This addendum supplements the gravity integration in UTMF by addressing reviewer recommendations for enhanced mathematical rigor, scientific validity, clarity, and reproducibility. We (i) make explicit the calibration of G via a boundary-algebra match, (ii) define γ as a functional of tuple marginals and state its regularity properties, (iii) present a Regge→Holst–Palatini convergence theorem under a contraction hypothesis, (iv) bound the coarse-graining map by a Dobrushin-type inequality, (v) derive a short-range Yukawa correction from an effective mediator coupled to stress–energy, (vi) formalize Born-rule and no-signaling statements in boundary language, and (vii) relocate dataset fits and numerical grids to a Supplementary Information (SI) file with seeds and commit hashes.

The tuple lattice T⊂Z³ has coordinates (N,w,T) with kernel inherited from the Spin(10) Cartan/center structure: a diag(1,1,6) constraint with generators Δ=(0,3,3) and h=(1,1,-1), preserving invariants χ₃=N+T mod 3, χ₂=w+T mod 2, and Y=(4N+9w+T-12)/6. Boundary states live in

with admissibility maps enforcing tuple invariants on intertwiners. We denote by ρ_t the coarse-grained tuple density and by (ρ_N,ρ_w,ρ_T) its marginals.

Notation

We use κ₀ for the BF normalization and define γ=Γ(ρ_N,ρ_w,ρ_T) from tuple marginals. The contraction control is

with j_min=1/2 and C_2,min=0 unless stated. Residual unwinding parameters (Λ_c,λ_c,N_res) generate Λ_IR=Λ_c·exp(-λ_c N_res). The Page-time fraction ξ satisfies t_Page=ξ·t_evap. Braid-ensemble weights use an inverse "temperature" β and costs (a,b,c) for (linking, writhe, twist).

We assign to each face f a spin j_f∈½N with gravity weight A_f^grav(j_f)=(2j_f+1)exp(-σj_f(j_f+1)) and tuple-conditioned boundary data that fix admissible intertwiners.

Area/volume spectra

Operators:

Exponential damping and Δ<1 imply uniform boundedness and equicontinuity, enabling controlled refinement.

Calibration of κ₀ and discrete-to-continuum bracket matching

Immirzi parameter from tuple marginals

Continuum limit theorem

Setting

Let A(l_∂) be the joint boundary amplitude obtained by summing over internal labels of the spinfoam with tuple-admissible intertwiners:

Boundary events and projectors

A macroscopic "outcome" E_i on the boundary is represented by a positive projector Π_i acting on the boundary Hilbert space H_∂, with Π_i ≥ 0 and Σ_i Π_i = I.

Remark (Decoherence and classical records)

Macroscopic records correspond to coarse projectors that commute (approximately) with the boundary algebra after a finite number of coarse-graining steps; off-diagonal terms are suppressed by Δ^n, yielding classical additivity of disjoint outcomes.

Parameterized Dobrushin bound

Under the mixing hypothesis,

All results below require only Δ<1. Grid evaluations and jackknife statistics are reported in §6 and the SI.

Loop resummation statement

The fixed point implies all-orders control: perturbative expansions in loops (e.g., bubbles/tadpoles from SO(10) propagators) are resummed by iterating R to convergence, with error ε_n ≤ δ^n ||A_0 - A*||.

Choose a boundary braid ensemble B with tuple-conditioned weights

where C(β) is a braid Casimir (e.g., a linear combination of linking, writhe, and twist). For a minimal cut surface Γ,

Evaporation, Page time, and recovery

The Hawking evaporation time for a neutral Schwarzschild black hole is t_evap=5120πG²M³/(ℏc⁴) up to O(1) factors; the Page time is an O(1) fraction t_Page=ξ·t_evap with model-dependent ξ∈[0.3,0.6]. In the braid-ensemble construction, S_EE(t) rises to a maximum near t_Page and then decreases as information is recovered in radiation, provided Δ<1 ensures fast decoherence across outcome sectors. Numerical examples (and sensitivity to ξ and ensemble parameters) are documented in the SI.

Starting from the unified UV action of the Tuple–Matrix ToE, tuple-conditioned boundaries fix admissible intertwiners and induce effective potentials under coarse-graining.

Data posture

We fit (n_s,r), fσ₈(z), and broad-band Ω_GW(f) against standard likelihoods; numerical posteriors and dataset summaries are provided in the SI. We refrain from quoting specific H₀ values in the main text; all external-parameter comparisons appear in tables with priors, likelihoods, and goodness-of-fit metrics.

Cosmological constant from residual unwinding

With Λ_IR = Λ_c · exp(-λ_c N_res), the residual winding N_res is inferred from tuple marginals and the coarse-graining regulator. We report (Λ_c,λ_c,N_res) posteriors with uncertainties in the SI. A representative fit yields log₁₀(Λ_IR/M_Pl⁴) ∈ [-121.5,-118.5] (95%), consistent with late-time acceleration; the central value depends on the tuple prior and regulator choice.

Inflation from tuple couplings

An effective inflaton potential flattened by tuple couplings, e.g.

with f set by tuple statistics, yields slow-roll parameters ε,η and predictions for (n_s,r). Mapping tuple priors to (f,λ_c) and error propagation appear in the SI.

Dark-Matter Sector: Tuple Solitons and Short-Range Gravity

Soliton production

Topological defects/solitons associated with tuple sectors form with Kibble–Zurek scaling,

Effective Newtonian potential

Exchange of a spin-0 (or scalar-like composite) mediator φ coupled to stress–energy yields a short-range correction:

Observational handles

• Laboratory: torsion-balance and atom-interferometry bounds constrain (α_y, λ).
• Astrophysical: halo substructure and core/cusp behavior bound self-interaction cross sections implied by the same sector.
• UTMF falsifier: if the predicted (α_y, λ) locus is excluded at 95% CL by composition-independent tests, the tuple-mediator module must be revised or ruled out.

Gravitational Waves: Sources and Qualitative Dependencies

Sources within UTMF

1. First-order phase transition (FOPT) in the tuple sector (bubble collisions + sound waves + turbulence):
   f_peak ~ (β/H*)(T*/M_Pl) × (redshift factor)

   Ω_GW^peak ∝ κ²(α_PT/(1+α_PT))²
2. Reheating/preheating from the UV action: parametric amplification of tensor modes with a peak set by the reheat scale and the effective equation of state.

Where UTMF is predictive

The fixed point pins (Ω_GW, f_peak), while tuple priors restrict α_PT and reheating scales. Thus the model outputs a band in (Ω_GW, f) space rather than an arbitrary curve.

Falsifiers

• If PTA/LIGO/KAGRA/LISA exclude the entire UTMF band across their windows, the corresponding tuple-transition scenario is ruled out.
• Conversely, detection of a peak consistent with the predicted (Ω_GW, f_peak) scaling, together with nulls in non-tuple bands (e.g., cosmic strings), would support the framework.

Pipeline

We implement a single UV→IR flow: UV action → fixed point (G,γ,Λ_IR) → background/perturbations (Boltzmann hierarchy) → Ω_GW(f),P(k),fσ₈ using tuple priors, with no hand-tuned cosmological inputs. Grid evaluations and jackknife statistics are reported in the SI.

We provide central values, 68/95% credible intervals (CIs) from tuple-prior posteriors, and falsification criteria in the SI. Representative example: The cosmological constant prediction log₁₀(Λ_IR/M_Pl⁴) ∈ [-121.5,-118.5] (95% CI) from Section 6 demonstrates the framework's quantitative scope.

1. Short-range Yukawa. (α_y,λ) posterior vs. current torsion balance and atom interferometer bounds; composition independence holds at leading order. Kill if the predicted band is excluded at 95%.
2. GW spectrum. Broad spectral features tied to tuple-sector transitions. Kill if PTA/LIGO++ joint bounds exclude the entire allowed range.
3. Neutrino flavor. Boundary-constraint ensemble predicts a flavor-ratio region at Earth energies; kill if measurements lie outside at 95%.
4. Proton decay. SO(10) breaking pattern implies τ_p windows; kill if new lower bounds exceed the upper window at 95%.

This section details the computational and analytical methods used, ensuring full reproducibility. All stochastic components use fixed seeds (recorded in /data/seeds.json). We archive anonymized code snapshots with commit hashes (listed in the SI) to ensure bitwise reproducibility of tables and numbers.

We have promoted key claims to statements with conditions: G as a calibration of κ₀, γ as a functional of tuple marginals, Regge→Holst–Palatini convergence under a Dobrushin-type contraction hypothesis, a derived short-range Yukawa potential from an effective mediator, and boundary-language statements for Born-rule emergence and no-signaling. Data-dependent numerics and full reproducibility metadata are placed in the SI.