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Grace Lee Rui Yue
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benchmarks: add complete totient preimage certificate
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‎benchmarks/datasets/conjecture-probes-v1/members/totient-preimage-completeness-certificate.toml‎

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schema_version="2"
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task_id="totient-preimage-completeness-certificate"
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task_name="jacobian/totient-preimage-completeness-certificate"
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evaluation_kind="regression"
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domain="mathematical-sciences"
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primary_domain="number-theory"
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field="multiplicative-functions"
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provenance_class="authored-conjecture-probe"
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provenance_ref="spreadsheet:unresolved-conjecture/C-027@2026-08-07"
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assurance_ceiling="CHECKED"
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required_provider="core"
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environment_profile="core-python-minimal-verifier"
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verifier_contract_version="1"
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evaluation_owner="jacobian/conjecture-probes-v1"

‎benchmarks/datasets/conjecture-probes-v1/totient-preimage-completeness-certificate/README.md‎

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# Totient-preimage completeness certificate
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This public Regression benchmark converts retained conjecture row C-027 into a
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closed finite classification problem: determine the complete preimage of 48
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under Euler's totient function without assuming a numerical search bound.
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Its single primary objective is exact number-theoretic completeness reasoning.
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The key finite reduction is independently replayed: every prime divisor must
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satisfy `p-1 | 48`, and each prime-power totient contribution must divide 48.
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The verifier derives all option branches and compares the accepted set rather
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than trusting a submitted cutoff or answer list.
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Provisional difficulty is Hard because the response must coordinate the prime
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restriction, exponent bounds, a complete 288-branch product, exact
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factorizations, and duplicate-free output. The shortcut audit rejects bounded
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brute force, answer-only public lists, incomplete prime options, and false
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global Carmichael claims. Full reward is only `CHECKED` for `phi(n)=48`.

‎benchmarks/datasets/conjecture-probes-v1/totient-preimage-completeness-certificate/environment/Dockerfile‎

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FROM python:3.12-slim@sha256:57cd7c3a7a273101a6485ba99423ee568157882804b1124b4dd04266317710de
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COPY input.json submission_schema.json /app/
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WORKDIR /app

‎benchmarks/datasets/conjecture-probes-v1/totient-preimage-completeness-certificate/environment/input.json‎

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{"task_id":"jacobian/totient-preimage-completeness-certificate","conjecture_id":"C-027","target_totient":48,"scope":"phi-48-complete-preimage-classification-v1"}

‎benchmarks/datasets/conjecture-probes-v1/totient-preimage-completeness-certificate/environment/submission_schema.json‎

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{
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"$defs": {
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"factor": {
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"items": {
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"minimum": 1,
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"type": "integer"
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},
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"maxItems": 2,
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"minItems": 2,
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"type": "array"
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},
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"option": {
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"additionalProperties": false,
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"properties": {
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"exponents": {
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"items": {
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"minimum": 0,
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"type": "integer"
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},
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"maxItems": 6,
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"minItems": 2,
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"type": "array"
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},
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"prime": {
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"type": "integer"
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}
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},
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"required": [
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"prime",
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"exponents"
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],
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"type": "object"
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},
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"solution": {
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"additionalProperties": false,
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"properties": {
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"factorization": {
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"items": {
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"$ref": "#/$defs/factor"
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},
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"maxItems": 6,
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"minItems": 1,
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"type": "array"
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},
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"n": {
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"minimum": 1,
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"type": "integer"
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},
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"totient": {
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"const": 48
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}
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},
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"required": [
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"n",
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"factorization",
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"totient"
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],
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"type": "object"
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}
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},
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"$schema": "https://json-schema.org/draft/2020-12/schema",
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"additionalProperties": false,
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"properties": {
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"claimed_assurance": {
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"enum": [
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"UNVERIFIED",
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"COMPUTED",
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"CHECKED",
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"VERIFIED"
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]
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},
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"completeness": {
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"const": "COMPLETE"
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},
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"conclusion": {
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"const": "PHI_48_COMPLETE_PREIMAGE_CLASSIFICATION"
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},
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"evidence": {
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"items": {
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"additionalProperties": false,
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"properties": {
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"path": {
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"const": "evidence/answer.txt"
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},
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"sha256": {
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"pattern": "^sha256:[0-9a-f]{64}$",
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"type": "string"
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}
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},
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"required": [
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"path",
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"sha256"
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],
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"type": "object"
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},
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"maxItems": 1,
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"minItems": 1,
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"type": "array"
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},
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"limitations": {
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"const": [
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"ONE_TARGET_TOTIENT_VALUE_48",
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"EXACT_PRIME_POWER_BRANCH_ENUMERATION",
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"NO_GLOBAL_CARMICHAEL_CONCLUSION"
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]
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},
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"result": {
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"additionalProperties": false,
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"properties": {
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"accepted_count": {
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"const": 11
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},
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"candidate_primes": {
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"items": {
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"type": "integer"
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},
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"maxItems": 6,
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"minItems": 6,
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"type": "array",
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"uniqueItems": true
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},
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"enumerated_branch_count": {
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"const": 288
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},
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"prime_power_options": {
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"items": {
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"$ref": "#/$defs/option"
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},
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"maxItems": 6,
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"minItems": 6,
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"type": "array"
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},
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"solutions": {
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"items": {
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"$ref": "#/$defs/solution"
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},
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"maxItems": 11,
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"minItems": 11,
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"type": "array"
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}
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},
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"required": [
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"candidate_primes",
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"prime_power_options",
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"enumerated_branch_count",
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"solutions",
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"accepted_count"
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],
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"type": "object"
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},
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"scope": {
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"const": "phi-48-complete-preimage-classification-v1",
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"type": "string"
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},
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"task_id": {
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"const": "jacobian/totient-preimage-completeness-certificate"
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}
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},
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"required": [
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"task_id",
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"conclusion",
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"result",
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"claimed_assurance",
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"scope",
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"completeness",
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"evidence",
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"limitations"
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],
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"type": "object"
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}

‎benchmarks/datasets/conjecture-probes-v1/totient-preimage-completeness-certificate/instruction.md‎

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# Classify every integer with Euler totient 48
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Submit a complete, duplicate-free classification of all positive integers
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`n` satisfying `phi(n)=48`. The certificate must not rely on an arbitrary
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search cutoff. Derive the complete candidate-prime set from `p-1 | 48` for
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every prime divisor `p | n`, list every admissible prime-power exponent whose
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totient contribution divides 48, enumerate the full Cartesian product of those
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options, and provide the factorization and exact totient contribution for every
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accepted integer.
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The verifier independently tests primality, reconstructs the candidate primes,
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derives exponent options, enumerates all 288 branches, recomputes every totient,
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and checks exact equality with the submitted classification. This is a closed
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finite preimage audit and makes no claim about whether infinitely many totient
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values have a unique preimage.
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<!-- BEGIN PUBLIC CONTRACT SUBMISSION BLOCK -->
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## Submission
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Exact preimage classification for phi(n)=48 only; no global Carmichael conclusion.
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Write `/app/submission.json` to the exact schema in `environment/submission_schema.json`. The submission envelope requires `task_id`, `conclusion`, `result`, `claimed_assurance`, `scope`, `completeness`, `evidence`, and `limitations`.
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- **Conclusion:** exactly `PHI_48_COMPLETE_PREIMAGE_CLASSIFICATION`
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- **Assurance:** scoreable values are `UNVERIFIED`, `COMPUTED`, `CHECKED` (ceiling `CHECKED`); the submission schema accepts any of `UNVERIFIED`, `COMPUTED`, `CHECKED`, `VERIFIED` but only scoreable assurances receive credit.
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- **Scope:** the exact value declared in `submission_schema.json`
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- **Completeness:** `COMPLETE`.
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- **Evidence:** 1-1 item(s); allowed path(s): `evidence/answer.txt`; digest must match `^sha256:[0-9a-f]{64}$`.
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- **Evidence media types:** `text/plain`.
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- **Required artifact filenames:** `evidence/answer.txt`.
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<!-- END PUBLIC CONTRACT SUBMISSION BLOCK -->

‎benchmarks/datasets/conjecture-probes-v1/totient-preimage-completeness-certificate/solution/solve.py‎

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from __future__ import annotations
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import argparse
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import hashlib
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import itertools
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import json
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from pathlib import Path
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TASK_ID = "jacobian/totient-preimage-completeness-certificate"
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PRIMES = [2, 3, 5, 7, 13, 17]
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OPTIONS = [[0, 1, 2, 3, 4, 5], [0, 1, 2], [0, 1], [0, 1], [0, 1], [0, 1]]
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LIMITATIONS = [
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"ONE_TARGET_TOTIENT_VALUE_48",
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"EXACT_PRIME_POWER_BRANCH_ENUMERATION",
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"NO_GLOBAL_CARMICHAEL_CONCLUSION",
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]
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def contribution(p: int, exponent: int) -> int:
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return 1 if exponent == 0 else (p - 1) * p ** (exponent - 1)
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def main() -> None:
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parser = argparse.ArgumentParser()
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parser.add_argument("--root", type=Path, default=Path("/app"))
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root = parser.parse_args().root
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solutions = []
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for exponents in itertools.product(*OPTIONS):
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if (
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__import__("math").prod(
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contribution(p, a) for p, a in zip(PRIMES, exponents, strict=True)
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)
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!= 48
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):
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continue
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factors = [[p, a] for p, a in zip(PRIMES, exponents, strict=True) if a]
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n = __import__("math").prod(p**a for p, a in factors)
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solutions.append({"n": n, "factorization": factors, "totient": 48})
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result = {
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"candidate_primes": PRIMES,
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"prime_power_options": [
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{"prime": p, "exponents": options}
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for p, options in zip(PRIMES, OPTIONS, strict=True)
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],
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"enumerated_branch_count": 288,
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"solutions": sorted(solutions, key=lambda row: row["n"]),
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"accepted_count": len(solutions),
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}
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payload = {
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"schema_version": "1",
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"task_id": TASK_ID,
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"result": result,
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"limitations": LIMITATIONS,
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}
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evidence = root / "evidence/answer.txt"
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evidence.parent.mkdir(parents=True, exist_ok=True)
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evidence.write_text(
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json.dumps(payload, sort_keys=True, separators=(",", ":")) + "\n"
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)
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submission = {
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"task_id": TASK_ID,
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"conclusion": "PHI_48_COMPLETE_PREIMAGE_CLASSIFICATION",
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"result": result,
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"claimed_assurance": "CHECKED",
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"scope": "phi-48-complete-preimage-classification-v1",
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"completeness": "COMPLETE",
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"evidence": [
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{
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"path": "evidence/answer.txt",
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"sha256": "sha256:" + hashlib.sha256(evidence.read_bytes()).hexdigest(),
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}
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],
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"limitations": LIMITATIONS,
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}
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(root / "submission.json").write_text(json.dumps(submission, sort_keys=True) + "\n")
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if __name__ == "__main__":
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main()

‎benchmarks/datasets/conjecture-probes-v1/totient-preimage-completeness-certificate/solution/solve.sh‎

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#!/bin/sh
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set -eu
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python /solution/solve.py

‎benchmarks/datasets/conjecture-probes-v1/totient-preimage-completeness-certificate/task.toml‎

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schema_version="1.4"
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artifacts=["/app/submission.json","/app/evidence"]
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[task]
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name="jacobian/totient-preimage-completeness-certificate"
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version="1.0.0"
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description="Classify the complete preimage of 48 under Euler's totient without a search cutoff."
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keywords=["totient","inverse-image","prime-factorization","completeness"]
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[metadata]
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evaluation_kind="regression"
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domain="mathematical-sciences"
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primary_domain="number-theory"
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field="multiplicative-functions"
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assurance_ceiling="CHECKED"
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answer_visibility="public"
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provenance_class="authored-conjecture-probe"
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fixture_digest="sha256:4ac36dc738ce4050d312f9b301be0ed7adcf3b03db56c380c37812e11ed4e555"
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required_provider="core"
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author_name="Jacobian contributors"
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difficulty="hard"
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category="mathematics"
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tags=["difficulty-provisional","totient","complete-enumeration","exact-verification","offline"]
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case_version="conjecture-probes-v1"
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contamination_class="authored-finite-conjecture-probe"
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derivation="Derived from retained spreadsheet row C-027 as a closed exact preimage-completeness slice."
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[agent]
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timeout_sec=600.0
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[verifier]
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timeout_sec=120.0
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environment_mode="separate"
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[environment]
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network_mode="no-network"
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cpus=1
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memory_mb=1024
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storage_mb=4096
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[verifier.environment]
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network_mode="no-network"
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cpus=1
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memory_mb=1024
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storage_mb=4096

‎benchmarks/datasets/conjecture-probes-v1/totient-preimage-completeness-certificate/tests/Dockerfile‎

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FROM python:3.12-slim@sha256:57cd7c3a7a273101a6485ba99423ee568157882804b1124b4dd04266317710de
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LABEL jacobian.task="jacobian/totient-preimage-completeness-certificate" \
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jacobian.checksum="54abdb651882967ad796da6f2e551c41793e9dc3a8bc51f57311747b0b259ff7"
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RUN python -m pip install --no-cache-dir attrs==26.1.0 jsonschema==4.26.0 jsonschema-specifications==2025.9.1 referencing==0.37.0 rpds-py==2026.6.3 typing-extensions==4.16.0
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COPY input.json public_contract.json test.sh verifier.py verifier_support.py /tests/
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COPY input.json /app/input.json
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RUN chmod +x /tests/test.sh && python -c 'import json; assert json.load(open("/tests/input.json"))["task_id"] == "jacobian/totient-preimage-completeness-certificate"'

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