The frequency response

Give each element n the real number

Since cos is even, a channel residue and its negative give the same term, so λ(n) sees each channel only up to sign: ⌊p/2⌋ + 1 values per channel, and ∏(⌊p_i/2⌋ + 1) = 2·2·3·4·6·7·9 = 18,144 classes at k = 7. The classes are the orbits of the group that flips signs independently in each channel — order 2^k = 128, not one involution, and the global map n ↦ −n alone would leave 255,256 orbits rather than 18,144. The action is far from free, so the collapse is not uniform: orbit sizes run 1, 2, 4, …, 64, and 510,510 / 18,144 = 28.1 is their average and nothing more.

The fingerprint is that those 18,144 classes land on 18,144 distinct reals — at this rung and at every other: distinctness holds for all k, proved in three steps. Suppose two classes tie. First, average the vanishing difference over the symmetries of the cyclotomic field — the rationals extended by a primitive N-th root of unity, which holds every cosine in the sum — that fix one chosen channel: the averaging keeps that channel's terms and replaces every other channel's by a rational number: its trace — the sum of its images under its own channel's symmetries — divided by the symmetry count, so each channel's own difference must itself be rational. Second, a rational difference of two class cosines at a channel p ≥ 5 is zero: every nonzero class value has trace −2 whatever the class, so a rational difference of two nonzero values equals its own averaged trace, 0; a rational difference against class 0 would make 2cos(2πc/p) itself rational, which its degree (p−1)/2 ≥ 2 forbids; and cos is strictly decreasing on the class range, so zero difference means equal classes. Third, channels 2 and 3, whose cosines are rational (±2, and −1), leave differences in 4Z and 3Z, which meet at nothing of magnitude ≤ 4. The middle step's computational content is checked exactly, no floats in the verdict path: with ψ_p — the monic integer polynomial whose roots are the channel's class cosines — certified irreducible, a rational pair difference is precisely a constant remainder when the two classes' integer polynomials are subtracted mod ψ_p, and that integer scan is silent at every channel through p = 53 while firing at 2 and 3, where rational differences provably exist. Nothing in the float computation is close either: the nearest pair sits 1.9 × 10⁻⁷ apart at this normalization, eight orders of magnitude above the double-precision spacing at these values (1.8 × 10⁻¹⁵), so the separation is arithmetic and not rounding. One real number recovers an element's whole residue tuple up to a per-channel sign.

Distinctness survives on the powered rung — the same primes with the small channels raised: 8, 9, 25, 49, 11, 13, 17 — on a strictly thinner margin, because the middle step breaks there: channel 8 carries three rational class values, 2, 0 and −2, and channel 9 two, 2 and −1, so a rational difference no longer forces equal classes inside those channels. The same exact scan, with each ψ now folded from the prime-power cyclotomic polynomial and certified irreducible the same way, enumerates each powered channel's set of rational class differences — {0, ±2, ±4} at 8, {0, ±3} at 9, empty at 25, 49, 11, 13 and 17 — and no choice of one difference per channel, not all zero, sums to zero. That closes the question in both directions, because per-channel differences are realized independently by the CRT: a nonzero choice summing to zero would itself be an explicit collision. So all 3,071,250 classes of the powered rung are distinct, the whole weight carried by one clash — even differences of magnitude at most 4 against multiples of 3. And the clash belongs to the primes, not to this rung. A rational difference of two irrational class values is zero at any prime-power channel: sharing a rational difference, the two values generate the same field, which puts them at one level of the channel's ladder of subfields, and averaging over the symmetries there sends both to one shared value, so the difference equals its own average, zero. Rational differences therefore run only between rational class values, and cos is rational only at angles with denominator 1, 2, 3, 4 or 6 — of which a prime power admits 2 and 4 only at p = 2, 3 only at p = 3, and 6 never. So a 2-power channel's differences sit within {0, ±2, ±4}, a 3-power's within {0, ±3}, every channel on a higher prime is silent whatever the exponent, and the same clash closes every powered tower: any distinct primes, any exponents, any number of channels.

Two readings of the same list. Embed each channel's circle in the plane and measure an element by the squared straight-line distances its residues stand from 0 — the chord reading, dominated by the largest circle where a count of differing channels is dominated by the smallest. The eigenvalue is that reading in disguise, λ(n) = 2k − ∑4sin²(πr_i/p_i) — so the top of the eigenvalue range is where the chords vanish and the largest channel sets the gap below it: 4sin²(π/17) ≈ 0.135, some 8.2 times the gap the powered rung manages, where 7² = 49 controls it. And the mass sits low. The bottom half of the eigenvalue range holds 43.9% of the classes but 58.4% of the elements, and the 60 lowest classes every one carry the largest orbit size, 2⁶ = 64 — a lean toward the floor, not a concentration at it. The level spacings show no repulsion at all. Measured after unfolding — dividing each gap by the average of its neighbours, since the values lie far denser low than high and a single global scale would swamp the comparison — they have variance 0.98 against 1 for a Poisson process and 0.273 for a random matrix. The rung is a crystal, not a glass, and CRT independence is why.

Scope. The count formula and distinctness are proved for every squarefree rung; the exact integer scan behind the middle step has no precision ceiling and runs silent through p = 53 (k = 16). Distinctness on the powered towers is a theorem — any distinct primes, any exponents, any k, the squarefree case its exponents-one face — with the per-channel sets cross-checked by the exact scan through exponent 5. The exhaustive float enumeration at k = 3..8 stands as an independent cross-check; its ceiling — the minimum separation falls about three orders per rung (2.2 × 10⁻⁵ at k = 5, 1.9 × 10⁻⁷ at k = 7, 1.8 × 10⁻⁹ at k = 8), double precision out near k = 10 — is the instrument's, not the theorem's. Every figure here uses the normalization above, with the factor 2 that makes the chord identity read cleanly; the plain-cosine convention halves each of them, the separation included. The chord identity is an algebraic rearrangement. The spacing statistics are an observation at this rung, and they require local unfolding — a globally normalized reading of the same list is dominated by the varying local density and says nothing about repulsion.

verifier: explore_spectral_theorem.py, explore_powered_fingerprint.py, explore_powered_theorem.py, explore_eigenvalue_landscape.py

Write F(n) = c_N(n)/φ(N). For squarefree N it is a product of independent binary gates, one per channel:

A channel is on when it divides the frequency and off otherwise, with contrast ratio (p−1) : 1, so larger channels gate more sharply. The extremes are the whole dynamic range: a frequency coprime to N gives 1/φ, about 10⁻⁵ at k = 7, and a multiple of N gives 1.

The off-value is not an artifact of the normalization. It is exactly the inclusion–exclusion weight of the prime p, so the frequency response of the rung is the Möbius function of the sieve — the same identification as the sieve identity, read in the frequency domain.

Scope. Proved for every squarefree modulus; the identification with the sieve is term-by-term.

verifier: explore_moment_resonance.py

Read statically, the response is transparency-blind: the gate −1/(p−1) is fixed by the channel alone and never by whether (p−1) divides λ, so the static response cannot report whether the prime the rung just gained was transparent. That is precisely the geometric side of the split.

Read dynamically it is not blind. Evaluate F at the frequencies n = 1..λ — one full period of the power maps — and collect the distinct values, the rung's spectrum. A jump extends that range, admitting frequencies that did not exist below, and many new values arrive; a plateau leaves the range fixed and few do. Jumps widen the spectrum and plateaus refine it, and both raise its entropy, the jumps by about three times as much.

Scope. The static half is immediate from the gate formula. The dynamical half is computed k = 3..22, with the entropy comparison over k ≤ 10; what a plateau gains is the subject of the two claims below.

verifier: explore_resonance_transparency.py

Call the gate set of a frequency n the set S of channels dividing it — the gates that are on. Three exact pieces settle what a plateau gains.

Which gate sets occur. The set S is realized among n = 1..λ if and only if ∏S ≤ λ: the product itself realizes it, and any frequency with that gate set is a multiple of the product. So a gate set costs the product of its channels and λ is the budget it must fit under. This replaces a sweep over λ frequencies with a pruned enumeration of subsets, which is what puts rungs far past k = 14 in reach — k = 22 takes seconds.

What the spectrum becomes. At a plateau λ does not move, so the gate sets available below are unchanged and each frequency either is or is not divisible by the new prime. Writing V_k for the spectrum at rung k and c = −1/(p_k−1) for the new off-gate,

where the window W is the old spectrum seen through the multiples of p_k — the gate sets that fit the reduced budget λ/p_k. Rescaling by c is injective and contributes nothing new, so the gain is exactly the part of the window the rescaled old spectrum fails to cover.

Which window values die. A window value collides with a rescaled old one if and only if there are disjoint sets A, B of older channels with |A| + |B| odd, each side realized by a gate set inside the budget, and

Collisions are therefore multiplicative relations among the shifted primes — the same objects that decide transparency in the first place. A transparent prime can be redundant twice over: dynamically, because its orbit already embeds, and spectrally, because its values do. At k = 11..14 the windows are nearly the same size — 133, 135, 135 and 139 values — so the gains 60, 20, 41 and 70 are almost entirely collision rate: 55%, 85%, 70% and 50% of the window dies.

Scope. All three pieces are proved; the decomposition and the collision criterion are verified at k = 6, 11..14 and 18..22. The k = 6 plateau gains exactly zero, its three-value window being covered entire.

verifier: explore_plateau_collisions.py

The criterion above says which values die; it does not say how many, and answering that by enumeration costs a spectrum. It can be had from the divisor structure of p_k − 1 instead. A window gate set S dies via the relation (A, B) exactly when B ⊆ S, A is disjoint from S, and ∏S ≤ λ·∏B/∏A, so the collision set is a union of explicit sets over the relations of p_k − 1. Fix an on/off assignment of the channels appearing in any relation — a cell — and within it those thresholds nest, collapsing to one. The rate over gate sets is then a closed sum of subset counts, one term per cell, with no spectrum enumerated at all, and it matches the measured gate-set rate exactly at every k ≤ 22.

The prime 2 is the trap in enumerating those relations. It contributes a factor 1 to either side and so looks like a free parity toggle, but it still costs 2 in the budget, and the budget can refuse it — first at k = 11, where fixing the parity needs a gate set of product 102,102 against a λ of 55,440.

Against distinct values the cell sum is one-sided rather than exact. Collided values carry more representing gate sets than survivors do, so it over-predicts, by 0.00 to 0.24 across the 18 plateau rungs from k = 6 to k = 37. That is still enough to forecast: at k = 37 a cell sum of 0.98 predicted a value rate between 0.74 and 0.98 before any k = 37 spectrum had been computed, and the measured rate was 0.92.

What carries the rate is the whole budget-weighted union of those sets and not any single relation. Rungs 31, 34 and 36 admit no relation with B empty at all — no way to write p_k − 1 as a product of shifted primes outright — and still collide 83%, 64% and 84%, because a rich window is full of gate sets that already carry a nonempty B.

Scope. The condition under which a relation kills a gate set is proved. The rate formula is exact over gate sets and verified k ≤ 22; the over-prediction against values is an observation over the 18-rung census, with the rungs past k = 30 computed at a cap of 300 relations, under which that over-prediction can only widen.

verifier: explore_plateau_rate.py