Resolved-Mysteries Concordance

A navigational companion to the NWT paper series (Papers 1–23): it maps open problems and “mysteries” posed in earlier papers to where they were later resolved, in both directions (forward: early → late; backward: late → the problem it closed).

Why this is a separate document. The papers are frozen published records — each carries a Zenodo DOI, so editing them would break citation and re-versioning. This concordance provides the cross-referencing value without touching the published papers. It is a living, additive artifact; the papers stay as published.

What it shows. The point is not just navigation. A problem posed by an early paper and solved later from the same substrate algebra is evidence of the program’s progressive coherence — closure, not ad-hoc patching.

Provenance. Compiled from a cross-paper self-consistency review of Papers 1–23, anchored on the open-source nwt-substrate library. Cited at paper granularity; a deeper per-paper “future-work / conjecture” extraction would extend it (see Extending this concordance below).

1. Resolved — a posed problem, later solved from the substrate

# Mystery / open problem Posed in Resolved in Resolution
R1 The SO(7) substrate origin. The τ-mass anchor mτ = (20/21)αΛ has 21 = k(2k+1) = 3×7, and the paper flags “2k+1 = 7 = dim of the fundamental rep of SO(7); its appearance warrants further investigation.” Paper 1 §7.4 Papers 7–22 The entire K7 / so(7) / Spin(7) / Cl(0,7) substrate algebra. The single most consequential forward-reference in the corpus — a τ-formula footnote became the framework’s foundation.
R2 Carrier-knot nq assignment (the “fitting critique”). nq ∈ {0, 2, 3, 5} entering the Paper-6 mass formula was assigned empirically per sector, not derived. Papers 6, 8 Paper 21a nq = Fn via the (2, Fn) Fibonacci torus-knot carrier family + Murasugi’s determinant formula; per-walk sector from the cosmogenic Z3 = AGL(1,7) rule (x → 2x mod 7). Closed-form, no fitting.
R3 sin2θW from first principles. Posed as 3/13 from a 13-mode count, with “a rigorous derivation from the spectral geometry of the torus deferred to future work.” Paper 1 §8.1 / Paper 2 Papers 13, 14 sin2θW = (2 + α)/(dim 𝕆 + 1) = (2 + α)/9 = 0.22303. (This is the on-shell angle — matches 1 − MW2/MZ2 to 0.009%; the effective-angle running is a separate open item.)
R4 The reactor angle θ13. Historically taken ≈ 0; its non-zero value was the open neutrino-mixing question. (pre-NWT / Paper 1 PMNS) Paper 20 θ13 = arcsin√(3α) = √(3α) ≈ 8.5°, from Spin(7) ⊂ Spin(8) breaking — matches NuFIT (0.7%). A genuine prediction of the non-zero reactor angle.
R5 Why three fermion generations. Paper 1 ties “3” to k = 3 Z3 harmonic modes — conjectural. Paper 1 §3.1 Papers 20, 21a The cosmogenic Z3 = AGL(1,7) (σ: x → 2x mod 7, QR cycle 1→2→4) / G2 → SU(3) breaking via the S6 stabilizer. The same Z3 fixes δCP and increments the Fibonacci carrier index.
R6 The CP phase δCP. Paper 1 gives δCP = π − 2 as a knot-deficit angle — ad hoc. Paper 1 §5.3 Paper 20 §7.5 δCP = −2π/3 = −120° from the Z3 = π1(PSU(3)) winding number — the same Z3 as the generations (R5).
R7 Proton mass from first principles. Paper 1 §13 “Known limitations”: the proton mass used two un-derived variational coefficients (3/2 kinetic, 2 Coulomb). Paper 1 §9.5, §13 Papers 6, 13 The (1, 3, 5, 5) nucleon-tuple correction + cinquefoil nq = 5 carrier; proton to ~0.1% with no variational fudge.

2. Cross-paper unifications — one structure, several observables

# Structure Appears in As
U1 The trefoil Pythagorean 13 = (p2 + q2) for T(2,3) Paper 13 and Paper 20 the charged-lepton mass ratio mτ/mμ = 13 + 4 − 25α (Paper 13) and the PMNS NLO rotation Uℓ = R(13α/2) (Paper 20 §7.6). One trefoil, two independent observables — a substrate-monism signature.
U2 The denominator-7 / so(7) thread Paper 1 → 7–22 the 7 first surfacing in Paper 1’s τ-anchor (R1) recurs as k+2 = 7 (SU(2)5 level), |V(K7)| = 7, νRR = 5/7 filling, and π/7 per-face flux — all forced by so(7) → k = 5.

3. Refined / superseded — the story moved on (historical drift, record-only)

These early forms were replaced by cleaner later ones (not always flagged as “open” — they simply evolved). Recorded so the lineage is legible; the papers stand as published.

Quantity Early form Current / late form
α closed form Skilton √(1372 + π2) (Paper 1) → 1/(√2·π4) = 137.76 (P2, P4, P8a, P9, P10) 1/(25π√3 + 1) = 137.0350 (P11+, library)
sin2θW 3/13 = 0.2308 (P2); 0.222 (P8/P9) (2 + α)/9 = 0.22303 (P13, P14)
κ (torus aspect ratio) π2 = 9.870 (P4–P9); 12/√7 (P5) κMacken = 9.844 (P11+)
Nucleon carrier knot trefoil nq = 3 (P15, P20) cinquefoil nq = 5 (P13, library)
nq meaning (“3 lobes = 3 quarks”) constituent count = # co-confined quarks; a baryon’s 3 “quarks” = the trefoil’s 3 crossings (P6 §“constituents”; P8 hedges to scare-quoted “quarks”) knot determinant = dimension of the carrier’s SU(2) rep, nq = det T(2, Fn) = Fn (P21a: “nq denotes the carrier determinant Fn, not Paper 6’s constituent count”). The carrier is a substrate-algebra label, not a 3-D lobed object (its walk’s spatial embedding is mostly the unknot), so “lobes-as-quarks” is retired. Integer preserved because crossing = det = n on T(2, n); they diverge off it (figure-eight 41: 4 crossings, det 5). Decisive falsifier: the proton (3 valence quarks) now sits on the cinquefoil nq = 5.
τ-lepton topology lepton / unknot T(2,1) (P2, P6, P8) stealth-baryon T(3,4) (P7, P11, P12)
αs 16α = p4α (Paper 1, derived, 0.97%) PDG input 0.1179 (library)
mp/me 6π5 = 1836 (Paper 1, Lenz; correct) — (P14 §5.3’s 9/(2α)·nq3 is off ~9×: a frozen-paper erratum)
BPS self-dual coupling Paper 16 quotes λ = e2/2, μBPS = π (v=1) — but these are the Ginzburg–Landau ½-kinetic code values (½|Dψ|2 + ½|B|2 + ⅛(…)2; local gauged_higgs, nwt-substrate), printed under Paper 16’s relativistic Lagrangian |Dψ|2 − ¼F2 − (λ/4)(…)2 ⇒ internally inconsistent (its own table shows μ = π at v=1 vs stated 2πv2 = 2π). λ = 2e2, μ = 2πv2 — correct for that relativistic Lagrangian (Paper 11; jax-solitons library PR #9); Paper 16’s own BPS equation B = e(|ψ|2 − v2) already implies 2e2. Same physics via ψ = √2 φ; the physical invariants (mH = mgauge, κGL = 1/√2, flux 2πn, helicity 8π2·lk) and the Paper-8a α-lock (the aspect κ = π2, a different κ) are unchanged. Action: correct Paper 16’s quoted λ → 2e2, μ → 2πv2 (or reprint its Lagrangian in GL form). Published as v2 — version DOI 10.5281/zenodo.20683110; concept 10.5281/zenodo.19710845 (always-latest).

3b. By paper — stale-value index (check before citing a paper’s value)

Per-paper view of §3 (→ = current value; see §3 for detail).

4. Still open — flagged, not yet resolved

Honest record of mysteries the corpus poses that remain open (so the concordance doesn’t read as “all solved”).

# Open problem Status
O1 PMNS Uℓ first-principles derivation (Paper 20 §7.6). Mechanism identified: the Z3 = AGL(1,7) inter-generation structure, with Uℓ and δCP sharing that Z3. But a uniform R(13α/2) rotation requires the charged-lepton off-diagonals to track the mass-squared gaps — not forced by the Z3 alone. The amplitude 13α/2 must emerge from the K8 edge structure, not be inserted.
O2 Mass-formula light-meson residual (Paper 6). π/K/ρ over-predicted ~+2–4%; the small-β / Kelvin fat-ring correction hypothesis was tested and refuted (residual doesn’t track β; worst case K+ at large β). ~1% median is the formula’s intrinsic zero-parameter accuracy.
O3 13-mode-counting derivation (Paper 1 §8.1). The promised spectral-geometry derivation of “13 modes” was never done — it was bypassed by (2 + α)/9 (R3), not rigorously derived.
O4 Native-so(7) Hamiltonian / SU(2)5 lattice realization. The chiral central charge c = 15/7 has been derived three independent ways and reproduced on IBM Heron hardware; but constructing the full modular-S / anyon-spin (θw) structure of SU(2)5 directly on the K7 lattice (a Z7-vs-Z2 mismatch in the Wilson-loop basis) remains an open construction problem.

Extending this concordance

This first pass matches problems posed in early papers to their later resolutions. A comprehensive version would read each paper’s “Future work / Open problems / Conjecture” sections directly and match every flagged item to its resolution — a per-paper pass. Add entries here as problems are posed or closed; for future (not-yet-frozen) papers, include backward-references at authoring time.

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