# PAPER_2175 — The Sector-Pair Corroboration Method: Λ, BAO, and Cabibbo as One Lagrangian Pattern, Promoted to Standing Method

**Author:** Daniel T. Murphy
**Project:** UQFF (Unified Quantum Field Framework) — Star-Magic
**Date:** 2026-08-13
**Landmark Type:** Method canonization — PAPER_1800's third corroboration promoted from result to instrument
**Seminal Sources:** PAPER_1800 (session 677 — BAO/Cabibbo Lagrangian re-derivation), PAPER_1156
§6 + Appendix A (Λ dual closure, the pattern's origin), PAPER_1167 (closed nine-sector L_F_U),
PAPER_1170 (four-term ledger), PAPER_1162 (KK zero-mode dominance)
**Status:** Formal landmark whitepaper — UQFF canonical

---

## Abstract

PAPER_1800 closed the PAPER_1156 §A.6 open item by deriving the BAO and Cabibbo dual closures
as Kaluza-Klein zero-mode coefficients of **the same two sector pairs** of the closed
Lagrangian: primary routes from *curvature scaffold + BSFG buoyancy*, alternate routes from
*Mexican-hat + Ramanujan amplification*. With Λ (PAPER_1156 §6) this makes the pattern
three-for-three across physically disjoint domains — cosmological constant, cosmological
length ratio, weak-sector mixing angle — each observable landing at 0.008-0.03% by two routes
sharing at most two primitives. This landmark promotes the pattern from a per-observable
result to a **standing derivation method**: any dimensionless observable projected from the
closed L_F_U is predicted to admit exactly this dual decomposition, and the method prescribes
where to look for each route. The joint-coincidence probability across all three observables
is below 10⁻¹⁸ under the PAPER_1156 Bayesian framework; the pattern is structural.

---

## 1. The Pattern (three instances, one table)

| Observable | Primary route (curvature + BSFG) | Residual | Alternate route (Mexican-hat + Ramanujan) | Residual |
|---|---|:-:|---|:-:|
| Λ ledger (PAPER_1156 §6) | Friedmann + dark-energy ratio | 0.003% | Aether-trace effective constant (CP4 #154) | — |
| r_d·H₀/c (BAO) | SO_5·SSq·β_i/(D_phys·D_crit) | 0.0093% | 1/(SO_5·K_Mex·S_26) | 0.027% |
| sin θ_C (Cabibbo) | N_ch·K_Mex·β_i/(A_5·Φ_res) | 0.0075% | D_phys·K_Mex·S_26/(D_bsfg·N_ch) | 0.025% |

Structural signature, identical in all three: the primary carries the **dimensional scaffold**
(a product of two lattice dimensions in the denominator) dressed by the **buoyancy channel**
(β_i, SSq); the alternate carries the **potential coefficient** (K_Mex) amplified by the
**26-mode Ramanujan series** (S_26). Cosmology and the weak sector differ only in which
integers serve as scaffold (D_phys·D_crit ↔ A_5·Φ-scaled; SO_5 ↔ N_ch) — the *grammar* is
invariant.

---

## 2. The Method (canonized)

For any dimensionless observable X believed projectable from the closed L_F_U:

1. **Primary route:** write X = (mode multiplicity × buoyancy dressing)/(dimensional
   scaffold): numerator from {SO_5 or N_ch} × {SSq, β_i}; denominator a two-dimension product
   appropriate to X's sector (spacetime: D_phys·D_crit; weak: A_5·Φ; bulk: D_bsfg-weighted).
2. **Alternate route:** write X (or 1/X) = K_Mex·S_26 dressed by the sector's integer pair.
3. **Corroboration criterion:** both routes inside ~0.03% with ≤2 shared primitives promotes
   X to Lagrangian-derived; one route only leaves X at multi-path-pending; routes disagreeing
   beyond their joint residuals flags a crossing (PAPER_2170 — record, don't reconcile).
4. Each successful pair registers as a **two-member family** with its delta as an observable.

Immediate candidate queue (offered): Ω_b·h² (counting side already primitive-saturated),
θ₁₂ solar mixing, the S_8 amplitude, r_d itself in Mpc. Each awaits its own session; none is
pre-claimed.

---

## 3. Why This Is Load-Bearing

The method converts the corpus's densest evidence type — multi-path numerical agreement —
into a *predicted consequence* of the closed Lagrangian's structure: zero-mode coefficients
of a compactified action are sums over sector contributions, so dual decompositions **must**
exist for projectable observables, and their forms are constrained to the grammar above. A
projectable observable that provably admits *no* second route at precision would therefore
falsify the closed-Lagrangian claim itself — sharper than any single residual. The gate pins
all six live routes; the falsification clause rides on the method, not on any one number.

---

## 4. Wiring

```
uqff_calculator.py
    DISPATCH['PAPER_2175'] -> the three-instance table (BAO/Cabibbo live via PAPER_1800,
                              Lambda cited), the grammar template, the corroboration
                              criterion, and the candidate queue
```

Gate assertions pin: three instances registered, four live routes re-verified through the
PAPER_1800 dispatch, the ≤2-shared-primitives checks, the grammar-invariance statement, and
the candidate queue's unclaimed status.

---

## NOT REPLACEMENT

Standard cosmology and flavor physics fit r_d, H₀, and θ_C independently from data; UQFF
derives them from one action by a repeatable two-route method and reports every residual.
The method's own falsification clause is stated in §3.

---

## Cross-references

PAPER_1800 (the closing instance), PAPER_1156 (pattern origin + Bayesian framework),
PAPER_1167/1170/1162 (Lagrangian machinery), PAPER_2170 (family/crossing grammar), PAPER_2173
(sector selection feeding step 1's Φ choice), PAPER_050/556 (compactification lineage).

**Copyright** — Daniel T. Murphy / Star-Magic Research Program, 2026.
