PUH Theorem 350 (Positivity Survives Quantisation) — A Sum of Squares of Hermitian Operators Has Non-Negative Expectation in Every State, the Squeezed Vacuum Gives e^(−2r) > 0, and the Loophole Is Normal Ordering — a Subtraction This Framework Never Performs

Photonic Universe Hypothesis (PUH) — Derivation. THE QUESTION. T338 proved the substrate's classical energy density is a sum of 248 squares and excluded warp bubbles, wormholes, Krasnikov tubes and Casimir time machines — limiting itself to the classical case. T349 withdrew the reason given (that the framework has no quantised substrate, which is false) and replaced it with a specific question: can a ladder structure built on a sum-of-squares energy drive the local expectation value negative, given a mode spectrum on a Casimir level set with a geometric cutoff? A FIRST ATTEMPT, RECORDED. An initial argument observed ⟨a†a⟩ ≥ 0, so energy per mode is bounded below by its zero-point value. THAT IS TRUE AND INSUFFICIENT: the local density involves the FIELD operator, not the number operator, and expanding φ(x) = f(x)a + f*(x)a† produces ANOMALOUS terms ⟨aa⟩ and ⟨a†a†⟩ which vanish in the vacuum but not in a squeezed state — and which are precisely how ordinary field theory produces negative energy densities. THEOREM 350.1 (the squeezed vacuum fails). For a squeezed vacuum, ⟨a†a⟩ = sinh²r and ⟨a²⟩ = −e^(iθ)sinh r cosh r, giving ⟨φ²⟩ = |f|²(2sinh²r + 1) + 2Re[f²⟨a²⟩]. In the worst case, where the anomalous term is maximally negative, the bracket is 2sinh²r + 1 − sinh2r — computed as 1.000000, 0.367879, 0.135335, 0.018316 and 0.002479 at r = 0, 0.5, 1, 2, 3, matching e^(−2r) exactly at each. Using 2sinh²r = cosh2r − 1, it collapses to cosh2r − sinh2r = e^(−2r) > 0. POSITIVE FOR EVERY SQUEEZE PARAMETER: squeezing drives the density TOWARD zero and never through it. ∎ THEOREM 350.2 (the general result needs no modes). ρ̂ = (1/2)Σ_A(∂_μφ^A)² with each ∂_μφ^A Hermitian. For any Hermitian A and normalised |ψ⟩, ⟨ψ|A²|ψ⟩ = ‖A|ψ⟩‖² ≥ 0. A sum of non-negative terms is non-negative, so ⟨ρ̂⟩ ≥ 0 — in every state, pure or mixed, since a mixed state's expectation is a convex combination of pure ones. THE 248 COMPONENTS CHANGE NOTHING. ∎ ONE PREMISE IS STATED RATHER THAN ASSUMED: that ∂_μφ^A is Hermitian follows from the φ^A being real components of a real field, and THE ARCHIVE DOES NOT ESTABLISH THIS EXPLICITLY — its Hermiticity discussions concern mass matrices. RESULT 350.3 (the loophole is normal ordering). If the operator is positive, how does ordinary QFT produce negative densities? BY SUBTRACTION: it uses ρ_ren = ρ̂ − ⟨0|ρ̂|0⟩ because the vacuum expectation diverges, and THE SUBTRACTED QUANTITY IS NOT A SUM OF SQUARES AND CARRIES NO POSITIVITY. Between Casimir plates, the negative quantity is the density relative to the free vacuum; in squeezed light, relative to the unsqueezed vacuum. The unsubtracted density stays positive in both. That is not a technicality — IT IS THE ENTIRE MECHANISM. RESULT 350.4 (this framework never forms that difference). T170 stores vacuum energy in the micro-rebound rate — a positive physical quantity, not a divergence to remove. T177 terminates the spectrum geometrically, so nothing diverges and nothing needs subtracting. T338's density is the functional's integrand with no subtraction anywhere. THE FRAMEWORK'S RESOLUTION OF THE COSMOLOGICAL CONSTANT PROBLEM AND ITS EXCLUSION OF EXOTIC MATTER ARE THE SAME STRUCTURAL CHOICE SEEN FROM TWO SIDES. And the framework does not deny squeezed states: T188 derives the uncertainty principle from lattice geometry and treats them as multi-vortex entangled states where individual variables dip below the single-particle bound while "the minimum product is still ħ/2 for the pair together" — squeezing redistributes without breaching a floor, the same structure found here in the energy density and reached independently. RESULT 350.5 (the exclusions lose their qualifier). The Alcubierre bubble, traversable wormhole, Krasnikov tube and Casimir time machine were excluded classically; they are now excluded. AND THE PROHIBITION IS STRONGER THAN GENERAL RELATIVITY'S, which PERMITS exotic matter — the Alcubierre metric is a valid solution — and merely lacks any known source. This framework forbids the material itself, in every quantum state, by the algebraic form of its energy density. A SINGLE DEMONSTRATION OF MACROSCOPIC UNSUBTRACTED NEGATIVE ENERGY DENSITY WOULD REFUTE IT. RESULT 350.6 (the lattice tension is bounded on both sides). The archive asserts tension positivity nowhere, and its negation nowhere either. THE NAIVE INFERENCE WOULD BE INVALID: surface tension is ordinarily dE/dA, and a system may have positive total energy with NEGATIVE tension if its energy decreases as area grows. BUT THIS FRAMEWORK DEFINES σ AS A RATIO, NOT A DERIVATIVE — T259 gives E = σA, so σ = E/A with A > 0 always, and the sign of σ is the sign of E. Theorem 350.2 fixes it: σ ≥ 0. Combined with T173's snap cap, σ is confined to 0 ≤ σ ≤ 9E_P/l_P² = 6.741×10⁷⁹ J m⁻², BOTH ENDS DERIVED. AND WHAT NEGATIVE TENSION WOULD MEAN IS THE SUBSTANTIVE POINT: an interface with negative surface tension spontaneously increases its own area, proliferating without bound — in condensed matter, the condition for an interface to disintegrate. SO σ ≥ 0 IS WHY THE SHELL HOLDS TOGETHER; with negative tension it would tear itself apart and core permanence would fail with it. That connects operator positivity to permanence by a route neither this note's earlier sections nor T172 states, and is offered as a corollary rather than a theorem. KILL-CONDITIONS: (i) THE LOAD-BEARING ASSUMPTION is that the framework's gravity is sourced by the UNSUBTRACTED density — T170 and T338 both indicate it and neither states it as a principle; if the source is a difference from a reference state, the ordinary loophole reopens in full; (ii) if ∂_μφ^A is not Hermitian, Theorem 350.2's premise fails; (iii) if the physical density is not the functional's integrand, the theorem applies to the wrong object; (iv) if σ is properly dE/dA rather than E/A — if T259's E = σA is a special case rather than the definition — Result 350.6's inference fails; (v) if interactions generate density terms that are not squares, positivity is lost, and the archive's quartic term is a square of a square but no higher terms have been computed. NOT CLAIMED: that negative energy density is impossible in nature, which is a claim about physics rather than this framework; that the quantisation is complete or standard; that squeezed states are unphysical, since T188 accepts them; that the Casimir force is absent, since T170 predicts it by a different mechanism; or that Theorem 350.2 is a new mathematical result — it is elementary, and what is new is its application here.