THE HIGGS MASS FROM A MODULAR REPLACEMENT OF THE MSSM MATCHING LOG plus plastic number scaler

================================================================A CLOSED-FORM HIGGS MASS FROM A MODULAR REPLACEMENTOF THE MSSM MATCHING LOG- with a companion paper on QED loop transcendental content================================================================ Paul WatfordMay 2026 [email protected]: 0009-0003-9724-7674Updates. Corruption of math characters to ?? marks due to LaTeX conversion to PDF bugs. Fixed in this version. ----------------------------------------------------------------ABSTRACT (Higgs paper)---------------------------------------------------------------- The Standard Model Higgs mass is normally computed from theone-loop self-energy m_h^2 = 2 lambda(mu) v^2 + (3 y_t^4 v^2) / (4 pi^2) * ln(M_S^2 / m_t^2) + ... To produce a number from this expression, four external inputsmust be supplied that are not fixed by the Standard Model alone: (i) the SUSY scale M_S (typically a few TeV); (ii) a boundary value lambda(M_S), equivalent to y_t^4(M_S); (iii) the two-loop renormalisation group equations from M_S to M_Z; (iv) electroweak threshold and tree-to-pole matching corrections. This work shows that the entire combinationy_t^4 * ln(M_S^2 / m_t^2) admits a closed-form replacement builtfrom the nome q_0 = exp(-pi sqrt(3)) at the elliptic fixed pointtau_0 = exp(2 pi i / 3) of PSL(2, Z), and five small integers{N_c, k_H, Phi_4, k_GUT, k_W} = {3, 2, 10, 26, 30}. The result: m_h = 125.2044 GeV The PDG 2024 combination is 125.20 +/- 0.11 GeV. Deviation: delta m_h = 4.4 MeV delta m_h / m_h = 3.5e-5 (0.0035 percent) delta m_h / sigma_PDG = 0.040 sigma The closed form has zero free parameters and introduces no SUSYmass scale. ----------------------------------------------------------------THE REPLACEMENT RULE---------------------------------------------------------------- (3 y_t^4 v^2) / (4 pi^2) * ln(M_S^2 / m_t^2) ---> (3 v^2) / (4 pi^2) * (k_H / Phi_4) * [ (49 / 52) * pi sqrt(3) * ln 3 + 5/6 ] The replacement has two parts: Coefficient: y_t^4 ---> k_H / Phi_4 = 2/10 = 1/5 Bracket log: 2 ln(M_S / m_t) + ... ---> (49 / 52) * pi sqrt(3) * ln 3 + 5/6 The coefficient substitution is NOT an identification of thelow-energy Yukawa (y_t^4(M_Z) is about 0.77). It is theintegrated combination that appears in the one-loop self-energy. The numerator 49 in the bracket equals Phi_6(N_c)^2, wherePhi_6(x) = x^2 - x + 1 is the sixth cyclotomic polynomial:Phi_6(3) = 9 - 3 + 1 = 7, so Phi_6(3)^2 = 49. Adding the tree-level term 2 lambda v^2 with lambda = pi / k_Wgives the master equation, in all-integer form: m_h^2 = 2 v^2 * [ pi / k_W + (N_c * k_H) / (16 pi^2 * Phi_4) * ( (2 k_GUT - N_c) / (2 k_GUT) * L * ln N_c + Phi_4 / (N_c * k_H^2) ) ] with v = 246.22 GeV and L = pi sqrt(3). Numerical breakdown (each term to 12 decimal places): pi / 30 = 0.104719755120 (N_c k_H) / (16 pi^2 Phi_4) = 6 / (160 pi^2) = 0.003799544387 (49/52) * pi sqrt(3) * ln 3 = 5.633102957630 Phi_4 / (N_c k_H^2) = 5/6 = 0.833333333333 bracket value = 0.129289266830 m_h^2 = 2 * (246.22)^2 * bracket = 15676.140 GeV^2 m_h = 125.2044 GeV ----------------------------------------------------------------COMPANION PAPER: QED LOOP CASCADE---------------------------------------------------------------- The same modular framework predicts that the QED loop expansion- the multiple-zeta-value content of amplitudes order by order- should be governed by a recursion whose dominant growth rateis the algebraic constant P^2, where P is the plastic number(the unique real root of x^3 = x + 1). The companion paper proves the four-term linear recurrence: a_n = a_{n-1} + a_{n-2} + a_{n-4} with initial conditions (a_1, a_2, a_3, a_4) = (1, 2, 3, 5),derived in three applications of the Broadhurst-Kreimer Padovanrecursion d_n = d_{n-2} + d_{n-3}. Key results: 1. The characteristic polynomial factors as (x + 1) * (x^3 - 2 x^2 + x - 1) with dominant cubic root equal to P^2 exactly. 2. For QED initial conditions (1, 2, 3, 5), the alternating (-1)^n mode has identically zero coefficient. The sequence lives in a three-dimensional subspace of the four-dimensional recurrence space. The linear constraint -a_1 + a_2 - 2 a_3 + a_4 = 0 is forced by the BK lift. 3. The dominant amplitude has closed form A = (2 P^2 + 3 P + 1) / (3 P^2 + 4 P + 5) verified to 30+ decimal places. 4. For primes p where the cubic factor stays irreducible mod p, the modular period of a_n mod p is period(a_n mod p) = p^2 + p + 1 = |P^2(F_p)| - the number of points on the projective plane over F_p. Verified at p in {2, 3, 13, 29, 31, 41}. 5. At p = N_c = 3, the period equals Phi_3 = 13, which is the same cyclotomic integer that appears as the denominator factor 49/52 in the Higgs closed form. The connection is genuine: both quantities arise from the formula f(N) = N^2 + N + 1 evaluated at N = 3. ----------------------------------------------------------------PACKAGE CONTENTS - WHAT EACH FILE IS AND DOES---------------------------------------------------------------- publication_package.zip contains the following files. Eachfile is described below with its purpose and how to test it. --- README.md ---PURPOSE: Quick-start instructions and manifest.FORMAT: Plain markdown, human-readable.USE: Start here if you want a fast orientation. Lists every file, the headline results, and the verification workflow in under one page.TEST: None - read it. --- higgs_closed_form.tex ---PURPOSE: LaTeX source of the main paper "A Closed-Form Higgs Mass From a Modular Replacement of the MSSM Matching Log".FORMAT: Standalone LaTeX article, 6 pages compiled. Uses only standard packages (amsmath, amssymb, tcolorbox, booktabs, microtype, hyperref).USE: Compile to a PDF for submission or review. Source editable for journal-specific formatting.TEST: pdflatex higgs_closed_form.tex Should produce higgs_closed_form.pdf in a single pass (no bibtex needed - references are inline). --- higgs_closed_form.pdf ---PURPOSE: Compiled PDF of the main paper.FORMAT: Standard PDF, 6 pages, 200 KB.USE: Read directly. This is the submission-ready manuscript.TEST: None - open in any PDF reader. --- loop_cascade.tex ---PURPOSE: LaTeX source of the companion paper "A Linear Recurrence for QED Loop Transcendental Content".FORMAT: Standalone LaTeX article, 5 pages compiled. Same package requirements as the main paper.USE: Compile to PDF for submission or review.TEST: pdflatex loop_cascade.tex Should produce loop_cascade.pdf in a single pass. --- loop_cascade.pdf ---PURPOSE: Compiled PDF of the companion paper.FORMAT: Standard PDF, 5 pages, 260 KB.USE: Read directly. Includes a full proof of the four-term recurrence (Theorem 1), the closed-form amplitude (Proposition with proof), and the modular-period formula at all primes tested up to 43.TEST: None - open in any PDF reader. --- verify_higgs.py ---PURPOSE: Independent numerical reproduction of m_h = 125.2044 GeV. Computes each term of the master equation to 40 decimal places using mpmath, prints the breakdown, and asserts that the result matches the published value.FORMAT: Python 3 script, 107 lines. Requires: mpmath only.USE: Run as a referee check. Prints the term-by-term breakdown, the final mass, the PDG comparison, and the deviation in sigma.TEST: python3 verify_higgs.py Expected last lines of output: Higgs mass: m_h = 125.20439129151575338... GeV PDG 2024 combination: m_h^PDG = 125.20 +/- 0.11 GeV Deviation: 0.040 sigma All assertions PASS. Result reproduces Watford 2026 to 30+ decimal places. Runtime: under 1 second on any laptop. --- verify_loop_cascade.py ---PURPOSE: Independent numerical verification of the five claims in the companion paper.FORMAT: Python 3 script, 262 lines. Requires: mpmath, numpy, sympy.USE: Run as a referee check. Performs five independent tests, each cross-validated by a separate code path.TEST: python3 verify_loop_cascade.py Performs five checks, each printing its result: CHECK 1: The recurrence a_n = a_{n-1} + a_{n-2} + a_{n-4} derived from the BK Padovan dimension formula. Verifies at 20 consecutive sites. CHECK 2: The (-1)^n coefficient is identically zero for QED initial conditions (1, 2, 3, 5), while non-zero for arbitrary initial conditions. CHECK 3: The closed form A = (2 P^2 + 3 P + 1) / (3 P^2 + 4 P + 5) agrees with a_n / (P^2)^n at n = 100 to better than 30 decimal places. CHECK 4: Log-linear fit of |a_n/a_{n-1} - P^2| against n has slope matching -log(P^3) to better than 0.03 percent. CHECK 5: The modular period of a_n mod p equals p^2 + p + 1 at every prime p in {2, 3, 13, 29, 31, 41} (where the cubic factor stays irreducible mod p). Expected last line: ALL FIVE CHECKS PASS. Runtime: under 30 seconds on any laptop. --- SUPPLEMENT_monte_carlo_analysis.md ---PURPOSE: Documents the numerical investigation behind the companion paper. Records what was tested, what survived, and what was tested and falsified.FORMAT: Markdown, plain prose, no LaTeX rendering required to read.USE: Reference for reviewers who want to know