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 paulwatford@gmail. comORCID: 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 mₕ² = 2 lambda (mu) v² + (3 yₜ⁴ v²) / (4 pi²) * ln (MS² / mₜ²) +. . . 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 MS (typically a few TeV) ; (ii) a boundary value lambda (MS), equivalent to yₜ⁴ (MS) ; (iii) the two-loop renormalisation group equations from MS to MZ; (iv) electroweak threshold and tree-to-pole matching corrections. This work shows that the entire combinationyₜ⁴ * ln (MS² / mₜ²) admits a closed-form replacement builtfrom the nome q₀ = exp (-pi sqrt (3) ) at the elliptic fixed pointtau₀ = exp (2 pi i / 3) of PSL (2, Z), and five small integersNc, kH, Phi₄, kGUT, kW = 3, 2, 10, 26, 30. The result: mₕ = 125. 2044 GeV The PDG 2024 combination is 125. 20 +/- 0. 11 GeV. Deviation: delta mₕ = 4. 4 MeV delta mₕ / mₕ = 3. 5e-5 (0. 0035 percent) delta mₕ / sigmaPDG = 0. 040 sigma The closed form has zero free parameters and introduces no SUSYmass scale. ----------------------------------------------------------------THE REPLACEMENT RULE---------------------------------------------------------------- (3 yₜ⁴ v²) / (4 pi²) * ln (MS² / mₜ²) ---> (3 v²) / (4 pi²) * (kH / Phi₄) * (49 / 52) * pi sqrt (3) * ln 3 + 5/6 The replacement has two parts: Coefficient: yₜ⁴ ---> kH / Phi₄ = 2/10 = 1/5 Bracket log: 2 ln (MS / mₜ) +. . . ---> (49 / 52) * pi sqrt (3) * ln 3 + 5/6 The coefficient substitution is NOT an identification of thelow-energy Yukawa (yₜ⁴ (MZ) is about 0. 77). It is theintegrated combination that appears in the one-loop self-energy. The numerator 49 in the bracket equals Phi₆ (Nc) ², wherePhi₆ (x) = x² - x + 1 is the sixth cyclotomic polynomial: Phi₆ (3) = 9 - 3 + 1 = 7, so Phi₆ (3) ² = 49. Adding the tree-level term 2 lambda v² with lambda = pi / kWgives the master equation, in all-integer form: mₕ² = 2 v² * pi / kW + (Nc * kH) / (16 pi² * Phi₄) * ( (2 kGUT - Nc) / (2 kGUT) * L * ln Nc + Phi₄ / (Nc * kH²) ) with v = 246. 22 GeV and L = pi sqrt (3). Numerical breakdown (each term to 12 decimal places): pi / 30 = 0. 104719755120 (Nc kH) / (16 pi² Phi₄) = 6 / (160 pi²) = 0. 003799544387 (49/52) * pi sqrt (3) * ln 3 = 5. 633102957630 Phi₄ / (Nc kH²) = 5/6 = 0. 833333333333 bracket value = 0. 129289266830 mₕ² = 2 * (246. 22) ² * bracket = 15676. 140 GeV² mₕ = 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 P², where P is the plastic number (the unique real root of x³ = x + 1). The companion paper proves the four-term linear recurrence: aₙ = a₍-₁ + a₍-₂ + a₍-₄ with initial conditions (a₁, a₂, a₃, a₄) = (1, 2, 3, 5), derived in three applications of the Broadhurst-Kreimer Padovanrecursion dₙ = d₍-₂ + d₍-₃. Key results: 1. The characteristic polynomial factors as (x + 1) * (x³ - 2 x² + x - 1) with dominant cubic root equal to P² 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 -a₁ + a₂ - 2 a₃ + a₄ = 0 is forced by the BK lift. 3. The dominant amplitude has closed form A = (2 P² + 3 P + 1) / (3 P² + 4 P + 5) verified to 30+ decimal places. 4. For primes p where the cubic factor stays irreducible mod p, the modular period of aₙ mod p is period (aₙ mod p) = p² + p + 1 = |P² (Fₚ) | - the number of points on the projective plane over Fₚ. Verified at p in 2, 3, 13, 29, 31, 41. 5. At p = Nc = 3, the period equals Phi₃ = 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) = N² + N + 1 evaluated at N = 3. ----------------------------------------------------------------PACKAGE CONTENTS - WHAT EACH FILE IS AND DOES---------------------------------------------------------------- publicationₚackage. 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. --- higgsclosedform. 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 higgsclosedform. tex Should produce higgsclosedform. pdf in a single pass (no bibtex needed - references are inline). --- higgsclosedform. 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. --- loopcascade. 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 loopcascade. tex Should produce loopcascade. pdf in a single pass. --- loopcascade. 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. --- verifyₕiggs. py ---PURPOSE: Independent numerical reproduction of mₕ = 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 verifyₕiggs. py Expected last lines of output: Higgs mass: mₕ = 125. 20439129151575338. . . GeV PDG 2024 combination: mₕPDG = 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. --- verifyₗoopcascade. 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 verifyₗoopcascade. py Performs five checks, each printing its result: CHECK 1: The recurrence aₙ = a₍-₁ + a₍-₂ + a₍-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 P² + 3 P + 1) / (3 P² + 4 P + 5) agrees with aₙ / (P²) ⁿ at n = 100 to better than 30 decimal places. CHECK 4: Log-linear fit of |aₙ/a₍-₁ - P²| against n has slope matching -log (P³) to better than 0. 03 percent. CHECK 5: The modular period of aₙ mod p equals p² + 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. --- SUPPLEMENTₘontecarloₐnalysis. 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