MATHSOLVE Seed: RM-DIO-004

This page is the reader-facing MATHSOLVE campaign seed produced by Fixture 004. It begins where MATHFORGE must stop: with a preserved source, audited uncertainty, and a bounded first action.

Result-status box

Field	Value
Work Package seed	WP-RM-DIO-004
Source fixture	RM-DIO-004
Imported status	unknown
MATHFORGE status	STATUS_UNVERIFIED_UNKNOWN
MATHSOLVE status	READY_FOR_TRIAGE
Theorem claim	none
Certification claim	none
First executable step	finite exact sanity screen

Motivating object

The object is the integer-point problem on the affine plane curve:

x^2 - x - y^5 + y = 0.

The source question asks for all integer pairs (x, y) satisfying the original equation.

Obstruction

The obstruction is not writing down the polynomial. The obstruction is complete integer classification.

Several weaker objects are easier to produce but do not solve the problem:

• complex solutions;
• rational points;
• finite windows of integer points;
• symbolic manipulation of the polynomial curve;
• dataset status metadata.

MATHSOLVE must therefore make every support boundary explicit.

Theorem spine seed

The first campaign spine should begin with four modest obligations:

1. Source reconstruction lemma
   The problem statement and imported status are faithfully reconstructed from the source row.

2. Canonical encoding lemma
   The equation x^2 - x = y^5 - y is equivalent to x^2 - x - y^5 + y = 0 over integer pairs.

3. Finite screen lemma
   For a declared finite set of y values, the discriminant test produces exactly the lifted integer x branches recorded in the ledger.

4. Status preservation lemma
   The finite screen does not promote the problem to solved, certified, or globally classified.

First executable step

The first step is a sanity screen, not an attack on the full theorem:

Input: finite set Y = {-1, 0, 1, 2, 3}
For each y in Y:
  D := 1 + 4(y^5 - y)
  If D is a nonnegative square:
    x := (1 ± sqrt(D)) / 2
    retain integer x values
Output: finite-screen ledger

This step is useful because it is exact, bounded, explainable, and directly derived from the polynomial form.

It is insufficient because it gives no completeness theorem outside the declared finite set.

Proof-debt register

Debt	Owner	Promotion condition
Independent literature/status audit	MATHFORGE + MATHSOLVE	source trail reconstructed beyond the dataset row
Source PDF reconstruction	MATHFORGE	cited problem list checked and summarized
Finite-screen implementation	MATHSOLVE	exact ledger generated and checked
Global arithmetic route selection	MATHSOLVE	number-theoretic method chosen or ruled out
Completeness criterion	MATHSOLVE + MATHCERT	theorem statement precise enough for certification handoff

Claim ledger seed

The campaign may initially create only these claim classes:

Claim class	Allowed status	Forbidden promotion
source reconstruction	AUDITED	CERTIFIED
canonical encoding	AUDITED or CHECKED after exact verification	global solution claim
finite screen	COMPUTED_EXACTLY for the declared finite set	completeness
route selection	PROVISIONAL	theorem proof
status audit	AUDITED	global open/solved authority without independent source review

Negative-result protocol

A failed screen is still evidence if it is bounded and reproducible.

MATHSOLVE should record:

• the finite set searched;
• the arithmetic test applied;
• all retained branches;
• all rejected y values;
• the exact reason for rejection;
• the statement that no conclusion outside the finite set follows.

Escalation gate

The Work Package may escalate beyond intake only when it has:

1. a finite-screen ledger;
2. a clear statement of what the screen did not prove;
3. at least one plausible global method candidate;
4. a proof-debt register with owners;
5. a proposed certification target narrower than the original open problem.

Handoff boundary

MATHSOLVE may use this seed to begin a campaign. It may not declare the ResearchMath problem solved. It may not treat unknown as independently verified open status. It may not ask MATHCERT to certify anything until a precise local theorem or exact certificate exists.

The first meaningful MATHSOLVE success is not a solution. It is a disciplined Work Package whose ledger makes the next honest mathematical move unavoidable.