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

Huckstead, Jeffery

doi:10.5281/zenodo.23839329

T_nDex · paper

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 · Version 4.1 · reader revised 2 November 2026 · DOI 10.5281/zenodo.23839329

Read the paper · PDF, 54 pages Open the interactive reader Zenodo record

The address

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. The paper derives its two inverse formulas, characterizes the admissible chamber-offset wedge (m, V), and proves the quadratic address families traced by affine cell paths.

Three reflections

Hold 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 − ½)/m and the nonlinear square-root phase κ_root = √(2N) − m + ½, 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. 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.

Branches and carriers

For scaled-square paths N_k = A(Qk)², a floor correction ℓ_k partitions the path into exact polynomial branches, and identifies norm-one triangular seams, algebraic returns at k = 2P, and complete negative-Pell diagonal squares. Applying the midpoint fold defines an upper-semicircle carrier map whose rational points are parameterized uniquely by odd squares, with primitive first chamber m₀ = (U² + W²)/2 ≡ 1 (mod 4). Its least nontrivial chamber is m₀ = 5, and its 45° Pell record sequence 1, 5, 29, 169, 985, … maps bijectively to the diagonal chambers 1, 7, 41, 239, 1393, …, which carry square addresses.

The odd square

On the address sheet, 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. 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 rational-carrier chamber-budget theorems and the all-direction union-shadow theory of the three-dimensional odd-square stack Ω_n.

What is proved, and what stays open

The theorems of §§1–6, §8 and Appendices A–B are proved in the paper, and no proof step depends on executing code. Finite statements carry explicit bounds: the odd-chamber mirror-prime verification covers 1403 ≤ m ≤ 20001. All-index mirror-prime existence, all-index non-intersection of the a = 2 and 45° record-chamber families beyond m₀ ∈ {1, 5, 29}, two-point prime occupancy and the analytic questions of §8.6 remain open.

Interactive reader

The reader is a self-contained HTML companion to the published PDF: the paper's text and 15 figures, with geometric exhibits and formula evaluators you can move and run. The Shape Court page holds more shapes to move by hand, and How N becomes (m, r) walks any address onto the triangle.

Figures

Figure 1: Positive Quadrant Enumeration, Admissible Chamber Wedge, and Inverse Bounds — Figure 1 (Positive quadrant address, chamber wedge, and inverse recovery).

Figure 2: The Five Rails on the Notebook Grid, Straight Rays in (m, r-tilde), and Fixed-V Parity 4-Cycle — Figure 2 (The five rails, straight odd-row rays vs. quadratic chamber addresses, and fixed- $V$ parity $4$ -cycle).

Figure 3: Three Chamber Reflections, Single-Cell N=5 Witness, and Phase Discrepancy Band — Figure 3 (Three chamber reflections, distinguishing witness $N=5$ , and root–midpoint phase discrepancy).

Figure 3b: Scaled-Square 181/128 Floor Branch, Phase-Linearity Shift, and Pell Seams / Descent — Figure 3b (Scaled-square $181/128$ floor branch, phase-linearity shift, and Pell seams/descent).

Figure 4: Bidirectional Carrier-to-Diagonal Square Bridge — Figure 4 (Rational carriers and the carrier-to-diagonal bridge).

Figure 5: Primitive Columns on the Address Sheet and Exact Cell Recovery — Figure 5 (Source map & coordinate recovery).

Figure 6: Diagonal Staircase Crossings, Active Span Rescaling, and Mandelbrot Cusp Fixed Points — Figure 6 (Geometric locus vs. rescaled fixed points).

Figure 7: Discrete Stack Fiber Multiplicity and Finite Atomic PMFs vs. Rescaled Weak-Limit Densities — Figure 7 (Stack projection & declared sampling measures).

Figure 9: Theorem 6.5 Synthesis — Common Core, Sector Congruence, Euler PGF, Gamma Family, and Stack Pushforward — Figure 9 (Theorem 6.5 across four panels).

Figure 12: Positive Lorentz Sheet, Balanced-Split Defects, Odd-Chamber Midpoint vs. Landau's n^2+1, and the Euler-Transform S_+ Series — Figure 12 (Geometric and analytic scope separations across §§6.4–6.6 & §8.2).

Figure 8: Odd-Chamber Mirror-Prime Symmetry Across the Diagonal and Record-Set Intersection on E: C^2 = (A-2)(A^2-200) — Figure 8 (Odd-chamber mirror-prime pairs and record-set intersection on $E$ ).

Figure 10: Divisor-Shell Numerator Diagnostics, Three Jordan-Totient Zeta Limits, the Factorial-Residual Dickman Density, and Odd-Prime Wheel Midpoint Counts — Figure 10 (Arithmetic structures on the Grid addresses across four panels).

Figure A.1: All-Budget 45-Degree Pell Staircase, Excluded Target (3,4) Budget-17 Exception, and Two-Chart One-Chamber Advance — Figure A.1 (Rational-carrier chamber budgets and two-chart advance).

Figure B.1: Solid Odd-Square Stack Quotient Shadow, Square Hiding Cone, and Shifted Tail Monotonicity — Figure B.1 (Three-dimensional odd-square stack $Ω n$ shadows, square hiding cone, and shifted tail monotonicity).

Figure 11: Static Geometric Summary — Straightened Parity Wedge, Address-Sheet Primitive Columns & Hull Incircle, Oblique Fiber Depth, and Solid vs. Display-Gap Domain — Figure 11 (Static geometric summary across four panels).

Versions

version	date	DOI
4.1 (this page): the same paper, with the reader, checker, casebook and final figure files	2 November 2026	10.5281/zenodo.23839329
4.0: the paper	October 2026	10.5281/zenodo.23632948
1.5: earlier working-paper deposit	October 2026	10.5281/zenodo.23459201

Cite

@misc{huckstead2026grid,
  author    = {Huckstead, Jeffery},
  title     = {The Notebook Grid: Exact Addresses, Three Reflections, Pell Branches, and Rational Carriers},
  year      = {2026},
  version   = {4.1},
  publisher = {Zenodo},
  doi       = {10.5281/zenodo.23839329},
  url       = {https://doi.org/10.5281/zenodo.23839329}
}