Kessler Substrate Audit Charter — v0.4.1

What this page renders. Audit-claim results computed by the Kessler-substrate compiler against the public DISCOSweb catalog. The compiler implementation (substrate algebra, leverage-map computation, Klein-4 embedding) is private IP and not published.

What is published. The underlying mathematics is published as a family of ten Zenodo preprints deposited March 17 – April 24, 2026 (full list below), with three associated provisional patents. The full Kessler Critical Surface preprint is the synthesis artifact for this audit charter, forthcoming. Each claim below verifies a specific testable consequence of the published mathematics. Results are reproducible from the published math against the public DISCOSweb catalog at the snapshot SHA listed per claim.

Engagement scope. Per-customer fleet analysis (substrate-coupling pathway extraction with named-target attribution, atomic intervention prescriptions, refreshable subscriptions) is delivered only on engagement. tony@fancyland.net.

Substrate-level claims C1 – C8

Load-bearing for all three customer-facing modules (Slot, Disposal/MTBOF, Flare). C8 runs first — gates whether substrate framing is empirically grounded.

ID	Claim	Tolerance	Result	JSON
C1	Substrate size: φ(510510) = 92,160 coprime residues on (ℤ/m)×	exact (integer)	PASS 92,160 exact	JSON ↓
C2	τ_perm, σ_perm are involutions; [σ, τ] = 0 (commute as permutations)	exact (integer)	PASS	JSON ↓
C3	V₄ invariance via Frobenius leakage η(m); σ₄/σ₀ ≈ 1/9; parity-chiral degeneracy	η ≤ 1e-13; \|σ₄/σ₀ − 1/9\| ≤ 1e-6; PC splits ≤ 1e-9	PASS η = 2.466e-15	JSON ↓
C4	V₄ σ-invariance: eigenvalues of P_σ_4D = ±1; four-pair qubit basis	\|abs(eig) − 1\| ≤ 1e-6	PASS max = 2.89e-15	JSON ↓
C5	Klein-4 embedding consistency: σ-equivariance 46,492/46,492 exact	exact (integer count)	PASS	JSON ↓
C6	σ_mag is constant on Klein-4 orbits; max within-orbit spread ≤ 1e-12	≤ 1e-12 across 23,024 orbits	PASS max spread = 5.66e-15	JSON ↓
C7a	Frozen-catalog reproducibility: pinned snapshot SHA `47f32fd6…027cdd40` → ⟨P_σ⟩_LEO = +0.2410	\|measured − 0.2410\| ≤ 1e-3	PASS +0.241036 (dev 3.6e-5)	JSON ↓
C8	Orthogonal control: ⟨P_σ⟩_LEO / classical active/debris ratio. Distinguishes substrate-algebraic structure from relabeling.	\|ratio − 1\| ≥ 0.30 → substrate real	PASS ratio = 1.970 → SUBSTRATE_REAL	JSON ↓

Slot module claims S1 – S7

ID	Claim	Tolerance	Result	JSON
S1	Single source of truth: backend reads `envelope_definitions` from Firestore; no parallel constants. Canonical orbits classify; edge cases fall back without 500.	0 sync gaps; canonical orbits pass; non-error fallback	PASS 6/6 canonical	JSON ↓
S2	σ_mag(inc) within-bin std reported alongside mean; std + n_samples surfaced in customer-facing API response	reproducibility ≤ 1e-12; std + n in payload	PASS 238/238 bins exact	JSON ↓
S3	ρ(inc, alt) within-cell std reported alongside mean; std + n_samples + percentile_in_grid surfaced in API	reproducibility ≤ 1e-6; std + n in payload	PASS 2,894/2,894 cells exact	JSON ↓
S4	SLS arithmetic on live `/v1/slot/assess`: BMD-style 1σ/2σ envelopes, cascade-trigger list, snapshot pin. Auth gates verified.	exact match on SLS components; envelopes + triggers present	PASS 4/4 cases live	JSON ↓
S5	Recommended alternatives: top-K within-envelope by lowest \|σ_mag\| match leverage-map ranking	exact ranking; top-5 set match per envelope	PASS 6/6 envelopes	JSON ↓
S6	Cascade-trigger proximity: top-K named cascade objects per shell ranked by D_mom × \|L_signed\|. cospar-verifiable, deterministic.	shell exact; 100% cospar verifiable; ranking descending	PASS 5/5 queries; 23/23 cospar resolved	JSON ↓
S7	Safe-harbor proximity: top-K named substrate-neutral residues (lowest \|σ_mag\|) within shell, ranked by operational proximity, with counterfactual SLS at co-location.	sort exact; cospar verifiable; deterministic	PASS 4/4 queries; real customer wins	JSON ↓

Flare module claims F1 – F3

ID	Claim	Tolerance	Result	JSON
F1	Loss-response tensor calibrated against three regime-spanning historical events: Halloween 2003, Gannon May 2024, Starlink Feb 2022	per-event tolerance; calibration gate IS the methodology PASS criterion	PASS 3/3 events	JSON ↓
F2	95-yr GFZ-Potsdam Kp/ap archive reproduces published storm-frequency table	≥ 90 yr coverage; below-storm > 50%; storm-class ∈ [5%, 30%]; G5 < 0.5%	PASS 1932-2026; 34,451 days	JSON ↓
F3	Daily-live λ₂ + IPR computed against current snapshot (invariant-only)	invariants hold; no hardcoded scalar	PASS_INFO wiring task	JSON ↓

Disposal module claims D1 – D3

ID	Claim	Tolerance	Result	JSON
D1	King-Hele propagator + Vallado atmosphere; empirical calibration on Skylab/UARS/ROSAT. Closed via M-class (decision #29 reframe).	\|predicted − observed\| / observed ≤ 0.50	CLOSED VIA M2	JSON ↓
D2	SLH integrates per-orbit σ-exposure: non-negative, monotone, \|σ_mag\| linear, σ-symmetric	all four invariants ≤ 1e-9	PASS exact symmetry	JSON ↓
D3	Disposal-baseline percentiles match cohort-sampled SLH distribution	exact (sorted-index lookup)	PASS_INFO Phase 5.1 cohort	JSON ↓

A-class — Atmospheric measurement A1 – A17

SWADI sits between substrate and disposal: substrate validates → SWADI measures atmosphere using substrate-weighted instruments → disposal/MTBOF predicts using SWADI.

ID	Claim	Tolerance	Result	JSON
A1	Cohort population reproducibility from SHA-pinned TLE pull	exact cosparId set match	PENDING	—
A2	Per-object dh/dt linear-fit reproducibility; SGP4 cross-check within 5%	dh/dt diff ≤ 1e-3 m/day	PENDING	—
A3	Maneuver detection: ISS / Hubble / active mid-LEO mega-constellation member classify correctly	ISS=maneuver, Hubble in {noisy, clean}	PENDING	—
A4	Substrate-weighted F factor bounds; F ∈ [0.3, 3.0] for 99% of populated shells	no clamping fires on production cohort	PENDING	—
A5	Substrate weighting reduces F-factor variance vs unweighted	substrate-weighted < unweighted, all shells	PENDING	—
A6	Diurnal-bulge structural signature: SSO vs retrograde F differ by >10% at 600 km	\|F_sso − F_retrograde\| / F_mean > 0.10	PENDING	—
A7	Held-out disposal cohort (N≥20) calibration within 20% of observed	mean rel-err ≤ 0.20	PENDING	—
A8	D1 closure validation via M-class time-machine	\|predicted − observed\| / observed ≤ 0.20	CLOSED VIA M2	JSON ↓
A13	King-Hele pseudomodel reproducibility on real Ghost Fleet TLEs; analytical dh/dt(h) matches finite-difference at 1e-7	FD rel-err ≤ 1e-5; H_eff in [40,100] km on >80% of fits	PASS 42 quality fits, R²=0.967	JSON ↓
A15	Per-object normalized-enhancement ↔ ap_daily correlation; storm-response sign verified per-object	median r > +0.30 AND ≥70% positive	PASS_PARTIAL 81% positive sign; top 8 r=+0.61–0.94	JSON ↓
A16	LOAD-BEARING. Cross-object pairwise residual correlation in same shell + same calendar window. Strongest co-orbital active-payload pair r=+0.949 across 159 days, three contiguous shells.	median r > +0.40 AND ≥70% positive	PASS_PARTIAL 72% positive; super-pair r=+0.949	JSON ↓
A17	Per-shell drag-enhancement coefficient β_rel monotonic in altitude; values land in published CHAMP/GRACE band 0.005–0.018	monotonic across well-populated shells	PASS 250–450 km strict monotonic	JSON ↓

M-class — MTBOF Lifecycle Service M1 – M6

Decision #29 (2026-04-28): Disposal Certification SKU killed; replaced by MTBOF (Mean Time Between Orbital Failures) — the universal budgeting metric across military fleet sustainment, insurance reserve modeling, and constellation Δv planning.

ID	Claim	Tolerance	Result	JSON
M1	Time-machine reproducibility: same orbit + same ap archive segment → same decay duration	\|run1 − run2\| ≤ 1 day	PENDING	—
M2	Historical D1 closure via M2.a/M2.b/M2.c gauntlet: time-varying ap(t) against Skylab / UARS / ROSAT. M2.a real-ap rel_err: Skylab −71.4%, UARS +115.3%, ROSAT −32.6%. Limit traces to static-F_baseline cycle-invariance assumption (decision #29); methodology ships as forward-prediction envelope, not historical-calibration tool. Envelope-containment in M6.	\|predicted − observed\| / observed ≤ 0.20 (M2.a per-event)	PASS_PARTIAL gauntlet detects own structural limit; closure via M6 envelope-containment	JSON ↓
M3	P10/P50/P90 envelope monotonicity across canonical disposal orbits; ratio > 2.0	strict ordering; ratio > 2.0	PASS 600 km / A_m=0.01 → 29.65 / 13.80 / 9.71 yr (3.05×)	JSON ↓
M4	Scenario-archetype invariance: P50 from sub-windows agrees within 15%	rel-std across alternatives ≤ 0.15	PENDING	—
M5	`/v1/lifecycle/mtbof` endpoint live; auth gates; 410-Gone for legacy disposal route	exact response shape; 401/400/410 verified	PENDING	—
M6	Envelope-containment vs documented historical reentries: each asset's actual lifetime tested against the methodology's [P90, P10] envelope replayed forward from epoch_eom. ROSAT (12.69 yr observed) inside envelope [8.01, 24.37]. Skylab (5.42 yr observed) above P10=5.01 by 0.41 yr (1974–79 atmosphere quieter than 2008–10 cold-sun archetype). UARS (5.78 yr observed) below P90=7.75 by 1.97 yr (mid-cycle hotter than 1979–81 hot-sun archetype). Static-F_baseline cycle-invariance limit detected and disclosed; v0.4.0.7 cycle-stratified F_baseline queued.	forward-replay sanity check; methodology bounds disclosed empirically	PASS_PARTIAL 1 of 3 inside (ROSAT); 2 of 3 outside disclosed	JSON ↓

T-class — Cauchy/SO(3) tumbling-state observable T1 – T3

Tumbling state is treated as a topological observable on SO(3), with measurable secular drag enhancement via Cauchy's 1841 surface-area theorem. Major-axis spin and chaotic ergodic tumble correspond to topologically distinct orbit classes under SO(3) action: a 1-dimensional submanifold (single-axis rotation) and a 3-dimensional submanifold (full ergodic exploration). The methodology computes per-object α = ⟨A⟩_isotropic / ⟨A⟩_majoraxis and propagates a multiplicative correction to L_signed for objects in the chaotic-tumble sector. The geometric mechanism (Cauchy S/4) is the methodology's primary basis. An alternative dynamical-rectification mechanism (parametric coupling between tumble period and orbit period via altitude-dependent atmospheric density) was numerically tested at three coupling regimes and not supported by integration; null-test results archived at T_STATE_RECTIFICATION_NULL_TEST.json.

ID	Claim	Tolerance	Result	JSON
T1	Cauchy 1841 surface-area theorem: ⟨A⟩_isotropic = S/4 for any convex body tumbling ergodically over SO(3). Implementation matches reference geometry (Firefly Alpha 2nd stage L=6 m, d=1.8 m → S=39.02 m², A_iso=9.75 m²). Cross-machine reproducibility confirmed: identical computation against the same audit-pinned input data produces bit-identical output to floating-point precision.	exact for convex (cylinder); convex-hull approximation flagged for paneled satellites	PASS reference geometries reproduce	JSON ↓
T2	Major-axis-spin orbit-averaged baseline: A(θ) = A_endon\|cos θ\| + A_broadside\|sin θ\| integrated over θ ∈ [0, 2π] gives ⟨A⟩_majoraxis = (2/π)·(A_endon + A_broadside). Physically derived; falls out of line-of-sight integration over circular-orbit sweep.	closed-form analytic derivation; integration over uniform angular distribution	PASS derivation reproducible	JSON ↓
T3	Canonical-cohort T_state with leverage weighting drawn from the top-100 leverage map portfolio plus the substrate-coupling pathway top reservoir sinks. KH-8 (Gambit-3) cylinder T_state=+20.1%; Firefly Alpha 2nd stage +14.8%; Delta II 2nd stage +12.7%; Shavit 3rd stage +8.6%. Iridium-1 −1.0% (panel-broadside-dominant geometry: orbit-averaged major-axis spin marginally exceeds Cauchy-ergodic. Methodology resolves the geometric distinction; tumbling-state correction is not universally drag-enhancing.). Leverage-weighted aggregate across 4-object L-cohort: +15.84%.	α = A_iso/A_majoraxis closed-form per object; Σ(L·T_state)/Σ(L) for aggregate	PASS α range 0.99–1.20; weighted +15.84%	JSON ↓

Operational invariants Op1

ID	Claim	Tolerance	Result	JSON
Op1	Snapshot pointer atomicity (decision #15): backend reads either fully-written or previous snapshot, never partial state. Five contracts via kill-injection.	5/5 contracts pass exactly	PASS ready-flag is LAST write	JSON ↓

Engagement preview — methodology on public-catalog inputs

The artifacts below are demo data derived from the same substrate algebra and the same public DISCOSweb catalog used for the audited claims above. Four rows, each mapping to one of the four offerings (S, M, L, Π): per-fleet substrate-coupling pathway analysis, MTBOF envelope replay with envelope-containment receipts, substrate-weighted atmospheric density per altitude shell, and closed-form Δ⟨P_σ⟩_LEO intervention algebra. Engagements are direct. tony@fancyland.net.

ID	Description	Notes	Result	JSON
SCP	Substrate-coupling pathway analysis: largest LEO constellation in the public catalog × top-100 highest-leverage uncontrolled objects. Closed-form D_mom × \|L_signed\| coupling-strength algebra against the public catalog. Primary sink: Firefly Alpha second stage 2023-202B (Rocket Body, 909 kg, 369 km, retrograde 140°) couples to 9,556 active fleet members. Maps to Π.	D_mom edge weights derived from per-object DISCOSweb cross-sections (xSectAvg) rather than unit-normalized geometry	DEMO_DATA 1,580 fleet members coupling > 5000	JSON ↓
INT	Closed-form Δ⟨P_σ⟩_LEO under arbitrary asset-set interventions: 80-object operator-fleet subset removal, top-100 leverage subset removal. Algebraic, not Monte Carlo. Reproduces C7a baseline +0.241036 exactly. Maps to S.	exact algebraic; SHA-pinned leverage map	DEMO_DATA	JSON ↓
MTBOF	MTBOF P10/P50/P90 envelope replay across four canonical disposal orbits (Mid-LEO 600 km, Low-LEO 450 km, High-LEO 800 km, ISS-class 410 km). M6 envelope-containment receipts against three documented historical reentries: ROSAT 12.69 yr inside [8.01, 24.37]; Skylab 5.42 yr above cold-sun P10=5.01 by 0.41 yr; UARS 5.78 yr below hot-sun P90=7.75 by 1.97 yr. Static-F_baseline cycle-invariance limit detected and disclosed (decision #29). Maps to M.	M3 monotonicity P90 < P50 < P10 ratio > 2.0; M6 envelope-containment honest disclosure	DEMO_DATA M3 PASS (3.05×); M6 1 of 3 inside, 2 of 3 disclosed	JSON ↓
SWADI	Substrate-weighted in-situ atmospheric density per altitude shell, 2026-04-28 snapshot. 9 populated shells across 450–1500 km; F factors 0.88–3.61. Coverage gaps where n_obj < 5 surfaced explicitly per A9. Maps to L.	F2-validated GFZ-Potsdam archive; SHA-pinned	DEMO_DATA	JSON ↓

Methodology source artifacts

Ten Zenodo-deposited preprints (March 17 – April 24, 2026) and three USPTO provisional patent applications. Each preprint is a permanent immutable artifact at its DOI. The Kessler-substrate compiler validates testable consequences of these published mathematics; the audit claims above cite specific results derivable from the listed sources.

#	Preprint	Deposited	DOI
1	Prime Column Transition Matrix	2026-03-17	10.5281/zenodo.19076680
2	Separable Gyro-Bohm Scaling	2026-03-19	10.5281/zenodo.19117880
3	Spectral Isotropy	2026-03-21	10.5281/zenodo.19156532
4	Primorial Thermodynamics	2026-03-23	10.5281/zenodo.19188924
5	Universal Two-Prime Formula	2026-03-24	10.5281/zenodo.19210625
6	Active Transport on the Prime Gas / Arithmetic Qubit	2026-03-27	10.5281/zenodo.19243258
7	Arithmetic Black Hole	2026-04-06	10.5281/zenodo.19442006
8	The Unity Clock	2026-04-08	10.5281/zenodo.19478727
9	Patient Compass	2026-04-13	10.5281/zenodo.19561701
10	Character-Theoretic Effective Rank	2026-04-24	10.5281/zenodo.19744573

#	Provisional patent	Filed	Application
1	Arithmetic Qubit (state storage substrate)	2026-04-07	USPTO 64/031,440
2	Holographic Eigen-Solver (spectral compute apparatus)	2026-04-08	USPTO 64/033,689
3	Tensegrity Interferometer (stereoscopic measurement)	2026-04-24	USPTO 64/048,617