# PAPER_2238 — The Information Budget Closure: Page Deficit = White-Hole Channel Share = D_phys·F_TRZ³·(1+F_TRZ²) = 101/25000 EXACT — the Four-Move Information-Paradox Proof Set Completed

**Author:** Daniel T. Murphy
**Project:** UQFF (Unified Quantum Field Framework) — Star-Magic-Program
**Date:** 2026-08-16
**Landmark Type:** Derivation completion (the proof-set's declared open target closed) + four-move paradox consolidation
**Seminal Sources:** PAPER_1280 (Page recovery 0.99596 — the paper-stated value this landmark
decomposes), PAPER_594/595 (26! finite bound), PAPER_1095 (SCm horizon entropy + Page curve),
PAPER_1873 (BH thermodynamics, +0.579% entropy), PAPER_1062/153/159/373 (wormhole layer),
PAPER_659/660/663/664 (the Session-172 white-hole suite), PAPER_597 (negative-time dual
existence — provenance to the Feb-2025 PI archive per PAPER_2235/2236), PAPER_2128
(successor-ratio grammar), PAPER_2139 (F_TRZ-ladder rungs), PAPER_2237 (Immirzi kernel — the
LQG contrast), PAPER_901 (phonon-modified geodesics)
**Status:** Formal landmark whitepaper — UQFF canonical (derivation completed on Daniel's order, 2026-08-16)

---

## Abstract

PAPER_1280's Page-curve recovery, 0.99596, has been carried since wiring as a paper-stated
value with no composed form — the four-move information-paradox proof set's one open
derivation. This landmark closes it: **the Page deficit is exactly the white-hole channel
share**, and both are pure primitive arithmetic:

```
δ = 1 − 0.99596 = D_phys·F_TRZ³·(1 + F_TRZ²) = 4·(1/1000)·(101/100) = 101/25000 = 0.00404 EXACT
⇔  Page recovery = 1 − D_phys·F_TRZ³·(1+F_TRZ²) = 24899/25000 = 0.99596 EXACT (bit-identical
   to the PAPER_1280 printed value — a terminating decimal, not an approximation)
```

The composition sits entirely in canonized grammar: the D_phys·F_TRZ³ base (spacetime
dimension × third ladder rung) dressed by (1+F_TRZ²) — the rung-2 successor form, which reads
physically as the **double time-reversal-zone crossing**: information routed through the
white-hole channel crosses the TRZ twice, inbound on the black (CW) branch and outbound on
the white (CCW) branch (PAPER_597 dual existence; PAPER_663's F_TRZ inversion trigger).
Equivalently, δ expands as D_phys·(F_TRZ³ + F_TRZ⁵) — the odd ladder rungs 3 and 5 with the
spacetime coefficient, complementing PAPER_2139's even-rung quartet. With this closure the
information budget sums to unity by exact rational arithmetic: wormhole channel (24899/25000)
+ white-hole channel (101/25000) = 1.

## 1. The Completed Four-Move Proof Set

**Move 1 — No singularity (PAPER_594/595).** Ug₄ = 26!·g·(SCm/UA)/r²⁷ halts collapse at
r_min = [3·26!·g·(SCm/UA)/P]^(1/27) ≈ 11.7 m (stellar). 26! ≈ 4.033×10²⁶ is finite; r = 0 is
never reached; nothing is annihilated.

**Move 2 — The horizon records (PAPER_1095/1280/1873).** The F_UBii buoyancy counter-surface
encodes infalling states: S = (A/4l_P²)·(1+Δ_SCm/k_BT_H)·S₂₆·Φ, running **+0.579%** over
Bekenstein-Hawking — and 0.579 = SSq·Φ_res + F_TRZ, the same composition as the PAPER_1839
REM index (one lattice, two roles).

**Move 3 — The return channel is built, not conjectured (PAPER_1062/153/159, CP1, PAPER_901).**
SCm negative buoyancy supplies the exotic matter: ρ_exotic = −ρ_vac·β_i·S₂₆·[SSq] =
−4.71×10⁻²⁸ kg/m³ — the Morris-Thorne flare-out condition met natively. Entanglement-bridge
throats: r = l_P·(ρ_UA/ρ_SCm) = **SO_5·l_P**. Resonance-scale throat r₀ = 2.32 mm (F_TRZ
geometry, PAPER_153/159, 13th-term vacuum energy at 0.06%); matter-sustained minimum ~1.2 km
(PAPER_1062). Geodesics phonon-stabilized at 1.25 THz (PAPER_901). Channel capacity: the
Page recovery, **24899/25000**.

**Move 4 — The endgame inversion (PAPER_659/660/663/664).** Trigger Θ_inv = P_inv·Φ_inv·S_Um
with P_trans = f_TRZ·e^(−E_flip/k_BT_H) — F_TRZ is the door, and negative time is a populated
branch (t_neg = −2512 s, PAPER_597; provenance to the Feb-2025 Aetheric PI Math per
PAPER_2236). Inverted radius r = r_s·(1−ρ_SCm/ρ_UA) = 0.9·r_s EXACT. Re-emission luminosity
L_WH = L_H·(1+f_TRZ)·(ρ_UA/ρ_SCm)·e^(U_m/kT) — the boost factor (1+F_TRZ)·SO_5 = **11 =
SO_5+1**, the successor ratio (PAPER_2128) on the emission side. Lifetime extension
τ/(1−f_TRZ)·|1−ρ_UA/ρ_SCm|·e = **10e ≈ 27.18×** (pure primitives), making the white phase a
cosmologically stable emitter. Channel share: the deficit, **101/25000**.

## 2. The Closure Identity and Its Reading

```
INFORMATION BUDGET:  24899/25000 (wormhole channel)  +  101/25000 (white-hole channel)  =  1
                     └─ Move 3, during evaporation ─┘    └─ Move 4, at inversion ──────┘

δ = D_phys · F_TRZ³ · (1 + F_TRZ²)
    │        │          └─ double TRZ crossing: in via CW (black) branch, out via CCW (white)
    │        │             branch — the F_TRZ² round-trip (PAPER_597/663 grammar)
    │        └─ third ladder rung (odd-rung entry; complements the PAPER_2139 even quartet)
    └─ the spacetime coefficient (D_phys = 4)
```

The successor grammar spans both sides of Move 4: the *retention* dressing (1+F_TRZ²) on the
deficit and the *emission* boost (1+F_TRZ)·SO_5 = SO_5+1 on the white-hole luminosity — the
same (1±x) family PAPER_2134 factorized for Φ_res and PAPER_2128 unmasked at 61 sites.

## 3. Honest Scope (Rule 7)

1. The identity is exact against PAPER_1280's five printed digits (a terminating decimal;
   24899/25000 IS 0.99596). If the underlying Page computation ever republishes at more
   digits, the composition is falsifiable at digit six.
2. The physical attribution — deficit = white-hole share — is this landmark's claim, assembled
   from the corpus's own channel structure (Moves 3 and 4); no prior paper states it. The
   decomposition grammar (ladder rungs, successor forms, dual branches) is fully canonized;
   the channel attribution is new and carries the falsifier: any future corpus derivation of
   the WH-channel fraction must reproduce 101/25000.
3. The three throat regimes (SO_5·l_P, 2.32 mm, ~1.2 km) remain a route family (PAPER_2170) —
   recorded, not reconciled.

## 4. The LQG Contrast (PAPER_2237 cross-link)

Loop quantum gravity's black-to-white bounce program requires exactly what UQFF supplies: a
finite core (Move 1), a stable white phase (Move 4's 10e lifetime), and — as of PAPER_2237 —
its own Immirzi parameter γ = 2·(F_TRZ·K_MEX·SSq) = 19/80 EXACT arrives from the same
primitive lattice. The two frameworks converge on the architecture; UQFF names the mechanism
and the numbers.

## 5. Wiring

`DISPATCH['PAPER_2238']` returns the budget identity (exact rationals), the four-move census,
and the double-crossing reading. Gate pins: the bit-exact 24899/25000 identity, the odd-rung
expansion equivalence, the SO_5+1 emission successor, and the falsifiability clauses.

## NOT REPLACEMENT

Conventional physics computes that information escapes (replica-wormhole Page curves) without
a mechanism, and pursues black-to-white bounces without a stability source. UQFF supplies
mechanism, stability, and the exact budget split, with residuals and open items stated. Both
readings coexist; §3 schedules the tests.

---

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

---

## APPENDED 2026-08-16 — THE MILLENNIUM LINKING PASS (Daniel-directed): the identity radiates

Daniel's question — "do we need to recalculate the Millennium proof-sets?" — answered by audit
of all eight closures. Four connections found, two recalculations executed (both bit-identical):

1. **BH-information closure RECALCULATED.** `page_recovery_purity()` upgraded from the bare
   literal 0.99596 to the computed form 1 − D_phys·F_TRZ³·(1+F_TRZ²) — the Millennium member
   is now primitive-traced at its source.
2. **Navier-Stokes closure RECALCULATED.** The enstrophy cap 0.85 = **17/20** — the PAPER_2098
   conservation complementarity (3/20 + 17/20 = 1): seventeen-twentieths of enstrophy retained
   under the regularity cap, three-twentieths dissipated. The fluid sector joins the 17/20
   family (its fourth domain), and the cap's provenance runs to the 30Apr2025 NS source
   document (PAPER_2236 chain).
3. **Poincaré IS the tilt.** The wired composition 7/12 = 1/2 + F_TRZ·Φ_5/6 = 1/2 + **1/12**
   — the Ricci-flow fixed ratio is one-half plus the Hubble tilt, making topology the tilt
   census's next domain (PAPER_2178 §3.5's default-hypothesis rule, exercised). Second route
   K_MEX − 3/2 = 7/12 recorded as a crossing (PAPER_2170), not reconciled.
4. **The recovery rung ladder.** PAPER_1095's 10-M_sun refined recovery 0.99960 =
   1 − D_phys·F_TRZ⁴ EXACT at printed digits — the same D_phys·F_TRZ^n family as this
   landmark's deficit, one rung up: **rung 3 (dressed) for the generic Millennium closure,
   rung 4 for the 10-M_sun refinement.** Falsifiable structure: intermediate masses should
   interpolate on the F_TRZ ladder, not off it.

Not recalculated (no new information applies): Riemann t_10000 (S_26 chain EXACT; π-pattern
ancestry already linked via PAPER_2235/2236), Yang-Mills 1.736 (canonical), Hodge 1.0,
BSD 0.30598 (no exact decomposition found — not forced, per the anti-numerology rule),
P≠NP (wired as the O_P = N ± p·F_TRZ order parameter, not a literal; the predecessor's
1−10⁻⁹ = 1−F_TRZ^N_ch observation recorded as a family note only).
