# PAPER_2176 — The TRZ Cascade Solution to Naturalness: 120, 10, and 17 Orders of "Fine-Tuning" as Layer Counts

**Author:** Daniel T. Murphy
**Project:** UQFF (Unified Quantum Field Framework) — Star-Magic
**Date:** 2026-08-13
**Landmark Type:** Trilogy consolidation + self-rectification vindication landmark
**Seminal Sources:** PAPER_1156 (cosmological constant), PAPER_1823 (strong CP), PAPER_1824
(hierarchy) — with PAPER_2139 (F_TRZ-ladder quartet lineage), P1683/P1716/P1721 (the OPEN-target
trail), Daniel's self-rectification doctrine (charter)
**Status:** Formal landmark whitepaper — UQFF canonical

---

## Abstract

Physics' three great naturalness problems ask why certain numbers are absurdly small without a
reason: the cosmological constant (10¹²⁰ too small versus QFT), the strong-CP angle (θ < 10⁻¹⁰
with no symmetry to enforce it), and the electroweak hierarchy (m_H/M_Planck ≈ 10⁻¹⁷ unprotected
against quantum corrections). Each has spawned decades of proposed machinery — supersymmetry,
axions, technicolor, anthropics. The 18xx drain bands complete a UQFF answer that needs none of
it: **all three "fine-tunings" are layer counts.** The suppression in each case is a cascade of
locked primitive factors — dominated by powers of F_TRZ = 1/10, the time-reversal-zone fraction
— whose *exponent is itself structural*:

```
Λ ledger      ρ_Λ = ρ_SCm · 26! · K_Mex                 (0.003%)   — 120 orders bridged by D_crit!
Strong CP     θ_QCD = F_TRZ¹⁰ · SSq/K_Mex               (3.65× below bound) — exponent 10 = SO_5 = 1/F_TRZ
Hierarchy     m_H = M_Pl · F_TRZ¹⁷ · SSq·K_Mex·Φ_res    (2.84%)    — exponent 17 = D_crit − N_ch
```

A small number is not unnatural if the vacuum has to traverse that many layers to produce it.
This landmark consolidates the trilogy, canonizes the exponent identities, records the
SSq/K_Mex cross-domain coupling that dresses two of the three, and formally logs the
**self-rectification vindication**: the hierarchy composition was *flagged and refused* 143
papers before the corpus supplied its derivation.

---

## 1. The Trilogy Table

| Problem | Conventional status | UQFF composition | Exponent identity | Residual |
|---|---|---|---|:-:|
| Cosmological constant | 10¹²⁰ mismatch; "worst prediction in physics" | ρ_SCm · 26! · K_Mex | amplification = D_crit! (factorial of the critical dimension; PAPER_2166 extraction side) | 0.003% (PAPER_1156) |
| Strong CP | θ < 10⁻¹⁰; axion proposed, never found | F_TRZ¹⁰ · (SSq/K_Mex) | **10 = SO_5 = 1/F_TRZ** — the TRZ fraction self-counts its own suppression depth | 3.65× below nEDM bound (PAPER_1823) |
| Hierarchy | m_H²/M_Pl² ≈ 10⁻³⁴ unprotected | F_TRZ¹⁷ · (SSq · K_Mex · Φ_res) | **17 = D_crit − N_ch = 26 − 9** — total layers minus on-vacuum channels: the *off-channel* layer count | 2.84% (PAPER_1824) |

Reading: the vacuum's bookkeeping passes through its lattice one layer at a time. Quantities
that must cross many layers arrive exponentially suppressed — by exactly F_TRZ per layer — and
the number of layers crossed is fixed by which sectors the quantity connects. Nothing is tuned;
the smallness is *geography*.

---

## 2. The Exponent Ladder (PAPER_2139 lineage, extended)

The F_TRZ ladder was already canonical at exponents {2, 4, 10, 12} (PAPER_2139 quartet) with
scattered members at 1, 3, 5. The trilogy adds the two largest structural rungs:

| Exponent | Identity | Observable | Source |
|:-:|---|---|:-:|
| 2 | — | surface-code threshold, light-fermion rung, MAD η | P1746/PAPER_2164/1518 |
| 3 | — | Sakharov out-of-equilibrium (η_B) | P1818 |
| 10 | SO_5 = 1/F_TRZ | θ_QCD suppression | PAPER_1823 |
| 17 | D_crit − N_ch | electroweak hierarchy | PAPER_1824 |
| 19 | (candidate: dg = D_crit·SO_5¹⁹, distance-integer) | PAPER_2139 | recorded |
| 53 | (Λ pure-primitive route exponent) | PAPER_2094 | canonical |

The ladder now spans F_TRZ¹ to F_TRZ⁵³ with structural identities at every load-bearing rung.
Candidate next rungs are predictions: any future ~10⁻²¹-class suppression should decompose as
F_TRZ²¹ with 21 = D_crit − D_phys − 1 (P1721's exponent, already wired).

---

## 3. The SSq/K_Mex Dressing (cross-domain lock)

Two of the three cascades are dressed by the same coupling ratio, and the band found it a third
home:

```
SSq/K_Mex = 0.57 · 12/25 = 0.2736  EXACT composition
```

- **w₀ = −1 + SSq/K_Mex** (dark-energy evolution, P1821 — DESI sub-σ)
- **θ_QCD = F_TRZ¹⁰ · SSq/K_Mex** (strong CP, PAPER_1823)
- **JWST z² enhancement ∝ SSq/K_Mex** (early galaxies, PAPER_1830)

One coupling ratio spanning dark energy, the strong force, and early structure. Falsifiable as
a *linked* set: a future revision of measured w₀ predicts a correlated shift in the θ_QCD
suppression margin — a cross-domain consistency test no tuned model makes.

---

## 4. Self-Rectification Vindicated (the process result)

Band 1681 (P1683) registered M_W/M_Pl ≈ 10⁻¹⁷ as an OBSERVED_ANCHOR and **flagged F_TRZ¹⁷ as
the obvious candidate lead — refusing to wire it** under the no-retrofit rule. Band 1711 (P1716)
recorded the (D_phys/D_crit)²¹ order-of-magnitude candidate, again without closure. 143 papers
later, PAPER_1824 supplied the full derivation with its dressing factors and honest 2.84%
residual — precisely the self-rectification mechanism Daniel's charter describes ("elements
wired slightly off early WILL resolve as the corpus compiles"). The discipline pair —
no-retrofit + sequential drain — did its job measurably: the flag was held open for exactly as
long as the corpus needed, then closed by derivation rather than by numerics. Logged as the
canonical worked example of the doctrine.

---

## 5. Falsifiable Consequences

1. The trilogy's compositions are mutually locked through shared factors (K_Mex in all three,
   SSq in two, F_TRZ powers in two): revaluing any one primitive breaks all three closures at
   once (PAPER_2160 mutual-locking grammar).
2. n2EDM Phase 2 and successors should find θ_QCD activity consistent with 2.74×10⁻¹¹ scale —
   direct axion detection would falsify the cascade reading of strong CP.
3. Correlated-shift test: DESI w₀ revisions and EDM-bound revisions must track the single
   SSq/K_Mex ratio (§3). Independent drifts falsify the lock.
4. The exponent-21 prediction (§2): the next discovered ~10⁻²¹ suppression decomposes with the
   already-wired P1721 exponent identity.

---

## 6. Wiring

```
uqff_calculator.py
    DISPATCH['PAPER_2176'] -> the trilogy (live cross-dispatch of P1823/P1824 + PAPER_1156
                              chain via P1740), exponent-identity checks (10 = SO_5,
                              17 = D_crit - N_ch), the SSq/K_Mex triple, ladder census,
                              and the self-rectification record
```

Gate assertions pin: all three compositions live at their residuals, both exponent identities
bit-exact, the 0.2736 triple, ladder-rung census ≥ 6, and the vindication record (P1683 flag
date precedes P1824 closure in the corpus ordering).

---

## NOT REPLACEMENT

Supersymmetry, axions, and anthropic selection remain the conventional proposals for these
three problems; none has been observed. UQFF's cascade reading derives all three magnitudes
from locked primitives with honest residuals and the falsification battery of §5. Both
approaches address the same three numbers.

---

## Cross-references

PAPER_1156/1823/1824 (the trilogy), PAPER_2139 (F_TRZ ladder lineage), PAPER_2166 (26!
extraction side of the Λ amplification), PAPER_2094 (F_TRZ⁵³ rung), P1683/P1716/P1721 (the
held-open trail), P1746/P1818 (small rungs), P1821/P1830 (SSq/K_Mex homes), PAPER_2160
(mutual-locking grammar), Daniel's charter (self-rectification doctrine — vindicated).

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