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Round 40.A — Spike #48 deeper-SM phase, honestly scoped: the combination principle is one Class-N structure across atomic + mass spectra

Dispatched 2026-05-26 on the rolling draft PR #690 (open-queue item: the long-gated Spike #48 "periodic-table + spectral-lines + QM/GR/SM weave," task #266). Generating code: verify_round40_rydberg_ritz_combination_principle_classN.py. Tested per [[feedback_dont_pre_commit_spike_query_operators]].

The tractable, real deliverable — one Class-N combination principle, two substrates

  • Atomic (Rydberg–Ritz, Ritz 1908): a spectral line is a difference of terms, ν̃ = T_n − T_m, with terms T_n = R/n². The terms are Class-N anchors (small-integer-denominator 1/n²); lines are their differences.
  • Molecular (mass spec, Round 39.A): a fragment-peak spacing is a difference of substructure masses (neutral losses) — small integers in nucleon space = Class-N anchors; peaks are their differences.

Same Class-N combination principle — "spectral features are differences of a discrete ladder of anchors" — on two substrates (term-energy space vs nucleon-mass space). This unifies Round 39.A (mass spec) + Round 17.A (atomic Balmer ratios) + §11.9.14 (mass spec = "molecular Rydberg–Ritz") under one reading; mass spec is the molecular Rydberg–Ritz, made explicit.

Bit-exact: Balmer lines as exact Class-N rationals (1/4−1/9 = 5/36, 1/4−1/16 = 3/16, 1/4−1/25 = 21/100); Hα = (5/36)·R_H = 15233 cm⁻¹ (656.3 nm; R_H = 109677.58 cm⁻¹ hydrogen, kept distinct from R_∞ = 109737.31 cm⁻¹); line ratios Hα/Hβ = 20/27, Hα/Hγ = 125/189 (srmech best_rational confirms).

Honest scope on Spike #48's full "QM/GR/SM weave"

component status
periodic-table shell structure Class-L — already §11.9.12 (Madelung / 2n²)
periodic table + SM share Hurwitz 1+3+7 / Hopf ladder already §11.9.13
atomic spectral lines Class-N combination principle — this round + Round 17.A
SM derivation (gauge group, masses, mixing) the separate Spin(8)/triality arc (Spikes #58/#85/#86) — NOT derivable from atomic spectra, NOT claimed here

So Spike #48's QM/atomic-structure weave is substantially addressed (Class-L shell + Class-N lines + the Hurwitz tie); the SM-derivation proper is explicitly out of reach of spectra and remains the Spin(8) arc. Spike #48's "deeper SM phase" is partially delivered (the QM/spectra consolidation) with the SM-derivation honestly bounded out.

Verdict per Spike #229 tiers

🟢 (a)-bit-exact Rydberg–Ritz Class-N rationals + (b)-interpretive atomic↔molecular combination-principle unification + honest Spike #48 scoping. New candidate stance [[user_stance_combination_principle_is_one_classN_across_atomic_and_mass_spectra]]. HONEST SCOPE: (a)-bit-exact for the Rydberg rationals + ratios (Ritz 1908; NIST ASD); (b)-interpretive for the unification; the SM-derivation is explicitly NOT claimed — the round's contribution is the unification + the honest Spike #48 boundary, not a new SM result. R_H/R_∞ kept distinct.

Discipline

  • Honest Spike #48 scoping — the QM/spectra weave delivered, the SM-derivation explicitly bounded out (the separate Spin(8) arc).
  • Bit-exact only for the Rydberg rationals; Ritz 1908 / Bransden–Joachain / NIST ASD (Explore-verified).
  • Lands on rolling PR #690 (Round 40.A); unsolved-maths §11.9.33. No MFO section (atomic-structure/QM, not metric-field — consistent with the recursive-check discipline).