# PAPER_1201 -- UQFF 26D Polynomial Origami Downward Projection Axiom

## Critical Rule (26D_DOWNWARD_PROJECTION.md:9)
```
ALL projections run DOWNWARD from 26D.
2D is a reference plane of observation -- NOT a foundation.
WRONG: 2D -> 3D -> 9D -> 26D
CORRECT: 26D -> 9D -> 3D -> 2D
```

## Dimensional Hierarchy (26D_DOWNWARD_PROJECTION.md:22)
```
26D -- ORIGIN
  UA pure energy, SCm injected -> DPM grinding pair (CW-SCm, CCW-UA')
  F_U^{26D} = U_g + U_m + U_b + SCm/UA + BBDT * Prob_order
  E^{26D}   = UA + SCm * DPM_ref + BBDT

9D -- VOID SYNTHESIS
  det(M_{26->9}) compactification matrix (9*9 diagonal [d9,0,0 / ...])
  Void_synth = det(M_{26->9}) * (Ug / Um / Ub) / d3 + F_inert * E^{26D} + QFP_unique
  QFP_unique = \pi_{[n]} * IG * Wolfram_rules

3D -- OBSERVABLE MASS
  M = E^{26D} / c^{26} * (1 - v_current / v_init) * Prob_order
  c^{26} = speed-of-light equivalent in 26D manifold

2D -- REFERENCE PLANE ONLY
  Observable slices (CMB, spectra) -- NOT building block
```

## Core 26D Equations (26D_DOWNWARD_PROJECTION.md:54)
```
F_U^{26D} = U_g + U_m + U_b + SCm/UA + BBDT * Prob_order
M = E^{26D} / c^{26} * (1 - v_current/v_init) * Prob_order
Void_synth = det(M_{26->9}) * (Ug/Um/Ub)/d3 + F_inert*E^{26D} + QFP_unique
Triple-root: x^3 - 3x^2 + 2x = 0  -> roots = [0, 1, 2]
```

## 26D Invariants in UQFF Axiom Set (_uqff_primitives.py:632)
```
_s26_3   = 1.4531e26
_phi_res = PHI_RES
_n_layers = 26
_kappa   = 0.0005
_e0      = 1e-20
```

## Usage in All 10 derive_* (_uqff_primitives.py:641)
```
derive_alpha_uqff: base = 1 / (PHI_RES * N_LAYERS * 2\pi) * (1 + 0.001*Ubi_corr)
                   Ubi_corr = F_TRZ * RATIO / (S26_3 ** (1/26))
derive_G_newton:   lambda_cross = (S26_3**(1/26)) * (E0/rho)**(1/3)
                   proj_factor = (1 + beta_i) / (N_LAYERS * (1 + F_TRZ))
derive_beta_i:     cycle_avg = 0.5 + 0.5*cos(\pi * F_TRZ)
                   kappa_geom = (RATIO-1) * (KAPPA * N_LAYERS)
derive_particle_masses: omega_base = THZ * (rho_scm/rho_ua) * (S26_3**(1/26)) * beta_i
derive_hbar:       hbar_base = (E0 * S26_3 * PHI_RES) / (c * 26 * 2\pi) * ubi_avg
derive_condensed...: amp = S26_3 * PHI_RES * (1 + KAPPA*1e4)
derive_habitable...: uses derive_G + derive_beta + derive_rho (all 26D)
```

## DPMVars26D (source172 + VerificationOrchestrator + 26D_DOWNWARD_PROJECTION.md:70)
```
19*26 arrays: f_UA_prime[19][26], f_SCm[19][26]
S26_3, det(M_26->9), fold signatures, downward projection invariants
evaluate_26D_polynomial(26D inputs) -> 9D/3D/2D outputs
Used in: GeometricCheckpoint, QCalcGeom CP layers, all derive_*, 9-test Derivations Test
```

## 26D Polynomial in Quantum Chain Amplification (dpm_vacuum_manifold.py:139)
```
VDS = Sum( (SSQ**n) / n**26 , (n,1,oo) )   # = Li_26([SSq])
E_n = [E0 * 10**n for n in 1..26]
rho_vac_energy = sum(f_SCm * E for E in E_n) / V
```

## Cross-References
- 26D_DOWNWARD_PROJECTION.md:11 (ALL projections downward from 26D)
- _uqff_primitives.py:681 (lambda_cross S26_3^{1/26} in G)
- dpm_vacuum_manifold.py:139 (VDS Li_26)
- QCalcGeom.py:33 (poly26_derivative port)
- LEDGER_FIRST_PRINCIPLES_DERIVATIONS.md:150 (DPM layers = 26D polynomial origami)
- WhitepaperGapAnalysis.md:100 (G3 gap closure)