The Thermodynamic Floor from Degrees of Freedom and Heat — The Conditional Witness of Irreversible Discard
Abstract
Physical systems are modelled by the counting invariant total = distinct · 2^free. An irreversible erasure that decrements free by one halves the microstate volume consistent with the record and reduces system entropy by k_B ln 2. Under the undischarged premise P1 (the whole merges no two states) the lost bit of content must be exported. The second law is the external thermodynamic reading that identifies that export, for an isolated system-plus-bath, with heat Q ≥ k_B T ln 2. The implication is theorem-conditional on P1. The measured floor (Landauer 1961; Bérut et al. 2012; Jun et al. 2014) corroborates the conditional statement and supplies none of the chain. The kernel records the counting and the arithmetic of the bound; it derives neither P1 nor the second law. Domains remain distinct.
1. Historical development
Maxwell’s demon (1867) raised the possibility of a sorting process that appeared to decrease entropy without compensation. Szilard (1929) identified the information cost of measurement. Landauer (1961) stated the minimal heat price for irreversible erasure of one bit. Bennett (1982) clarified the role of logical irreversibility. Experimental saturation of the bound followed (Bérut et al. 2012; Jun et al. 2014). The present arc isolates the counting step, marks P1 as the load-bearing premise, and keeps the second-law reading external under the orthodoxy firewall.
2. Counting model
A system carries distinct macrostates distinguished by a record and free unresolved binary labels. The invariant is
total = distinct · 2^free.
An EraseProcess decrements free by one while preserving distinct, so before.total = 2 · after.total. System entropy (scaled) decreases by ln 2. This step is internal and settles at L3m.
3. Premise P1 and the export
P1 states that the whole merges no two states. Under P1 a lost bit of content cannot remain inside the system; it must be exported. This is the load-bearing premise. It is not derived from RA or from the one arrow.
4. Thermodynamic reading
The second law is the external statement that the total entropy of an isolated system-plus-bath cannot decrease. When the export is realized in a thermal bath at temperature T, the compensating entropy increase requires heat Q ≥ k_B T ln 2. This reading enters marked [consensus]. It is not restated as the framework’s own.
5. Conditional implication
The count-to-heat implication is theorem-conditional on P1:
a merge that exports a bit costs at least k_B T ln 2.
The kernel proves only the existential witness that a number meeting the bound exists. The physical content is carried in the prose.
6. Measured corroboration
Laboratory experiments saturate the bound in the slow-erasure limit. Cosmological residual heat flux, after standard subtractions, is consistent with zero at present sensitivity. These measurements corroborate the conditional implication and supply none of the chain. Consensus weighs zero.
7. Formal kernel (excerpt)
The Lean binding records the structures, the entropy decrease, the existential heat witness, and the arithmetic that a null residual forces zero density. It declares no axiom. It derives neither P1 nor the second law. The master seal is the conjunction of the counting and arithmetic claims.
Declaration
The second law remains an external premise. The thermodynamic floor is the conditional implication under P1 plus that external reading. No element of the seat or the Bridge depends on it. The arc is closed under the same rules as the least-erasure paper. ΔM = 0.