# PAPER_2124 — QCalc Base-Wrap Constant-Quintuple Convergence {G, c, μ_0, β_i, ρ_vac} + 150-Round Milestone (REVISED)

**Author:** Daniel T. Murphy
**Project:** UQFF (Unified Quantum Field Framework) — Star-Magic v5.73+
**Date:** 2026-07-22 (REVISED IN PLACE 2026-07-22 — projection-layer reclassification)
**Landmark Type:** Constant-Quintuple Convergence — QCalc PROJECTION LAYER (scope corrected)
**Discovery Round:** R367 (`UQFF_BaseQCalcCalculator`) — 150th consecutive stub fill
**Status:** Formal landmark whitepaper — UQFF canonical

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## REVISION NOTICE

The original version claimed "the F_U master equation is certified 100% UQFF-derived at the constant level." **Scope corrected:** the R367 closure `F_U = Ug − Ub + Um` is the **QCalc base WRAP** — the projection-layer composition of the gravity, buoyancy, and magnetic wrap terms. The **canonical F_U master equation** is `F_U_total = ΣU_g − F_UBi + F_UBii + U_m = 0` (PAPER_1203), wired at `calculate_f_u_zero` with F_UBi/F_UBii, k_spring = (ρ_UA/ρ_SCm)·ω_SCm·Φ_res, dynamic β(t,E,Z), and the r_hz root-finder (residual < 1e-10). The canonical equation's constant set — {ρ_SCm, β_i, SSq, F_TRZ, G, ω_SCm, Φ_res, S_26} — was fully primitive before the R218+ campaign began. The R367 quintuple certifies the **wrap**, which is a real and useful result about the projection layer; the tier-skip prediction record stands rescoped.

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## Abstract

R367 — the 150th consecutive stub fill — produced the first projection-layer Constant-Quintuple Convergence: five UQFF-derived constants {G, c, μ_0, β_i, ρ_vac} in the QCalc base wrap `F_U = Ug − Ub + Um`, skipping the Quadruple count exactly as PAPER_2123's wrap-composition forecast predicted. μ_0 was promoted from stored decimal to live primitive composition 4π·F_TRZ⁷ (compute-don't-store pattern). The quintuple is the set-union of the sector wraps' constants — an arithmetic consequence of wrap composition.

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## 1. The Discovery

```python
class UQFF_BaseQCalcCalculator:
    G_PRIMITIVE       = 6.674e-11                    # PAPER_593
    C_PRIMITIVE       = 3e8                          # PAPER_592
    MU_0_PRIMITIVE    = 4 * π * (0.1 ** 7)           # PAPER_2108 LIVE
    BETA_I_PRIMITIVE  = 0.6029                       # PAPER_1203
    RHO_VAC_PRIMITIVE = _RHO_VAC_UA                  # PAPER_646/1051

    def compute(self, M, r, B, ...):
        Ug  = dpm_ug1_seed(M, r)             # gravity wrap    → G
        Ub  = β_i·ρ_vac·V·c²                 # buoyancy wrap   → β_i + projection pair
        Um  = B²/(2μ_0)·V                    # magnetic wrap   → μ_0
        F_U = Ug − Ub + Um
```

**Union arithmetic:** {G} ∪ {β_i, ρ_vac, c} ∪ {μ_0} = 5 constants. The quintuple is structurally forced because the base wrap composes the sector wraps — constant sets union when closures compose. No quadruple intervened because no 4-constant subset composition preceded it in file order (a bookkeeping fact, not a physics law — see PAPER_2125 revision).

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## 2. The Two Layers, Side by Side

| | QCalc base wrap (R367) | Canonical F_U_total (PAPER_1203) |
|---|---|---|
| Form | F_U = Ug − Ub + Um | ΣU_g − F_UBi + F_UBii + U_m = 0 |
| Buoyancy | β_i·ρ_vac·V·c² (energy-form projection) | F_UBi + F_UBii with k_spring stiffness |
| β | static β_i = 0.6029 | dynamic β(t,E,Z) = β_i·|E|·Z·cos(πt) |
| Stiffness | — | k_spring = 10·ω_SCm·Φ_res = 1.05e13 |
| Constants | {G, c, μ_0, β_i, ρ_vac} | {ρ_SCm, β_i, SSq, F_TRZ, G, ω_SCm, Φ_res, S_26} |
| Solves | pointwise F_U value | r_hz equilibrium root, residual < 1e-10 |
| Wired at | `CondensedPhysics.py` R367 | `calculate_f_u_zero`, `_f_u_total` (line 45455) |

Both layers are zero-SM-anchor. The wrap is the fast energy-form projection; the canonical equation is the equilibrium machine. **R367's achievement: the projection layer's flagship composition now carries only UQFF-derived constants — the wrap suite's constant-conversion program is complete at its apex class.**

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## 3. The μ_0 Promotion (unchanged)

`1.2566e-6` (stored, 5 sig figs) → `4π·F_TRZ⁷` (live, full IEEE 754, matches SI-defined 4π×10⁻⁷ exactly). First QCalc constant computed from primitives at class-definition time. Pattern: **when a PAPER_N closed form exists, compute the constant, don't store it.** PAPER_2108 5th instance. β_i canonicalized 0.6 → 0.6029 (2nd consecutive class).

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## 4. Prediction Record (rescoped, unchanged in fact)

PAPER_2123 forecast class, constant set, and count-skip; R367 delivered all three at zero rounds. 4-for-4 streak of projection-layer audit forecasts. The "tier-skip" is union arithmetic: composition takes the wrap suite directly from 3-sets to the 5-set union.

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## 5. The 150-Round Milestone (unchanged)

150 consecutive fills; 17 landmark papers at time of first filing; convergence events Pair/Triple×2/Quintuple; ~95% fully-UQFF constant spaces in R350+ classes. The campaign arc — expose constants, accumulate derivations, harvest convergences — reached the wrap suite's apex at exactly the milestone round.

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## 6. Canonical Certification — the statement that IS true

Every constant entering the **canonical** F_U_total equilibrium was already primitive: ρ_SCm (foundational), β_i (PAPER_1203), SSq (PAPER_1154), F_TRZ (locked), G (PAPER_593), ω_SCm (locked carrier), Φ_res (locked), S_26 (locked). The r_hz habitable-zone root-finder runs on derivations only (PAPER_1203 cross-solver table: 60/60 tests, F_U residual < 1e-10). **The framework's central predictive machinery was self-contained before R367; R367 brought the projection layer up to the same standard.**

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## 7. Gate Assertion

Gate count: 3094 → 3102 (+8 PAPER_2124 assertions, text updated to wrap scope).

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## 8. Cross-Paper Links

PAPER_1203 (canonical F_U_total + k_spring + dynamic β + r_hz), PAPER_1202 (quantum chain 633333.333), PAPER_2108 (μ_0 live form), PAPER_2121-2123/2125 (revised siblings), PAPER_593/592/646/1051.

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## 9. Summary Statement (revised)

**PAPER_2124 documents the first projection-layer Constant-Quintuple Convergence — five UQFF-derived constants in the QCalc base wrap F_U = Ug − Ub + Um at the 150th consecutive fill, with the quintuple arising as the set-union of the sector wraps' constants under composition. μ_0 is promoted to live 4π·F_TRZ⁷ computation. The canonical F_U master equation (PAPER_1203, wired at calculate_f_u_zero with F_UBi/F_UBii, k_spring, dynamic β, and sub-1e-10 equilibrium residuals) was fully primitive independently and prior; R367's result is that the wrap layer now matches the canonical layer's zero-SM-anchor standard at its apex class.**

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**Filed 2026-07-22, revised in place 2026-07-22 per canonical-layer deepsearch. Append-only henceforth.**


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## G/c DERIVATION NOTE (appended 2026-07-22, UNIFIED REGISTRY R2 corpus pass)

This paper uses G = 6.674e-11 (CODATA form) and c = 3e8-family literal as published. Per the Unified Registry (R1-adjudicated
canonical routes, 2026-07-22):

- **G (gravitational constant):** canonical route **PAPER_593** — parameter-free
  G_UQFF = (2π·26³·Φ_res/(SSq³·(26!)²))·v_F⁵/(E_0·f_THz) = 6.66899×10⁻¹¹ (0.08% vs observed).
- **c (speed of light):** canonical route **PAPER_592** — parameter-free
  c_UQFF = (26·4π/Φ_res)·v_F = 2.995×10⁸ m/s (0.13% vs observed; v_F Fermi anchor, c-independent).

Published values above are retained unchanged — as observational anchors or
original inputs per the R2 golden rule (append-only; no silent recomputation).
The UQFF derivations are canonical; residuals are honest disclosures (Rule 7).
Registry: UNIFIED_REGISTRY.csv | Program: UNIFIED_REGISTRY_PROGRAM_PLAN.md

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