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MIND-QX-Order-Selective-Erasure-Calculus

# Proof-assistant roadmap

No Lean, Isabelle, Coq, or other kernel-checked proof is supplied. This plan describes concrete obligations for a future formalization; it is not itself verification.

## Finite foundation

Represent the universe as a finite tagged type containing mandatory marker labels and allocated ordered-pair records. States are finite subsets ordered by inclusion. Define every eraser from its input state using the exact simultaneous guard rule.

Prove deflationarity, monotonicity, and idempotence for all states. Express the distinct-label protocol property explicitly. Define trace composition, fixed upward observers, common reset, and extinction time without identifying the observer with the entire state.

## Dependency order

| Stage | Claim IDs | Formal obligation |
|---|---|---|
| A | C01 | Operator axioms on arbitrary subsets |
| B | C02/C03 | Inductive survivor formula, count, injectivity, antichain, and subsequence observer semantics |
| C | C04/C05 | Adjacent-swap state comparison; missing-pair and singly represented-pair obstructions; exceptional constructions |
| D | C06 | Set-valued parity rails, balanced reduction, unresolved partial words, reset, and extinction formula |
| E | C07 | Prefix-set invariance under commuting maps and parity-reversing fixed-point-free involutions |
| F | C08 | Live-term partial-order closure, mixed parity of nonunique extensions, adjacent chain necessity |
| G | C09-C11 | Disjoint block actions, local product sign, second-block context reversal, and tensor certificates |
| H | C12 | Strict-loss and representational pigeonhole bounds |

Keep arithmetic cell bounds separate from observer-circuit costs. A proof of a finite exhaustive case is not a formal proof of the theorem for all $n,m$.

## Statements that must remain distinct

- The positive-DNF lower bound on saturated reachable states.
- The compact monotone circuit's total function on every subset.
- Their agreement only on the specified protocol domain.
- The existence of a pair-guard implementation.
- The minimality of records only within that architecture.

Export exact assumptions and theorem statements with any kernel-checked release. Include the proof-assistant version, imported libraries, build command, and logs. Do not set `proof_assistant_checked` to true while a proof relies on placeholders or untrusted axioms for the main result.

	# Proof-assistant roadmap

	No Lean, Isabelle, Coq, or other kernel-checked proof is supplied. This plan describes concrete obligations for a future formalization; it is not itself verification.

	## Finite foundation

	Represent the universe as a finite tagged type containing mandatory marker labels and allocated ordered-pair records. States are finite subsets ordered by inclusion. Define every eraser from its input state using the exact simultaneous guard rule.

	Prove deflationarity, monotonicity, and idempotence for all states. Express the distinct-label protocol property explicitly. Define trace composition, fixed upward observers, common reset, and extinction time without identifying the observer with the entire state.

	## Dependency order

	\| Stage \| Claim IDs \| Formal obligation \|
	\|---\|---\|---\|
	\| A \| C01 \| Operator axioms on arbitrary subsets \|
	\| B \| C02/C03 \| Inductive survivor formula, count, injectivity, antichain, and subsequence observer semantics \|
	\| C \| C04/C05 \| Adjacent-swap state comparison; missing-pair and singly represented-pair obstructions; exceptional constructions \|
	\| D \| C06 \| Set-valued parity rails, balanced reduction, unresolved partial words, reset, and extinction formula \|
	\| E \| C07 \| Prefix-set invariance under commuting maps and parity-reversing fixed-point-free involutions \|
	\| F \| C08 \| Live-term partial-order closure, mixed parity of nonunique extensions, adjacent chain necessity \|
	\| G \| C09-C11 \| Disjoint block actions, local product sign, second-block context reversal, and tensor certificates \|
	\| H \| C12 \| Strict-loss and representational pigeonhole bounds \|

	Keep arithmetic cell bounds separate from observer-circuit costs. A proof of a finite exhaustive case is not a formal proof of the theorem for all $n,m$.

	## Statements that must remain distinct

	- The positive-DNF lower bound on saturated reachable states.
	- The compact monotone circuit's total function on every subset.
	- Their agreement only on the specified protocol domain.
	- The existence of a pair-guard implementation.
	- The minimality of records only within that architecture.

	Export exact assumptions and theorem statements with any kernel-checked release. Include the proof-assistant version, imported libraries, build command, and logs. Do not set `proof_assistant_checked` to true while a proof relies on placeholders or untrusted axioms for the main result.