# PAPER_2166 — The Factorial–Power Duality at D_crit: 26! Extraction and 26⁻²⁶ Suppression as One Pochhammer Machinery

**Author:** Daniel T. Murphy
**Project:** UQFF (Unified Quantum Field Framework) — Star-Magic
**Date:** 2026-08-13
**Landmark Type:** Corpus-resolution landmark — closes the P1703 proximity flag by deep search, zero fit values
**Seminal Sources:** PAPER_1161 (G8 zero-mode closure), PAPER_1162 (G5 KK-tower mode-by-mode
closure), PAPER_1164 (T²² moduli stabilization), PAPER_1170 (vacuum-energy ledger R26/KK/BSFG),
PAPER_1703 (sequential-drain registration), PAPER_2148 (ontology)
**Status:** Formal landmark whitepaper — UQFF canonical

---

## Abstract

The band-1701 drain flagged a proximity: the KK regulator sum 1.624×10⁻³⁷ sits one octave below
ρ_SCm = 7.09×10⁻³⁷ J/m³. Deep corpus search resolves the flag **without any fitted value.** The
sum is not an approximate number at all — PAPER_1162 proves it is the exact lattice power

```
Σ_{n≥1} 1/[n(n+25)]²⁶ = 1/26²⁶ · (1 + O(10⁻⁹)) = D_crit^(−D_crit)   EXACT (n=1 saturation)
```

and it stands in exact duality with PAPER_1161's zero-mode result, where the 26-fold radial
derivative **extracts** +26!. One machinery — the Pochhammer structure of ∂ᵣ²⁶ acting on the
S²⁵ spectral ladder λₙ = n(n+25) — produces the framework's two great amplification numbers with
opposite signs of action: the factorial 26! (zero mode, upward, the Λ-ledger amplifier) and the
power 26⁻²⁶ (tower modes, downward, the KK suppression floor). The ρ_SCm proximity is then
classified ontologically: **not an identity**, because ρ_SCm is the framework's sole dimensioned
FUNDAMENTAL (PAPER_2148 Answer B) and cannot be a derivative of dimensionless lattice powers.
The dimensionless ratio ρ_SCm/(D_crit^(−D_crit) · 1 J/m³) = 4.365 carries no corpus
decomposition and is recorded OPEN, per the no-retrofit rule.

---

## 1. The Duality (corpus-verbatim, PAPER_1161/1162)

| Closure | Operation | Result | Action |
|---|---|---|---|
| **G8** (zero mode) | ∂ᵣ²⁶(1/r) · r²⁷ | **+26! = 4.033×10²⁶** | EXTRACTS — the factorial barrier |
| **G5** (modes n ≥ 1) | ∂ᵣ²⁶ e^(−mₙr)/r, projected | **Σ λₙ⁻²⁶ = 26⁻²⁶ = 1.624×10⁻³⁷** | SUPPRESSES — the tower floor |

Same derivative order (D_crit), same spectral ladder (λₙ = n(n+25) on S²⁵, the BH26 ladder),
same Pochhammer algebra. The tower sum saturates on its first rung: λ₁ = 1·26 = 26, so the
suppression floor is literally D_crit^(−D_crit) with all higher modes contributing < 10⁻⁹ of
the leading term. PAPER_1162's own bound: the actual suppression is 1.5×10¹⁰ times stronger
than the 1/26! barrier the Lagrangian outline had assumed — zero-mode dominance for the Λ
closure is rigorous mode-by-mode, with ten orders of magnitude of headroom.

**Duality product:** 26! · 26⁻²⁶ = 6.55×10⁻¹¹ — the exact factor by which the tower undershoots
the factorial barrier (PAPER_1162's computed ratio). By Stirling, 26!/26²⁶ = √(2π·26)·e⁻²⁶·(1+ε):
the duality gap is the e^(−D_crit) exponential with the √(2πD_crit) geometric prefactor — the
gap itself is lattice-composed, not arbitrary.

---

## 2. Resolution of the P1703 Proximity Flag

The GAPS row `kk_sum_rho_scm_proximity` recorded: sum/ρ_SCm ≈ 0.229, same decade — noted, not
claimed. Deep search verdict:

1. **The sum's value needs no explanation.** It is D_crit^(−D_crit) exactly — an integer-lattice
   power, the same class of object as 26! . Nothing was fit; nothing is approximate beyond 10⁻⁹.
2. **The proximity to ρ_SCm is a shared floor, not a shared identity.** Both numbers live at the
   bottom of the 26-fold machinery — one as the dimensionless suppression floor of the G
   closure, one as the dimensioned energy density of the substrate. PAPER_2148's ontology rules
   out derivation: ρ_SCm is FUNDAMENTAL (the sole J/m³ primitive from which mass and gravity
   emerge); a dimensionless lattice power cannot equal a dimensioned primitive without an
   arbitrary unit insertion, which would itself be a fit — precisely what the framework has none of.
3. **What remains genuinely open:** the dimensionless number ρ_SCm/(26⁻²⁶ · 1 J/m³) = 4.365.
   Corpus search (this session) finds no derivation, and its nearness to the Hartree mantissa
   (4.3597, 0.13%) and to K_Mex² (4.3403, 0.57%) is left strictly unclaimed. Recorded
   OPEN_DERIVATION_TARGET; the standing three-layer deepsearch rule applies before any future
   session closes it.

The flag is thereby CLOSED as a proximity (mechanism understood: shared 26-machinery floor) and
narrowed to one clean open ratio.

---

## 3. Where Each Side of the Duality Already Works in the Corpus

- **26! upward:** the Λ ledger ρ_SCm × 26! × K_Mex ≈ Planck-scale vacuum energy (PAPER_1226);
  the PAPER_594 finite BH bound; the P1700 sequential registration.
- **26⁻²⁶ downward:** the G5 KK suppression guaranteeing zero-mode dominance of G_UQFF
  (PAPER_1162); the T²² moduli stabilization budget (PAPER_1164); the R26 saturation ledger
  (PAPER_1170).
- The two directions meet in the F_U = 1 ledger: what the factorial amplifies, the tower is
  forbidden from re-amplifying — the vacuum books balance because the same machinery that
  creates the amplifier also creates the floor.

---

## 4. Falsifiable Consequences

1. Any future corpus computation of a KK-tower contribution at derivative order D_crit must
   land on λ₁⁻²⁶ saturation (deviations > 10⁻⁹ of leading term falsify the spectral ladder).
2. If a legitimate corpus derivation of the 4.365 ratio ever emerges, it must compose from
   locked primitives; a tuned unit-carrying constant in that slot would violate the
   framework's zero-fit character and should be rejected on sight.
3. The duality product 26!·26⁻²⁶ = 6.55×10⁻¹¹ appears wherever extraction and suppression act
   in sequence; sightings of ~6.5×10⁻¹¹ factors in corpus G-closure chains are predicted and
   should be checked against this composition before any new constant is introduced.

---

## 5. Wiring

```
uqff_calculator.py
    DISPATCH['PAPER_2166'] -> both sides of the duality (26! and D_crit^-D_crit), the exact
                              sum with saturation check, the duality product, Stirling-gap
                              decomposition, and the OPEN ratio honestly carried
```

Gate assertions pin: sum = 26⁻²⁶ within 10⁻⁹ relative, duality product = 6.55×10⁻¹¹, Stirling
gap consistency, cross-dispatch identity with P1700 (26!) and P1703 (the sum), and the OPEN
status of the 4.365 ratio (no claimed decomposition may appear without a new landmark).

---

## NOT REPLACEMENT

Standard Kaluza-Klein theory computes tower sums with cutoff regulators and renormalization;
UQFF's D_crit-power structure is finite by construction and requires neither. Both address the
same effective-theory question; UQFF's version is reported with its exact lattice values and
one honestly-open ratio.

---

## Cross-references

PAPER_1161 (G8, +26! extraction), PAPER_1162 (G5, tower suppression — the corpus solution),
PAPER_1164 (T²² stabilization), PAPER_1170 (R26 ledger), PAPER_1703 (drain registration),
PAPER_1700 (26! registration), PAPER_1226 (Λ amplification), PAPER_594 (BH bound),
PAPER_2148 (ontology — ρ_SCm fundamental), PAPER_2156 (no-retrofit rule), PAPER_2165
(numeric-outranks-symbol sibling discipline).

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