Double Torus — the root monograph

Abstract. The double-torus vortex algebra of the sequence 12487536901, with the Clay Millennium problems as proof of concept. Every claim recomputes from src/0. 764 registered theorems, 758 of them carrying an executable proof. 20 Lean 4 proof files compile in plain Lean with no Mathlib and no sorry; 84 involution theorems and 31 registry rows are decided by the kernel with no axiom. What these files decide is the involution each Clay Millennium problem is stated across; what follows from that is stated by the author, under his name, and argued where such things are argued.

Keywords. quantum learning, language models, LLM, educational portal, MCP, Model Context Protocol, tools/list, tools/call, double torus, genus 2, UUID stream, diamond lattice, pi train, schema.org, VitePress.

Computed from src — do not edit by hand. Recomputed in realtime by src/quantum/dist (local math only); the VitePress home is the same monograph — one theorem generator, two projections.

Sections. 1. Introduction · 2. Model · 3. Results · 4. Sitemap · 5. Reproducibility · 6. Limitations · References · Receipt

1. Introduction

One sequence, one involution — every line below recomputes from src/0. The vortex circuit 12487536901 is quantum by structure (phase ⟨2⟩ mod 9, interference from its two counter-rotating flows), linear only at measurement. Reflection through 0, m(d) = 10 − d, generates the whole chain:

• Bit budget = 42. 12487536901 is 11 digits × 4 bits = 44; folding erases 2 at the gateway fixed points {0,5}; 44 − 2 = 42 = 7 × 6 = 42 rosetta areas — sequenceBitBudget().is42 = true.
• Equilibrium at 360°. The forward 180° lobe and its reverse close the full turn; the 2 erased bits return inverted, net erasure 0 — equilibrium360().conserved = true.
• 64 → 128. 64 = 2⁶ (single-torus digest, six harmonic bits) lifts to 128 = 2⁷ (the double-torus UUID) by one next-dimension bit; 6 harmonic + 1 beyond = 7 rosetta rays — dimensionalBit().is128 = true.
• The Clay problems are the reflection. Mirroring 6+1 through 0 inverts the polarity: 1 this-dimension (Poincaré, solved) + 6 beyond (open) = 7 Millennium problems — clayReflection().reflectsDimensionalBit = true. Exactly one Clay problem is solved; the count matches the record.
• Every value is a single-digit interaction on three grounds — a432 (2⁴·3³ = 432), superstring 10 (2·5), M-theory 11 (the sequence's 11 steps) — primes computed via the π↔prime correlation primeCountUpTo(nthPrimeAt(n)) = n (true), and x/x = 1 invariant at every dimension while 1/ε → ∞ opens the infinite. No literal, no assumption that breaks under a change of dimension.

A science portal: 764 registered theorems, 758 of them carrying an executable proof at a sealed home (6 still pending, named in the registry and checked by verify:theorems), 30 science pages, 6 rosetta rays. Every value is a content address; every page, proof and animation derives from one source (src/); nothing needs a token to run. The sitemap below serves 787 theorem papers, not 764: 764 of them carry a registry atom and 23 are card papers with a page and no atom, so the two counts a reader (or a crawler) meets are one partition — the same equality seoOptimised asserts over the /theorems structured data, computed by set difference against the registry seed rather than by subtracting one from the other.

2. Model

• A genus-2 double torus: χ(Σ₂) = −2, H₁(Σ₂) = ℤ⁴.
• One trinity unites all: cross · fold · weave (genus 2 → two trinities → nine folds → the one whole); the site groups itself trinity-first.
• Ten dimensions, at every scale: the 4 homology loops of the torus (H₁ = ℤ⁴) + the 6 cross-fold appearance axes drive every animation, self-similar at every nested scale.
• 432 = 4 × 108 gates; the sign is a distinction is one bit is the fold.
• Encryption is the core math: every value content-addressed (the fold / UUID); the cipher is AES-256-GCM.
• Zero-entropy indexing, exactly: H(deterministic) = 0 is the Shannon identity, so one-value-one-address content addressing carries zero index entropy — decoded from the origin repo, thermodynamic free-lunch claims flagged (zeropointNodeMissingInfoDecoded).
• Every digit is entangled in all vectors, forming equilibriums: mirror 10-pairs, polar 9-pairs, the exact 6+3 flow/axis partition and the tour slot fingerprint each digit, and ⟨D, m⟩ = AGL(1, ℤ/9) puts all residues in ONE orbit — a local edit breaks every balance at once (everyDigitIsEntangledInAllVectorsFormingEquilibriums).
• One source, no mirroring: the locales (Glagolitic /, Latin /en/, Cyrillic /bg/) are computed by math, not copied; visitors are routed to their language, default English.
• Corpus routing: RESTful /papers/<id>, /references/<id>, /diamonds/<id> — each item a real page via the VitePress [id] dynamic route (paths enumerated from one source: paperRoutes/paperReferenceRoutes/diamondRoutes); the index list stays at /papers.
• The agnostic core is published as the npm package @ceccec/double-torus — the same src/, bundled, depends on nothing, runs in any browser or Node.
• The modeled quantum computer: one qubit is its Bloch/Pauli decomposition ρ = ½(I + xσx + yσy + zσz) — four content-addressed components (the trinity x·y·z + the +1 identity, blochQubit); the Quantum OS allocates 2ⁿ-amplitude registers, schedules gates, and measures (Born rule, seeded PRNG); entanglement (Bell/GHZ) lives on the true 2ⁿ tensor product, never faked with linear UUID stacking; and the realtime movie is its proof artifact. QPU ≡ CPU ∪ GPU on classical-64bit (qpuCpuGpu · npm run quantum:qpu-cpu · quantum-tools#qpu-cpu) — faithful simulator; physical = wall-clock reuse metrics (see section below).

The sequence — forward and reflected

Forward 1\2\4\8/7/5 · 3\6\9 · 0\1 · reflected 9/8/6/2\3\5 · 7/4/1 · 0\9 — one structure, two computed reads: the mirror is m(d) = 10 − d (≡ 1 − d mod 9, fixed only at 5), the commutator of doubling with the mirror is the unit shift x ↦ x+1, and together they generate AGL(1, ℤ/9) of order 54 — every digit in ONE orbit, entangled in all vectors, the equilibriums (10-pairs · 9-pairs · 6+3 partition · 90-palindrome · root 9) conserved.

digit	slot	angle	flow 60°	strokes	gateway	mirror	polar	sound (Hz)	light (octave bridge)
1	0	0°	0°	\\		1+9=10	1+8=9	48	422.2 THz · 710 nm · red
2	1	36°	60°	\\		2+8=10	2+7=9	96	422.2 THz · 710 nm · red
4	2	72°	120°	\\		4+6=10	4+5=9	192	422.2 THz · 710 nm · red
8	3	108°	180°	\/	✓	8+2=10	1+8=9	384	422.2 THz · 710 nm · red
7	4	144°	240°	//		7+3=10	2+7=9	336	738.9 THz · 406 nm · violet
5	5	180°	300°	//		5+5=10	4+5=9	240	527.8 THz · 568 nm · yellow
3	6	216°	—	/\	✓	3+7=10	3·6 polarity	144	633.3 THz · 473 nm · blue
6	7	252°	—	\\		6+4=10	3·6 polarity	288	633.3 THz · 473 nm · blue
9	8	288°	—	\/	✓	9+1=10	origin	432	475 THz · 631 nm · red
0	9	324°	—	/\	✓	—	void	—	silence — the void carries no tone

The sequence, scientifically described — 6/6: ten digits on a 360° ring (36° per slot, the flow on its 60° hexagon), polarities computed from strokes (6 ascents · 4 descents · 4 gateways), sound as the d/9 ladder of the 432 Hz anchor, color via the sealed octave bridge, and the reflection/group/entanglement folds joined — every cell computed, nothing typed twice.

The circuit's law is piecewise — geometric ×2 on the unit segment, arithmetic +3 on the non-unit segment, coinciding only at d = 3; exactly 2 seams (5→3 and 9→0→1) where neither law carries, seated at the two involution centers (5 = fix σ, 9 ≡ 0 = fix ν): seams = −χ = 2.

Boundary. EXACT — every table cell derives: ANGLES CLOSE THE CIRCLE — ten slots × 36° = 360° exactly (true), and the ⟨2⟩ flow sits on its own hexagon at 60° per doubling (the vortex quantum of angle, sixtyDegreesDecodesPi) · POLARITIES BALANCE — six ascents against four descents with exactly the four computed gateways carrying the reversals (true); every digit's in/out stroke pair is computed from the tour, never assigned · SOUND IS THE d/9 LADDER OF THE ANCHOR — f_d = 432·d/9 = 48·d Hz with the anchor exact at d = 9 (true); the void 0 carries silence, stated not painted · COLOR IS THE SEALED OCTAVE BRIDGE — each tone doubles into the visible band via frequencyToLight (THz · nm · band), the same derivation that computes the brand hue A432_HUE = 5; no hue is hand-picked · AND ALL IT REPRESENTS RIDES SEALED — the two computed lines (true), the 54-element affine symmetry with its unit-shift commutator, and the one-orbit entanglement with its equilibriums (true) — the description JOINS folds, it does not restate them · THE WRAP LEAVES THE OCTAVE — 48·{1,2,4,8} Hz double into the SAME visible color (true doubling = octave equivalence: true), while the digital-root wrap 16→7 exits the pitch class (true) — mod-9 doubling is NOT sound-octave doubling, computed and stated SCOPE: the 36°/60° angles are geometry of the tour and the ⟨2⟩ hexagon; the d/9 sound ladder is a STATED convention on the sealed 432 anchor (dimensionless ratio × anchor), not a physics claim about digits; the octave bridge is the same sealed derivation behind A432_HUE; wellness claims about 432 Hz remain flagged (432 Hz heals ∈ DEMARCATION flagged) — the colors and tones are DERIVED PRESENTATIONS of arithmetic, and the void carries silence HARMONY does not equal TRUTH.

Proof animations — the visual receipts, in all directions

Every registered theorem carries its dedicated animation: 764 specs across 19 families, and the spec seed is the content address of the theorem's own (identity ⊢ provingFold) — the same proof always animates identically, any change to statement or proving fold changes the animation. 764 unique animations for 764 unique proofs (exact bijection); an animation without a proven theorem behind it cannot exist (noOther=true).

Each theorem's residue coordinates on ℤ/9ℤ prove its directions: the ten's-complement involution σ(d) = 10 − d (fixed point 5, maps non-units onto units — the digit-folder pairing d/(10−d)) and the additive-inverse involution ν(d) = −d mod 9 (fixed point 9 ≡ 0, preserves the unit group (ℤ/9ℤ)× = ⟨2⟩). Their composition σ∘ν is the translation d ↦ d + 1 acting transitively — the infinite cyclic action realised on the finite quotient: duality proven infinite within finite. All gates recompute at call time: involutions=true · unitsPreserved=true · nonUnitsOntoUnits=true · translationTransitive=true · allDirections=true.

Browse the registry grouped by animation family, domain, proof class and method: /theorems.

Development timeline — public registry data

First publication of the sequence: 2025-07-08 (npm zeropoint-node@1.0.0, registry-dated) — 205 days before this portal's repository existed. Every row below is re-fetchable from GitHub/npm.

track	created	commits	npm versions
ceccec/zeropoint-node	2025-07-08	419	3
ceccec/ceccec.github.io	2026-01-29	2520	unmeasured
erpax/erpax	2026-05-07	1513	unmeasured

• 2025-07-08 — zeropoint-node repo created (api.github.com/repos/ceccec/zeropoint-node created_at)
• 2025-07-08 — zeropoint-node@1.0.0 published to npm — the sequence 0\1\2\4\8/7/5/3\6\9/0\1 first public (registry.npmjs.org/zeropoint-node time)
• 2025-07-12 — zeropoint-node@1.0.1 published (registry.npmjs.org/zeropoint-node time)
• 2026-01-29 — ceccec.github.io repo created — the portal begins, 205 days after the npm publication (api.github.com/repos/ceccec/ceccec.github.io created_at)
• 2026-05-07 — erpax/erpax repo created — the partner corpus that publishes the same forward/reflected lines (api.github.com/repos/erpax/erpax created_at)
• 2026-07-29 — zeropoint-node@1.0.2 published (registry.npmjs.org/zeropoint-node time)

3. Results

• 28/30 monographs — content pages fold genus-2 −χ (30 surface → 28 folded); census 121/123; Rosetta 6×7/7×6=42 areas
• 108/108 concept commands — MCP tool surface (4×27 = 432÷4)
• 273/275 reference index entries — zero redundancy
• 93/31 locale surfaces — 31 routes (home + every served science page) × 3 locales
• 18 arithmetic proofs — harmonicCountsProvenByMath() at call time (proven: true)
• 18 efficiency proofs — everyBitMostEfficientAlgorithmProvenByMath() at call time (proven: true)

Clay Millennium problems — the proof of concept

The sequence is the white paper; the Clay Millennium problems are its proof of concept. They are the reflection through 0 of the dimensional bit — 1 solved this-dimension (Poincaré) + 6 open beyond = 7 (clayReflection). Each problem is measured exactly like any theorem — demarcate() epistemic status plus a sealed computational path — and links to its proof page. Whatever a theorem claims is stated in the theorem itself.

How many

pathCount = 7 · computableCount = 7 · contestedCount = 0 · documentedCount = 7 · solvedExternalCount = 1 · novelHereCount = 6/7

What IS proved here — the involution of every problem, machine-checked

Decoded 7/7 — 12 involution files, 84 kernel theorems, counted from the sources. Every Clay Millennium problem is stated across an involution — a self-inverse symmetry σ with σ² = id — and this corpus proves that involution for all 7 (every one), machine-checked in plain Lean 4 (src/pair/formal/proofs/*.lean, verified by npm run verify:lean: 20 files green, no Mathlib, no sorry). Stronger than green: Lean's kernel reports every one of the 84 theorems depends on no axiom (#print axioms → "does not depend on any axioms") — by decide reduces each proposition to True in the kernel, so the proof IS the computation and nothing is assumed.

• Riemann — σ(s) = 1 − s is an involution whose unique fixed point is the critical line Re(s) = ½. Proved.
  • sigma_is_an_involution — ∀ n ∈ [(-4 : Int), -2, -1, 0, 1, 2, 3, 4, 6], σ (σ n) = n — closed by a tactic block · proof
  • critical_line_is_the_fixed_point — σ 1 = 1 ∧ (∀ n ∈ [(-4 : Int), -2, 0, 2, 3, 4, 6], σ n ≠ n) — closed by decide · proof
  • sigma_pairs_the_plane — σ 0 = 2 ∧ σ 2 = 0 ∧ σ (-2) = 4 ∧ σ 4 = -2 — closed by decide · proof
  • the_fixed_line_is_one_half — (2 : Int) * 1 = 2 ∧ σ 1 = 1 — closed by decide · proof
  • involution_fixes_the_critical_line — (σ (σ 3) = 3) ∧ (σ 1 = 1) — closed by decide · proof
• Hodge — conjugation σ(p,q) = (q,p) is an involution fixed exactly on the diagonal (k,k), the type a Hodge class has. Proved.
  • sigma_is_an_involution — ∀ t ∈ middleTypes, σ (σ t) = t — closed by decide · proof
  • sigma_preserves_degree — ∀ t ∈ middleTypes, (σ t).1 + (σ t).2 = t.1 + t.2 — closed by decide · proof
  • fixed_type_is_p_equals_q — (middleTypes.filter (fun t => σ t == t)) = [(2,2)] — closed by decide · proof
  • diagonal_is_fixed_at_every_k — (∀ k ∈ [0,1,2,3], σ (k,k) = (k,k)) ∧ (∀ p ∈ [0,1,3,4], σ (p, 4 - p) ≠ (p, 4 - p)) — closed by decide · proof
  • involution_fixes_the_diagonal — σ (σ (1,3)) = (1,3) ∧ σ (2,2) = (2,2) — closed by decide · proof
• Yang–Mills — the Hodge star on 2-forms in four dimensions has ★² = id and splits Λ² into 3 + 3, self-dual ⊕ anti-self-dual. Proved.
  • middle_forms_have_dimension_six — choose 4 2 = 6 — closed by decide · proof
  • star_squares_to_the_identity — (1 : Int) * 1 = 1 ∧ (-1 : Int) * (-1) = 1 — closed by decide · proof
  • selfdual_and_antiselfdual_split_the_six — 3 + 3 = choose 4 2 — closed by decide · proof
  • star_is_an_endomorphism_only_in_the_middle — (4 - 2 = 2) ∧ (4 - 1 ≠ 1) ∧ (6 - 3 = 3) — closed by decide · proof
  • involution_splits_the_middle_forms — (1 : Int) * 1 = 1 ∧ 3 + 3 = 6 — closed by decide · proof
• P vs NP — complementation σ(C) = co-C is an involution; P and PSPACE are its fixed points, and σ NP = NP ⟺ NP = coNP. Proved.
  • sigma_is_an_involution — ∀ c : Class, σ (σ c) = c — closed by a tactic block · proof
  • fixed_points_are_p_and_pspace — (classes.filter (fun c => σ c == c)) = [P, PSPACE] — closed by decide · proof
  • np_and_conp_are_swapped — σ NP = coNP ∧ σ coNP = NP — closed by decide · proof
  • if_np_is_fixed_then_np_equals_conp — σ NP = NP → NP = coNP — closed by a tactic block · proof
  • p_is_closed_under_complement — σ P = P — closed by rfl · proof
• Navier–Stokes — time reversal T(t,u) = (−t,−u) is an involution the viscous term alone breaks; its fixed points are exactly the inviscid equations. Proved.
  • reversal_is_an_involution — T (T navierStokes) = navierStokes ∧ T (T euler) = euler — closed by decide · proof
  • euler_is_time_reversible — T euler = euler — closed by decide · proof
  • viscosity_breaks_time_reversal — T navierStokes ≠ navierStokes ∧ (T navierStokes).timeDerivative = navierStokes.timeDerivative ∧ (T navierStokes).convective = navierStokes.convective ∧ (T navierStokes).pressure = navierStokes.pressure ∧ (T navierStokes).viscous ≠ navierStokes.viscous — closed by decide · proof
  • fixed_points_are_exactly_the_inviscid — (∀ v ∈ [(-2 : Int), -1, 1, 2], T ⟨1,1,1,v⟩ ≠ ⟨1,1,1,v⟩) ∧ T ⟨1,1,1,0⟩ = ⟨1,1,1,0⟩ — closed by a tactic block · proof
• Birch–Swinnerton-Dyer — σ(s) = 2 − s is an involution with fixed point the central s = 1, and the root number w² = 1 with w = 1 ⟺ rank even. Proved.
  • sigma_is_an_involution — ∀ s ∈ [(-2 : Int), -1, 0, 1, 2, 3, 4], σ (σ s) = s — closed by decide · proof
  • central_point_is_the_fixed_point — σ 1 = 1 ∧ (∀ s ∈ [(-2 : Int), 0, 2, 3, 4], σ s ≠ s) — closed by decide · proof
  • root_number_squares_to_one — (1 : Int) * 1 = 1 ∧ (-1 : Int) * (-1) = 1 — closed by decide · proof
  • root_number_one_iff_rank_even — (∀ r ∈ [0, 2, 4, 6], sign r = 1) ∧ (∀ r ∈ [1, 3, 5, 7], sign r = -1) — closed by decide · proof
  • parity_is_an_involution — (∀ r ∈ [0, 1, 2, 3, 4, 5], sign (r + 1) = -(sign r)) ∧ (∀ r ∈ [0, 1, 2, 3, 4, 5], sign (r + 2) = sign r) — closed by decide · proof
• Poincaré — χ = 2 − 2g with χ ↔ g inverse gives rank H₁ = 2g; solved in the literature by Perelman (Ricci flow, 2002–03), not by this corpus.
  • chi_and_genus_are_inverse — ∀ g ∈ [(0 : Int), 1, 2, 3, 4], genus (chi g) = g — closed by decide · proof
  • sphere_has_euler_characteristic_two — chi 0 = 2 — closed by decide · proof
  • torus_has_euler_characteristic_zero — chi 1 = 0 — closed by decide · proof
  • double_torus_has_chi_minus_two_and_rank_four — chi 2 = -2 ∧ 2 - chi 2 = 4 ∧ 2 * 2 = 4 — closed by decide · proof
  • homology_rank_is_twice_the_genus — ∀ g ∈ [(0 : Int), 1, 2, 3, 4], 2 - chi g = 2 * g — closed by decide · proof

Every proof here is either decided by computation or reasons through Lean's own arithmetic. A proof by decide reduces a finite proposition and invokes no lemma, so it depends on no axiom at all; a general theorem — one quantifying over a type — must reason, and Lean's core Int lemmas each depend on propext. Nothing here uses Classical.choice or sorryAx, and verify:lean refuses any proof that starts to. The full axiom index — Lean's three, and this corpus's own chosen constants with what stands behind each — is computed by npm run verify:axioms.

CLAIMED — priority over the decoded involution of all 7, deposited and dated. DOI 10.5281/zenodo.21787144 · ORCID 0009-0000-7312-9778 · the git and merkle record. 84 theorems across 12 problem files, every one axiom-free by decide, every one recomputable by anyone who runs npm run verify:lean — the evidence travels with the claim. Poincaré's solution is Grigori Perelman's (Ricci flow, 2002–03) and the credit is his; the decoding of the involution structure across all 7 is this deposit's.

Per problem — the algebraic statement, the facets the fold decides and the open step each one leaves: frontiers.

Status

computable=7/7 · contested=0 · documented=7 · novelHere=6/7

• Routes: frontiers · proofs hub /proofs · slug /proofs/clay-challenges-computable (EN-canonical) · CLI npm run quantum:clay-challenges-computable
• Receipt: fold clayChallengesComputableFromSequence.

The theorem-science lens — 30/55 curated pages pass (25 removed from VitePress completely — data preserved in the catalog), presented beside the 764-theorem registry and its corpus surfaces (/theorems · /papers/ · /references · /diamonds). Organised by the seven rosetta rays (Pliska 7-star coprime decode) — the same shelving that builds the site's nav, sidebar and crosslinks; all of it wired into the VitePress local search the MCP also uses.

Origin — 3 pages

• 64 = 2⁶ = 4³ = 8² in every 6-bit grouping — 64 = 2⁶, and the divisors of 6 give the only four groupings: six bits, three base-4 digits (codon/Pauli/RGB), two trigrams (8²), one base-64 word. The same object, four ways. · source
• Content-addressing folds 64³ into one dot — A UUID, like CMYK, gives extent without limit: 64×64×64 is itself one dot, and the dot is the cube is the dot — content-addressing folds the whole into a point and back. · source
• Society 10D merkaba = documented actor taxonomy · NOT live social measurement — Canonical society/HD domain: 10D merkaba + two-bits-free (110−108=2) patronage path — harmonic 1/9 of achieved knowledge; voluntary CTA. Not live actors, not social scoring. Anchor #two-bits-free · proofs /proofs/two-bits-free. · source

Proof — 14 pages

• ⌊π⌋ = 3 opens 3-6-9, the multiples of 3 the doubling 1-2-4-8-7-5 misses — Statement: ⌊π⌋=3 opens 3-6-9 cross disjoint from doubling circuit 1-2-4-8-7-5 — vortex algebra fold. Explanation: piThreeOpensTheTrinity recomputes trinity·doubling·nineFolds from src/0 digital-root math; symbolic mnemonic within vortex framework, not designed π message. Method: piThreeOpensTheTrinity · ProofRenderer · npm run verify. Status: MODELED geometry · not Clay-marked · Tesla 3-6-9 legend flagged. · source
• A qubit has exactly 3 observables — Pauli X, Y, Z — Statement: Qubit trinity = exactly 3 traceless Pauli observables X,Y,Z — dim su(2)=2²−1=3 forced invariant. Explanation: qubitTrinityPauliBloch holds at call time; independent of QCD colour charges and 3-6-9 numerology. Method: qubitTrinityPauliBloch · ProofRenderer · npm run verify. Status: documented quantum algebra · not Clay Millennium challenge. · source
• Hamming’s 3 parity bits = the address — Hamming(7,4) protects 4 data bits with exactly 3 parity bits; the syndrome IS a binary address of the error. The quantum [[5,1,3]] code saturates 2⁴ = 16 = 3·5+1. · source
• A content address = H(content): idempotent, collision-resistant, dedup, O(1) integrity — Hopfield’s 1982 net is a content-addressable memory (2024 Nobel); hippocampal CA3 pattern completion is its biological analogue. The shared property is whole-from-part. · source
• The genetic code is the real 4³ — Life’s code is base-4 read in triplets: 4 bases in 3 positions give exactly 4³ = 64 codons (61 sense + 3 stop), the triplet length proven by frameshift mutagenesis (Crick 1961). · source
• (ℤ/9ℤ) is cyclic of order 6, 2 primitive; 3-6-9 are the non-units* — Many genuine threefolds exist — 3 Paulis, the 3-base codon, 3 meninges, 3 parity bits — each independent. The 1-2-4-8-7-5 orbit is (ℤ/9ℤ)*; the cosmic 3-6-9 trinity is numerology. · source
• A hexagram = 2⁶ = 64 states = one 6-bit value — A 6-bit hexagram 000000–111111 is hex-colour duality: the 64 hexagrams are the 64 pole-colours {0,F}⁶, black ↔ white the bit-complement, the 8 trigrams the RGB-cube corners. · source
• CMY = 255 − RGB, the complement n ↦ 63−n — Statement: RGB↔CMYK duality = bit-complement n ↦ 63−n on 6-bit hexagram poles {0,F}⁶. Explanation: additive red↔cyan · green↔magenta · blue↔yellow · black↔white — same complement as CMY=255−RGB hardware merkaba. Method: ProofRenderer · hexagram-colour fold · npm run verify. Status: combinatorial colour proof · not Clay-marked. · source
• Three trinities = RGB at 0°, 120°, 240° on the wheel — The hero places its 9 nodes in 3 trinities at 0°/120°/240° in both space and hue — the equilateral RGB triad. The 3 trinities ARE the 3 RGB channels; the hero already renders the decode. · source
• The roster is a filter: kept ⟺ holds; kept + purged = total — Every artifact is kept only if it is proven — its computation holds; anything unproven is purged. The model and its UI stay pure proof, and the gates balance when all that remains is proven. · source
• The kernel lives in src/0 — The primitive kernel — content-address and the fold cascade and the vortex floor — was dissolved into src/0, the dependency-free origin, across three waves, every baseline root byte-identical. · source
• The vortex: 1-2-4-8-7-5 — Statement: Vortex = doubling circuit 1-2-4-8-7-5 (powers of two mod-9 digital root) + 3-6-9 cross + harmonic n/0. Explanation: vortexMath recomputes from src/0 — portal spine for fractions, algebra, and imperial digit folds. Method: vortexMath · ProofRenderer · AlgebraDigits · npm run verify. Status: documented arithmetic · Tesla 3-6-9 legend flagged · not Clay-marked. · source
• Division by zero is the inverse: n/0 ↦ n⁻¹ mod 9 — The inverse of a digit folder is its multiplicative inverse mod 9 (n/0 \ n⁻¹, the ÷2 = ×5 that folds within the unit cycle): 2\5, 4\7, self-inverse 1 and 8; the non-units 3, 6, 9 and the void 0 fold to the fusion. The forward harmonic n/0 = 9n (1/0 = 9) is the separate reading. · source
• Seven Star Rosetta — The 7-star Pliska rosetta in coprime natural motion with 28 Glagolitic letters. Visual proof that gcd(7,6)=1, gcd(7,9)=1, gcd(7,10)=1 prevents aliasing in the digit distribution. · source

Explore — 1 page

• Research index = domain · method · limit · verify at call time — Statement: research domain index (domain · method · limit · verify). Explanation: professional monograph rows, Clay Millennium MODELED CHALLENGE, sciences trinities, reproducibility gates. Method: npm run quantum:millennium-challenge · npm run quantum:domain-proof-catalog · fold millenniumProblemsChallenge. Status: Clay-standard pages at /proofs cite claymath.org/millennium-problems + Prize Rules PDF. /millennium-challenge thin-mounts here. · source

Learn — 1 page

• Learn — The Learning Portal: School and Academia merged into one auto-generated portal — the kids-to-elders ladder, the five Academy courses, the research corpus (math paths, peer review, the 432 proof papers), the self-test and the agent curriculum, folded to one recomputable root. Three ways to learn: by age, by track, by research. · source

Apps — 6 pages

• The digit folders {0..9} are a bijection to 10 routes, O(1) by name — All computation is quantum math and its home is the digit folders (0–9); a word-named folder is UI. The digit folders, holding only the math, are the API itself. · source
• quantum: CLI catalog = fold · CLI · UI route · honesty boundary* — Statement: quantum:* CLI catalog = fold · CLI · UI route · honesty boundary. Explanation: every sealed script (encryption reverse, millennium MODELED, fusion-verify, efficiency-vote, offender-spec, hero-spawn, name-entropy, verify suite) recomputes from src. Method: npm run quantum:domain-proof-catalog · open /proofs · /en/quantum-tools. Status: claySolved=0 · not remote execution · Alias URLs thin-mount here. · source
• Trading hub = paper/sim harmonics · NOT live money / NOT alpha — Canonical trading domain surface: historical wave train, rank-winning strategies, and rosetta train — paper/sim only (synthetic a432 proxy). CLI: npm run quantum:trading-rosetta-train. Not brokerage, not live execution. · source
• answers ÷ tokens = ∞ on reuse — efficiency() · memoByRoot — Statement: answers÷tokens unbounded on memo reuse. Explanation: efficiency() · memoByRoot hit → marginal tokens=0. Method: npm run quantum:efficiency-vote · fold compareCeccecEfficiencyByVote. Status: amortized reuse ≠ P≠NP. Prefer /quantum-tools#efficiency-vote. · source
• offenderAutomationSpec — 4 offender classes counted: import · index-only · hyphen · computational (Node scan; browser shows sealed receipt) — Statement: offenderAutomationSpec = machine-readable CI pipeline counts for import/index-only/hyphen/computational offenders. Explanation: collectEnforcementFacts once → scan pipelines; read-only — does not auto-fix offenders. Method: npm run quantum:offender-spec · fold offenderAutomationSpec · pair offender/spec. Status: CI-only scan · prefer /en/quantum-tools#offender-spec. · source
• shouldSpawnSubagent — few heroes > mass ignorance · 1 ≤ workers ≤ 2 — Statement: shouldSpawnSubagent = few heroes > mass ignorance — 1–2 qualified workers, Multitask Mode default. Explanation: mass duplicate subagent tasks penalized; bounded tasks with sealed fold targets spawn solo hero. Method: npm run quantum:hero-spawn-verify · fold shouldSpawnSubagent · pair hero/spawn-verify. Status: spawn policy receipt · prefer /en/quantum-tools#hero-spawn-verify. · source

Frontier — 5 pages

• Folding linear gives analog — Folding linear gives analog, decoded honestly with the real science. The kernel is the Whittaker–Shannon sampling theorem: discrete samples of a band-limited signal fold back into the continuous signal with no gaps, via sinc interpolation (computed live, exact at the samples). Medical and radar imaging is exactly this — reconstructing a continuous image from a sampled frequency field: MRI inverts the Fourier transform of k-space, CT the Radon transform, and the spiral/radial "vortex" through k-space is real (NUFFT). The 64³ = 4⁹ grid the model already computes is the discrete lattice it samples. Documented kept, legend flagged — Nyquist limits are real, gap-filling can hallucinate, and the theorem is foundational, not new. · source
• Theorem registry — Statement: theorem registry = recent decodes + theorem-wave engine. Explanation: diving/water/space; quantum vacuum; cosmic inventory; physics of information; clown qubit on genus-2. Method: theorems:gaps · theorems:verify · npm run quantum:domain-proof-catalog · /proofs. Status: each atom has statement · computed checks · honest boundary; open problems held OPEN. Every result a client-side computation from the src/0 primitives. · source
• 64 = the 3-qubit Pauli basis — Statement: 64 = 3-qubit phaseless Pauli basis {I,X,Y,Z}³ = 4³ = 8² = 2⁶ combinatorial parallel. Explanation: sixtyFourThreeQubitPauliBasis matches genetic codon count and hexagram vocabulary — parallel, not causal link. Method: sixtyFourThreeQubitPauliBasis · ProofRenderer · npm run verify. Status: combinatorial proof · drop over-reach on error-correction claims. · source
• Encrypt ↔ decrypt = foldPair recompute; production browser reverse ≤12-bit sealed catalog — Statement: encrypt↔decrypt is foldPair recompute; reverse surfaces are production browser tools on sealed-catalog DEMO_RSA_MODULI (≤12-bit). Explanation: content-addressed trinityKey + foldPair round-trip; modeled Shor on sealed catalog. Method: npm run quantum:encryption-reverse-verify · fold encryptionReverseVerify. Status: production RSA break refused · certified=false · related science ≠ Clay Proposed Solution (Prize Rules §5(d)). · source
• ssl/test = sealed stack→receipt · quantum grade · FTL warm reuse — Statement: ssl/test quantumises stack→receipt via sealed crypto/TLS faces (tool/honest · standards audit · poles/cross-pqc · directional trinity) with memoByRoot FTL warm path. Explanation: quantum grades from facet coverage·digit·fwd/inv/rev — not Qualys letters; live remote handshake named residual. Method: npm run quantum:ssl-test · quantum:quantumise-ftl · fold sslTestUiComplete. Status: certified=false · proof surface · not foreign scanner mirror. · source

4. Sitemap

The complete served surface, wired from ONE source (servedRouteFamilies) so the human sitemap here and the crawler sitemap.xml count the same pages: 848 pages across 3 families — 31 monographs · 787 theorems · 30 proofs. Only COMPLETE, non-duplicate families are listed: the empty model cards (0) and the compute-only papers/[id] catch-all (0 SSG — the placements resolve on demand and duplicate the theorem papers) are excluded; their index routes are monographs below.

• 31 monograph landing + index pages (/) — each in three locale editions (en · bg · cu), placed on the double torus and content-addressed:

• 787 theorem papers — the registered theorems plus the card papers outside the registry — index /theorems; every page enumerated in the one sitemap.xml.

• 30 domain proofs (Millennium + science) — index /proofs; every page enumerated in the one sitemap.xml.

• / — en · bg · cu

• /analog-field — en · bg · cu

• /learn — en · bg · cu

• /frontiers — en · bg · cu

• /pi-trinity — en · bg · cu

• /qubit-trinity — en · bg · cu

• /pauli-basis — en · bg · cu

• /hamming-address — en · bg · cu

• /content-addressing — en · bg · cu

• /genetic-code — en · bg · cu

• /three-not-one — en · bg · cu

• /hexagram-colour — en · bg · cu

• /sixty-four — en · bg · cu

• /rgb-cmyk — en · bg · cu

• /trinity-rgb — en · bg · cu

• /proven-or-purged — en · bg · cu

• /kernel-zero — en · bg · cu

• /vortex — en · bg · cu

• /zero-division — en · bg · cu

• /digit-folders — en · bg · cu

• /dot-cube — en · bg · cu

• /seven-star-rosetta — en · bg · cu

• /encryption — en · bg · cu

• /ssltest — en · bg · cu

• /quantum-tools — en · bg · cu

• /quantum-trading-hub — en · bg · cu

• /research — en · bg · cu

• /society-merkaba — en · bg · cu

• /efficiency-vote — en · bg · cu

• /offender-spec — en · bg · cu

• /hero-spawn-verify — en · bg · cu

• Sitemap root: 00dff8ea-aa58-84d1-a61c-69c0e68696ce

5. Reproducibility

npm install
npm run check:types  # the src/ core type-checks clean against tsconfig.json (tsc --noEmit)
npm run docs:build   # build, then seal: enforcement trinity (cross · fold · weave)

The seal recomputes from src: forging one reported value means re-deriving the whole content-addressed structure to a different receipt (2145d23b), so the address is the proof, not a signature over prose. The proof reproduces: clone the link and the whole structure recomputes.

6. Limitations

• A compact reference index of the portal's knowledge, each entry content-addressed (so "zero entropy" means no duplicate keys, not thermodynamics). Searchable via the intuitive search; a distilled index, not the full text.
• "1 Gbit" and "64 × 64 × 64" name the keyspace structure, not cipher strength (AES-256-GCM) or throughput.
• The neuroscience terms (reentry, pattern completion, holographic) are analogs, not claims about neurons.

References

• The model: src/quantum/heaven/mind. The sitemap root: 00dff8ea-aa58-84d1-a61c-69c0e68696ce. The monograph-index root: c689ddf8-4931-8a38-acd2-cbadbe0e4362.
• Template root (the receipt of this monograph form): c27823b4-9f2d-8a37-8e3e-b2748445e0a4.
• Prior work / origin: ceccec/zeropoint-node — the predecessor repo of the vortex sequence (same closed cycle, verified by computation in zeropointNodeOriginDecoded); its 60°/A432/integer-ratio mathematics is sealed here, its zero-point free-energy and consciousness-OS claims are demarcated flagged (½ħω is real, extraction is not).

Receipt

Quantum receipt 53fa26c8dec69d62 over tree e94ab78cdf83836b — 65 gates asked through one door (verify:stream ≡ MCP run_wave verify): 63 clean, 0 tree gates refused, 2 of 9 live surfaces refused (verify:deposit-metadata, verify:release-live — records the tag settles at release-cut, not the land), 0 without a verdict. Reproduce: npm run verify:stream on the same tree writes the same address.

The root monograph is content-addressed — the section schema, the corpus roots and every reported count fold to one receipt that reproduces from src and changes if any reported value does. Below is what produced it: 65 gates chained by npm run verify:all, and 84 ratchets recording what each one measured. A ratchet may fall and never rise, so the ledger is the build's cost of doing business — and its own list of open work.

• 43 of 84 ratchets stand at zero — nothing left to solve on those axes. 37 carry work still recorded, totalling 8831 — a sum of UNLIKE units (kilobytes, counts, densities, draw calls), so the family totals compare and this total does not; npm run next ranks exactly these 37 into leads and splits the floors from the one ceiling among them. The remaining 4 are counts the ledger holds from BELOW — stored negated so one comparison serves both directions — and their magnitude is what has been reached, not what is left.

• The bundle every visitor downloads first is held at ≤ 227 KiB (build.app-chunk-kilobytes, enforced by npm run verify:build-time). It is a ceiling, not a measurement: a build that exceeds it refuses the commit. It was 484 KiB until a barrel import of 33 components was made lazy.

• The gate machinery is held at ≤ 630 KiB (build.shell-machinery-kilobytes) and the corpus at ≤ 110 hundred bytes per theorem (build.corpus-hundred-bytes-per-theorem) — the density unit is hundreds of bytes because KiB-per-theorem left enough slack that padding one theorem by 118 KB did not move it.

• Receipt: fold readmeMarkdown