PUH Theorem 324 (Both Couplings Determined) — The Field Normalisation Is a Units Convention, Not an Unknown; J/κ² and K/κ⁴ Are the Only Physical Combinations and Both Are Now Fixed; and the Lattice Goes Nonlinear at 114 Times the Schwinger Field

Abstract

Photonic Universe Hypothesis (PUH) — Determination. THE SITUATION INHERITED. T320 added a nearest-neighbour coupling, quadratic in the difference between neighbouring cells, with one free coefficient J. T321 showed a quadratic coupling contributes EXACTLY NOTHING to nonlinear optics, so vacuum birefringence requires a quartic term — unique in form, since E8 has no independent degree-4 invariant — introducing a second coefficient K. T323 showed T320's proposed calibration CANNOT determine J, because both stiffness and inertial density scale with the field normalisation κ while the wave speed sees only their ratio. THREE NOTES, AND THE COUNT OF UNDETERMINED QUANTITIES HAD RISEN FROM ONE TO THREE. THEOREM 324.1 (the resolution: the count was wrong). J, K and κ are not separately physical; only J/κ² and K/κ⁴ are. PROOF: the lattice field φ is dimensionless — its scale is a choice of units. Under φ → λφ the bond energy (J/2)⟨Δφ,Δφ⟩ + (K/4)⟨Δφ,Δφ⟩² is unchanged only if J → J/λ² and K → K/λ⁴, and the correspondence A = κφ from T323 forces κ → κ/λ. Then J/κ² and K/κ⁴ are invariant, and no other independent combination is. ∎ SO κ WAS NEVER AN UNKNOWN TO BE FOUND. It is a convention, and a convention must drop out of everything measurable — which is precisely what T323 observed when it cancelled from the wave speed. What looked there like an obstruction was the normalisation behaving correctly. THE FIRST COMBINATION, FROM THE WAVE SPEED. T323's potential-term matching gave 60J = κ²/μ₀ directly, so J/κ² = 1/(60μ₀) = 1.326291×10⁴ (SI). No residual freedom: what the wave speed fixes is exactly the invariant combination, which is all there was to fix. THEOREM 324.2 (birefringence emerges from the quartic term, rather than being inserted). Write s = ⟨Δφ,Δφ⟩, so E(s) = (J/2)s + (K/4)s². Split the bond difference into a strong background b and a weak probe p: s = ⟨b,b⟩ + 2⟨b,p⟩ + ⟨p,p⟩. Expanding to second order in the probe — which IS the probe's effective wave operator — gives a ⟨p,p⟩ term with coefficient J/2 + K⟨b,b⟩/2, isotropic; and a ⟨b,p⟩² term with coefficient K, which is MAXIMAL FOR A PROBE POLARISED ALONG b AND ZERO ACROSS IT. That angular dependence is birefringence, arriving from the algebra. With the index going as the inverse square root of stiffness and x = K⟨b,b⟩/J, the difference expands as n∥ − n⊥ = −x + 3x² + …, leading term −(K/J)⟨b,b⟩. THE INDEX DIFFERENCE IS LINEAR IN THE BACKGROUND'S ENERGY DENSITY AND HENCE QUADRATIC IN B — the dependence quantum electrodynamics gives. The structures agree before any number is computed, and that agreement is a check the construction could have failed. RESULT 324.3 (the second combination). Converting through A = κφ gives Δn = −(K/J)B²/κ². Matching the parameter-free Δn = (α/30π)(B/B_c)² yields K/(Jκ²) = α/(30πB_c²) = 3.981228×10⁻²⁴ T⁻², and substituting J = κ²/(60μ₀) gives K/κ⁴ = α/(1800πμ₀B_c²) = 5.280267×10⁻²⁰ (SI). BOTH PHYSICAL COMBINATIONS ARE DETERMINED. NO FREE PARAMETER REMAINS IN THE COUPLING. Two coefficients, two independent measurements, neither fitted in the ordinary sense — each set by a quantity the framework did not choose and cannot adjust. RESULT 324.4 (a scale that was not put in). The ratio has dimensions of inverse squared field, and its reciprocal square root is the field at which the quartic term rivals the quadratic — where the lattice ceases to be linear: B_nonlinear = 5.0118×10¹¹ T = 113.6 × B_c. THIS NUMBER WAS NOT FITTED. It follows from matching a parameter-free prediction to a structure whose form the algebra fixed, and it could have landed anywhere. Had it come out at laboratory strengths the framework would predict nonlinear optics where none is seen; had it come out at the Planck field — roughly 10⁴⁴ times larger — the lattice would be linear far beyond any regime where the vacuum is known not to be. Instead it sits two orders above the scale at which the quantum vacuum's own nonlinearity sets in. THIS IS NOT A DERIVATION OF THE CRITICAL FIELD AND IS NOT CLAIMED AS ONE — the matching used B_c as input. What is not automatic is that the resulting scale should be of the same order rather than forty orders away. A CORRECTION RECORDED. The first pass concluded that the birefringence gave one equation in three unknowns and could not close the system. THAT WAS WRONG, for the reason in Theorem 324.1, and it is recorded rather than removed. The error is instructive: a normalisation looks exactly like an unknown — it is a number, it appears in the equations, and one does not know its value. What distinguishes it is that asking for its value is not well posed, because it changes under a relabelling that leaves all physics unaltered. THE COUNT OF FREE PARAMETERS IS NOT THE COUNT OF SYMBOLS. KILL-CONDITIONS: (i) if the lattice field is NOT dimensionless — if its scale is fixed by something in the framework — Theorem 324.1 fails and κ becomes a genuine unknown again; (ii) if the quartic term is not the leading anharmonicity, the matching fixes a different combination; (iii) if the index difference has additional contributions at the same order — from the potential's expansion about the vacuum, which T321 identified as an alternative nonlinearity source — the matching attributes to K what belongs elsewhere, and both K/κ⁴ and the nonlinear field are OVERESTIMATES; (iv) if birefringence is measured to differ from the QED value, the input changes and both derived numbers move. NOT CLAIMED: that the critical field has been derived — it is an INPUT, and Result 324.4 observes only that the output lands near it; that the framework predicts birefringence, since it was matched TO it; that J, K or κ individually have values, which Theorem 324.1 denies; that the index-difference coefficient is exact, since a geometric factor of order unity depending on the background's orientation relative to the bond directions has not been evaluated; or that the potential's contribution has been excluded, which kill-condition (iii) states it has not.