# Paper 4 v0.2 Draft ## Precision Data Testing of Candidate J₃(𝕆ₛ) Relations Against PDG 2024 and NuFIT 6.0 **Author:** Tom O'Sieg **Drafting support:** ChatGPT **Date:** 2026-03-29 **Companions:** Paper 1 v1.50, Paper 2 v3.11, Paper 3 v0.1 --- ## §1. Introduction Paper 4 is the data paper. Its job remains narrower than Papers 1–3 and 5: it introduces no new theory, no new derivations, and no new extension routes. It takes the formulas already on the table, extracts every numerically testable relation that the coalition has promoted or explicitly carried forward, and compares them against current benchmark data. Version 0.2 differs from v0.1 in one major respect: it incorporates the Session 9 updates now reflected in Paper 2 v3.9–v3.11. The most important of these is the **PMNS pentagonal moiré ansatz**, a non-polynomial flavor-sector escape route that produces a numerically strong four-row PMNS package against NuFIT 6.0 IC19. The other two Session 9 updates do not change numerical values, but they do change how two rows should be read: `m_t` now carries a proposed structural `y_t = 1` interpretation via the TKK/E₇ line, and `δ_CKM` now has a deeper algebraic provenance through the cubic Jordan norm. Three ground rules govern the audit. First, **status labels are derivational, not empirical**. A relation can be `CLOSED` because the algebraic chain is complete and still fail badly against a modern benchmark. Conversely, a `CONJECTURAL` or `PROPOSED` relation can look numerically excellent and still remain derivationally incomplete. Second, **all σ-gaps in this paper are experimental-only σ-gaps unless explicitly marked as proxy values**. The framework does not yet supply a theory covariance matrix. As a result, some relations with small percent gaps still miss by many standard deviations because the underlying experimental quantity is measured far more precisely than the theory is. Third, **benchmark discipline matters**. For PMNS quantities, NuFIT 6.0 provides materially different benchmark sets in its published Table 1: IC19 without Super-K atmospheric data and IC24 with SK atmospheric data. For the weak mixing angle, PDG 2024 quotes more than one fit-dependent benchmark. Wherever a benchmark choice materially changes the verdict, both are shown. Paper 4 v0.2 therefore has a double purpose. It is a scoreboard, but it is also a test of whether the Session 9 improvements really survive modern benchmark discipline. In the PMNS sector, they do — partially, strongly, and with caveats. --- ## §2. Input Catalog The framework input ledger used for the precision audit is: ```text Quantity Value Type Role Counted as input? α 1/137.035999084 physical input fine-structure input used by the framework yes v 246.220 GeV physical input electroweak VEV yes M_Pl 1.22090×10^19 GeV physical input used in hierarchy-sector formulas yes β_δ 11/(6π) derived structural coefficient split-octonion/G₂(2) torsion coefficient yes (derived structural) φ (1+√5)/2 mathematical constant not a measured physics input no y_t = 1 unit-normalized top Yukawa assumption required for m_t and inherited heavy-quark relations conditional input AX1 minimal-geodesic vacuum postulate postulate fixes the vacuum ratio used in Paper 1 assumption D_IV(n)→Peirce map J12↔D_IV(5), J13↔D_IV(4), J23↔D_IV(3) assumption sector-to-domain assignment assumption μ0 / matching-scale choices e.g. μ0 = φ·m_b convention needed for matched-scale comparisons convention ``` Two bookkeeping remarks remain essential. 1. `φ` is a mathematical constant, not a measured particle-physics input. 2. Session 9 does **not** remove `y_t = 1` from the input ledger. It only upgrades the status note: the latest Paper 2 text now treats `y_t = 1` as a **proposed structural insight** rather than a completely naked choice, but HP12 remains open. --- ## §3. Tier 1 — CLOSED / DERIVED / Inherited Structural Checks Tier 1 rows are the algebraically strongest relations. This does **not** mean they are automatically the best current fits. ```text id quantity pred_str benchmark_label obs_str pct_str sigma_str status P70 sin²θ_W 0.231422 PDG 2024 s_Z² (LHC fit) 0.231220 +0.088% 2.53σ CLOSED P70a sin²θ_W 0.231422 PDG 2024 s_Z² (all-data fit) 0.231290 +0.057% 3.31σ CLOSED P70b sin²θ_W 0.231220 PDG 2024 s_Z² (LHC fit) 0.231220 +0.000% 0.00σ CLOSED / OPEN-SCALE G2 M_Z 91.4786 PDG 2024 M_Z 91.1876 +0.319% 139σ CLOSED P2-MW M_W 80.1980 PDG 2024 M_W 80.3692 -0.213% 12.87σ DERIVED G1 m_μ 0.105807 PDG 2024 m_μ 0.105658 +0.141% ≫10^4σ CLOSED Shilov-23 θ23^PMNS 49.2157 NuFIT 6.0 NO IC19 θ23 48.5000 +1.476% 1.02σ CLOSED Shilov-23b θ23^PMNS 49.2157 NuFIT 6.0 NO IC24+SK θ23 43.3000 +13.662% 5.92σ CLOSED P71 θ_C 13.1494 PDG 2024 θ_C from λ 13.0035 +1.123% 3.65σ CLOSED ``` ### Tier-1 reading notes - **P70 remains percent-level good but not σ-level safe.** Against the PDG 2024 effective-angle benchmark used in Papers 1–2, the formula is only +0.088%, but that still corresponds to **2.53σ**. Using the tighter all-data global-fit benchmark worsens the tension to **3.31σ**. - **The matched-scale P70 row is numerically exact only because the scale is chosen to make it exact.** That row remains worth keeping as a consistency check, not as a free-standing prediction. - **G2 and the Peirce-derived M_W cascade are the clearest examples of percent-vs-sigma divergence.** Their percent gaps are only +0.319% and −0.213%, respectively, yet they miss by **139σ** and **12.9σ** because the benchmark errors are now tiny. - **The muon formula is a similar case.** The percent gap is only +0.141%, but the σ-gap is effectively enormous because the experimental muon-mass uncertainty is microscopic. - **The Shilov θ23 relation is benchmark-sensitive.** Against NuFIT 6.0 NO IC19 it is a respectable **1.02σ** result; against NO IC24+SK it jumps to **5.92σ**. - **P71 remains algebraically closed but no longer a numerically elite Cabibbo fit on current data.** Using the PDG 2024 CKM-fit λ converted to θ_C gives a **3.65σ** tension. --- ## §4. Tier 2 — CONDITIONAL Relations These relations close only after granting an extra assumption, typically `y_t = 1`, and in the heavy-quark sector they also inherit renormalization-scheme caveats. ```text id quantity pred_str benchmark_label obs_str pct_str sigma_str status m_t m_t 174.104 PDG 2024 m_t (direct) 172.570 +0.889% 5.29σ CONDITIONAL P67 m_c 1.2798 PDG 2024 m_c 1.2730 +0.537% 2.44σ CONDITIONAL m_b m_b 4.2133 PDG 2024 m_b 4.1830 +0.724% 7.09σ CONDITIONAL ``` ### Tier-2 reading notes - All three heavy-quark rows remain **sub-1% in percent gap**, which is why they were historically among the strongest-looking numerical successes. - But once current PDG uncertainties are used, the same rows look sharper: - `m_t = v/√2` sits at **5.29σ** against the direct top-mass benchmark. - `m_c = m_t α/(1−α)` sits at **2.44σ**. - `m_b` from the power-law branch sits at **7.09σ**. - Session 9 changes the **interpretive note** on `m_t`, not the number: `y_t = 1` is now treated as a **proposed E₇/TKK structural insight** rather than a fully arbitrary input, but the closure claim is still open. - For `m_c` and `m_b`, the PDG summary-table uncertainties are quoted at **90% CL**. For a uniform σ-reporting scheme, this paper keeps the v0.1 convention of converting them to approximate Gaussian 1σ errors by dividing by 1.64. --- ## §5. Tier 3 — PROPOSED / PLAUSIBLE / OPEN Numerical Relations Tier 3 remains the widest category: formulas that exist, often look interesting numerically, but whose derivation is incomplete, dataset-sensitive, or structurally blocked from promotion. ```text id quantity pred_str benchmark_label obs_str pct_str sigma_str status H1 m_H 124.083 PDG 2024 m_H 125.200 -0.892% 10.15σ CONJECTURAL H2 m_H 123.611 PDG 2024 m_H 125.200 -1.269% 14.44σ CONJECTURAL C3-τ m_τ 1.7780 PDG 2024 m_τ 1.7769 +0.058% 11.43σ PROPOSED C3-e m_e 0.000513184 PDG 2024 m_e 0.000510999 +0.428% ≫10^6σ PROPOSED P46 m_c 1.2491 PDG 2024 m_c 1.2730 -1.874% 8.50σ PROPOSED PMNS-12 θ12^PMNS 33.2631 NuFIT 6.0 NO IC19 θ12 33.6800 -1.238% 0.60σ PROPOSED P45 θ13^PMNS 8.6700 NuFIT 6.0 NO IC19 θ13 8.5200 +1.760% 1.36σ PROPOSED P44a δ_CP^PMNS 226.643 NuFIT 6.0 NO IC19 δCP 177.000 +28.047% 2.61σ OPEN P44b δ_CP^PMNS 226.643 NuFIT 6.0 NO IC24+SK δCP 212.000 +6.907% 0.56σ OPEN P73 θ_C 13.0502 PDG 2024 θ_C from λ 13.0035 +0.360% 1.17σ PLAUSIBLE δCKM γ = δ_CKM 70.1674 PDG 2024 γ 65.7000 +6.800% 1.49σ PROPOSED Jhyb J_CKM 3.01e-05 PDG 2024 J_CKM 3.12e-05 -3.526% 0.92σ PROPOSED Vcb |V_cb| 0.031512 PDG 2024 |V_cb| 0.041830 -24.666% 14.95σ PROPOSED Vub |V_ub| 0.007116 PDG 2024 |V_ub| 0.003732 +90.666% 37.60σ PROPOSED Hier M_Pl/M_Z 1.3465e+17 M_Pl(full)/PDG 2024 M_Z 1.3389e+17 +0.569% — PROPOSED alpha_s α_s(M_Z) seed 0.031533 PDG 2024 world average α_s(M_Z) 0.118000 -73.277% 96.07σ OPEN Rb R_b 0.381966 PDG 2024 R_b (ρ̄,η̄) 0.386559 -1.188% 0.60σ PROPOSED Rubcb |Vub/Vcb| 0.090170 PDG 2024 |V_ub/V_cb| 0.089218 +1.067% 0.36σ PROPOSED ``` ### Tier-3 reading notes #### Surviving numerical positives Restricting attention to the legacy Tier-3 block (i.e. excluding the new Session 9 pentagonal rows), the strongest surviving proposed/plausible/open rows on current benchmarks are: - `Rubcb`: **0.36σ** - `P44b`: **0.56σ** - `PMNS-12`: **0.60σ** - `Rb`: **0.60σ** - `Jhyb`: **0.92σ** - `P73`: **1.17σ** - `P45`: **1.36σ** - `δ_CKM`: **1.49σ** #### Mixed or benchmark-sensitive rows - **Legacy P44 PMNS δ_CP** remains highly benchmark-sensitive. - Against NuFIT 6.0 **IC19 without SK-atm**: **2.61σ** - Against NuFIT 6.0 **IC24 with SK-atm**: **0.56σ** - **δ_CKM** keeps a decent numerical showing (**1.49σ**), and Session 9 improves its algebraic story: the formula is now tied to `arg N(X_off)` rather than being carried only as an arctan torsion ansatz. #### Weak or strongly disfavored rows - The absolute CKM element formulas requested in the original dispatch, - `|V_cb| = sin²θ_C/φ` - `|V_ub| = sin³θ_C/φ`, remain poor current-data rows: **14.95σ** and **37.60σ**, respectively. - The **α_s seed** is still far from a usable prediction. - Both Higgs-mass ansätze remain double-digit-σ misses. #### Kernel-era lepton-tower caveat The electron and tau cylinder formulas are numerically close in percent terms, but their σ-gaps are not meaningful theory-quality pulls because the experimental charged-lepton masses are measured orders of magnitude more precisely than the framework specifies them. --- ## §5A. Tier 3A — CONJECTURAL: Pentagonal Moiré Ansatz (Session 9) Session 9 adds a new block that materially changes the flavor audit. The pentagonal moiré ansatz is **not** a prediction in the strict Paper 1 sense. It is a four-integer fit to four PMNS observables: - `θ12 = π/5 − 4C` - `θ23 = 3π/10 − 7C` - `θ13 = π/10 − 14C` - `δ_CP = −π + 25C` with - `C = arctan(αφ)`, and the integers `(4, 7, 14, 25)` are **motivated but not derived**. Because this is a non-polynomial escape route from the Paper 3 no-go, it belongs in its own tier. It is the strongest new numerical block in the update, but the `CONJECTURAL` label is mandatory. ```text id quantity pred_str benchmark_label obs_str pct_str sigma_str status Pent-12 θ12^PMNS 33.2941 NuFIT 6.0 NO IC19 θ12 33.6800 -1.146% 0.55σ CONJECTURAL Pent-23 θ23^PMNS 49.2646 NuFIT 6.0 NO IC19 θ23 48.5000 +1.577% 1.09σ CONJECTURAL Pent-23b θ23^PMNS 49.2646 NuFIT 6.0 NO IC24+SK θ23 43.3000 +13.775% 5.96σ CONJECTURAL Pent-13 θ13^PMNS 8.5293 NuFIT 6.0 NO IC19 θ13 8.5200 +0.109% 0.08σ CONJECTURAL Pent-δ δ_CP^PMNS 196.912 NuFIT 6.0 NO IC19 δCP 177.000 — 1.05σ CONJECTURAL Pent-δb δ_CP^PMNS 196.912 NuFIT 6.0 NO IC24+SK δCP 212.000 — 0.58σ CONJECTURAL ``` ### Tier-3A reading notes - **Three angle rows against NuFIT 6.0 IC19 are excellent by the standards of the rest of the framework:** - `Pent-13`: **0.08σ** - `Pent-12`: **0.55σ** - `Pent-23`: **1.09σ** - **The pentagonal δ row is also good, but less spectacular than the Session 9 note suggested.** - Using circular phase distance and the published NuFIT 6.0 IC19 best fit `177^{+19}_{−20}°`, the correct pull is about **1.05σ**, **not 0.01σ**. - **The atmospheric-data sensitivity does not disappear.** - `Pent-23` goes from **1.09σ** on IC19 to **5.97σ (proxy)** on IC24+SK. - `Pent-δ` goes from **1.05σ** on IC19 to **0.58σ** on IC24+SK. - Numerically, the pentagonal block **supersedes the legacy P44 PMNS-phase row** and the older standalone PMNS-angle formulas. - Structurally, it does **not** supersede them yet, because the integers are not derived. If the integer coefficients are ever promoted from fit parameters to derived combinatorial data, this block would instantly become the strongest flavor-sector result in the framework. --- ## §6. Tier 4 — DEAD Relations Paper 3 already killed several relations structurally. Paper 4 shows what that looks like numerically when a benchmark exists. ```text id quantity pred_str benchmark_label obs_str pct_str sigma_str status λdead Wolfenstein λ 0.236068 PDG 2024 CKM λ 0.225010 +4.914% 16.26σ DEAD Adead Wolfenstein A 0.786151 PDG 2024 CKM A 0.826000 -4.824% 2.66σ DEAD MF23 θ23^PMNS 43.0500 NuFIT 6.0 NO IC19 θ23 48.5000 -11.237% 6.06σ DEAD / STRUCTURAL MF12 θ12^PMNS 15.1100 NuFIT 6.0 NO IC19 θ12 33.6800 -55.137% 26.53σ DEAD / STRUCTURAL MF13 θ13^PMNS 14.6100 NuFIT 6.0 NO IC19 θ13 8.5200 +71.479% 55.36σ DEAD / STRUCTURAL ``` ### Dead rows with numerical benchmarks - `λ = φ^-3` is dead on current PDG 2024 CKM-fit data at **16.26σ**. - `A = φ^-1/2` is dead at **2.66σ** numerically, but the decisive reason it is dead is algebraic: Paper 3 rules out the required Jordan-polynomial mechanism. - The bare `M_Freud` PMNS angles are quantitatively excluded. ### Structural dead results without a single-number score Paper 3 also kills several structural routes that are not best represented by one benchmark row: - polynomial CKM generator inside strict `J₃(𝕆ₛ)`; - PMNS parameter derivation from `J₃(𝕆ₛ)` constants alone; - Wolfenstein-`A` derivation from Jordan-polynomial constructions. These stay dead regardless of small benchmark shifts. --- ## §7. Summary Scorecard A compact summary of the full audit: - Active non-dead framework rows audited (excluding the inherited `P55` input-baseline row): **36** - Active rows within 1%: **15** - Active rows within 2σ: **15** - Dead rows explicitly scored: **5** The full master scorecard is: ```text id quantity formula pred_str benchmark_label obs_str pct_str sigma_str status tier P55 1/α_em (9/8π^4)(π^5/1920)^(1/4) inverted 137.036 PDG/CODATA baseline 137.036 +0.000% — INHERITED Structural / inherited P70 sin²θ_W 1/φ³ − 2α/π 0.231422 PDG 2024 s_Z² (LHC fit) 0.231220 +0.088% 2.53σ CLOSED Tier 1 P70a sin²θ_W 1/φ³ − 2α/π 0.231422 PDG 2024 s_Z² (all-data fit) 0.231290 +0.057% 3.31σ CLOSED Tier 1 P70b sin²θ_W Matched-scale value at μ0 = φ·m_b 0.231220 PDG 2024 s_Z² (LHC fit) 0.231220 +0.000% 0.00σ CLOSED / OPEN-SCALE Tier 1 G2 M_Z v_- / √2 · (1−α/4π) 91.4786 PDG 2024 M_Z 91.1876 +0.319% 139σ CLOSED Tier 1 P2-MW M_W M_Z^route · √(1−sin²θ_W) 80.1980 PDG 2024 M_W 80.3692 -0.213% 12.87σ DERIVED Tier 1 G1 m_μ α^(5/4)(3/16)^(3/4) v/√2 0.105807 PDG 2024 m_μ 0.105658 +0.141% ≫10^4σ CLOSED Tier 1 Shilov-23 θ23^PMNS arctan(1 + 1/(2π)) 49.2157 NuFIT 6.0 NO IC19 θ23 48.5000 +1.476% 1.02σ CLOSED Tier 1 Shilov-23b θ23^PMNS arctan(1 + 1/(2π)) 49.2157 NuFIT 6.0 NO IC24+SK θ23 43.3000 +13.662% 5.92σ CLOSED Tier 1 P71 θ_C arctan(1/φ³) − arctan(α/π) 13.1494 PDG 2024 θ_C from λ 13.0035 +1.123% 3.65σ CLOSED Tier 1 m_t m_t v/√2 174.104 PDG 2024 m_t (direct) 172.570 +0.889% 5.29σ CONDITIONAL Tier 2 P67 m_c m_t·α/(1−α) 1.2798 PDG 2024 m_c 1.2730 +0.537% 2.44σ CONDITIONAL Tier 2 m_b m_b [(m_t·π/2)]^(2/3) (MeV-scaled power law) 4.2133 PDG 2024 m_b 4.1830 +0.724% 7.09σ CONDITIONAL Tier 2 H1 m_H v·√(16/63) 124.083 PDG 2024 m_H 125.200 -0.892% 10.15σ CONJECTURAL Tier 3 H2 m_H 3v / [π√(φ²+1)] 123.611 PDG 2024 m_H 125.200 -1.269% 14.44σ CONJECTURAL Tier 3 C3-τ m_τ m_μ·R23^(19/√2), R23=π²/8 1.7780 PDG 2024 m_τ 1.7769 +0.058% 11.43σ PROPOSED Tier 3 C3-e m_e m_μ / R12^(2√2), R12=2π²/3 0.000513184 PDG 2024 m_e 0.000510999 +0.428% ≫10^6σ PROPOSED Tier 3 P46 m_c m_μ·(m_t/m_μ)^(1/3) 1.2491 PDG 2024 m_c 1.2730 -1.874% 8.50σ PROPOSED Tier 3 PMNS-12 θ12^PMNS 45° − θ_C(P73) + arctan(απ) 33.2631 NuFIT 6.0 NO IC19 θ12 33.6800 -1.238% 0.60σ PROPOSED Tier 3 P45 θ13^PMNS arctan(α^(1/3)/√φ) 8.6700 NuFIT 6.0 NO IC19 θ13 8.5200 +1.760% 1.36σ PROPOSED Tier 3 P44a δ_CP^PMNS −π + arctan[1/(φβ_δ)] 226.643 NuFIT 6.0 NO IC19 δCP 177.000 +28.047% 2.61σ OPEN Tier 3 P44b δ_CP^PMNS −π + arctan[1/(φβ_δ)] 226.643 NuFIT 6.0 NO IC24+SK δCP 212.000 +6.907% 0.56σ OPEN Tier 3 P73 θ_C tan θ_C − sin²θ_W = α/(2π²) 13.0502 PDG 2024 θ_C from λ 13.0035 +0.360% 1.17σ PLAUSIBLE Tier 3 δCKM γ = δ_CKM arctan(φ/β_δ) 70.1674 PDG 2024 γ 65.7000 +6.800% 1.49σ PROPOSED Tier 3 Jhyb J_CKM Hybrid value quoted in Paper 2 3.01e-05 PDG 2024 J_CKM 3.12e-05 -3.526% 0.92σ PROPOSED Tier 3 Vcb |V_cb| sin²θ_C(P73)/φ 0.031512 PDG 2024 |V_cb| 0.041830 -24.666% 14.95σ PROPOSED Tier 3 Vub |V_ub| sin³θ_C(P73)/φ 0.007116 PDG 2024 |V_ub| 0.003732 +90.666% 37.60σ PROPOSED Tier 3 Hier M_Pl/M_Z (1/27)·exp(π√27·φ²) 1.3465e+17 M_Pl(full)/PDG 2024 M_Z 1.3389e+17 +0.569% — PROPOSED Tier 3 alpha_s α_s(M_Z) seed α / sin²θ_W 0.031533 PDG 2024 world average α_s(M_Z) 0.118000 -73.277% 96.07σ OPEN Tier 3 Rb R_b φ^-2 0.381966 PDG 2024 R_b (ρ̄,η̄) 0.386559 -1.188% 0.60σ PROPOSED Tier 3 Rubcb |Vub/Vcb| φ^-5 0.090170 PDG 2024 |V_ub/V_cb| 0.089218 +1.067% 0.36σ PROPOSED Tier 3 Pent-12 θ12^PMNS π/5 − 4·arctan(αφ) 33.2941 NuFIT 6.0 NO IC19 θ12 33.6800 -1.146% 0.55σ CONJECTURAL Tier 3A Pent-23 θ23^PMNS 3π/10 − 7·arctan(αφ) 49.2646 NuFIT 6.0 NO IC19 θ23 48.5000 +1.577% 1.09σ CONJECTURAL Tier 3A Pent-23b θ23^PMNS 3π/10 − 7·arctan(αφ) 49.2646 NuFIT 6.0 NO IC24+SK θ23 43.3000 +13.775% 5.96σ CONJECTURAL Tier 3A Pent-13 θ13^PMNS π/10 − 14·arctan(αφ) 8.5293 NuFIT 6.0 NO IC19 θ13 8.5200 +0.109% 0.08σ CONJECTURAL Tier 3A Pent-δ δ_CP^PMNS −π + 25·arctan(αφ) 196.912 NuFIT 6.0 NO IC19 δCP 177.000 — 1.05σ CONJECTURAL Tier 3A Pent-δb δ_CP^PMNS −π + 25·arctan(αφ) 196.912 NuFIT 6.0 NO IC24+SK δCP 212.000 — 0.58σ CONJECTURAL Tier 3A λdead Wolfenstein λ φ^-3 0.236068 PDG 2024 CKM λ 0.225010 +4.914% 16.26σ DEAD Tier 4 Adead Wolfenstein A φ^-1/2 0.786151 PDG 2024 CKM A 0.826000 -4.824% 2.66σ DEAD Tier 4 MF23 θ23^PMNS Bare M_Freud diagonalization 43.0500 NuFIT 6.0 NO IC19 θ23 48.5000 -11.237% 6.06σ DEAD / STRUCTURAL Tier 4 MF12 θ12^PMNS Bare M_Freud diagonalization 15.1100 NuFIT 6.0 NO IC19 θ12 33.6800 -55.137% 26.53σ DEAD / STRUCTURAL Tier 4 MF13 θ13^PMNS Bare M_Freud diagonalization 14.6100 NuFIT 6.0 NO IC19 θ13 8.5200 +71.479% 55.36σ DEAD / STRUCTURAL Tier 4 ``` ### What survives best on current data The numerically strongest non-closed rows are now concentrated in the flavor sector, especially after the Session 9 update: - `Pent-13`, - `Pent-12`, - `Pent-δb`, - `PMNS-12`, - `R_b`, - `J_CKM` hybrid, - `Pent-δ`, - `Pent-23`, - `P73`. ### What now looks weakest The most severely stressed rows remain: - G2 `M_Z`, - the Peirce-derived `M_W` cascade, - both Higgs ansätze, - the `α_s` seed, - the absolute `|V_cb|` and `|V_ub|` formulas, - and the bare `M_Freud` PMNS closure. --- ## §8. Sensitivity Analysis The rows most likely to flip interpretation under modest benchmark updates are listed below. ```text Relation Benchmark shift Old result New result Impact P70 sin²θ_W PDG 2024 s_Z²: 0.23122±0.00008 → 0.23129±0.00004 0.088%, 2.53σ 0.057%, 3.31σ Percent gap improves slightly; σ tension worsens because the global-fit error is tighter. Shilov θ23 NuFIT 6.0 NO IC19 → NO IC24+SK 1.48%, 1.02σ 13.66%, 5.92σ The legacy atmospheric-angle row flips from acceptable to strongly disfavored. Pentagonal θ23 NuFIT 6.0 NO IC19 → NO IC24+SK 1.58%, 1.09σ 13.78%, 5.97σ (proxy) The strongest new PMNS angle row is still highly atmospheric-data sensitive. P44 legacy δ_CP NuFIT 6.0 NO IC19 → NO IC24+SK 2.61σ 0.56σ Legacy phase row remains benchmark-fragile. Pentagonal δ_CP NuFIT 6.0 NO IC19 → NO IC24+SK 1.05σ 0.58σ Much stabler numerically than the legacy row, but still benchmark-dependent. P73 Cabibbo Internal 13.049° → PDG 2024 λ-derived 13.0035° 0.03σ (internal) 1.17σ Still one of the best surviving non-closed relations, but not an exact hit on current data. δ_CKM Older γ benchmark → PDG 2024 γ = 65.7±3.0° 0.40σ 1.49σ Still viable, but the sub-σ narrative is gone. M_W route Older world average → PDG 2024 80.3692±0.0133 GeV 5.39σ 12.87σ Percent gap is stable; σ tension rises as the benchmark tightens. ``` ### Sensitivity summary The most fragile rows are now: 1. **Legacy and pentagonal PMNS θ23 rows**, because the IC19 vs IC24+SK split is large. 2. **Legacy and pentagonal PMNS δ_CP rows**, for the same reason. 3. **Cabibbo-angle claims**, because the coalition’s older internal angle benchmark is not the same as the current PDG 2024 λ-derived value. 4. **δ_CKM**, because the modern PDG γ average is lower than the older coalition benchmark. 5. **Precision boson masses**, where percent gaps stay small but σ-gaps explode as the benchmark uncertainties shrink. The most stable survivors are: - the hierarchy formula (percent-only comparison), - `R_b`, - `|V_ub/V_cb|`, - and the dead verdicts for exact `λ`, exact `A`, and bare `M_Freud` closure. --- ## §9. The DUNE Window A correction from v0.1 remains important: the phrase “68°–72° framework prediction” refers to the **CKM γ** prediction, which is a **B-physics observable**, not a DUNE observable. DUNE probes the **leptonic** CP phase. Paper 4 v0.2 therefore carries **two** PMNS-channel phase targets: - **Legacy P44:** `δ_CP ≈ −133.4°` (equivalently `226.6°`) - **Pentagonal moiré (Session 9):** `δ_CP ≈ −163.1°` (equivalently `196.9°`) These two phase targets are separated by about **29.7°**, which is large enough that DUNE Phase II can plausibly discriminate between them if its published `δ_CP` precision goals are achieved. Against current NuFIT 6.0: - the **legacy P44** row is benchmark-fragile, - the **pentagonal** row is numerically much stronger, - but the pentagonal row remains `CONJECTURAL` rather than predictive. So the DUNE-facing statement in v0.2 is: - if the legacy P44 route is the correct PMNS phase mechanism, DUNE should begin to push it hard; - if the pentagonal moiré route is the correct PMNS phase mechanism, DUNE should see a value much closer to `−163°` than to `−133°`. DUNE therefore does not test the CKM 68°–72° window. It tests whether the leptonic phase lands near the legacy Jordan-torsion row, near the new pentagonal row, or somewhere else entirely. --- ## §10. Conclusions Paper 4 v0.2 changes the balance of the precision audit in a meaningful way. Version 0.1 already showed that the framework had a cluster of interesting percent-level relations, but that many of them weakened sharply when modern PDG 2024 and NuFIT 6.0 uncertainties were applied. Version 0.2 keeps that verdict for the electroweak and heavy-quark sectors. It does **not** soften the pressure on `P70`, `G2`, the `M_W` cascade, or the heavy-quark block. What changes is the flavor picture. The Session 9 pentagonal moiré ansatz adds a new PMNS package that is numerically stronger than any previous non-closed PMNS block in the framework. Against NuFIT 6.0 IC19, it gives: - `θ12` at **0.55σ**, - `θ23` at **1.09σ**, - `θ13` at **0.08σ**, - and `δ_CP` at about **1.05σ**. That is a real upgrade over the legacy PMNS rows. But it is **not** a promotion in status. The integers `(4, 7, 14, 25)` are still fit-level ingredients, not derived data. So the block is strong numerically and still conjectural methodologically. The cleanest takeaways are therefore these: 1. **Percent-level success still survives in many places, but σ-level precision success remains selective.** 2. **The electroweak and boson-mass rows remain empirically strained even when algebraically closed.** 3. **The Cabibbo sector is still one of the strongest structurally motivated pieces of the framework.** 4. **The PMNS sector is now bifurcated:** the legacy formulas are benchmark-fragile, while the Session 9 pentagonal block is numerically strong but derivationally incomplete. 5. **Paper 3’s dead rows stay dead.** No Session 9 update revives the exact Wolfenstein `λ`, exact `A`, or polynomial CKM-generator routes. So Paper 4 v0.2 sharpens the program rather than rescuing it. It now contains: - a few closed rows under real precision stress, - a handful of respectable conditional rows, - one notably stronger conjectural PMNS package, - and a clean set of dead relations that should stay retired. If the pentagonal integers are ever derived, the flavor balance of the entire framework changes. If they are not, then the numerical success should be read as a highly organized fit, not yet as a prediction. --- ## Self-Score (/140) Using the coalition rubric: - **math_rigor:** 8/10 ×3 = 24 - **honest_labeling:** 10/10 ×3 = 30 - **numerical_accuracy:** 9/10 ×2 = 18 - **physics_bridge:** 5/10 ×2 = 10 - **self_consistency:** 9/10 ×2 = 18 - **completeness:** 9/10 ×1 = 9 - **falsifiability:** 9/10 ×1 = 9 **Total: 118/140** Compared with v0.1, the gain comes almost entirely from improved completeness and better benchmark handling of the PMNS sector. --- ## Sensitivity Matrix (compact verdict form) - **Very high sensitivity:** legacy and pentagonal `θ23` rows; legacy and pentagonal `δ_CP` rows - **High sensitivity:** `P71`/`P73` Cabibbo against current CKM-fit `λ`, `δ_CKM` against updated `γ` - **Moderate sensitivity:** `P70` benchmark choice, the `M_W` route - **Low sensitivity:** hierarchy formula, `R_b`, `|V_ub/V_cb|`, dead verdicts for `λ`, `A`, and bare `M_Freud` --- ## New Library Entries (L331+) - **L331** — Pentagonal moiré PMNS block against NuFIT 6.0 IC19 - **L332** — Pentagonal atmospheric bifurcation: IC19 vs IC24+SK - **L333** — Two-target DUNE phase ledger: legacy `P44` vs pentagonal `δ_CP` - **L334** — `y_t = 1` structural-note ledger: E₇/TKK proposal vs HP12 still open - **L335** — Cubic-norm provenance note for `δ_CKM = arctan(φ/β_δ)` - **L336** — Precision rank-ordering after Session 9 --- ## New Proof / Audit Prompts - **PP4-1:** Derive a theory-error model for the kernel-era charged-lepton tower so that charged-lepton rows can be quoted as meaningful pulls rather than experimental-only σ values. - **PP4-2:** Produce a benchmark map between the framework’s `sin²θ_W` object and the specific PDG effective-angle definitions to remove the present `P70` scheme bifurcation. - **PP4-3:** Clarify which current experimental object is the correct Cabibbo benchmark for `P71`/`P73`: direct `|V_us|`, CKM-fit `λ`, or a dedicated angle extraction. - **PP4-4:** Explain, within the framework, why the legacy PMNS sector is so sensitive to the IC19 vs IC24+SK split. - **PP4-5:** Derive a direct `M_W` route independent of the `M_Z` cascade. - **PP4-6:** Derive the pentagonal integers `(4, 7, 14, 25)` from framework combinatorics rather than fitting them. - **PP4-7:** Formalize the non-polynomial escape route behind the pentagonal ansatz so that its relation to the Paper 3 no-go theorems is mathematically explicit. - **PP4-8:** Settle the TKK/E₇ normalization issue behind `y_t = 1`, including why the insight improves the structural story but does not yet close HP12. - **PP4-9:** Turn the cubic-norm origin of `δ_CKM` into a basis-independent CKM observable map rather than a provenance note alone. --- ## Reference Data Used Data references used in this draft: - **PDG 2024 Review of Particle Physics:** Electroweak Model and Constraints on New Physics; Gauge & Higgs summary tables; Lepton summary tables; Quark summary tables; CKM Quark-Mixing Matrix. - **NuFIT 6.0 (2024):** Table 1, using the NO IC19 benchmark set without SK atmospheric data and the NO IC24 benchmark set with SK atmospheric data. - **The DUNE Science Program (2025):** Table 2 for projected `δ_CP` precision and CP-violation sensitivity. - **Session 9 internal formula source:** Paper 2 v3.9–v3.11 update stream (pentagonal moiré ansatz, TKK `y_t` note, cubic-norm `δ_CKM` provenance). Method note: - For phase rows, circular phase distance is used. - For the `Pent-23b` row, the σ value is explicitly a **proxy** carried in the Session 9 / v0.1 comparison convention.