(m-1)!, so Q(T_m) = 1. If m+1 is an odd prime (m ≥ 2), gcd(m+1, 2(m-1)!) = 1 so Q(T_m) = m+1. If m+1 ≥ 6 is composite, either m+1 = ab with 2 ≤ a < b ≤ (m+1)/2 ≤ m-1 (so ab | (m-1)!) or m+1 = p² for an odd prime p ≥ 3, where 2p ≤ p² - 3 < p² - 1 = m for p ≥ 5 and at p = 3 (m = 8) 3 · 6 | 7!, so p² | (m-1)! and Q(T_m) = 1.

(3) By (1), Q(N) = 1 iff P⁺(N) ≤ m(N) and N ∉ {p² : p∈ℙ} ∪ {27}. Every p² has m(p²) ≥ p = P⁺(p²) and 27 has m(27) = 7 > 3, so for M ≥ 7, S_sm(M) - C_res(M) = π(√T_M) + 1 = O(M/log M). Because | m(N) - √(2N)| < 1/2, replacing P⁺(N) ≤ m(N) by P⁺(N) ≤ √(2T_M) alters at most O(M²/log M) = o(T_M) integers (since N ≤ ε T_M contributes ≤ ε T_M and on ε T_M < N ≤ T_M primes in [√(2ε T_M)-1, √(2T_M)+1] contribute ≪ T_M/log M near the boundary). Applying de Bruijn's theorem Ψ(A, A^1/2) = (1 - log 2)A + O(A/log A) [13] at A = T_M gives C_res(M)/T_M → 1 - log 2, so A_res(M)/T_M → log 2. Partial summation on ∑_m=1^M F_res(m) = ((log 2)/2 + o(1))M² yields (s-2)ℛ(s) → log 2 as s ↓ 2.

(4) ∑_i=1^M 1/T_i = ∑_i=1^M (2/i - 2/(i+1)) = 2 - 2/(M+1), and substituting a = 1/2 in T^-1(a) = (√(8a+1)-1)/2 gives (√5-1)/2.

(5) Let g_n = gcd(e₁(n),…,e_n-1(n)) for n ≥ 2, and suppose a prime p divides g_n. If p ≤ n, then p | n! = e_n(n) as well, so Π_n(y) = 1 + ∑_k=1^n-1 e_k(n)y^k + n! yⁿ ≡ 1 (mod p) as polynomials; evaluating at y = -1 gives 0 = Π_n(-1) ≡ 1 (mod p) (the factor j = 1 vanishes), a contradiction. Hence no prime p ≤ n divides g_n. Since g_n | e₁(n) = T_n = (n(n+1))/2 and every prime factor of n (and 2, since n ≥ 2) is ≤ n, g_n must divide n+1 and be odd. If n+1 is composite, every prime factor of n+1 is ≤ (n+1)/2 < n, forcing g_n = 1 = a_n. If n+1 = p is an odd prime (n = p-1), ∏_j=1^p-1(1+jy) ≡ 1 - y^p-1 (mod p) in 𝔽_p[y], so p | e_k(p-1) for 1 ≤ k ≤ p-2; since v_p(T_p-1) = 1, g_p-1 = p = a_n. Next, T_n/n! = (n+1)/(2(n-1)!) = a_n/(2(n-1)!/d_n) gives b_n = (2(n-1)!)/d_n, so for n ≥ 3 (where n and n+1 cannot both be prime), D_n = b_n/b_n-1 = (n-1)d_n-1/d_n yields the three stated cases.

(6) Bijectivity of (k,d,h) ↦ N(kh, k(d-h)+1) follows from Proposition 1.2 and Theorem 8.1. Since Λ_j/((j!)²) = (C(2j, j))/(2 j!), Kummer's theorem (v_pC(2j, j) = C_p(j), the base-p carry count of j + j) gives δ₂(j) = C₂(j) - 1 - v₂(j!) = 2 wt₂(j) - 1 - j ≤ 0 for all j ≥ 2, where wt₂(j) counts the ones in the binary expansion of j (δ₂(2)=δ₂(3)=0, and 2 wt₂(j) ≤ 2⌊log₂ j⌋ + 2 ≤ j + 1 for j ≥ 4), and δ_p(j) = C_p(j) - v_p(j!) for odd primes p ≤ j. Let L = ⌊log_p j⌋ + 1 be the number of base-p digits of j, so C_p(j) ≤ L. If j ≥ p² (L ≥ 3), Legendre's sum gives v_p(j!) ≥ ∑_i=1^L-1 p^L-1-i = (p^L-1-1)/(p-1) ≥ (3^L-1-1)/2 > L ≥ C_p(j), so δ_p(j) < 0. If p ≤ j < p² (L = 2), write j = ap + b (1 ≤ a ≤ p-1, 0 ≤ b ≤ p-1): then v_p(j!) = a and C_p(j) ≤ 2, so δ_p(j) > 0 forces a = 1 and C_p(j) = 2. Adding (1, b)_p + (1, b)_p produces two base-p carries iff 2b ≥ p and 1 + 1 + 1 ≥ p, forcing p = 3, b = 2, and j = 5 (δ₃(5) = 2 - 1 = +1, Λ₅/((5!)²) = 21/20). □

8.4a Entire-Function Realization of the Triangular Coefficients

The factorial-weighted triangular sequence T_n/n! of Theorem 8.3(5) admits an exact entire-function realization via the Euler differential operator [15]. Define on all z ∈ ℂ the entire series and differential operators

Here z ∈ ℂ is an analytic variable and n ≥ 1 a Taylor coefficient index. In the notation of [20, §6.2], E_Ei(z) = -Ein(-z); on any domain in ℂ∖{0} with a consistent branch of the logarithm, E_Ei(z) = Ei(z) - γ - Log z, and the power series in (8.1) is the branch-independent entire definition used throughout.

Proposition 8.4 (Triangular Coefficient Lift and Centered ODE). For every z ∈ ℂ,

Consequently, G₊(z) + G_-(z) = z(z+1)e^z and G₊(z) - G_-(z) = z e^z, so n = n² and n = n for all n ≥ 1. For every integer n ≥ 2, Theorem 8.3(5) gives

Moreover, the centered entire function Ê(z) := e^-z/2E_Ei(z) satisfies Ê(0) = 0 and the linear differential equation

where the right-hand side takes its removable value 1 at z = 0.

Proof. Because E_Ei(z) has infinite radius of convergence, termwise differentiation holds on ℂ. Since Θ_z(zⁿ) = n zⁿ, applying 𝒰 and ℒ multiplies zⁿ/(n n!) by 1/2 n²(n± 1), producing (n(n+1))/(2 n!) zⁿ = T_n/(n!) zⁿ and (n(n-1))/(2 n!) zⁿ = T_n-1/(n!) zⁿ. Using ∑_{n≥ 1}n/(n!) zⁿ = z e^z and ∑_{n≥ 1}(n(n-1))/(n!) zⁿ = z² e^z gives ∑_{n≥ 1}T_n/(n!) zⁿ = 1/2 (z² e^z + z e^z) = (z(z+2))/2 e^z and ∑_{n≥ 1}T_n-1/(n!) zⁿ = z²/2 e^z, from which G₊ ± G_- and the n ≥ 2 coefficient reduction (via Theorem 8.3(5), since [zⁿ]G₊ = T_n/n!) follow immediately (noting that at n = 1, [z¹]G₊ = 1 with n+1 = 2, so the reduced-numerator dichotomy requires n ≥ 2). Finally, E_Ei'(z) = ∑_{n≥ 1}z^n-1/(n!) = (e^z - 1)/z (equal to 1 at z = 0), so Ê'(z) + 1/2 Ê(z) = e^-z/2E_Ei'(z) = (e^z/2 - e^-z/2)/z = (2 sinh(z/2))/z. □

Scope. Proposition 8.4 is an exact analytic realization of the triangular Taylor coefficients T_n/n! and the sums n² and n. Passing from a rational Taylor coefficient [zⁿ]G₊ to its lowest-terms numerator num(T_n/n!) is a nonlinear arithmetic operation on the discrete index n ≥ 2; no interchange with infinite zeta-zero explicit-formula sums, prime-distribution theorem, or statement concerning the Riemann hypothesis is implied.

8.5 Residue Sieves, Odd-Prime Wheel Counts, and the Quadratic Characters of the Diagonal and the Subdiagonal

Comparing primitive lattice visibility (§8.3), shell-numerator progressions (§8.2), and symmetric prime-pair configurations (§7.1) isolates the distinction between local congruence obstruction and global prime occupancy [12]:

1. One-Point Visibility vs. Two-Point Odd-Prime Wheels (3 ≤ p ≤ β): Removing ν_p forbidden residue classes modulo each prime p leaves local survival factor 1 - ν_p/p (or 1 - p^-k on ℤ^k). For one-point primitive lattice visibility (k ≥ 2), inclusion-exclusion converges unconditionally to 1/ζ(k) (6/π² at k=2). For two-point prime patterns—twin pairs {n, n+2} (ν_p = 2 for odd primes p ≥ 3) and Goldbach midpoint pairs {M-k, M+k} (ν_ℓ = 2 for odd primes ℓ ∤ M, collapsing to ν_ℓ = 1 when ℓ | M)—the Chinese Remainder Theorem gives the exact survivor counts modulo the odd-prime wheel P^odd_β = ∏_{3≤ p≤ β} p:

recovering the exact local factor (ℓ-1)/(ℓ-2) of the Hardy–Littlewood singular series 𝔖(2M) = 2C₂∏_{ℓ | M,ℓ≥ 3}(ℓ-1)/(ℓ-2), while global prime occupancy remains obstructed by the classical sieve parity barrier.

1. Twin-Prime Wheel-Exception Rule: Because every twin prime pair (n, n+2) with n ≥ 3 already has n odd, the parity modulus p = 2 excludes no twin prime pair; (3,5) and (5,7) fail the odd-prime wheel (3 ≤ p ≤ β) only when p ∈ {3, 5} divides one of the primes themselves, leaving n ≡ 5 (mod 6) for all twin pairs beyond (3,5).
2. The Diagonal Midpoint Sum 2N_c = m²+1 vs. the Subdiagonal 2a²+1: In an odd chamber m, a mirror-prime pair {N_c-k, N_c+k} around N_c = (m²+1)/2 (§7.1) is a Goldbach midpoint representation of 2N_c = m² + 1 = (N_c-k) + (N_c+k) restricted to 1 ≤ k ≤ (m-1)/2 = O(√N_c). The quadratic residue conditions on the diagonal and the subdiagonal carry distinct Legendre characters:

• The diagonal midpoint sum 2N_c = m² + 1 and n² + 1: Odd prime divisors p | (n²+1) satisfy ((-1)/p) = +1 ⇔ p ≡ 1 (mod 4) (ν_p = 2 for p ≡ 1 (mod 4), ν_p = 0 for p ≡ 3 (mod 4)).
• The subdiagonal N_sub(a) = 2a² + 1 at (a+1, a) in even chamber m = 2a: Odd prime divisors p | (2a²+1) satisfy (2a)² ≡ -2 (mod p), so ((-2)/p) = +1 ⇔ p ≡ 1 or 3 (mod 8) (ν_p = 2 for p ≡ 1,3 (mod 8), ν_p = 0 for p ≡ 5,7 (mod 8), with a = 1 ↦ N_sub(1) = 3 prime, a = 2 ↦ 9 = 3², a = 3 ↦ 19 prime, and a = 11 ↦ 243 = 3⁵).

8.6 Mathematical Synthesis and Remaining Open Problems

Taken together, §§1–8 establish a clean structural boundary across the Notebook Grid:

1. Unconditionally Proved Coordinate, Carrier, Measure, and One-Point / Smooth Laws: The quadrant address bijection and its parity wedge (§1), affine quadratic paths and period-4 offset parity (§2), three chamber reflections and the anti-diagonal pair correspondence (§3), floor-selected Pell branches and diagonal square descent (§4), odd-square rational semicircle carriers and chamber budgets (§5, Appendix A), address-sheet geometry, three-dimensional directional stack shadows, and the common (2n+1)² = 8T_n + 1 balanced-split / PGF / Gamma / stack synthesis (§6, Appendix B), divisor-shell uniqueness and the three Jordan-totient zeta limits (§§8.2–8.3), and the factorial-residual dichotomy Q(N) ∈ {1}∪ℙ with Dickman density log 2 and entire coefficient lift G_±(z) (§§8.4–8.4a) all hold unconditionally with self-contained proofs.
2. Explicitly Open All-Index and Analytic Questions: Five mathematical problems encountered along the boundary of these theorems remain open and are recorded without speculative promotion:

• (a) All-Index Odd-Chamber Mirror Primes (§7.1): Proving that every odd chamber m ≥ 1403 contains a prime pair {N_c-k, N_c+k} (1 ≤ k ≤ (m-1)/2) around N_c = (m²+1)/2, beyond the verified finite range 1403 ≤ m ≤ 20001.
• (b) All-Index Record-Chamber Exclusion on E(ℤ) (§7.2): Proving over all indices j ≥ 0 that m₀ ∈ {1, 5, 29} (E₀, E₁, E₂) are the only positive intersections between the 45^∘ Pell record chambers and the a = 2 excluded-target record chambers on E : C² = (A - 2)(A² - 200).
• (c) Two-Point Prime Occupancy vs. Sieve Parity (§8.5): Passing from exact finite odd-prime wheel survivor counts A_β, A_β(2M) and quadratic characters ((-1)/p), ((-2)/p) to infinite prime occupancy along n²+1, 2a²+1, or short chamber intervals.
• (d) Optimal Multi-Prime Divisor-Window Constant (§8.2): Determining the sharp multi-prime supremum constant refining sup_{0≤ β≤ 1}|Δ_m,D(β)| ≤ ∑_{d∈ D} 2^ω(d) in Theorem 8.1(1).
• (e) Local and Meromorphic Structure of the Residual Dirichlet Series ℛ(s) (§8.4): Determining whether ℛ(s) = ∑_{m≥ 1} F_res(m) m^-s admits meromorphic continuation left of Re s = 2 beyond its leading asymptotic (log 2)/(s - 2), and characterizing the chamber-by-chamber variance and sign changes of F_res(m) - m log 2.

Appendix A. Complete Chamber-Budget Theory for Rational Carriers

This appendix proves the complete chamber-budget approximation theorems for centered and leg-sum rational carriers on the upper unit semicircle under an integer chamber bound m ≤ M [9].

A.1 The 45^∘ Centered Chamber-Budget Theorem and Stepwise Error

Fix an integer chamber budget M ≥ 1, and restrict the admissible set to positive integer cells (r,c) with m = r+c-1 ≤ M whose centered carrier point 𝒞(r,c) in (5.1) is rational. We minimize Euclidean chord distance to the 45^∘ target p_* = (1/√2, 1/√2). With m₀, ℓ₁, ℓ₂, G = ℓ₁ + ℓ₂, and δ_leg = ℓ₁ - ℓ₂ defined as in (5.7),

Theorem A.1 (All-Budget 45^∘ Optimum and Complete Minimizing Cell List). Let (U_j, W_j) be the Pell carrier sequence (5.6) with first chambers m_0,j = (U_j² + W_j²)/2 ∈ {1, 5, 29, 169, 985, …}. For any integer budget M ≥ 1, let j ≥ 0 be the unique index such that m_0,j ≤ M < m_0,j+1. Then:

1. The unique minimizing circle point among all rational centered carriers with m ≤ M is p(U_j, W_j).
2. The complete set of minimizing cells in the grid with m ≤ M consists of its odd-scale repeats:

and the number of such minimizing cells is ⌊ (⌊ M/m_0,j ⌋ + 1)/2 ⌋.

1. Let E₄₅(M) = min_{m ≤ M} ‖ p(U,W) - p_* ‖₂. Along record chambers, m_0,j E₄₅(m_0,j) → 1/√2, while over all integer budgets M ≥ 1,

Proof. Since δ_leg = (U² - W²)/2 - UW is odd and G² + δ_leg² = 2m₀², any odd prime p | gcd(δ_leg, m₀) would divide G and hence divide ℓ₁ = (G+δ_leg)/2 and ℓ₂ = UW, contradicting gcd(U,W)=1. Moreover, for any odd prime p | δ_leg, coprimality gives p ∤ m₀, and G² + δ_leg² = 2m₀² reduces to G² ≡ 2m₀² (mod p). Thus 2 must be a quadratic residue modulo p, which excludes p = 3 and p = 5. Consequently every primitive carrier ray has either | δ_leg | = 1 or | δ_leg | ≥ 7.

Every | δ_leg | = 1 pair belongs to (5.6): the base is (1,1), and if U < W then δ_leg < -UW ≤ -3, while U = W is primitive only at (1,1). For any nonbase solution, U > W and (U - W)² - 2W² = ± 2, which forces 2W < U ≤ 3W (for W ≥ 3, U ≤ 2W makes the left side ≤ -W² < -2, while at W = 1 the unique positive odd nonbase solution is U = 3; conversely U > 3W makes the left side > 2W² ≥ 2). Hence (U', W') = (W, U - 2W) is positive, odd, and coprime, preserves | δ_leg | = 1, and strictly decreases U + W until (1,1) is reached. The record chambers satisfy m_0,j+1 = 6m_0,j - m_0,j-1 for j ≥ 1, and the growth bound m_0,j+1 < 6m_0,j holds for all j ≥ 0 (using m_0,1 = 5 < 6m_0,0 = 6 at j = 0). Any non-record primitive ray available at budget M < m_0,j+1 has | δ_leg | ≥ 7 and m₀ ≤ M < m_0,j+1 < 6m_0,j, so

Since G/m₀ = √(2 - (δ_leg/m₀)²) for G > 0 (and G ≤ 0 cannot beat j=0), p(U_j, W_j) strictly beats every non-record ray and every earlier record i < j. Finally, √(2 - G_j/m_0,j · √2) ∼ 1/(√2 m_0,j) and lim_j→∞ m_0,j+1/m_0,j = 3 + 2√2 give (A.3). □

A.2 Excluded Opposite-Parity Rational Targets and the Finite-Prefix Exception

Let A, C ∈ ℤ_{≥ 1} be coprime integers of opposite parity, and set

Because A, C have opposite parity, p_* is never attained on the centered rational chart 𝒞(r,c). For any primitive centered carrier pair (U,W), define the nonzero odd determinant ε = CU - AW. Direct expansion gives the exact chord-distance identity

Let b₊ = (U₊, W₊) be the unique positive odd pair in 0 < U₊ < 2A, 0 < W₊ < 2C with CU₊ - AW₊ = +1, and set b_- = (2A - U₊, 2C - W₊), which satisfies C(2A-U₊) - A(2C-W₊) = -1. Then all positive odd pairs with | ε | = 1 form the two canonical Bézout progressions b_± + 2k(A,C) (k ≥ 0). Define the explicit tail cutoff

Theorem A.2 (Bézout Tail Theorem and Necessary Finite-Prefix Exception). For every integer budget M ≥ M_cut, the unique minimizing point for p_* on the centered chart is the last available | ε | = 1 Bézout ray with m₀ ≤ M, and E_*(M) = min_{m ≤ M} ‖ p(U,W) - p_* ‖₂ satisfies

On the finite initial prefix 1 ≤ M < M_cut, however, non-Bézout rays with | ε | ≥ 3 can win: for (A,C) = (3,4) (D₀ = 25), the initial | ε | = 1 rays are b₊ = (1,1) (m₀ = 1) and b_- = (5,7) (m₀ = 37 = M_cut), and at budget M = 17 the unique winner is the non-Bézout pair (U,W) = (3,5) (m₀ = 17, ε = -3), which achieves ε²/m₀ = 9/17 < 1/1 and squared distance 18/425 < 2/25.

Proof. Write p(U,W) = ((U²-W²)/(U²+W²), (2UW)/(U²+W²)) and p_* = ((A²-C²)/D₀, (2AC)/D₀) with D₀ = A²+C². Since both points lie on the unit circle,

where ε = CU - AW and m₀ = (U²+W²)/2. Because U, W are both odd and A, C have opposite parity, CU and AW have opposite parity, so ε is always an odd integer (| ε | ∈ {1, 3, 5, …}).

1. Unique Canonical Odd Bézout Pair Construction: To construct the unique positive odd pair b₊ = (U₊, W₊) in the rectangle 0 < U₊ < 2A, 0 < W₊ < 2C with CU₊ - AW₊ = +1, start with any integer Bézout solution (a₀, b₀) to Ca₀ - Ab₀ = +1. Since A, C have opposite parity, one coordinate of (a₀, b₀) is already forced odd by Ca₀ - Ab₀ = 1; adding (A, C) if necessary makes the other coordinate odd while preserving the determinant. All odd integer solutions then differ by integer multiples of 2(A, C). Reducing the first coordinate modulo 2A by such shifts yields a unique odd representative U₊ ∈ (0, 2A). The corresponding second coordinate W₊ = (CU₊ - 1)/A is an odd integer; since CU₊ - 1 ≥ 0, W₊ cannot be negative, and because W₊ is odd it cannot be zero, while U₊ < 2A implies W₊ < 2C. This proves existence and uniqueness of b₊ = (U₊, W₊). The complementary pair b_- = (2A - U₊, 2C - W₊) is likewise positive and odd in (0, 2A) × (0, 2C) with determinant C(2A-U₊) - A(2C-W₊) = -1. All positive odd solutions with | ε | = 1 are therefore b₊ + 2k(A, C) and b_- + 2k(A, C) (k ≥ 0, since k < 0 makes both coordinates negative), and each is automatically coprime because any common divisor divides | ε | = 1. Since U₊ ≤ 2A - 1 and W₊ ≤ 2C - 1, both initial chambers satisfy m_{±, 0} ≤ ((2A-1)² + (2C-1)²)/2 = 2D₀ - 2(A + C) + 1, giving (A.6).

1. Complementary Projection Interleaving and Uniform Ratio Bound < 9: Set g = (A, C) and let ν = (g· b₊)/D₀ ∈ (0, 2); the complementary projection of b_- is (g· b_-)/D₀ = 2 - ν. Because | det(g, b_±) | = 1, orthogonal decomposition along and perpendicular to g gives

The value ν = 1 is impossible: if ν = 1, then b₊ - g would be a nonzero integer vector orthogonal to the primitive vector g, hence an integer multiple of (C, -A), whose determinant with g is a multiple of D₀ ≥ 5 rather than ± 1. Define β = min(ν, 2 - ν) ∈ (0, 1). Sorting the two complementary progressions by chamber size (without assuming the +1 ray is shorter), their positive projections interleave strictly as

with corresponding Bézout-ray chambers m̂₀ < m̂₁ < m̂₂ < ⋯, where m̂₁ = M_cut = max(m_+,0, m_-,0) has projection 2 - β > 1. For every index j ≥ 1, the current projection ν_j ≥ 2 - β > 1 and the next projection satisfies ν_j+1 < ν_j + 2, so the successive chamber ratio is strictly bounded by 9:

because 9(ν² + D₀^-2) - ((ν+2)² + D₀^-2) = 8ν² - 4ν - 4 + 8D₀^-2 = 4(ν-1)(2ν+1) + 8D₀^-2 > 0 for all ν ≥ 1. Moreover, ν_j+1/ν_j → 1 implies m̂_j+1/m̂_j → 1.

1. All-Budget Tail Optimality, Asymptotic Law, and Finite-Prefix Counterexample: For any integer budget M ≥ M_cut = m̂₁, choose the last available Bézout ray j ≥ 1 so that m̂_j ≤ M < m̂_j+1. Any primitive odd pair outside the two Bézout progressions has odd determinant | ε | ≥ 3; if it is available at budget M, its chamber satisfies m₀ ≤ M < m̂_j+1, so

and it loses strictly to the j-th Bézout ray. Earlier Bézout rays (i < j) also lose strictly since m̂_i < m̂_j and ε² = 1. Because m̂_j+1/m̂_j → 1 forces m̂_j/M → 1, the minimum squared chord error E_*(M)² = 2/(D₀ m̂_j) satisfies M E_*(M)² = 2/D₀ M/m̂_j → 2/D₀, proving (A.7). Finally, at (A,C) = (3,4) (D₀ = 25), b₊ = (1,1) (m̂₀ = 1) and b_- = (5,7) (m̂₁ = M_cut = 37); enumerating the seven primitive ordered odd pairs with m₀ ≤ 17—namely (1,1), (1,3), (3,1), (1,5), (5,1), (3,5), (5,3)—shows that at budget M = 17, (U,W) = (3,5) (m₀ = 17, ε = -3) achieves ε²/m₀ = 9/17 < 1 = 1/m̂₀ and squared distance 18/425 < 2/25. □

Remark A.2a (Sharp Dichotomy for the Minimal Tail Onset M_tail(A,C)). Put D₀ = A² + C², ν = (AU+CW)/D₀, ε = CU - AW, and m̂ = m₀ = (U²+W²)/2. Orthogonal decomposition D₀(U²+W²) = (AU+CW)² + ε² gives the exact normalized chamber formula m̂ = (D₀ ν² + ε²/D₀)/2, so the two initial | ε | = 1 rays r₀ = (U₀, W₀) (ν = β = p/D₀ with integer p = AU₀ + CW₀ ∈ [A+C, D₀-1]) and r₁ = (2A-U₀, 2C-W₀) (ν = 2 - β) have chambers m̂₀ = (D₀β² + D₀^-1)/2 and M_cut = m̂₁ = (D₀(2-β)² + D₀^-1)/2. On [m̂₀, M_cut - 1], only r₀ is available from | ε | = 1, while the minimum error among available | ε | ≥ 3 rays is non-increasing in M. A primitive positive odd pair (U,W) with | ε | ≥ 3 ties or beats r₀ before M_cut iff m̂ = (D₀ ν² + ε²/D₀)/2 ≥ ε² m̂₀ = (ε² D₀ β² + ε²/D₀)/2 ⇔ ν² ≥ ε²β² ⇔ ν ≥ | ε |β and m̂ < M_cut ⇒ ν < 2 - β. Since ν ≡ ± | ε |β (mod 2) and ν = | ε |β is the non-primitive multiple | ε |r₀, any primitive intruder has ν = 2 - | ε |β and (U,W) = w_{| ε |} = (2A - | ε | U₀, 2C - | ε | W₀), whose positivity and 2 - | ε |β ≥ | ε |β ⇔ β ≤ 1/| ε | imply the same conditions for | ε | = 3 (with 3β ≠ 1 since gcd(A,C)=1 ⇒ 3 ∤ D₀). Conversely, when w₃ = (2A - 3U₀, 2C - 3W₀) ∈ ℤ_>0² and 3p < D₀, w₃ is primitive (as gcd(2A-3U₀, 2C-3W₀) divides both C(2A-3U₀)-A(2C-3W₀)=± 3 and gcd(2A,2C)=2 if 3 divided both coordinates) and its chamber m̂(w₃) = (D₀(2-3β)² + 9/D₀)/2 satisfies the strict bounds m̂(w₃) - 9m̂₀ = 2D₀(1-3β) = 2(D₀-3p) > 0 and M_cut - m̂(w₃) = (4[p(D₀-p)-1])/D₀ > 0 (since p ≥ A+C ≥ 3 and D₀ - p ≥ 1 are positive integers). Thus w₃ enters at m̂(w₃) ≤ M_cut - 1 and strictly beats r₀ across [m̂(w₃), M_cut - 1], proving the universal two-value dichotomy: M_tail(A,C) = M_cut if 3U₀ < 2A, 3W₀ < 2C, 3(AU₀+CW₀) < D₀, and M_tail(A,C) = m̂₀ otherwise.

A.3 Adjoining the Positive Leg-Sum Chart κ_leg and the Explicit Leg-Ratio Involution

For any cell (r,c) ∈ ℤ_{≥ 1}², the positive leg ratio a := r/c ∈ ℚ_>0 determines the leg-sum phase through the map

Multiplicative inversion a ↦ 1/a = c/r (exchanging legs (r,c) ↦ (c,r) = R_mid(r,c) in chamber m = r+c-1) conjugates to phase reflection across 1/2:

which is the algebraic form of the complementary fraction pairing r/n ↔ (n-r)/n = c/n (n = r+c = m+1).

Using the leg-normalized phase κ_leg = r/(r+c) = r/(m+1) defines the alternative leg-sum carrier map

Its rational points correspond uniquely to (r,c) = k(A², C²) for k ∈ ℤ_{≥ 1} and coprime A, C ∈ ℤ_{≥ 1}, with first chamber m = A² + C² - 1 and repeats at m = k(A² + C²) - 1. In particular, every opposite-parity target (A.4) is realized exactly on the leg-sum chart at cell (A², C²) in chamber D₀ - 1 (for example, (3/5, 4/5) appears exactly at (4,1) in chamber m = 4).

Theorem A.3 (One-Chamber Advance Under the Combined Two-Chart Budget). When rational points from both 𝒞(r,c) and p_leg(r,c) with chamber m ≤ M are admitted to approximate p_* = (1/√2, 1/√2), let j ≥ 0 be chosen by m_0,j ≤ M + 1 < m_0,j+1. For j ≥ 1:

• At M = m_0,j - 1 (e.g., M = 4, 28, 168), the unique minimizing point is the coordinate swap of p(U_j, W_j), realized on the leg-sum chart at (A_j², C_j²) where A_j = (U_j+W_j)/2 and C_j = (U_j-W_j)/2.
• For m_0,j ≤ M < m_0,j+1 - 1, the minimizing points are precisely that leg-sum swapped point and the centered point p(U_j, W_j), which are distinct and equidistant from p_*.

Base Budgets and All Cell Realizations. For M ∈ {1, 2, 3} (j = 0), the unique minimizing point is p₀ = (0, 1), realized in either chart by the diagonal cells (r, r) with 2r - 1 ≤ M (the swapped boundary point (1, 0) is excluded because both charts require positive input cell parameters r, c ≥ 1, which forbids a zero leg). For j ≥ 1, the centered minimizing cells are the odd-scale repeats in (A.2), the leg-sum swapped point τ(p_j) is realized by (r, c) = k(A_j², C_j²) with k ∈ ℤ_{≥ 1} and km_0,j - 1 ≤ M, and the centered point p_j also has leg-sum realizations (r, c) = k(U_j², W_j²) whenever 2km_0,j - 1 ≤ M, giving ⌊(⌊ M/m_0,j⌋+1)/2⌋ + ⌊(M+1)/m_0,j⌋ + ⌊(M+1)/(2m_0,j)⌋ chart-labeled minimizing cells in total.

Proof. First, if any candidate cell (r, c) on either chart has r < c, its carrier point (p₁, p₂) has p₁ < 0 and p₂ > 0; transposing the cell (r, c) ↦ (c, r) on the same chart and at the exact same chamber m = r+c-1 produces (-p₁, p₂), which has strictly larger inner product (-p₁ + p₂)/√2 > (p₁ + p₂)/√2 with the target p_* = (1/√2, 1/√2). Thus every minimizing cell on either chart satisfies r ≥ c and p₁ ≥ 0.

On the positive leg-sum chart (A.8), p_leg(r,c) = ((r-c)/(r+c), (2√(rc))/(r+c)) is rational iff rc is a square, i.e., (r, c) = k(A², C²) for a unique primitive pair of coprime positive integers A, C ≥ 1 and scale k ∈ ℤ_{≥ 1}, with first leg-sum chamber m = A² + C² - 1, repeats at k(A² + C²) - 1, and circle point ((A²-C²)/(A²+C²), (2AC)/(A²+C²)) (where for arbitrary coprime A, C ≥ 1, setting U = A+C and W = | A-C | gives the signed identity A² - C² = sgn(A-C) UW). Restricting to the non-negative half-quadrant r ≥ c ⇔ A ≥ C ≥ 1 (p₁ ≥ 0), there are two parity cases:

1. Opposite Parity (A + C odd with A > C ≥ 1): The reduced denominator is m₀ = A² + C² ≡ 1 (mod 4), the first reduced numerator A² - C² > 0 is odd, and the second numerator 2AC > 0 is even. Setting U = A + C and W = A - C > 0 (coprime positive odd integers with (U²+W²)/2 = A²+C² = m₀), we have A² - C² = UW and 2AC = (U² - W²) / 2. Thus this point is τ(p(U,W)) ∈ τ(ℋ) (where τ(p₁, p₂) = (p₂, p₁)) and first appears on the leg-sum chart at chamber m = m₀ - 1.
2. Both Odd (A ≥ C ≥ 1 with A ≡ C ≡ 1 (mod 2)): Dividing numerators and denominator by 2, the reduced denominator is m₀ = (A² + C²)/2 ≡ 1 (mod 4), with reduced point (((A²-C²)/2)/m₀, (AC)/m₀) = p(A, C) already in the centered parity class ℋ. On the centered chart it is already available at chamber m₀ (via cell ((A²+1)/2, (C²+1)/2)), whereas its first realization on the leg-sum chart is (r,c) = (A², C²) at chamber A² + C² - 1 = 2m₀ - 1 ≥ m₀, with equality 2m₀ - 1 = m₀ if and only if m₀ = 1 at the diagonal base (A,C) = (1,1) (where both charts realize p₀ = (0,1) at (1,1) in chamber 1) and strict inequality 2m₀ - 1 > m₀ for all m₀ ≥ 5 (for example, (r,c) = (9,1) with (A,C) = (3,1) gives (4/5, 3/5) at leg-sum chamber 9 = 2(5) - 1 > 5, whereas the centered chart already realizes (4/5, 3/5) at (5,1), chamber 5).

By Cases 1 and 2, every admitted point with p₁ ≥ 0 at combined budget m ≤ M is either a non-negative centered point p(U,W) ∈ ℋ (U ≥ W) with m₀ ≤ M or the coordinate swap τ(p(U,W)) ∈ τ(ℋ) of a positive centered point (U > W) with m₀ - 1 ≤ M ⇔ m₀ ≤ M + 1. Since τ preserves the linear score p₁ + p₂, the maximum score p₁ + p₂ at combined budget M is bounded above by the centered-only optimum score S_j = G_j / m_0,j at hypothetical centered budget M + 1, where j ≥ 0 is defined by m_0,j ≤ M + 1 < m_0,j+1 (Theorem A.1).

For j = 0 (M ∈ {1, 2, 3}), p₀ = (0, 1) is admitted from chamber 1 on both charts at (r, r) (2r - 1 ≤ M), while its swap (1, 0) lies outside the positive-parameter domain r, c ≥ 1. For j ≥ 1, the integers A_j = (U_j + W_j)/2 and C_j = (U_j - W_j)/2 are positive, coprime, and of opposite parity with A_j² + C_j² = m_0,j, so the leg-sum cell (A_j², C_j²) has chamber m_0,j - 1 ≤ M and realizes τ(p(U_j, W_j)), attaining the upper bound S_j. The unswapped point p(U_j, W_j) is admitted iff m_0,j ≤ M (entering on the centered chart at m_0,j and on the leg-sum chart only at 2m_0,j - 1). Because a line p₁ + p₂ = S_j intersects the unit circle in at most the two points p(U_j, W_j) and τ(p(U_j, W_j))—which are distinct for all j ≥ 1 since U_j > W_j > 0 implies ℓ_1,j ≠ ℓ_2,j—no other point can tie. □

Appendix B. The Odd-Square Stack Ω_n and Its Directional Shadows

This appendix establishes the all-direction orthographic union-shadow, convex-hull, hiding-cone, and isotropic-mean theorems for the three-dimensional unit-height odd-square stack Ω_n, extending the tetrahedral sum ∑_j=0ⁿ (2j+1)² = C(2n+3, 3) to continuous solid projections [9].

B.1 Explicit Solid Model and Exact Union-Shadow Area Law

For each integer n ≥ 0, define the unit-height odd-square stack Ω_n ⊂ ℝ³ by

and let π_v : ℝ³ → v^⊥ denote orthogonal projection along a nonzero viewing vector v = (A, C, D) ≠ 0, where D is explicitly declared as the vertical third-axis component. (Orientation, Slab-Center Translation & Affine-Point vs. Linear-Camera Reflection Between Top-Widest Ω_n and Bottom-Widest Ω_n^bot / S_n: In (B.1), slab j ∈ {0,…,n} has side 2j+1 on height [j,j+1], so Ω_n is top-widest with widest slab j=n at [n,n+1]. Applying the affine point reflection R₃^aff(p₁,p₂,p₃) = (p₁,p₂,n+1-p₃) maps Ω_n isometrically onto the bottom-widest solid stack Ω_n^bot = R₃^aff(Ω_n) = ⋃_ℓ=0ⁿ [-n+ℓ-1/2, n-ℓ+1/2]² × [ℓ,ℓ+1] (ℓ = n-j). Each bottom slab ℓ ∈ {0,…,n} is a union of unit cubes centered at half-integer heights (p₁, p₂, ℓ + 1/2) with | p₁|,| p₂| ≤ n-ℓ; translating those slab centers downward by (0,0,1/2) via (p₁,p₂,ℓ+1/2) ↦ (p₁,p₂,ℓ) identifies them bijectively with the discrete integer point stack S_n = {(p₁,p₂,ℓ)∈ℤ³ : 0≤ℓ≤ n, | p₁|,| p₂|≤ n-ℓ} of §6.5 Proposition 6.5. On viewing directions v=(A,C,D), the induced linear directional reflection is the homogeneous differential R₃^lin(v) = dR₃^aff(v) = (A,C,-D) (distinct from the affine point map R₃^aff), which conjugates orthographic projection along v on Ω_n to projection along R₃^lin(v)=(A,C,-D) on Ω_n^bot and preserves 𝒜_n(v) and ℋ_n(v) identically since (B.2)–(B.8) depend only on | A|,| C|,| D|.)* Denote by 𝒜_n(v) = Area₂(π_v(Ω_n)) the actual union-shadow area and by ℋ_n(v) = Area₂(π_v(convΩ_n)) the projected convex-hull area.

Theorem B.1 (All-Direction Actual Union-Shadow Area Law). For every integer n ≥ 0 and v = (A, C, D) ≠ 0:

1. Horizontal Views (D = 0): Because the open slab height bands (j, j+1) are disjoint, their projected strips have disjoint interiors and sum to

1. Non-Horizontal Views (D ≠ 0): Set a = min(| A |, | C |)/| D | and b = max(| A |, | C |)/| D |. Then 𝒜_n(v) = (| D |)/(‖ v ‖₂) 𝒜̃_n(a, b), where 𝒜̃_n(a, 0) = (2n+1)² and for b > 0, with ν = b/max(1, a) and (τ)₊ = max(0, τ),

Writing k = min(n, ⌊ ν/2 ⌋), the first term (ν-1)₊² in (B.3) contributes across all n summands whenever ν > 1 while (ν - 2j - 2)₊² contributes only for 0 ≤ j < k, so the corner-notch correction reduces in O(1) arithmetic operations to

which holds identically across all regimes ν ≤ 1, 1 < ν < 2, and ν ≥ 2.

Negative Control Against Naive Positive-Part Extrapolation. At n = 1 and v = (1/3, 3/2, 1), formula (B.3) gives quotient union area 𝒜̃₁(1/3, 3/2) = 541/36 (Euclidean area 541/66), whereas the projected convex hull has quotient area 31/2 (Euclidean area 93/11) and omitting the final corner-notch sum in (B.3) gives the false value 15.

Proof.

1. Horizontal Views (D = 0): When D = 0, orthogonal projection onto (A, C, 0)^⊥ preserves the vertical coordinate p₃. For each slab interior j < p₃ < j + 1, the horizontal cross-section [-R_j, R_j]² (R_j = j + 1/2) projects to a segment of length (2j + 1)(| A | + | C |)/√(A² + C²). Because distinct slabs occupy disjoint open p₃-intervals (j, j+1) whose shared horizontal boundaries have planar measure zero, the projected union area is the sum of the individual strip areas:

establishing (B.2) directly for all n ≥ 0 without taking a limit as D → 0.

1. Quotient Projection and All-n Markov Intersection Nesting (D ≠ 0): For D ≠ 0, map each point (p₁, p₂, p₃) ∈ ℝ³ to the horizontal plane p₃ = 0 along v via (p₁, p₂, p₃) ↦ (p₁ - (A/D)p₃, p₂ - (C/D)p₃). A planar region of quotient area 𝒜̃ in p₃ = 0 lifts to an orthogonal projection of Euclidean area (| D |)/(‖ v ‖₂) 𝒜̃ in v^⊥ because the unit normal to p₃ = 0 makes cosine | D | / ‖ v ‖₂ with v / ‖ v ‖₂. By sign reflection and coordinate swap on the square base [-R_j, R_j]², we may take sorted nonnegative quotient parameters 0 ≤ a ≤ b with a = min(| A |, | C |)/| D | and b = max(| A |, | C |)/| D |. The j-th slab Ω^(j) = [-R_j, R_j]² × [j, j+1] then has quotient shadow P_j = {(p₁ - ap₃, p₂ - bp₃) : (p₁, p₂, p₃) ∈ Ω^(j)}, which is the Minkowski sum of [-R_j, R_j]² and the segment [-(j+1)(a,b), -j(a,b)], described explicitly by

(For a, b > 0 these inequalities cut out a convex hexagon; if a = 0 < b they reduce to a rectangle, and if a = b = 0 they give the square [-R_j, R_j]².) For any higher slab k > j, the lower bounds -R_k - a(k+1) < -R_j - a(j+1) and -R_k - b(k+1) < -R_j - b(j+1) strictly relax, and the diagonal strip | bp₁ - ap₂ | ≤ R_k(a + b) strictly widens (or is identically 0 ≤ 0 when a = b = 0). Thus the only constraints of P_k (k > j) that can cut P_j are the upper coordinate bounds

If a ≤ 1, then p̄₁(k) is nondecreasing in k, so p₁ ≤ p̄₁(k) is already implied on P_j by p₁ ≤ p̄₁(j); if a > 1, then p̄₁(k) strictly decreases with k. The identical dichotomy holds for b and p̄₂(k). Consequently, every active constraint from P_k (k > j) tightens monotonically with k, which proves the Markov intersection-nesting property

Therefore, the set of shadow points whose highest covering slab is j simplifies to Δ_j = P_j ∖ ⋃_{k > j} P_k = P_j ∖ P_j+1 for 0 ≤ j < n, with Δ_n = P_n. Since Δ₀, …, Δ_n partition ⋃_j=0ⁿ P_j without any convexity assumption on the union,

1. Local Difference Area Across All (a, b) Branches and O(1) Sum Reduction: Fix j and translate P_j by (R_j + a(j+1), R_j + b(j+1)) into local nonnegative coordinates (p'₁, p'₂) ∈ [0, 2j + 1 + a] × [0, 2j + 1 + b]. In (p'₁, p'₂), P_j is the bounding rectangle [0, 2j + 1 + a] × [0, 2j + 1 + b] minus the two disjoint open corner triangles p'₂ > 2j + 1 + (b/a)p'₁ and p'₁ > 2j + 1 + (a/b)p'₂ (for a > 0; each has area ab/2), giving Area₂(P_j) = (2j + 1 + a)(2j + 1 + b) - ab = (2j + 1)² + (2j + 1)(a + b) (and Area₂(P_n) = (2n+1)² + (2n+1)(a+b)). For P_j ∖ P_j+1:

• If b ≤ 1 (inside the hiding cone), both p̄₁(k) and p̄₂(k) are nondecreasing, so P_j ⊆ P_j+1 and Area₂(P_j ∖ P_j+1) = 0.
• If a = 0 < b and b > 1 (axis tilt outside the cone), the rectangle P_j = [0, 2j + 1] × [0, 2j + 1 + b] is cut at p'₂ ≤ 2j + 2, leaving a protruding strip of width 2j + 1 and height b - 1, so Area₂(P_j ∖ P_j+1) = (2j + 1)(b - 1).
• If 0 < a ≤ b and b > 1, put â = min(1, a) (covering both 0 < a ≤ 1 < b, where â = a, and 1 < a ≤ b, where â = 1). In the translated (p'₁, p'₂) coordinates, the upper bounds p₁ ≤ p̄₁(j+1) and p₂ ≤ p̄₂(j+1) become p'₁ ≤ 2j + 1 + â and p'₂ ≤ 2j + 2. Thus P_j ∩ P_j+1 is the clipped rectangle [0, 2j + 1 + â] × [0, 2j + 2] minus the portions of the two corner triangles lying inside it:
• The northwest triangle p'₂ > 2j + 1 + (b/a)p'₁ meets [0, 2j + 1 + â] × [0, 2j + 2] in a right triangle of legs a/b ≤ â and 1, with area a/(2b).
• The southeast triangle p'₁ > 2j + 1 + (a/b)p'₂ has horizontal width (2j + 1 + â) - (2j + 1 + (a/b)p'₂) = â - (a/b)p'₂ inside p'₁ ≤ 2j + 1 + â for 0 ≤ p'₂ ≤ min(bâ/a, 2j + 2); integrating ∫₀^{min(bâ/a, 2j+2)} (â - (a/b)p'₂) dp'₂ gives area

Subtracting Area₂(P_j ∩ P_j+1) = (2j + 1 + â)(2j + 2) - a/(2b) - (bâ²)/(2a) + a/(2b) ((bâ)/a - 2j - 2)₊² from Area₂(P_j) = (2j + 1)² + (2j + 1)(a + b) and using a + b - 1 - â = (a - 1)₊ + (b - 1)₊ together with the algebraic identity

gives Area₂(P_j ∖ P_j+1) = (2j + 1)[(a - 1)₊ + (b - 1)₊] + a/(2b) [(ν - 1)₊² - (ν - 2j - 2)₊²]. Summing ∑_j=0^n-1(2j+1) = n² proves (B.3), and indexing i = j + 1 ∈ {1, …, k} with k = min(n, ⌊ ν/2 ⌋) evaluates ∑_i=1^k (ν - 2i)² = kν² - 2ν k(k+1) + 2/3 k(k+1)(2k+1) in O(1) operations. □

B.2 The Square Hiding Cone, Convex Hull Gap, and Global Extrema

Theorem B.2 (Square Hiding Cone, Hull Area, and Extrema). Let δ₊ = (| A | - | D |)₊ + (| C | - | D |)₊. Then for every v = (A, C, D) ≠ 0:

1. Projected Convex Hull Area:

1. Exact Hiding Cone and Convexity Equivalence (n ≥ 1): The following four conditions are equivalent:

• All lower slabs hide inside the widest top slab's shadow: π_v(Ω_n) = π_v([-n-1/2, n+1/2]² × [n, n+1]).
• The actual shadow π_v(Ω_n) is a convex polygon.
• The hull gap vanishes: ℋ_n(v) - 𝒜_n(v) = 0.
• The viewing direction lies inside the square cone

In polar angle ω from the vertical axis at azimuth ψ, the hiding boundary is ω_max(ψ) = arctan(1/max(|cos ψ|, |sin ψ|)), which equals 45^∘ in the coordinate vertical planes and reaches arctan √2 ≈ 54.7356^∘ along the isometric cube diagonals (± 1, ± 1, ± 1).

1. Global Minimum, Isometric View, Global Maximum, and Sharp Hull Gaps:

Proof.

1. Three-Dimensional Convex Hull and Facet Projection Formula (B.4): Write 𝒫_n = {(p₁, p₂, p₃) : 0 ≤ p₃ ≤ n, | p₁ |, | p₂ | ≤ p₃}. Each slab Ω^(j) is the Minkowski sum of the base unit cube Ω₀ = [-1/2, 1/2]² × [0, 1] and the horizontal square [-j, j]² × {j}. Since convexification commutes with Minkowski addition and the convex hull of ⋃_j=0ⁿ ([-j,j]² × {j}) is the inverted square pyramid 𝒫_n, we have

For n > 0, convΩ_n is a square frustum of height n topped by a square prism of height 1: its boundary consists of a bottom square of area 1 (normal (0,0,-1)), a top square of area (2n+1)² (normal (0,0,1)), four vertical rectangles of area 2n + 1 (normals (± 1,0,0), (0,± 1,0)), and four slanted trapezoids of area √2 n(n+1) (normals (± 1,0,-1)/√2 and (0,± 1,-1)/√2). For any convex polyhedron, almost every line parallel to v piercing the shadow meets the boundary twice, so ℋ_n(v) = 1/(2‖ v ‖₂) ∑_F Area(F) | n_F · v |. Summing the ten facets and using 1/2 (| p - | D | | + | p + | D | |) = max(| p |, | D |) = | D | + (| p | - | D |)₊ together with (1 + (2n + 1)²)/2 + 2n(n+1) = (2n + 1)² yields (B.4).

1. Strict Gap Inequality Outside the Cone and Convexity Equivalence (n ≥ 1): For D ≠ 0, in normalized sorted coordinates 0 ≤ a ≤ b, put D₊ = (a - 1)₊ + (b - 1)₊ and 𝒞_n = ∑_j=0^n-1 C_j(a, b) where C_j(a, b) = a/(2b) [(ν - 1)₊² - (ν - 2j - 2)₊²]. Comparing (B.4) and (B.3) gives

Inside the cone (b ≤ 1), D₊ = 𝒞_n = 0. Outside the cone (b > 1, so D₊ > 0), each summand C_j(a,b) satisfies the strict bound 0 ≤ C_j(a,b) < D₊/2:

• If a = 0, then C_j = 0 < (b - 1)/2 = D₊/2.
• If 0 < a ≤ 1 < b, then ν = b and C_j ≤ (a(b-1)²)/(2b) < (b-1)/2 = D₊/2 since a(b-1) ≤ b - 1 < b.
• If 1 < a ≤ b, then ν = b/a and C_j ≤ (a(b/a - 1)²)/(2b) = ((b-a)²)/(2ab) < (a + b - 2)/2 = D₊/2 because ab(a + b - 2) - (b - a)² = a²(b - 1) + b²(a - 1) > 0.

Summing over 0 ≤ j < n (n ≥ 1) proves that for every non-horizontal view outside the cone,

while for horizontal views (D = 0, n ≥ 1), ℋ_n(A,C,0) - 𝒜_n(A,C,0) = (n(| A | + | C |))/(√(A²+C²)) > 0. Because orthogonal projection commutes with convex hull (π_v(convΩ_n) = convπ_v(Ω_n)), π_v(Ω_n) is convex iff ℋ_n(v) - 𝒜_n(v) = 0, which for n ≥ 1 holds iff max(| A |, | C |) ≤ | D |, where π_v(Ω_n) = π_v(Ω_n,top). (At n = 0, Ω₀ is a single cube and every gap vanishes.)

1. Global Minimum, Global Maximum Across All Four Linear Forms, and Sharp Gaps (B.6)–(B.8): Let M = (n+1)² = 2n + 1 + n² and write the naive lower-bound numerator as ℬ = | D | (2n + 1)² + (2n + 1)(| A | + | C |) + n²δ₊. By (B.3) and (B.2), ‖ v ‖₂ 𝒜_n(v) ≥ ℬ in every direction. Since | p | - (| p | - | D |)₊ = min(| p |, | D |),

whence 𝒜_n(v) ≥ (ℬ)/(‖ v ‖₂) ≥ M (| A | + | C | + | D |)/(√(A² + C² + D²)) ≥ M = (n+1)², with equality for n ≥ 1 iff D = 0 and AC = 0 (the horizontal coordinate axes).

For the global maximum, 𝒜_n(v) ≤ ℋ_n(v) everywhere. On the nonnegative octant (A, C, D) ≥ 0 with ‖ v ‖₂ = 1, ℋ_n(v) is the maximum of four linear forms w_i · (A, C, D) according to which positive parts in (A-D)₊ + (C-D)₊ are active:

Since ‖ w₀ ‖₂² = (2n + 1)⁴ + 2(2n + 1)², direct expansion gives

By Cauchy–Schwarz, every linear form is at most ‖ w₀ ‖₂ = (2n + 1)√((2n + 1)² + 2), so 𝒜_n(v) ≤ ℋ_n(v) ≤ (2n + 1)√((2n + 1)²+2) over the entire sphere S², not only inside the cone. Equality holds precisely when (| A | : | C | : | D |) = (2n + 1 : 2n + 1 : (2n + 1)²) = (1 : 1 : 2n+1), which lies inside the hiding cone (a = b = 1/(2n + 1) ≤ 1), where 𝒜_n = ℋ_n = (2n + 1)√((2n + 1)²+2). Finally, ℋ_n - 𝒜_n ≤ (nδ₊)/(‖ v ‖₂) ≤ (n(| A | + | C |))/(√(A²+C²+D²)) ≤ √2 n (attained at (1,1,0)), and for | A | + | C | > 0, using ‖ v ‖₂ 𝒜_n ≥ ℬ ≥ M(| A | + | C |) and δ₊ ≤ | A | + | C | gives

with equality at every horizontal view (D = 0). □

B.3 Isotropic Mean Shadow, Silver-Ratio Leading Term, and O(n^-5) Tail Brackets

Averaging 𝒜_n(v) over the unit sphere S² with respect to the uniform probability measure yields the isotropic mean shadow 𝒜_n. Because Ω_n is non-convex for n ≥ 1, its total exposed surface area is Area₂(∂Ω_n) = 2(2n + 1)² + 4∑_j=0ⁿ (2j+1) = 12n² + 16n + 6, and Cauchy's surface average Area₂(∂Ω_n)/4 = 3n² + 4n + 3/2 counts ray-piercing multiplicity, exceeding 𝒜_n by (2-√2)n² - χ n + 2/π ∑_i=1ⁿ I(2i) > 0. By contrast, the convex-hull mean is ℋ_n = ((2n + 1)²+2(2n + 1))/2 + n(n+1)(√2-1) = (1+√2)n² + (3+√2)n + 3/2 (using 𝔼| A | = 1/2 and 𝔼 max(| A |, | D |) = 1/(√2) on S²), which shares the quadratic leading term (1+√2)n² of 𝒜_n and differs only at linear order (√2-1-χ)n + 2/π ∑_i=1ⁿ I(2i).

Theorem B.3 (Exact Isotropic Mean Shadow, Closed Constant C_*, and Tail Asymptotics). For every integer n ≥ 0, the isotropic mean actual shadow area of Ω_n is

where I(β) = ∫_β^∞ ((p-β)²)/(p(p²+1)(p²+2)) dp (β ≥ 1), ε_n = 2/π ∑_{i > n} I(2i), and the exact constants are

To 50 rounded decimal places:

• 4 + χ = 4.03493671467655295067734553073116140549324173926145,
• C_* = 1.47999653090760042442100878465142982068794763080876,
• 𝒜₁ = 7.93754352744295952118684565150792784372616830715013 (vs. facet average Area₂(∂Ω₁)/4 = 8.5).

Moreover, for every integer n ≥ 1, writing P₃(n) = 1/(24π n) - 1/(48π n²) + 7/(1440π n³) and P₄(n) = P₃(n) + 1/(320π n⁴), the tail ε_n satisfies the global two-sided bounds

The shifted sequence (n + τ)ε_n strictly increases to 1/(24π) for all n ≥ 1 for every real shift τ ≤ 1/2 (with τ = 1/2 the largest shift having this property: 24π(n + 1/2)ε_n = 1 - 2/(15n²) + O(n^-3)). By contrast, the third-order remainder extractor G_n = n³(ε_n - 1/(24π n) + 1/(48π n²)) is not monotone, satisfying G₁ < G₂ > G₃.

Proof.

1. Equal Quotient-Chart Integrals and Exact Finite Mean Assembly: On the sorted domain 0 ≤ a ≤ b, define W_β(a, b) = a/(2b) (ν - β)₊² for β ≥ 1 (with ν = b/max(1,a) and W_β = 0 at b = 0), so that ∑_j=0^n-1 C_j(a,b) = n W₁(a,b) - ∑_i=1ⁿ W_2i(a,b). Combining the spherical surface Jacobian (1+a²+b²)^-3/2 with the orthogonal area factor (1+a²+b²)^-1/2 and the 16-fold sign/swap symmetry on S² gives the averaging operator 4/π ∫₀^∞ ∫₀^b (·)/((1+a²+b²)²) da db. Splitting at a = 1 by Tonelli's theorem:

• Chart 1 (0 ≤ a ≤ 1): Here ν = b ≥ β ≥ 1, leaving the full inner range 0 ≤ a ≤ 1:

• Chart 2 (a ≥ 1): Substitute b = aν (ν ≥ β, db = a dν, a/(2b) = 1/(2ν)):

Adding the two equal chart contributions and multiplying by 4/π proves 𝔼_S²[(| D |)/(‖ v ‖₂) W_β(a,b)] = 2/π I(β). Partial fractions ((p-β)²)/(p(p²+1)(p²+2)) = β²/(2p) + ((1-β²)p - 2β)/(p²+1) + ((β²/2 - 1)p + 2β)/(p²+2) integrate directly to

giving I(1) = (ln 3)/4 - π/2 + √2 arctan √2 and χ = 2/π I(1). Since 𝔼_S²[δ₊/‖ v ‖₂] = 2(𝔼 max(| A |, | D |) - 𝔼| D |) = √2 - 1, subtracting the mean gap 𝔼[ℋ_n - 𝒜_n] = n(√2-1) - n2/π I(1) + 2/π ∑_i=1ⁿ I(2i) from ℋ_n = (1+√2)n² + (3+√2)n + 3/2 establishes (B.9).

1. Closed Form for C_* via the Weighted Log-Sinh Integral: For β ≥ 2, expanding 1/((1+p^-2)(1+2p^-2)) = ∑_k=0^∞ (-1)^k(2^k+1-1)p^-2k and integrating termwise via ∫_β^∞ ((p-β)²)/p^2j+3 dp = (2β^-2j)/((2j)(2j+1)(2j+2)) gives the absolutely convergent series

Setting β = 2i and summing over i ≥ 1 gives R_∞ = 2/π ∑_i=1^∞ I(2i) = 4/π ∑_j=1^∞ ((-1)^j-1(2^j-1)ζ(2j))/((2j)(2j+1)(2j+2)4^j) = 4/π [F(1/√2) - F(1/2)], where F(z) = ∑_j=1^∞ ((-1)^j-1ζ(2j)z^2j)/((2j)(2j+1)(2j+2)) = 1/2 ∫₀¹ (1-ψ) ln((sinh(π z ψ))/(π z ψ)) dψ by the Euler product for sinh(π z ψ)/(π z ψ). Writing ln((sinh(kψ))/(kψ)) = kψ - ln(2k) - ln ψ + ln(1 - e^-2kψ) and integrating against (1-ψ) yields

Evaluating 2/π [L(π/√2) - L(π/2)] with k₁ = π/√2, k₂ = π/2, and ζ(2) = π²/6 combines the π-linear and 1/π terms to (k₁-k₂)/6 - (ζ(2))/2 (k₁^-1-k₂^-1) = (π(√2-1))/12 - (π(√2-2))/12 = π/12, which multiplies by 2/π to contribute 1/6 to R_∞ (while -1/2 ln(2k₁) + 1/2 ln(2k₂) = -(ln 2)/4 contributes -(ln 2)/(2π)), yielding R_∞ = 1/6 - (ln 2)/(2π) + (2Li₃(e^-π) - Li₃(e^{-√2 π}) - ζ(3))/π³ and C_* = 3/2 - R_∞ in (B.10).

1. Signed Power-Tail Bounds for (B.11): Since the alternating terms of I(2i) decrease in magnitude for i ≥ 1, 1/(48i²) - 1/(320i⁴) < I(2i) < 1/(48i²) - 1/(320i⁴) + 1/(1536i⁶), so with the Hurwitz zeta function [20, §25.11] ζ(k, n+1) = ∑_{i > n} i^-k,

In the Laplace representation ζ(k, n+1) = 1/((k-1)!) ∫₀^∞ (ψ^k-1e^-nψ)/(e^ψ - 1) dψ, the partial-fraction series 1/(e^ψ - 1) = 1/ψ - 1/2 + 2ψ∑_j=1^∞ 1/(ψ² + 4π² j²) has alternating geometric truncations for all ψ > 0:

Integrating against ψ^k-1e^-nψ/(k-1)! yields 1/n - 1/(2n²) + 1/(6n³) - 1/(30n⁵) < ζ(2, n+1) < 1/n - 1/(2n²) + 1/(6n³) - 1/(30n⁵) + 1/(42n⁷), 1/(3n³) - 1/(2n⁴) + 1/(3n⁵) - 1/(6n⁷) < ζ(4, n+1) < 1/(3n³) - 1/(2n⁴) + 1/(3n⁵), and ζ(6, n+1) < ∫_n^∞ ω^-6 dω = 1/(5n⁵). Substituting these signed brackets gives ε_n > P₄(n) - 1/(288π n⁵) and ε_n < P₄(n) - 37/(11520π n⁵) + 41/(20160π n⁷) ≤ P₄(n) - 19/(16128π n⁵) for all n ≥ 1 (since 37/11520 - 41/20160 = 19/16128), proving (B.11).

1. Exact Signed-Tail Proof of Shifted Monotonicity: Put Q_i = i² I(2i). Substituting p = βψ in I(β) gives

whose integrand is positive, strictly increases with β > 0, and is dominated by (ψ-1)²/ψ⁵ (of integral 1/12). By dominated convergence, Q_i = 1/4 (2i)² I(2i) strictly increases with i ≥ 1 toward lim_i→∞ Q_i = 1/48. For M = n + 1 and any shift τ ≤ 1/2, taking the forward difference gives

Because Q_i > Q_M for every i > M and the trapezoidal comparison on [i-1, i] (i ≥ M+1) gives the strict lower bound 1/M = ∫_M^∞ ω^-2 dω < 1/(2M²) + ∑_{i > M} i^-2 ⇒ ∑_{i > M} i^-2 > 1/M - 1/(2M²) (whereas the midpoint rule on [i-1/2, i+1/2] gives the upper bound ∑_{i > M} i^-2 < 1/(M+1/2)), we obtain

Since (n+τ)ε_n → 1/(24π), strict increase implies (n+τ)ε_n < 1/(24π) for all n ≥ 1 whenever τ ≤ 1/2 (yielding the sharp enclosure 0 < ε_n < 1/(24π(n+1/2)) at τ = 1/2). Conversely, for any τ > 1/2, the expansion (n+τ)ε_n = 1/(24π) + (τ - 1/2)/(24π n) + O(n^-2) lies strictly above its limit 1/(24π) for all large n, so it cannot be globally increasing. (Note that differentiating the asymptotic truncation (ω+τ)P₄(ω) cannot substitute for this exact discrete proof: at τ = 1/2, Φ(ω) = 24π(ω+1/2)P₄(ω) = 1 - 2/(15ω²) + 2/(15ω³) + 3/(80ω⁴) actually has Φ'(1) = -17/60 < 0 and Φ(1) = 83/80 > 757/768 = Φ(2), whereas the exact sequence (n+1/2)ε_n strictly increases for all n ≥ 1.) Finally, high-precision evaluation of I(2), I(4), I(6) and C_* from (B.10) gives G₁ ≈ 0.00176526 < G₂ ≈ 0.00180513 > G₃ ≈ 0.00176694. □

Remark B.3a (Slab-Center Sets, the Constant C_*, and Working Precision). In §B.1, let 𝒵_n^top = {(p₁,p₂,n+1/2-ℓ) : 0≤ℓ≤ n, | p₁|,| p₂|≤ n-ℓ} denote the cube-center set of the top-widest solid stack Ω_n. The vertical affine reflection R₃^aff(p₁,p₂,p₃) = (p₁,p₂,n+1-p₃) maps 𝒵_n^top onto the bottom-widest slab-center set 𝒵_n^bot = {(p₁,p₂,ℓ+1/2)}, and the translate 𝒵_n^bot - (0,0,1/2) is exactly the lattice point stack S_n of Proposition 6.5. Throughout Theorem B.3 and Figure B.1(D), the isotropic mean constant C_* and the tail sum R_∞ = 3/2 - C_* are defined by the closed trilogarithm formula (B.10):

(the numerical plots in Figure B.1(D) evaluate ε_n = R_∞ - 2/π ∑_i=1ⁿ I(2i) at 90-digit working precision across 1 ≤ n ≤ 15, while monotonicity for all n when τ ≤ 1/2 rests on the proof of Theorem B.3(4)).

Appendix C. Summary of Coordinates, Notation, and Static Geometric Constructions

C.1 Local Symbol and Coordinate Summary

All notation used in §§1–8 and Appendices A–B is defined locally at first appearance and collected below; the letters follow the notation register [7]:

C.2 Rail Definitions and Static Geometric Summary

1. Rails (§2.2): The five reference rays are the first-row edge (1,k) (orange; N = T_k-1+1, r̃=1), the first-column edge (k,1) (yellow; N = T_k, r̃=2m-1), the diagonal (k,k) (purple; V=0, N=(m²+1)/2), the superdiagonal (k,k+1) (teal; V=-1, N=2k²), and the subdiagonal (k+1,k) (pink; V=+1, N=2k²+1). The colors are figure keys only.
2. Ideal Gapless Solid Ω_n (gap = 0) vs. Schematic Display-Gap Render (gap > 0): Theorems B.1–B.3 apply strictly to the gapless union of unit-height slabs Ω_n (gap = 0, where each quotient shadow P_j = [-R_j,R_j]² + [-(j+1)(a,b), -j(a,b)] is a convex hexagon of area (2j+1)² + (2j+1)(a+b), smaller by ab than its axis-aligned bounding box). Shrinking displayed cells to side (1-gap)×pitch (gap = 0.14 in Figure 11(D)) is solely a visual aid for exposing interior layers and alters the projected silhouette.

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Acknowledgments and Computational Reproducibility

Author & Collaboration Note. Authored by Jeffery Huckstead (Cerebral Graphix, cerebralgraphix.com, ORCID 0009-0007-0234-2177). AI-assisted mathematical drafting, symbolic reorganization, proof checking, and computational verification tools were used under the author's direction during the preparation of Version 4.1; responsibility for all mathematical definitions, scope boundaries, and conclusions rests with the author.

Optional Computational & Interactive Supplements. Every theorem and proof in this paper is self-contained in print. For readers wishing to replay finite-range algebraic, rational, decimal, and sieve checks, the supplementary archive provides the standalone Python 3 standard-library script check_notebook_grid_v4_1.py alongside an optional offline single-file interactive HTML reader (Notebook_Grid_v4_1_Interactive_Reader.html).