Begin with the four core review papers. Topic collections below preserve each theorem manuscript, its version and its claim boundaries.
The foundation, reconstruction, collective mechanisms and prepared-controller papers form the main reading spine. Review remains scope-specific.
Full-input image and nonlinear terms enter an exact budget.16 constructor controls pass; twelve physical witness categories remain missing. Physical Promotion0.
Read theorem and proposed calibration specification →Timing and quadrature/error requirements connect digital complex samples to a conditional leakage bound. Exact synthetic RMS checks pass; outside-model evidence remains open. Physical promotion0.
Read theorem, certificate and synthetic fixture →An exact Gram bound covers the declared ±5 mHz/10 ppb uncertainty family. A near-collision counterexample exposes RMS-only limits. Physical source and monitor evidence remain open; promotion0.
Read robust theorem and exact certificate →An unexecuted reference-chain protocol, twelve blank receipts and a conditional finite-mode theorem expose the remaining measurement requirements. All physical evidence stays open; physical promotion0.
Read protocol, theorem and blank receipts →A full-state theorem couples spectral leakage and input uncertainty. The declared requirements pass conditionally; noise, gain, timing and baseline evidence remain open. Physical promotion0.
Read theorem, frontier and requirements →The unchanged kernel has exact-rational continuous-band bounds under explicit source and input-error premises. A 600 Hz unit tone is a certified failure. No empirical calibration or physical promotion.
Read theorem, requirements and exact certificate →A conditional theorem bounds quadrature error using source variation and input uncertainty. The fixed window uses measured samples only, passes2/6 synthetic cases and leaves the empirical0/891 result intact. Physical promotion0.
Read theorem, transfer response and limits →An explicit rescaling leaves apparatus data unchanged while changing native action. Independent support calibration and the readout’s primitive-period ratio are required. Exact conditional theorem and finite error interval; physical promotion 0.
Read theorem and same-carrier measurement requirements →Within a specified monomial same-carrier map, independent native energy and time anchors determine action; length and time alone do not. Eighteen typed equations and 2,952 anchor configurations pass scoped checks. QPB1 stays quarantined; EXIT-03 HOLD; physical promotion 0.
Read theorem and measurement entry requirements →Finite Tetrahedral Complexes, Skew Support Flow and Contact Readiness
Finite carriers, skew flow, unique-simplex gluing, spectra, ambient jumps and supplied readiness.
Projective Reconstruction, Complex Phase Receipts and Frozen-Tree Validation
Real and complex phase receipts, parity ambiguities, frozen-tree validation and covariance.
Tetrahedral Symmetry, Radial Spectra and Finite Collective Mechanisms
Signed symmetry, radial shells, prepared shapes, burden, retention and collective mechanisms.
Finite Prepared Quantum Controllers, Capture Windows and Correlation-Preserving Renewal
Finite apparatus, history clocks, capture windows, joint errors and correlation-preserving renewal.
Read from the action and constraints to reduced modes and nonlinear stability.
A flatness theorem and an exact phase-controlled spectrum.
Unitary transport on the 28 Hasse links preserves support and gauge covariance. Retaining the transport chain identity forces flatness on the filled tetrahedron. A one-link phase gives a conditional spectral splitting. Source-selected group, gauge-field action, continuum and physical correspondence remain open.
A constrained joint link–matter Hamiltonian with an exact neutral curvature seed.
A declared classical U(r) action preserves matrix Gauss law, total energy and both matter norms, with global temporal-gauge flow. A dynamical opposite-charge partner admits nonzero neutral initial states. Their backreaction can leave the flat-link sector. Action and species selection, physical group and correspondence remain open. Physical promotion 0.
Exact mode frequencies, persistent Jordan chains, and a stable vacuum gauge orbit.
The nonzero neutral flat family has 86 real reduced dimensions in the Abelian branch. Its imaginary spectrum retains nongauge secular growth for every positive diamond stiffness. A separate conserved-functional proof establishes nonlinear vacuum stability. Higher-rank slice modes retain their residual stabilizer constraint. Candidate action; independent review pending; physical promotion 0.
Finite matter departure from arbitrarily small admissible perturbations.
An additional conserved Gram matrix and a force-canceling sector enable a controlled averaging proof. The nonzero neutral flat family is nonlinearly orbitally unstable within the unchanged candidate action except at six positive parameter values left open. The zero-matter vacuum stability theorem is preserved. 42 exact constructor checks; independent review pending; physical promotion 0.
Exact symmetry sectors complete the nonlinear instability theorem.
An S3 vertex-stabilizer sector and a C4 cyclic sector provide exactly Gauss-admissible, fixed-norm nonlinear departure witnesses at the six formerly excluded values. Together with the preceding proof, the specified nonzero neutral flat family is unstable for every a>0 and κ≥0 at finite rank. 104 exact constructor checks and six numerical campaigns; independent review pending; unchanged candidate action; physical promotion 0.
Follow existence, admissible modes, phase branches, readout and finite-offset bounds.
The unchanged action admits a stable scalar background with negative diamond holonomy.
Links U=-ib and positive lowest-mode matter give an exactly neutral relative equilibrium. At scalar rank, a constrained energy proof establishes nonlinear orbital stability for an explicit positive stiffness interval. A separate finite-rank theorem guarantees a stable minimizing set and forces nonflat minima at sufficiently small κ/(rN). Individual-orbit stability at higher rank remains open. 40 exact checks and two numerical campaigns; independent review pending; physical promotion 0.
42 reduced mode pairs and a stable critical endpoint.
At fixed equal species norms the scalar reduction has 84 real dimensions: 14 neutral matter pairs and 28 link–charge pairs. Exact weighted stiffness gives κ*/N≈0.03760248457, improving the previous sufficient bound. The orbit is nonlinearly stable through equality and unstable above it. At equality, three zero Jordan chains coexist with stability because an exact integer interval certificate proves positive quartic restoring energy. 47 exact checks and three numerical campaigns; independent review pending; physical promotion 0.
New color directions preserve the upper endpoint and expose a zero-stiffness exception.
For r≥2 the specified one-active-color orbit is nonlinearly stable exactly for 0<κ≤rκ*(N), where κ*/N≈0.03760248457. Active-to-unused color modes have positive restoring energy and a vanishing magnetic Hessian. At κ=0 an unused color permits exact holonomy drift; above the upper boundary active scalar instability embeds. The residual U(r-1) Gauss condition is retained. 57 exact checks and two full matrix nonlinear campaigns; independent review pending; physical promotion 0.
The final residual constraint becomes an explicit matrix charge balance.
The remaining U(r-1) moment is a difference of color Gram matrices plus vacuum self-commutators. A nonzero isolated circular color mode fails its trace condition. Every trace-balanced color preparation admits a one-matrix vacuum completion. At rank two the color quotient is a rank-one matrix cone with 110 real dimensions on its open stratum; the full principal stratum has 222. General finite rank gives principal dimension 26r²+60r-2. Exact local constraint geometry is separated from quadratic mode dynamics. 70 exact constructor checks; independent review pending; physical promotion 0.
The curved branch has a complete local scalar bifurcation diagram.
For sufficiently small positive ε=κ/(rN)-κ*/N, there are 26 new labeled scalar branches: six balanced 2+2 minima, twelve saddles with one growing pair and eight with two. The balanced branches remain orbitally stable under every finite-rank embedding. Their amplitudes scale as √ε, energy gains as ε², and soft squared frequencies as ε. The original orbit's stability boundary is unchanged. Conserved energy prevents interpreting these minima as attractors. 66 exact checks and twelve full stationary-energy solves; independent review pending; physical promotion 0.
Oriented matter-dressed Wilson phases reconstruct all three critical amplitudes exactly on the nonlinear stationary sheet.
A scale-free fourth moment equals 1, 1/2 or 1/3 on the stable, index-one and index-two branch types. A deterministic phase-error margin guarantees class discrimination. Real-only and density-only channels lose the overall phase sign; six labeled real loops still give asymptotic class fingerprints. The stable soft-mode frequency ratio approaches 2.1787254. Includes a proposed measurement contract and a two-study data-compatibility screen; no experimental samples reanalyzed. 64 exact constructor checks, twelve full stationary solutions, 32 gauge and 600 bounded-noise fixtures. Independent review pending; physical promotion 0.
All 26 labeled phase branches continue throughout 0<ε≤10⁻⁵, with ε=κ/(rN)−k*. A 44-variable analytic rescaling removes the threshold singularity.
The relative radius-squared error is at most 25ε, fast-loop error at most 297ε³ᐟ², and six-real-loop classifier bias at most 41ε. Explicit phase and real-channel noise budgets guarantee class discrimination. In a=N=r=1, the soft-mode frequency ratio stays within 0.005 of its limiting value. Three global boxes, 1,536 nested interval boxes and 2,048 spectral cut tests passed. Stationary embeddings and readout bounds apply at every finite rank; full-rank finite-offset stability of the minima remains open. Independent review pending; physical promotion 0.
The six balanced phase-branch embeddings are nonlinearly orbitally stable throughout 0<ε≤10⁻⁵, with ε=κ/(rN)−k*, for every fixed finite rank and a,N>0.
A matter-aligned gauge cancels the mixed matter–color quadratic term. The bounded magnetic correction leaves a color-normal squared-frequency gap >aN/16 and an unused-vacuum gap >3aN/32. In a=N=1 these frequencies exceed the slow radial frequency by factors of 20 and 25. Twenty saddle embeddings retain their scalar instabilities. Twenty-eight exact/interval checks and four synthetic matrix fixtures passed. Oscillators precede the residual stabilizer constraint and quotient. The interval is sufficient; an explicit nonlinear perturbation radius remains open. Independent review pending; physical promotion 0.
Continue from nonlinear confinement to the input, error and source-work contracts.
For every finite rank, a,N>0 and 0<ε≤10⁻⁵, let m=min(a,Nε)/2 and J=max(a,25N). The trapping radius is R=3m/(62N), its boundary energy barrier is B=mR²/2, and the sufficient initial radius is δ=R√(m/(2J)).
Initial orbit distance ≤δ guarantees energy excess ≤B/2 and distance <R for all time. Distances use a declared Frobenius/unit-sphere product metric. A scalar Hessian floor passed on 64 interval cells: 211 exact/interval checks total, plus six synthetic matrix/sphere jet fixtures. An exact budget calculator returns safe rational initial-radius bounds. These are conservative guarantees under the frozen action; ε=0, maximal basins and physical calibration remain outside scope. Independent review pending; physical promotion 0.
For a target η within the certified nonlinear tube, the gradient budget ν*=mη/16 preserves all-time trapping under static gauge/species-phase invariant Hamiltonian errors. A compact invariant minimizing set exists within η/8 of the nominal balanced orbit.
The coefficient bound is Aη+N[(1+12η)K+14G+3D]≤ν*, with typed kinetic, normalized diamond, relative hopping and site-detuning errors. Time-dependent errors also require a cumulative gradient-variation budget. Complex loop errors are <21η; choosing η≤ε/1680 reserves half the real-channel classifier noise budget for state motion and half for the instrument. Thirty exact/symbolic checks and six synthetic gradient/readout fixtures passed. The nominal orbit need not remain invariant, and no unique shifted stationary branch is claimed. Independent review pending; physical promotion 0.
The zero-offset critical orbit now has a quantitative nonlinear stability modulus at every finite rank.
Inside an explicit sufficient radius, energy excess is at least N/1600 times the fourth power of orbit distance. The boundary barrier scales as η⁴ and the sufficient initial radius as η². Two rigorous Arb matrix bounds, the inherited directed-integer quartic replay and exact budget checks support the analytic proof. In a=N=1 the outer radius is about 2.36×10⁻¹⁴ in the declared metric: a conservative guarantee, not a basin estimate or laboratory tolerance. No model-error extension at zero offset is claimed. Independent review pending; physical promotion 0.
An exact energy reflection cancels the leading odd soft-direction losses.
A sixth-order remainder proof increases the sufficient radius by at least 50,000 for every a,N>0 at each finite rank, while preserving the quartic energy barrier and all-time trapping conclusion. At a=N=1 the inward-rounded radius is about 1.20549×10⁻⁹, with barrier 1.31988×10⁻³⁹ and sufficient initial radius 7.26603×10⁻²¹. Twenty-two exact conditions, four rational budget examples and four synthetic matrix fixtures pass. These are conservative mathematical bounds, with no physical calibration or sharp-basin claim. Independent review pending; physical promotion 0.
Generic forces and centered curvature errors have distinct sufficient budgets.
Static errors preserve finite-radius trapping when ν+ηβ≤Nη³/12800. A nearby compact minimizing invariant set lies within η/2, improving to η/4 for purely centered errors. Time dependence needs a cumulative gradient-variation bound. Typed coefficient and loop-readout requirements include preparation limits. A stiffness shift can pass the trapping budget while destabilizing the exact nominal orbit. Twenty-four exact checks, five rational budgets and four synthetic matrix fixtures pass. Independent review pending; physical promotion 0.
Twenty-seven invariant scalars give a constructive upper bound on distance to the critical orbit.
A spanning tree combines transported matter coherence, full cycle holonomies and electric energy. The score vanishes exactly on the reference orbit and has explicit local conditioning. Guaranteed interval inputs can certify entry into the frozen endpoint trapping ball. Exact zero-Gauss counterexamples show why active dressed loops alone cannot certify full preparation, even from their complete history at rank at least two. Forty-six exact checks and finite-rank floating diagnostics pass. Independent review pending; physical promotion 0. Precision budgets are sufficient mathematical bounds without laboratory calibration.
A joint geometry-and-energy gate admits preparations beyond the earlier sufficient distance-only ball.
Conserved norm errors spend a centered curvature budget. Ambient Hamiltonian confinement holds at nonzero Gauss levels, with an explicit residual envelope; unequal norms still preclude closed-carrier zero-source neutrality. A frozen residual-cone example rules out a linear correction estimate. An exact zero-Gauss witness is admitted at 2.5 times the earlier sufficient radius. Forty-nine exact checks and separately typed matrix diagnostics pass. Independent review pending; physical promotion 0. No constraint repair or physical calibration is implied.
Exact preparation-box majorants, state/tracking barrier, stronger fault gaps, series-loss plant and a nonempty clock interval. Five pages; symbolic and rational constructor checks. Catalogue-screened route; actual calibration and independent review OPEN; hardware HOLD; physical promotion 0.
Exact LC clock-product identity, absolute filter-history bound, conservative certificate obstruction, revised error allocation and identification plan. Four pages; ten exact controls and inherited-budget replay. Hardware HOLD; review pending; physical promotion 0.
Continuous tracking bound with sample age, input and command error, actuator lag, conditional preload and invariant output range. Four-page theorem; nine rational controls and Gauss-budget replay. Apparatus OPEN; independent review pending; physical promotion 0.
Finite-cover residual theorem and obstruction, active-port plant equations, controller-age budget and exact Gauss-drift fault gaps. Six pages; apparatus premises OPEN; independent review pending; physical promotion 0.
Common action and energy scales, opposite species brackets, reciprocal currents and a Gauss-source law. Eight-page manuscript with CD-1 channels, conditions, engineering gates and fault signatures. Sixty-four exact checks; independent review pending; physical promotion 0.
Norm variation, parameter drift and tangent defects share a cumulative confinement budget.
A moving-reference energy proof charges base norm gain, while pure loss still spends the amplitude allowance and can create Gauss charge. Defect work must be bounded at every time prefix; terminal cancellation is insufficient. The exact matrix source equation, raw-forcing conversion and constant-rate horizon inequality are supplied. Forty-three exact checks and separate matrix diagnostics pass. Independent review pending; physical promotion 0. This is a continuous trajectory contract, not a physical source construction, constraint repair or arbitrary-integrator guarantee.
The finite-model observation compilation links to the original packages; later estimator certificates live in the capture register.
Twenty-six accessible packages/captures cover boundary/coframe corrections, exact sector ranks, alternative readouts, finite conditioning, fitted calibration, field/Hankel compatibility, acquisition, nominal/noisy spectra and review intake. Each preserves its source scope; this compilation is not a new merged theorem or independent endorsement.
A separate 56-page theoretical synthesis by Benjamin Walker Mayes, dated 15 October 2026 UTC.
Ingested source: gluing and dynamic memory, noisy access, finite records, sampled events and receipt-error limits. Its claims retain their stated assumptions; companion replay packages were not supplied for this intake. Independent review remains pending. Physical promotion 0.
Every imported source retains its own constructor-check record. The new atlas additionally verifies three core incidence constructions with exact arithmetic; that check is limited to boundary identities, skew symmetry, spectra and simplex counts.
The earlier v1.0 package is retained as an earlier replay object, not relabeled as a full verifier for the later papers.