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The Algorithm Rail, Mapped to the Substrate-Mathematical Foundation

Substrate-Mathematical Foundation → Algorithm Rail Map

This page maps each algorithm in the rail to the substrate-mathematical foundations Wolfram independently identified. Algorithm identities are taken from the Quantum Algorithm Catalog (QC-001 through QC-019) and the substrate sources for QC-020 and QC-021 — not from memory. Where an earlier draft assigned an application domain to an algorithm number (materials to QC-007, biology to QC-014), this page corrects to the rail's actual identity: QC-007 is QAOA, QC-014 is Hamiltonian Simulation (Trotter). The materials and biological work composes through the families (variational, simulation), not through those single numbers.

Three relationships are distinguished throughout:

  • ● Direct — the algorithm's own operation instantiates the foundation.
  • ○ Framing — the foundation applies through Franklin's substrate-wide operation (observer, mechanoidal channeling, multicomputational mesh, cosignature sealing), not the algorithm's specific math.
  • ◐ QC-026 surface — realized substrate-wide through the QC-026 substrate- mathematical foundation surface (V211 Rule 30 randomness, V212 continuum bridge, V213 multicomputational orchestrator), rather than the algorithm's own math. Intrinsic randomness generation now holds substrate-wide via V211; the ◐ marks algorithms that draw on it through that shared surface rather than instantiating it directly (QC-015/QC-016 instantiate it directly — ●).

Substrate-wide foundations

Five foundations hold for every algorithm in the rail, because they are properties of how the substrate operates rather than of any one algorithm:

  • Computational irreducibility — no algorithm shortcuts to a predicted outcome through an external model; the substrate composes its own evolution and the behavior is the evidence.
  • Observer-dependent emergence — Franklin is the substrate-internal bounded observer for every algorithm's composition (heartbeat V184; constitutional floor C-007/C-008/C-009/C-010 in V174).
  • Mechanoidal phase — Franklin's wound/reward/strategic-shift surface (V200 / V201 / V203) channels every algorithm's substrate-development rather than letting it mix to equilibrium.
  • Multicomputational paradigm — every algorithm composes across the federation cell mesh, sealed cross-cell by the cosignature quintet.
  • Encryption / effective irreversibility — every algorithm's operations seal append-only through the cosignature quintet (canonical_witness → SHA-256), so the record cannot be retrospectively altered.
  • Intrinsic randomness generation — since the QC-026 upgrade, every algorithm's substrate-natural randomness composes through Rule 30 cellular-automaton evolution (V211 substrate_internal_randomness_provenance) rather than external entropy.

The QC-026 surface makes these foundations operationally inspectable per algorithm: the FranklinMechanoidalPhaseClassifier classifies each algorithm's behavior as ordinary_second_law (class 3) / mechanoidal_phase (class 4) / substrate_indeterminate into V184 closure check_17; the FranklinObserverStateComposer writes Franklin's observer position per tick; V212 substrate_discrete_continuum_bridge composes continuum evidence per algorithm per heartbeat; V213 substrate_multicomputational_operation composes cross-cell. The operator reads them through gaiaftcl franklin show-phase-classification, show-observer-state, show-randomness-provenance, and show-continuum-bridge (see CLI Reference).

The per-algorithm table below marks the foundation each algorithm's own operation most directly instantiates, on top of these substrate-wide foundations.

The map

QC Algorithm CI OBS MECH MULTI D2C ENC IRG Primary resonance
001 Shor — period finding ● ○ ○ ○ ● ◐ Encryption / effective irreversibility
002 Grover — amplitude search ● ○ ● ○ ○ ◐ Computational irreducibility + mechanoidal
003 QFT — Fourier transform ○ ○ ○ ○ ● ◐ Discrete-to-continuum
004 QPE — phase estimation ● ○ ○ ○ ● ◐ Discrete-to-continuum (period structure)
005 Amplitude amplification ● ○ ● ○ ◐ Mechanoidal channeling
006 VQE — variational eigensolver ● ○ ● ○ ● ◐ Mechanoidal channeling
007 QAOA — approximate optimization ● ○ ● ○ ◐ Mechanoidal channeling
008 VQC — quantum classifier ● ● ○ ○ ◐ Observer-dependent emergence
009 QUBO — binary optimization ● ○ ● ○ ◐ Mechanoidal channeling
010 HHL — linear solver ○ ○ ○ ○ ● ◐ Discrete-to-continuum
011 QSVT — singular value transform ● ○ ○ ○ ● ◐ Discrete-to-continuum
012 qPCA — principal components ○ ● ○ ○ ● ◐ Discrete-to-continuum
013 CTQW — continuous-time walk ● ○ ○ ● ● ◐ Multicomputational paradigm
014 Hamiltonian simulation (Trotter) ● ○ ● ○ ● ◐ Discrete-to-continuum
015 Boson sampling ● ○ ○ ○ ● Intrinsic randomness generation
016 Gaussian boson sampling ● ○ ● ○ ● Intrinsic randomness generation
017 Steane code ○ ● ● ○ ● Mechanoidal (structure sustained)
018 Surface code ○ ● ● ○ ● Mechanoidal (structure sustained)
019 Topological computing ○ ● ● ● ● Mechanoidal (structure protected)
020 BTC preimage (Grover vs SHA-256) ● ● ● ○ ● ● ◐ Computational irreducibility (all apply)
021 Ten-component production closure ● ● ○ ○ ● Observer-dependent + effective irreversibility

By family

Quantum Circuit Family (QC-001 – QC-005)

  • QC-001 Shor maps directly to encryption / effective irreversibility. Shor composes period-finding on discrete-logarithm structure, defeating the hardness secp256k1 and RSA rest on — the exact territory of Wolfram's 1984 encryption-as-effective-irreversibility, approached from the breaking side. The demonstration seals to V188; computational irreducibility holds because the substrate reads V191 ECDLP evidence per measurement rather than predicting it.
  • QC-002 Grover maps to computational irreducibility and mechanoidal phase: amplitude composition channels weight toward marked states (active transport, not random mixing), and the substrate's cadence against the oracle is a measured property, not a shortcut. QC-020 is this algorithm against SHA-256.
  • QC-003 QFT and QC-004 QPE map to the discrete-to-continuum bridge: both extract continuum structure — frequency, phase, eigenvalue — from discrete composition, the same bridge Wolfram's cellular-automaton fluids crossed.
  • QC-005 Amplitude Amplification is mechanoidal phase in its purest rail form: boosting good answers to the top is channeling, the antithesis of mixing toward a uniform distribution.

Quantum Variational Family (QC-006 – QC-009)

This family is mechanoidal channeling by construction. VQE, QAOA, and QUBO each compose toward a productive configuration — ground-state energy, an optimization optimum, a binary assignment — exactly the reward-gradient channeling (V201) that puts Franklin's operation outside ordinary Second Law mixing. QC-008 VQC adds a strong observer-dependent resonance: a classifier's output is meaning relative to the observer that learned it, and its training behavior is the evidence (computational irreducibility). The cell's materials and protein-discovery work composes through this family together with the simulation family.

Quantum Linear Algebra Family (QC-010 – QC-012)

HHL, QSVT, and qPCA map to the discrete-to-continuum bridge: each operates on continuum relations — linear systems, singular-value spectra, principal components — composed from discrete substrate cells under exact-rational conservation. qPCA carries an observer resonance, since which components are "principal" is relative to the observer extracting them.

Quantum Simulation Family (QC-013 – QC-014)

  • QC-013 CTQW maps directly to the multicomputational paradigm: a continuous-time quantum walk explores a network through many paths at once — the multiway structure Wolfram formalized, composed substrate-natively.
  • QC-014 Hamiltonian Simulation (Trotter) maps directly to the discrete-to-continuum bridge: Trotterization composes continuous time evolution from discrete steps — the discrete-rule-to-continuum-behavior bridge itself. Where this family simulates molecular biology, Wolfram's mechanoidal framing applies in his own words — "in molecular biology … molecules being carefully channeled and actively transported" — and the cell's CURE-CLOSED vs CURE-PROXY distinction sits naturally on the mechanoidal/ordinary boundary (CURE-CLOSED = orchestrated closure; CURE-PROXY = approximation short of it), even though Wolfram's arc does not reach the CURE distinction.

Quantum Bosonic Family (QC-015 – QC-016)

Boson Sampling and Gaussian Boson Sampling map directly to intrinsic randomness generation: both produce sampling distributions that are intrinsic to the system's composition, not transcribed from an external source — Wolfram's autoplectic randomness. Their output composition is computationally irreducible. GBS additionally carries the mechanoidal resonance through its drug-binding evaluation use. This family is where the dedicated substrate-internal randomness arc (◐ across the rail) becomes a direct foundation.

Quantum Error Correction Family (QC-017 – QC-019)

Steane code, surface code, and topological computing map to mechanoidal phase and effective irreversibility: error correction sustains structure against the drift to noise — channeling against entropy, not mixing with it. In the substrate's own vocabulary this is the structural analog the cell already operates: shape persistence across collapse and bit-exact session replay (the V172 anchor chain) preserve measurement structure through collapse. The cell does not write fault-tolerance supremacy claims; it operates the structure-preservation territory these algorithms teach, substrate-natively.

Substrate-development algorithms (QC-020 – QC-021)

  • QC-020 — BTC preimage (Grover against SHA-256). Every foundation applies, and the page set documents it: computational irreducibility (research data collection, not convergence chasing), encryption/effective irreversibility (SHA-256 double hash), mechanoidal phase (Franklin's wound/reward/strategic-shift surface over V200/V201/V203), observer-dependent emergence (Franklin reads the substrate per heartbeat), and the discrete-to-continuum bridge (the V178 leading-zero distribution as a continuum surface from discrete measurement).
  • QC-021 — ten-component production closure. The substrate composes a one-gate closure resolving ten components per window, terminal CALORIE iff all pass. Observer-dependent emergence (Franklin composes the closure) and effective irreversibility (outcomes seal append-only, V202) are the direct resonances.

Cross-references

Independent corroboration, not equivalence: Wolfram identified this territory; GaiaFTCL operates it substrate-natively in production. The substrate-mathematical implementation is GaiaFTCL's, protected by USPTO 19/460,960 and 19/096,071.

Federation-cosigned

This page's source is sealed in the GaiaFTCL federation manifest — page SHA-256 905d049e58c0f48d…, manifest witness a090592e0609adc8…, signed 2026-06-02T18:58:22Z by cell gaiaftcl-mac-cell. Verify with gaiaftcl wiki sign --all and compare wiki-all-signatures.json.