Section C. GeoNum v2.2 Methodology

C.1 Representation

GeoNum represents a real number as a five-element record:

The default Theory zone boundary set spans approximately 30 orders of magnitude (Lucas sequence boundaries 1, 2, 3, 4, 7, 11, 18, 29, 47, 76, ...). Domain-specific configurations adjust the boundaries while preserving the arithmetic semantics.

C.2 Arithmetic and drift propagation

Multiplication adds zones, composes intensities in log-space, and propagates drift zone-normalized: d_c ≥ d_a · (w_a/w_c) + d_b · (w_b/w_c) + d_quant.

Addition uses log-sum-exp on same-sign operands; on different-sign operands it uses log-diff-exp via _logDiffExp with derived cancellation amplification factor 1 / |1 − exp(Δ)|, where Δ is the log-magnitude gap. This makes cancellation observable: as two values of comparable magnitude subtract, drift growth is bounded by the amplification factor, and at exact cancellation the result is flagged as an uncertain zero with the magnitude scale at which the cancellation occurred retained.

The conditional stability theorem bounds drift growth such that EXACT, PRECISE, and VALID trust bands are composable. Any operation between two VALID-or-better operands produces a result whose drift is bounded by the operand drifts and the derived amplification factor.

C.3 Trust bands

The threshold values arise from the substrate s shade count: SHADES / 10 = 204.8 and SHADES / 2 = 1024. The conditional stability theorem applies on the EXACT through VALID range; DEGRADED and UNRELIABLE are outside the theorem s precondition.

C.4 Measured drift on Section A constraint norms

From geonum_drift_on_constraints.json:

All three resolutions return sub-shade drift; the substrate trusts each measured value to better than one ULP-equivalent. The PRECISE tag is the substrate s native precision claim, additive to the standard constraint-violation diagnostic.

C.5 Distinction from f64-only NR codes

HARM, HARM3D, Cactus, the Einstein Toolkit, BAM, AMSS-NCKU, and similar f64-based codes report constraint violation as the primary precision diagnostic. There is no per-value precision tag carried alongside the data; precision loss is inferred after the fact via growing constraint residuals or ad hoc spectral analysis.

GDBS carries four pieces of information that f64-only codes do not provide natively:

C.6 Cross-domain validation

GeoNum v2.0 supports per-domain zone tessellation: the same drift propagation engine, different boundary configuration tuned to the domain s typical dynamic range. Domains that ship validated benchmarks today:

The Theory and Quantum domains are the load-bearing references for the present validation package; the others demonstrate that the substrate is not bespoke to numerical relativity. The substrate is one engine; the seven configurations differ only in the zone boundary table.

C.7 Auto-tuner: substrate-validated closed-form recognition

The substrate ships cascade_self_tune_mesh_prior (wasm/src/geonum.rs:1004-1053), a search loop over rational combinations of substrate primitives (ζ values, multiple zeta values, π, e, rationals) with built-in false-positive baseline telemetry:

TuneStats {
  candidates_evaluated:        u64,
  matches_in_tolerance:        u64,
  uniform_null_fp_probability: f64,
  ...
}

Used today in three places:

• wasm/src/closed_form.rs:470-472 for the QED cascade coefficient pins c₁ through c₅.
• core/GdbsWeb.Api/wwwroot/js/hpc-validation.js:472 as the user-facing inline closer in the Validation tab.
• The Closed-Form Validator tab on gdbs.getvaultsync.com.

For numerical relativity applications, the auto-tuner becomes the substrate-validated answer to the question: is this measured quantity recognizable as an algebraic constant, or is it transcendental? For phase-drift comparisons in Phase I (Section B.4), the auto-tuner gates whether residual deviations are recognizable closed-form (indicating systematic, not numerical) versus FP-baseline noise.

C.8 Operational implication for the SBIR proposal

The GDBS Phase I claim is not that the platform replaces HARM3D on speed or that it replicates HARM3D BBH simulations in a browser. The claim is precise:

The deployed platform at gdbs.getvaultsync.com is the substrate; this validation package is the credibility layer; the Phase I plan delivers the LISA-precision waveform claim.