Reading orientation
Part I. The finite quantum seed calculus develops the packet language and states the precise scope of the archived formal proof.
Part II. The strengthened development extends this language through seventy-nine theorem, proposition, lemma, and corollary proofs.
Part III. Six supplements give the detailed mathematical constructions behind the formal proofs, followed by three alternative derivations.
Part IV. Forty worked visual pages connect the formulas to finite models and identify the hypotheses needed at each continuum boundary.
Reading this edition
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A seed is useful only together with the operators allowed to act on it. Paper VII follows that principle through symmetry packets, quantum circuits, moving projectors, channels, effective dynamics, and refinement. The finite calculus in Part I supplies the common language. The strengthened development in Part II gives broader bounded constructions and limit criteria. Part III retains the detailed proof completions rather than reducing them to a status summary.
The original finite item numbers keep the prefix I. The strengthened development has the prefix II, and the supplementary constructions have the prefix III. All nine mathematical source parts are retained. This edition revises their explanatory prose while preserving the mathematical expressions, hypotheses, proof steps, and statement inventory. The accompanying report records the changes. Overlapping statements and proofs are retained because their hypotheses and uses differ. Their total is not a count of independent discoveries.
The selected archived formal checkpoint covers nineteen scoped manuscript items. It records 323 explicit theorem declarations across twenty-five active Lean proof modules. These are different units. The strengthened Part II is outside that nineteen-item contract. The formal evidence comes from the archived checkpoint of 6 September 2026. The present prose revision does not report a fresh Lean compilation. Exact hypotheses, declarations, inherited compiler records, and reproduction instructions appear in the Lean companion.
Five explicitly named conjectures remain open in the finite core: anomaly-free refinement, uniform Feshbach control, Lorentzian reconstruction, the Einstein infrared limit, and massless spin-two packets. Bounded direct limits and norm limits provide useful criteria. Each physical target still needs the additional structure stated in its conjecture.
Four distinctions to keep in view
A seed's algebraic orbit is different from a unitary state-preparation orbit. Compression is different from an effective resolvent. A normalized channel is different from an unnormalized error family. A finite spectrum is different from a continuum particle interpretation. Keeping these distinctions visible makes the later results easier to use.
Read Part I first for the packet grammar and the finite statements. Turn to Part II for bounded operator transport, spectral and channel constructions, and refinement towers. Use Part III when the geometric bundle construction or a formal specialization needs more detail. The visual atlas follows the same topics through explicit examples. It illustrates the statements with finite data and does not replace their proofs.
Part I Corrected finite quantum seed calculus
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Reader roadmap
The finite core starts with seed rank, then resolves symmetry and circuit transfer into packets. Its geometric statements use the actual range bundle, including the Lie-bracket term. Its channel and Feshbach statements keep normalization and invertibility visible. The three examples below introduce those distinctions with exact finite data.
Read three exact finite models
Part II Strengthened quantum packet calculus and continuum towers
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Reader roadmap
This part broadens the packet calculus while keeping its hypotheses explicit. Read each result with its assumptions about nonzero spaces, a common ambient space, self-adjointness, finite address sets, and trace class operators. Its bounded limits do not establish the five open physical continuum targets.
Part III Mathematical proof completions and retained derivations
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Reader roadmap
The six supplements retain the substantive algebra and geometry of the formal completion. The final supplement constructs local range-bundle charts, their overlaps, and a fiber-valued connection. Three earlier derivations follow because they offer a distinct route through the same finite mathematics.
Read iii. exact packets, symmetry, and resolvents Read iii. bases, refinement, connections, and channels Read iii. orbits, schmidt rank, and character projectors Read iii. leakage, curvature, and syndrome recovery Read iii. finite error correction and recovery Read iii. range bundles, connections, and curvature Read iii. alternative derivations
Part IV Worked visual atlas
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References
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