# The fused window — three proven identities, the exact spectral forms, and the Anchored Typicality Measurement *(2026-08-14. Companion to `natal-cap-19-calm-lemma.js`, which verifies every identity below as exact integers/floats. Notation: level x, tile width W = x#, natal set N = N_x ⊂ [0, W) with indicator `ind`, |N| = N̄, mirror μ(r) = W−2−r with μ(N) = N (cap-09 P0 / cap-13). Scour prime q: gcd(q, W)=1, q² ≤ W; strike classes of rotation t are {t, t−2} mod q; q⁻¹ is the inverse of q mod W. Sibling A(m) = ind[(qm) mod W], sibling B(m) = ind[(qm−2) mod W] — cap-01's dilated kill-image patterns.)* Status of these results and of the anchored calm as a whole: [anchored-calm.md](anchored-calm.md). --- ## Lemma 1 (Mirror-Sibling Identity) **B(m) = A(−m mod W) for every m.** *Proof.* B(m) = ind[(qm−2) mod W] = ind[μ(qm−2) mod W] since μ(N) = N. μ(qm−2) = W−2−(qm−2) = W−qm ≡ −qm ≡ q·(−m) (mod W). So B(m) = ind[(q·(−m)) mod W] = A(−m mod W). ∎ The two dilated siblings are one pattern read in opposite directions. ## Lemma 2 (Fusion Identity — the anchor) Let lA = ⌈W/q⌉, lB = ⌊(W+1)/q⌋, L_q = lA + lB. Then **gross(0,q) = #( A ∩ [W−lB, W+lA) ) — a single cyclic window of length L_q containing m = 0**, where the window is read cyclically mod W. *Proof.* The class-0 strikes are the natal r = qm with 0 ≤ qm < W, i.e. m ∈ [0, lA); their count is Σ_{m=0}^{lA−1} A(m) (this is cap-01's certified branch-A identity). The class-(−2) strikes are the natal r = qm−2 with 0 ≤ qm−2 < W, i.e. m ∈ [1, lB] (q∤W excludes qm = W; qm = W+1 gives r = W−1 ∉ N and contributes 0 either way); their count is Σ_{m=1}^{lB} B(m) = Σ_{m=1}^{lB} A(−m) (Lemma 1) = Σ_{v=W−lB}^{W−1} A(v). The two index intervals [W−lB, W−1] and [0, lA−1] are cyclically adjacent across 0 and disjoint (L_q ≈ 2W/q < W), so the sum is one window. ∎ Since q·lA − W ∈ [1, q−1] and q·lB − W ∈ [2−q, 1], the fused length obeys **q·L_q = 2W + r_q with |r_q| < q**: the anchored window is within O(q/W) of exactly 2/q of the period — the resonance used in Lemma 4. ## Lemma 3 (Mirror-Phase Doubling Lemma — the two degenerate phases, concatenation and duplication) The mirror acts on strike classes by c ↦ (W−2−c) mod q, hence on rotation pairs by {t, t−2} ↦ {W−t, (W−t)−2}: rotation t's pair maps to rotation (W−t)'s pair — the proven ensemble symmetry stat(t) = stat(W−t). The pair is **self-paired iff 2t ≡ 0 (mod W)**, i.e. exactly t = 0 and t = W/2: - **t = 0 (concatenation).** The self-pairing swaps the two classes {0, −2}, and by Lemma 2 the two windows are mirror-adjacent: one window of doubled length. - **t = W/2 (duplication).** Here the self-pairing fixes each class: W−2−(W/2) ≡ W/2 − 2 (mod q) means the mirror maps class W/2−2 **onto** class W/2, so n_{W/2−2} = n_{W/2} and dev(W/2, q) = 2·(one class-window's deviation). (This is cap-13's variance-doubling, re-derived: dev in steps of 2, variance ≈ 2× the unfused pair.) For generic t the two windows of A sit at starts q⁻¹t and q⁻¹(t−2), separated by 2q⁻¹ mod W — generically far apart. **Refinement (resonant primes).** Adjacency of the two *literal* windows of a generic rotation requires q·l ≡ −2 (mod W) for the relevant window length l ≈ W/q, i.e. **q | W−2**: for such q every rotation's pair is adjacent and the whole ensemble is fused (measured @17: q = 23, 31, 179; varPair/V_fused = 1.00 exactly; the anchor has no advantage there, R ≈ 1). Dually q | W+1 forces r_q = 2, the tightest resonance (q = 19 @17: R = 0.15). ∎ ## Lemma 4 (Exact spectral forms) With S(k) = Σ_{r∈N} e(kr/W) (factored with CRT twist, cap-02): 1. **Phase alignment.** μ(N) = N gives S(k) = e(−2k/W)·conj(S(k)), so **T(k) = e(k/W)·S(k) is real** for all k (the pattern is symmetric about the two centers (W−2)/2 and W−1). 2. **Fused-window variance.** The variance of the count of A over all W cyclic starts of a length-L_q window is **V_fused(q) = (1/W²) Σ_{j≠0} |S(j)|² · sin²(πj·r_q/W) / sin²(π(qj mod W)/W)** (substitute k = qj in the sibling spectrum S_A(k) = S(q⁻¹k) and use q·L_q = 2W + r_q). Verified to relerr ≤ 7·10⁻¹²; the same with a_q = q·lA − W gives the single-head variance. 3. **Anchored Σ-form.** dev(0,q) = ρ·r_q/q + (1/W) Σ_{j≠0} S(j)·conj(E_I(qj)), E_I the fused window's geometric kernel; reproduces the integer counts to 2·10⁻¹¹. This is the sharpest exact expression for the anchored deviation: an incomplete twisted sum with resonant kernel — no closed form found, and no c·mean^{1/4} bound is visible in it (the budget sits in the diffuse support-≥2 spectral cloud, as in cap-02). ∎ ## What Lemmas 1–3 do and do not give Four separate objects, at three calibrations. Their statuses are held in [anchored-calm.md](anchored-calm.md), not here. - **Fusion Identity [PROVEN, every scour prime q, every level x].** dev(0,q) is the deviation of ONE cyclic sibling window of length L_q ≈ 2W/q (Lemmas 1–2); a generic rotation's deviation is the sum over TWO windows 2q⁻¹ apart. - **Mirror-Phase Doubling Lemma [PROVEN, every q, every x].** dev(W/2,q) = 2 × (one window's deviation) (Lemma 3) — the same symmetry that calms the anchor doubles W/2. - **The suppression, measured here and since made a theorem elsewhere.** One fused window is quieter than two: Σ_q V_fused / Σ_q VarRot = 0.666 @13, 0.589 @17, driven by adjacent-window anticorrelation Σ 2Cov_adj / Σ(V_A+V_B) = −0.39 / −0.40 (the window-scale face of the sieve's ρ_p(d) = p−4 < p−2 sub-Poisson correlations). The two ratios above are measurements at @13 and @17. The anticorrelation behind them is no longer one: the −1/2 constant is exact for every q coprime to 30 (Minus-Half Theorem, cap-26), and the aggregate bound that makes it negative is certified as an exact integer inequality at six levels @11 through @29 (Aggregate 30-Skeleton Bound, cap-30 and cap-36). - **Anchored Typicality Measurement [MEASURED].** The anchor is position-typical of fused windows: Σ dev(0,q)² / Σ V_fused = 0.935 @13, 0.939 @17; mean position percentile 47.9% / 47.8%. Together: measured VR(0) = 0.623 / 0.553 = suppression × typicality. **The anchored calm is fusion × anticorrelation, with a ≈0.94 typicality residual.** The heads' own percentiles (45–48%, not extreme) refute "the head window is specially thin/quiet" as a variance mechanism: the head's only gift is adjacency. ## What is not proven, precisely 1. **The uniform-in-q form of the anticorrelation is REFUTED, not open.** Cov_adj < 0 for *every* scour q is false: six counterexample primes are known across six levels, cataloged as skeleton resonances (cap-23's verdict table, cap-30 Prop D). Any correct general statement is aggregate, or all-but-sparse-exceptions. What remains open is the all-x aggregate form, which cap-30 reduces to G30_agg(x) < 1/2 and cap-36 names as the Skeleton Equidistribution Conjecture. 2. **The Anchored Typicality Measurement is the anchored-escape wall, one level down.** "The one arithmetic position is ensemble-typical" is exactly the kind of statement measure arguments cannot finish (natal5-variance reading 6) — but the unexplained residue has shrunk from the whole calm to a factor 0.94. Its home says the rest: no proof mechanism is in sight. @19 (full enumeration, 9,699,690 rotations, 435 primes): anchored VR rank 14/9,699,690, Z2 rank 6 — percentile ladder 1.84% → 3.94% → 0.0021% → 0.00014%; W/2 the single loudest rotation here, as it is at @11 and @17. The rank deepens because the ensemble sd shrinks ~K^(−1/2) while the suppression factor stays ≈0.55–0.70 — consistent with (iii) being a level-stable mechanism, not a fluctuation. **W/2 is not loudest at every level, and this sentence used to say it was.** It read "the single loudest rotation for the third level running", a streak that @13 breaks: `natal-cap-38-loudness-driver.js` enumerates VR(W/2) = 1.6652 at @13, rank 198 from the top of 30,030 — the 99.34th percentile, not the maximum, against a true max of 2.3518 at t = 3461. W/2 IS the loudest at @11, @17 and @19, all three at rank-from-top 0. What the Mirror-Phase Doubling Lemma proves is that dev(W/2, q) is twice one window's deviation, per prime; maximality of the sum over primes is a separate claim and is measured, not proven. `paper/anchored-note.md` already carried the correctly scoped version ("the loudest rotation of the entire ensemble at @11 and @17 and the 99.3rd percentile at @13") while this file and `history/CHRONICLE.md` did not.