The Freeze-Exponent Convention Corrigendum and the p-Theta Bridge (T263): Reconciling Two Apparently-Contradictory Statements of the Kurdyka-Lojasiewicz Exponent as One Physics in Two Conventions, and Deriving Theta = 1 - 1/(2p) Linking the Mobility

Photonic Universe Hypothesis (PUH) — T263 — CORRIGENDUM AND UNIFICATION NOTE. Two cascade papers state the late-time freeze exponent in apparently CONTRADICTORY directions: - T251 (Mobility-Collapse Exponent, 10.5281/zenodo.20651979): KL exponent theta > 1/2 for the slow power-law freeze; mobility exponent p = 8 (split) or p = 124 (compact); freeze laws t^(-1/7), t^(-1/123). - KL/cascade-rate note (10.5281/zenodo.20532621): theta NEAR ZERO for the slow freeze, with an internal "correction of a convention error" asserting theta in (0,1/2], theta near 0 = slow. Read literally these disagree on whether the freeze exponent is large or small for the SAME endpoint. This note resolves it: it is not a contradiction in physics but a clash of two STANDARD CONVENTIONS for the same Greek letter. THE TWO CONVENTIONS for the Lojasiewicz-Simon inequality |E(x)-E*|^(1-a) <= C||grad E||: - DESINGULARIZING convention (Attouch-Bolte-Svaiter; the form Li-Pong is usually quoted in): exponent theta in (1/2,1) for a degenerate flat minimum, =1/2 at Morse; flatter => LARGER theta. Slow freeze: theta > 1/2. - LOJASIEWICZ-SIMON convention: exponent a in (0,1/2], =1/2 at Morse, ->0 as minimum flattens; flatter => SMALLER a. Slow freeze: a near 0. Related by a = 1 - theta. T251 uses the desingularizing convention (theta>1/2). The KL note uses the Lojasiewicz-Simon convention but WRITES its exponent as theta (theta near 0). Both describe the IDENTICAL maximally-degenerate rank-zero endpoint of T238 and the IDENTICAL slowest-power-law freeze. The numbers differ only because the same symbol names two different, reciprocally-related exponents (a = 1-theta). THREE CORRECTIONS + ONE ADDITION: 1. Adopt the desingularizing convention throughout (theta in (1/2,1) for slow freeze), matching the cited Li-Pong classification; re-label the KL note's "theta near 0" as "a = 1-theta near 0" (equivalently theta near 1). 2. Identify the KL note's internal "convention correction" as the act that introduced the clash (switched conventions while keeping the symbol theta). 3. Confirm the PHYSICS of both papers (maximally degenerate endpoint, slowest power-law freeze, not exponential) is correct and UNCHANGED. 4. ADD the bridge: theta = 1 - 1/(2p), linking the desingularizing exponent to T251's mobility exponent. Verified at the Morse anchor (p=1 => theta=1/2); gives theta=0.9375 (split, p=8) and theta=0.9960 (compact, p=124). The two exponents the archive computed by independent routes are RIGIDLY LINKED, both placing the endpoint in the degenerate slow-freeze regime. UNCHANGED: every freeze law and physical conclusion. t^(-1/7), t^(-1/123), p=8, p=124, degenerate endpoint, power-law character all stand. CHANGED: the labelling (one convention now used consistently) plus the added bridge relation. Found via full-corpus search (read-first discipline at scale). The same kind of apparent-conflict-dissolves-into-two-correct-descriptions as T260.