bibkey: katz2015goldeninterfaces authors: Nicholas M. Katz year: 2015 title: Wieferich past and future doi: 10.1090/conm/632/12632 claim: Katz’s torus equidistribution conjecture has an exact fixed-golden specialization; macroscopic equidistribution alone does not force a WSS zero. strata_touched: [] license: citation-only triage: anchor

Original WSS arithmetic across geometric realizations

Verified locator

Nicholas M. Katz, Wieferich past and future, Contemporary Mathematics 632 (2015), 253-270, DOI 10.1090/conm/632/12632. Author-hosted paper: https://web.math.princeton.edu/~nmk/wieferich42.pdf

1. The geometric equidistribution conjecture

Source scope: Sections 2-6, including Conjectures 3.1, 4.7 and 6.1, the Lie-algebra exact sequence, and Section 5’s lattice warning.

Conjecture 4.7 concerns a torus over Z[1/N], a point generating a Zariski-dense cyclic subgroup, and an integral Lie lattice. It predicts Haar-equidistribution of the Wieferich fractions in the compact real torus of that lattice. Conjecture 6.1 gives a framed formulation. Section 5 cautions that an arbitrary change of lattice is not an automatic equidistribution-preserving operation.

CG.1 and CG.7 choose the norm-one torus of O=Z[phi], the SAME point u=phi/(1-phi)=-phi^2, and Lie lattice Z*sqrt(5). For p>5, N_p=p-(5/p), and original q_p=F_(N_p)/p mod p, the computed defect is

(u^N_p-1)/p = (5/p)*q_p*sqrt(5) mod p.

Hence the actual Katz fraction is c_p/p, where c_p is the least nonnegative residue of (5/p)*q_p. Its zero set is exactly the original WSS set. The point is nontorsion and therefore Zariski dense in this one-dimensional torus. This is a specialization of an existing conjecture, not a randomness theorem.

CG.7 proves that even prime-denominator equidistribution need not hit zero. Its artificial countermodel is not a Fibonacci sequence or a refutation of Katz. The exact Fourier zero-counting identity is recorded separately. Its full nonconstant-frequency error would need a genuinely new estimate on the log-log scale; no such estimate is supplied. Fixed-frequency macroscopic cancellation is not silently strengthened to shrinking-target control.

2. p-rationality and actual local geometry

Zakariae Bouazzaoui, Fibonacci Sequences And Real Quadratic p-Rational Fields, arXiv:1902.04795. https://arxiv.org/abs/1902.04795 https://arxiv.org/pdf/1902.04795

Source scope: Corollary 2.2, Remark 2.3, Proposition 3.2 and Theorem 3.4, with their discriminant and class-number conditions. For the fixed field Q(sqrt(5)), class number one and p>5 give: p is WSS iff the field is not p-rational. Corollary 2.2 also relates this to divisibility of L(2-p,chi_5). Nonvanishing of a p-adic logarithm is weaker than initial valuation exactly one and cannot settle WSS. In the split case O/pO is a product of fields; CG.2-CG.4 use its norm-one group, not a falsely cyclic full unit group.

CG.2-CG.4 prove their local formulas directly. Exactly one of the p lifts of a norm-one residue preserves the N_p return. The derivative U_(N_p)’(1) is -2/5 modulo p, always nonzero. The unique Hensel root a_p in 1+p Z_p has v_p(a_p-1)=h_p and first displacement (5/2)q_p. Varying the coefficient to this root does not produce a WSS prime for the fixed coefficient one. The local orbit closure has index p^(h_p-1) in the principal norm-one group. These are complete local descriptions retaining the unknown actual h_p.

Rigoberto Florez, Robinson A. Higuita and Alexander Ramirez, The Resultant, the Discriminant, and the Derivative of Generalized Fibonacci Polynomials, Journal of Integer Sequences 22 (2019), Article 19.4.4. https://cs.uwaterloo.ca/journals/JIS/VOL22/Florez/florez23.html

The paper provides classical polynomial-derivative context. CG.4 proves its displayed identity; no unverified theorem number or first-discovery claim is used.

3. An exact integral-basis transfer

Lenny Jones, A new condition for k-Wall-Sun-Sun primes, arXiv:2302.10357v4, revised July 15, 2023; author PDF dated July 18, 2023. https://arxiv.org/abs/2302.10357 https://arxiv.org/pdf/2302.10357

Source scope: the definition of polynomial monogenicity, Theorem 1.1, Proposition 2.5 and the index-criterion argument. At k=1 the hypotheses hold, giving

p is WSS iff X^(2p)-X^p-1 is non-monogenic.

For a root theta_p with theta_p^p=phi, K_p=Q(theta_p) contains the original fixed golden field. Non-monogenic here concerns the SPECIFIED power basis: Z[theta_p] is not the full ring of integers. It does not assert that no other generator of K_p has a power integral basis. The displayed polynomial discriminant has absolute value 5^p p^(2p), and equals the square of that index times disc(K_p). Its p-divisibility, and the reduction to (X^2-X-1)^p mod p, occur for every p and alone do not decide the index. No new global index constraint is proved here.

4. Arithmetic derivatives and cohomology

Alexandru Buium and Santiago R. Simanca, Arithmetic partial differential equations, arXiv:math/0605107v2. https://arxiv.org/abs/math/0605107

The abstract describes Fermat-quotient operators and usual derivations on arithmetic groups; the full technical development is outside this note’s source scope. CG.6 defines delta_p(x)=(sigma_p(x)-x^p)/p on the unramified golden algebra, proves its multiplication law, and computes its value at phi as a unit multiple of original q_p. No arithmetic Ricci-flow theorem or WSS existence result is attributed to this paper.

Gebhard Boeckle, David-A. Guiraud, Sudesh Kalyanswamy and Chandrashekhar Khare, Wieferich Primes and a mod p Leopoldt Conjecture, arXiv:1805.00131v2. https://arxiv.org/abs/1805.00131

The primary introduction and real-quadratic discussion connect regulator congruences with Galois representations, cohomology and deformation theory. No statistical hypothesis there is used as a proved golden-unit theorem.

5. The actual three-manifold bridge and its limit

CG.5 uses A=[[1,1],[1,0]], B=A^2 and S=2A-I. For every positive EVEN n, it proves B^n-I=F_n S A^n and computes the actual integer cokernel with Smith factors F_n and 5F_n. Abelianizing the actual torus mapping-torus group gives H1=Z plus that cokernel. At n=p-(5/p), its p-primary torsion is (Z/p^h_p)^2. This is a literal continuous three-manifold realization of the original WSS depth. No new bound on that depth follows merely from real hyperbolicity, entropy, or the topological identification.

6. Related results and boundaries

Nic Fellini and M. Ram Murty, Wieferich primes in number fields and the conjectures of Ankeny-Artin-Chowla and Mordell, arXiv:2508.08472v2. https://arxiv.org/html/2508.08472v2

Theorem 1.2 gives conditional non-Wieferich infinitude under number-field abc. Theorem 1.3 instead assumes finitely many super-Wieferich primes. Neither is an unconditional WSS-existence theorem. The same source reports AAC and Mordell counterexamples in VARYING quadratic fields; they are not fixed-golden WSS examples. The Pell-height route remains conditional and points primarily toward non-WSS results.

The p-rational and integral-basis formulations are established transfers, not additional independent problems counted on top of WSS. The remaining arithmetic obligation is a constraint on actual original q_p/h_p, a WSS witness, or a new global distribution theorem. Rephrasing the same unknown in multiple spaces does not supply such a result.

7. Predictive spacetime completion and the original square-level drift

The WSS owner’s SJC section reads Katz’s Section 4 exact Lie-kernel sequence on printed page 3 together with the actual repository task-quotient theory. The relevant repository inputs are docs/develop/theory/CONTEXTUAL_SPACETIME_ARITHMETIC_ML.md, Sections 1-3, at inspected dev b603c498c3cdc76478bfb37c2e3c4ea29edb86ed, and the draft docs/develop/theory/SYMPLECTIC_PREDICTIVE_COMPLETION.md, Section 2, at candidate commit29c5504b11a1685c220e0ad2a9705a987fc6cff4 in PR8891. The former distinguishes update closure from task sufficiency and requires actual joint images. The latter’s automatic symplectic and thermal results require their stated positive-energy hypotheses. Their interpretation as a physical model is not an arithmetic premise.

SJC keeps v=phi^2 and its prime-to-p residue order r, which is different from the Fibonacci rank. The full preimage of the residue orbit modulo p^2 has canonical multiplicative coordinates (j,z) in Z/r x F_p. The fixed update is exactly (j,z) mapping to (j+1,z-q_p). A fiber rephasing adds a telescoping difference to this drift and preserves its cycle sum -r*q_p. The torsion section is a group section only: characteristic p^2 prevents a unital ring section of the reduction map or its replacement by characteristic-p dual numbers. The fixed polynomial X^2-3X+1 selects the actual lift among p comparison lifts; choosing the zero-drift lift is not a WSS construction.

Two exact delayed traces have matrix [[2,1],[3,4]], determinant five, and recover the entire state modulo every p^a for p>5. The returned trace alone has zero first-order defect; the next trace reads -5r*q_p after division by p. These are ordinary finite-ring proofs, not extra Lean conclusions of the existing trace-Gram source. The real flow generating multiplication by v is symplectic but has indefinite quadratic energy, so positive Gibbs and positive-energy compression cannot be imported from PR8891 without new hypotheses.

This is a task-relative completion of information discarded by reduction. The integers with the usual absolute-value metric are already complete; the p-adic topology has its own completion Z_p. The integer WSS formula itself requires no new arithmetic or set-theoretic axiom. The drift and its higher precision remain the actual q_p and h_p. No new prime-family existence, exclusion, or cross-prime equidistribution follows from this choice of coordinates.

8. Cyclotomic norm calibration

Tyler Ross, Zhongyan Shen and Tianxin Cai, The p-adic Valuations of Mobius Duals of Lucas Sequences, arXiv:2512.03481v1. https://arxiv.org/html/2512.03481v1

The Introduction, Theorem 2.2(c), and Proposition 3.2 give the precise source scope used for calibration. The regular Lucas-sequence cyclotomic and entry-point valuations are explicitly credited there to Carmichael and classical valuation theory; the paper extends them to irregular sequences. No new integer WSS example is supplied by these statements.

SJC independently derives, for the prime-to-p period r of phi^2, the integer G_r=Psi_r(3), where Psi_r is the real r-cyclotomic polynomial, and proves v_p(G_r)=h_p using the canonical torsion trace. Its elementary height bound is 1<G_r<5^(EulerPhi(r)/2). This is a classical cyclotomic-norm interface, not a claimed new WSS family. This unramified-at-p coefficient field of conductor r must not be confused with the ramified p-power coefficient fields used for the conditional Maass families in SGN and HCR.

9. The published squarefree criterion and the stronger norm estimate

The complete primary HTML of Ross-Shen-Cai was rechecked, including its conclusion, Corollary 5.1. That corollary explicitly states equivalence between absence of WSS primes and squarefreeness of every Fibonacci Mobius dual M_n^F except n=6. The WSS owner’s SJC.9 proves the exact change of index between that source and its own norm: G_r=M_(iota(r))^F, where r odd gives iota(r)=2r, r=2 modulo4 gives iota(r)=r/2, and4|r gives iota(r)=r. Thus the source’s exceptional index6 is G_3=4. The resulting squarefree characterization is cited prior work, not a new discovery.

SJC.9 also records all prime valuations, including the small primes2,3,5. For p>5 an inherited factor at r=r_p*p^a has valuation exactly one; only the primitive r=r_p factor can have the initial depth h_p. Consequently any repeated prime divisor of G_r for r>=4 is an original WSS prime, without an additional coprimality assumption on that divisor and r.

SJC.10 is a separate elementary estimate for the actual positive norm. Writing R=rad(r), s=r/R and a=phi^(-2s), the Mobius logarithm is shown to have sign -mu(R) and magnitude less than -log(1-a). This yields the uniform bound phi^(EulerPhi(r)-1)<G_r<phi^(EulerPhi(r)+1) and the exact integer upper bound B(r)=L_(EulerPhi(r))-1 for mu(R)=1, or L_(EulerPhi(r)+1) for mu(R)=-1. The latter upper bound is attained at every odd prime index. This signed estimate is proved in the dossier rather than attributed to Ross-Shen-Cai or Katz; its independent priority has not been established.

SJC.11 deduces the quotient-independent implication p^2>B(r_p) => q_p!=0 and the joint depth budget for primes having the same actual period r. It does not prove that a new unbounded prime family satisfies this condition. The fixed113 example demonstrates an improvement over the former height bound, not a newly discovered non-WSS prime. No WSS existence, complete squarefreeness theorem, spectral nonvanishing or Lean certification follows from the size estimate. The remaining obligation is stated in SJC.12.

10. SIC dimension towers and a fixed-field obstruction

Gary McConnell, Some new infinite families of non-p-rational real quadratic fields, arXiv:2406.14632v1, June 20, 2024. https://arxiv.org/abs/2406.14632 https://arxiv.org/html/2406.14632v1

The exact source scope is Theorems 2.1-2.2, definition (2.1), and Lemma 3.3. The construction originates in SIC-POVM questions but constructs real quadratic fields with a varying squarefree discriminant parameter D. It does not construct a WSS prime in a fixed preselected golden field.

TBN.1 in the existing WSS owner imposes that fixed-field condition. It proves D(d)=5 for a square dimension d>=4 only at d=4, using the golden Pell classification and L_(2n)=L_n^2-2*(-1)^n. DCE.4 subsequently extends the exclusion to EVERY pure odd-prime-power seed p^a with a>=2. The odd index case uses the published D=3 Lebesgue-Nagell theorem; the even case uses coprime factors L_n-1 and L_n+1. This excludes construction inputs, not the rational prime p from the original WSS set.

TBN.2-TBN.6 use the actual companion blocks B_j=L_(23^j)+1=Psi_(3^(j+1))(3), j>=1. Every prime factor is split, has Fibonacci rank23^(j+1), and occurs with its original h_p. Different blocks are coprime. The product modulo4 and5 and modulo3^(j+2) gives the three simultaneous balances. These specialize the existing GP3 strategy to a different Lucas block, not the earlier L_(3^j)^2+1 block.

The classical regular valuation source has also appeared as Ross-Shen-Cai, The p-adic Valuations of Mobius Duals of Lucas Sequences, The Fibonacci Quarterly, published online July 21, 2026, DOI10.1080/00150517.2026.2656703: https://www.tandfonline.com/doi/full/10.1080/00150517.2026.2656703 The publisher metadata and abstract were checked; the detailed theorem locators above refer to the separately read primary arXiv version. No claim that its full journal text was inspected is made here.

11. Global perfect-power input and actual common-depth exclusion

The dedicated source note Library/notes/bugeaud2006lebesguenagell.md records the published theorem and the exact D=3 table locator. DCE.1-DCE.2 uses that external theorem to prove

gcd{h_p:p|L_(3^j)^2+3}=1 for every j>=1.

The factor exponents are the actual h_p by TBN3. The conclusion excludes every common-divisor depth pattern, including common odd divisors beyond three. For every fixed integer e>=2 it produces infinitely many distinct split primes of ranks2*3^s whose depths are not divisible by e. This does not choose between depth one and a larger depth in any unknown prime.

DCE.3 strengthens the coverage budget uniformly in H: all depths at least H>=2 require at least two prime factors and total multiplicity at least 2H+1. Equality gives two depths H,H+1. The smallest all-WSS pattern must therefore be P^2 Q^3, with Q=19 modulo40 and the displayed ternary class for P. The fourth fixed block has an actual factor with h_p<=4 by an exact size comparison, without factoring that block. It is not a newly found non-WSS prime or a proof that every block has a simple factor.

No novelty is claimed for the external exponential theorem or its formal instantiation. The new ordinary deductions narrow block-depth patterns; heterogeneous depths such as2,3 remain unexcluded. The existing Scribe reference records this arithmetic context without changing the Lean trace-image theorem, its formulas, or its certification status.

12. Exact Zeckendorf predictive states at the original prime-square scale

This section keeps the original Fibonacci numbers and the same WSS problem family. It connects the least-significant-first residue transducer to the completed-future viewpoint and to an explicitly specified probability law. The state count concerns an autonomous finite-state input reader. Reading time supplied by an external clock is a different resource model.

12.1 Actual words, tasks, and the observable state

Fix an integer M>=2. A finite binary word w=b_0…b_(n-1) is read from its least significant position, with value

$$V(w)=\sum _{j=0}^{n-1}{b}_j{F}_{j+2}.$$

It is legal if it has no adjacent ones. The empty word represents zero; arbitrary zero padding on the HIGH end, which is the right end in this orientation, is permitted. Invalid words are rejected. This is not the self-delimiting Fibonacci code with a terminal11 marker.

Let T=pi(M) be the least period of the pair (F_j,F_(j+1)) modulo M, and R=rho(M) the least positive index at which M divides F_R. Both exist: S(u,v)=(v,u+v) is invertible, with inverse (v-u,u), on a finite set. Returning the pair (0,1) gives the period. Necessarily T>=3. The actual reader state is

$$(r,k,l),r\in \mathbb{Z}/M,k\in \mathbb{Z}/T,l\in \{0,1\},$$

with next weights (u,v)=(F_(k+2),F_(k+3)) modulo M. Reading b updates

$$(r,u,v,l)⟼(r+bu,v,u+v,b),$$

unless l=b=1, which leads to one absorbing rejecting state. Start at (0,1,2,0). These are the recurrence coordinates of the existing D5/S1/Digit/ZeckendorfResidueTransducer.lean. That owner is formulated for prime moduli; the arithmetic here explicitly includes composite M.

Two tasks must be separated. The full-residue task outputs the entire r at every finite termination and an invalid symbol at the rejecting state. The divisibility task outputs only whether termination is legal and r=0. Two prefixes have the same task-complete future if every finite suffix produces the same corresponding output. This includes the empty suffix. It is stronger than equality of the current output.

12.2 A constructive supply of every legal arithmetic probe

Theorem ZP1. At any incoming boundary bit and any consecutive weight row (u,v), every coefficient pair (A,B) modulo M is realized by a finite legal continuation with added residue Au+Bv. The continuation can restore the entire weight clock. A word of length at most4T(M-1) suffices. An additional initial block0^T can reset the boundary bit when required.

Proof. The exact return F_T=0,F_(T+1)=1 implies S^T=I on every row. For example induction gives

$$S^{n+1}(u,v)=(F_{n}u+F_{n+1}v,F_{n+1}u+F_{n+2}v),$$

and F_(T-1)=1 follows from the recurrence. Define two literal words

$$B_0=0^{T}10^{T-1},B_1=0^{T+1}10^{T-2}.$$

Both have length2T, begin and end with zero, and contain a single one. They are legal after either boundary bit. The one in B_0 is at positionT and contributes u; the one in B_1 is at positionT+1 and contributes v. Both restore S’s clock. If A_0,B_0’ are the least representatives of A,B, concatenate A_0 copies of the first word and B_0’ copies of the second. Its value is Au+Bv for every initial row simultaneously, its clock returns, and its length is at most4T(M-1). The empty concatenation is legal too; prefixing0^T resets a possibly nonzero boundary without changing the value or clock. This proves unbounded controllability by actual legal words, not by arbitrary coefficient queries that were never realizable.

12.3 The exact future kernel over composite residue rings

A row (u,v) is unimodular if eu+fv=1 for some e,f modulo M. Every actual consecutive Fibonacci row is unimodular by consecutive coprimality.

Theorem ZP2. Two live reader states (r,u,v,l) and (r’,u’,v’,l’) have the same complete divisibility future if and only if

$$\boxed{l=l^{\prime }\text{and}(r^{\prime },u^{\prime },v^{\prime })=a(r,u,v)\text{ for some unit }a\mathrm{mod}M.}\text{\hspace{1em}(ZP1)}$$

This holds for every M>=2, including rings with zero divisors.

Proof. If the boundary bits differ, append a one to the state with boundary zero. That state remains legal; by ZP1 and unimodularity of its new row, an additional guarded continuation makes the terminal residue zero. The other state rejects already at the first one. Thus equal futures force equal boundary bits. Every live state has an accepting continuation by the same argument, so it is distinguishable from the absorbing rejector.

With a common boundary, ZP1 implies equality of the two affine zero sets

$$r+Au+Bv=0⟺r^{\prime }+A{u}^{\prime }+B{v}^{\prime }=0$$

for EVERY pair A,B in Z/M. Choose eu+fv=1 and define a=eu’+fv’. The pair (A,B)=(-re,-rf) forces r’=ar. The pair (-re+v,-rf-u) then forces vu’-uv’=0. Combining this relation with eu+fv=1 gives u’=au and v’=av. Choose e’u’+f’v’=1 for the second row; then

$$a(e^{\prime }u+f^{\prime }v)=1,$$

so a is a unit. No field division was used. Conversely, scaling all three coordinates by a unit commutes with the actual input recurrence and preserves a zero terminal residue. Common boundary bits give the same legality on every suffix. Induction on the word proves the reverse implication, and hence the exact future kernel.

The companion ZeckendorfFutureKernel.result constructs ZP1 from the literal Fibonacci return and proves the same-boundary equivalence with explicit Bezout certificates. Boundary separation and the counting consequences below are the ordinary deductions just displayed. This scope is not enlarged by calling every consequence kernel-certified.

12.4 Exact minimal autonomous memory, not merely an upper bound

Theorem ZP3. With the stated orientation and padding convention, the minimal COMPLETE deterministic observer has exactly

$$\boxed{2M\pi (M)+1\text{states for full residue output},}$$

$$\boxed{2M\rho (M)+1\text{states for divisibility}.}\text{\hspace{1em}(ZP2)}$$

In each expression the added one is the absorbing invalid-word state.

Proof of reachability. ZP1 from (1,2) can add any desired residue, restore phase zero and leave boundary zero. Append zeroes to choose any phase with final bit zero. To obtain phase k and final bit one, first choose phase k-1 modulo T by zeroes, and then append one; subtract its known weight in the originally programmed residue. Thus all2MT live states are reachable. The word11 reaches the rejector.

For the full output, the empty suffix identifies r. Suffix1 distinguishes the two boundary bits through validity. With boundary zero, suffixes1 and01 identify u and v. With boundary one, the legal suffixes01 and001 identify v and u+v, hence u. No two different live states can merge. There are exactly T different weight rows, proving the first count.

For the binary task, ZP2 describes every possible merger. The scalar subgroup of the clock consists precisely of shifts d for which F_d=0 modulo M. Indeed, at the initial row, scalarity is equivalent to

$$F_{d+3}-2F_{d+2}=-F_d=0.$$

Its scalar is then F_(d+1), a unit by consecutive coprimality. Invertibility of S transports this characterization to every other phase. Strong Fibonacci divisibility shows that the zero indices are exactly the multiples of R: gcd(F_R,F_d)=F_gcd(R,d), so minimality of R forces R|d at every zero, and the converse is ordinary index divisibility. Consequently the scalar clock subgroup has size kappa=T/R.

Each live future-equivalence class has exactly kappa members. Its action is free on the T distinct clock phases, and the residue scales by the same unit. There are therefore2MT/kappa=2MR live classes. These classes are distinct future languages and are reachable, so the minimality lower bound is exact, by the deterministic future-equivalence criterion. Adding the separate rejector proves the second formula.

The same kappa=pi(p)/rho(p) that organized the scalar stationary phases in the conic Fourier calculation is thus the EXACT compression factor between these two arithmetic reading tasks. This is a concrete common quotient; no assumption of independent observational channels is involved.

Orientation boundary. Charlier, Rampersad, Rigo and Waxweiler proved 2M^2 states for the trim minimal Fibonacci divisibility automaton with most-significant-first input. Moradi, Rampersad and Shallit’s2026 survey uses that orientation and recalls that result. The present language is read in the reverse direction and counts the rejector explicitly. Formula ZP2 does not contradict or improve the2M^2 theorem in its own model. Least-significant reading can require either more or fewer states, depending on rho(M). Merely forgetting to charge for a clock would change the comparison again.

12.5 A specified probability process preserves the same exact quotient

Theorem ZP4. Feed the reader with the legal golden Markov digit source

$$P=\left(\begin{array}{cc}{\varphi }^{-1}& {\varphi }^{-2}\\ 1& 0\end{array}\right).$$

Observe the actual future bits jointly with the terminal divisibility answer, for every finite horizon. The exact predictive equivalence of live states is still ZP1. The quotient has2M rho(M) states and a unique stationary probability law, with mass

$$\boxed{\mathrm{Pr}([(r,k,l)])=\frac{{\pi }_l}{M\rho (M)},{\pi }_0=\frac{{\varphi }^2}{{\varphi }^2+1},{\pi }_1=\frac{1}{{\varphi }^2+1}.}\text{\hspace{1em}(ZP3)}$$

Each expression [(r,k,l)] denotes one quotient state, not a low-prefix cylinder in an infinite Zeckendorf sequence.

Proof. Both allowed transitions out of zero have positive probability, and the one-to-zero transition has probability one. Every finite legal word therefore has positive probability. Two states with equal boundary and equal deterministic futures give the same joint bit-and-answer laws. If ZP1 fails with equal boundary, the separating legal word in ZP2 has positive probability, and its terminal answers differ. If boundaries are different, the very next bit laws already differ. Thus the equivalence is exact for this specified observed process. If only the digit process were observed and the divisibility answers were omitted, the conclusion would be different; only the boundary bit would be needed.

On the full live state space the stationary mass is pi_l/(MT). To verify it, fix a target phase k and bit b. Each allowed predecessor boundary has exactly one predecessor residue, since adding b times the old weight is a permutation of Z/M. The predecessor phase is k-1. Summing incoming weights therefore reduces to pi P=pi. Every scalar equivalence fibre has kappa elements and a fixed boundary bit, so its total mass is ZP3. Transition probabilities depend only on that boundary and the input bit, while the deterministic transitions preserve ZP1. Hence this is a valid quotient Markov chain. ZP1’s guarded controls, with phase adjustment, reach every live state from every other using positive-probability words. The finite chain and its quotient are irreducible, proving uniqueness. They can be periodic; time-by-time convergence from an arbitrary start is not asserted. Stationarity or Cesaro averaging is sufficient here.

Writing H(pi)=-sum_l pi_l log pi_l, the finite stationary entropies are

$$H(\text{full state})=\log (M\pi (M))+H(\pi ),$$

$$\boxed{H(\text{predictive state})=\log (M\rho (M))+H(\pi ),H(\text{full state}\mid \text{predictive state})=\log \kappa .}$$

The last equality follows because every fibre has kappa equal-mass states. It measures task-irrelevant information under the stated source, not an observer-independent physical entropy or a count of unknown WSS primes.

This source is positional digit generation. It is NOT the Bernoulli arithmetic-successor kernel (1-a)Id+aT studied in RRO Section36. That section’s no-autonomous-finite-prefix-kernel theorem remains unchanged. Here the finite state includes a modular accumulator and its clock, and is proved sufficient for a different, explicitly finite arithmetic task. The Parry source is also different from a uniform distribution on fixed- length finite legal words, which has endpoint-conditioned probabilities.

12.6 WSS becomes a two-regime exact memory growth law

Let p>5 be prime and h_p=v_p(F_(p-(5/p))). The classical initial-depth valuation formula and the exact rank imply

$$\rho (p^s)=\rho (p)p^{\max (0,s-h_p)}.$$

Indeed every zero index is a multiple of rho(p), and its Fibonacci valuation is h_p plus the valuation of that multiplier. The least positive multiplier attaining depth s is p^max(0,s-h_p). This reasoning retains arbitrary actual h_p; it does not use Wall’s conjectured growth.

Corollary ZP5. If K_s is the minimal number of LIVE divisibility states at modulus p^s, then

$$\boxed{K_s=2\rho (p)p^{s+\max (0,s-h_p)},\frac{K_{s+1}}{K_s}=\{\begin{array}{ll}p,& s<h_p,\\ p^2,& s\ge h_p.\end{array}}\text{\hspace{1em}(ZP4)}$$

In particular p is WSS if and only if K_2/K_1=p; for a non-WSS prime the ratio is p^2. The stationary predictive entropy increases by log p or2log p under the same two conditions. For p=7 the complete binary counts at7 and49 are113 and5489, so the live-state ratio is49; for p=11 they are221 and26621, with live-state ratio121.

This is an exact operational realization of the original depth. It does not independently constrain that depth: computing rho(p^2), or minimizing the full exact reader, may already require the same initial arithmetic. No faster WSS decision algorithm, original WSS witness or unbounded exception/exclusion family is claimed. The autonomous-clock convention is essential. If the current input position is given for free by an external clock, one may update a residue and boundary with a time-dependent transition using2M live states. That clock carries the missing arithmetic phase; the autonomous lower bound must not be cited for the uncharged model.

12.7 Finite completion versus infinite-prefix topology

Corollary ZP6. No continuous map from the infinite legal digit space K to the discrete set Z/M agrees with value modulo M on every finite Zeckendorf word followed by zeroes. In fact any such extension is continuous at no point. The analogous divisibility-valued extension has the same property.

Proof. Fix any legal finite prefix. Its current consecutive row is unimodular. After that prefix, ZP1 realizes every target final residue by an actual legal finite continuation, and padding its high end with zeroes gives a finite natural Zeckendorf row in the same cylinder. Thus every nonempty cylinder contains finite rows of all M residues, and contains both divisibility answers. A continuous map into a finite discrete target would be constant on some cylinder about a continuity point, a contradiction. This is compatible with a finite sequential reader: the reader answers at a supplied finite termination. It does not continuously decide an unterminated infinite input from a prefix.

This makes the link to the completed-future viewpoint concrete. A current residue alone does not predict all continuations; the clock and boundary must be retained until their task-preserving scalar quotient is taken. Conversely, an infinite-prefix topology and a time-complete task interface are different observations. The existing abstract predictive-memory quotient does not by itself compute this arithmetic kernel; ZP1-ZP3 do.

12.8 Source and formalization boundary

The actual current sources inspected are ZeckendorfResidueTransducer, PredictiveMemoryMinimalQuotient and the golden hard-core Markov discussion in the contextual spacetime theory. The actual RRO Section36 proof from #8335, mirrored by loning’s #8397, and the temporal-query Section47 from #8408 were read. Their successor, finite-prefix and time-query claims are not silently replaced by the positional digit dynamics used here.

Relevant primary literature: E. Charlier, N. Rampersad, M. Rigo and L. Waxweiler, The minimal automaton recognizing mN in a linear numeration system, Integers11B(2011),A4, https://arxiv.org/abs/1008.1668 ; D. Moradi, N. Rampersad and J. Shallit, Complexity of Linear Subsequences of Fibonacci-Automatic Sequences, arXiv:2603.21645v1, Section2 and Section3.2, https://arxiv.org/html/2603.21645v1 ; and I. Ben-Ari and S. J. Miller, A Probabilistic Approach to Generalized Zeckendorf Decompositions, https://arxiv.org/abs/1405.2379 . The last source’s finite-word conditioning is not identified with the stationary source ZP3. The WSS rank-lifting input is the classical valuation formula recorded by Medina-Rowland, https://arxiv.org/abs/0910.2907 .

Searches for least-significant-first Fibonacci divisibility, minimal states, reversed input, and rank of apparition located the known most-significant-first theorem and related automatic-sequence work. They did not establish priority of the exact combined formulation above; no first-ever mathematical claim or new externally solved open problem is attached. Myhill-Nerode equivalence, finite Markov stationarity and Fibonacci rank theory are classical. The contribution is the explicit legal-word construction and the exact task-specific arithmetic quotient, with its stated probability and WSS interfaces.

The new Lean/Scribe pair is D5/S3/Arith/ZeckendorfFutureKernel.lean and its Blueprint companion. It proves legal coefficient saturation and the full fixed-boundary future equivalence, rather than assuming that all affine tests can be realized. The state-count, stationary-law, entropy, and topology results above are complete ordinary proofs and are not additional kernel-certified declarations. No source or conclusion assumes h_p=1.

13. Probability-state completion: terminal rank and two-read tomography

Section7 classified individual deterministic states. It did not assert that mixtures on those distinct states have different future laws. This section computes the additional probability kernel and proves that a family of TWO successive arithmetic readings removes it. The number of readings per specified experiment is separated from the number of experiments, word lengths, and statistical samples needed to learn their probabilities.

13.1 The exact terminal response and its closed input update

Fix M=p^s with p prime and s>=1, and a known incoming digit boundary. Let D_s be the projective Fibonacci orbit: the distinct unit-scaling classes of rows (F_(k+2),F_(k+3)) modulo M. Choose a unimodular representative v_d=(u_d,v_d) for each d, scaling the accumulated residue at the same time. Write R_s=rho(p^s). By ZP2-ZP3, |D_s|=R_s and the fixed-boundary live quotient states are (d,r), d in D_s and r in Z/M. There are M R_s of them. Only in formulas where it is unambiguous, v_d denotes the entire row.

Let mu(d,r) be a real signed mass, with total S. For probabilities S=1. Define its one-terminal response table

$$f_{\mu }(A,B)=\sum _{d\in D_s}\mu (d,-A{u}_d-B{v}_d).\text{\hspace{1em}(PT1)}$$

This is the probability that the actual residue is zero after a word with coefficient pair (A,B). Every such pair is implemented by the guarded legal words of ZP1, and every legal word has some pair. Thus equality of PT1 is EXACTLY equality of all single-terminal acceptance predictions at this fixed boundary. For the specified golden Markov input source, equality is also equality of the joint word-and-terminal-answer laws: each legal word has the same known positive input probability for both initial mixtures. This statement includes no intermediate divisibility measurement.

Theorem PT1. After consuming an allowed bit b in {0,1}, the response of the pushed-forward mass satisfies

$$\boxed{f_{{\mu }^{+}}(A,B)=f_{\mu }(b+B,A+B).}\text{\hspace{1em}(PT2)}$$

The new boundary is b, and the terminal acceptance probability is f_mu(0,0). The response table is therefore a closed predictive state for this single-terminal task, with an explicitly invertible affine pullback.

Proof. The actual update sends (r,u,v) to (r+b u,v,u+v). Substituting in the next test gives r+(b+B)u+(A+B)v. Summing its indicator against the original mass proves PT2. Selecting other projective representatives only scales the whole equality by a unit and does not change its zero set. Illegal input is handled by the known boundary and the rejecting state. No hidden-state reconstruction is assumed in this update.

13.2 All prime-power probability blind directions

For a row mass write its unnormalized finite Fourier transform as

$${\hat{\mu }}_d(t)=\sum _{r\mathrm{mod}M}\mu (d,r)e_M(tr),e_M(x)=\exp (2\pi ix/M).$$

Use the normalized two-dimensional transform

$$\hat{f}(\xi )=M^{-2}\sum _{A,B\mathrm{mod}M}f(A,B)e_M(-\xi \cdot (A,B)).$$

Theorem PT2. The full terminal kernel has the exact description

$$\boxed{{\hat{f}}_{\mu }(\xi )=\frac{1}{M}\sum _{\begin{array}{c}d\in D_s,t\mathrm{mod}M\\ t{v}_d=\xi \end{array}}{\hat{\mu }}_d(t).}\text{\hspace{1em}(PT3)}$$

Its real linear rank is

$$\boxed{L_s=1+\sum _{j=1}^s(p-1)p^{j-1}\rho (p^j).}\text{\hspace{1em}(PT4)}$$

Consequently the invisible signed subspace has dimension M R_s-L_s. The image of the probability simplex has affine dimension L_s-1. This is a linear/affine dimension, not a number of classical states or a lower bound on arbitrary discontinuous encodings into real numbers.

Proof. Fourier inversion of each row mass at -v_d dot (A,B) gives M^(-1) sum_t muhat_d(t)e_M(t v_d dot (A,B)). Character orthogonality gives PT3. At frequency zero, t=0 for every unimodular row, so fhat(0)=S/M.

A nonzero frequency of additive order p^j has the unique form p^(s-j) eta, with eta primitive modulo p^j. It lies on a row’s Fourier line exactly when that row reduces to the projective class of eta modulo p^j. The actual projective orbit reduces ONTO D_j. To see its size and fibres directly, the determinant of two clock rows with index difference a is plus or minus F_a. Vanishing modulo p^j is therefore equivalent to rho(p^j)|a. Its reduction fibres have size R_s/rho(p^j). Each direction in D_j contains exactly (p-1)p^(j-1) primitive vector representatives, and these representatives are disjoint for distinct directions. Counting by j, together with zero, gives PT4.

Each coordinate muhat_d(t) occurs in exactly one output sum in PT3. Every output frequency in the counted set has a nonempty preimage. The map is therefore onto all these Fourier coordinates over C, with one independent equation per coordinate for its kernel. Its original matrix has real entries, so real and complex ranks agree. Conjugate symmetry gives the same real dimension directly. The total mass is recovered from fhat(0), and the positive simplex has nonempty relative interior in its mass-one hyperplane. Its image consequently has affine dimension L_s-1, as stated. The probability constraint does not remove the linear blind directions near a strictly positive prior.

PT3 retains every sum over coincident low-conductor directions. Replacing it by separate equations muhat_d(t)=0 would incorrectly discard those cancellations when different directions coincide modulo a lower p-power.

13.3 At prime precision: a constructive inverse and a stability identity

For M=p let m_d=sum_r mu(d,r). Distinct projective directions have nonzero determinant over F_p. Put

$${\mathcal{L}}_{d,r}=\{(A,B):A{u}_d+B{v}_d=-r\}.$$

Theorem PT3. The centered row probabilities are reconstructed by

$$\boxed{\mu (d,r)-m_d/p=\frac{1}{p}\sum _{(A,B)\in {\mathcal{L}}_{d,r}}{f}_{\mu }(A,B)-S/p.}\text{\hspace{1em}(PT5)}$$

In particular, two probability laws mu,nu have all the same one-terminal responses if and only if mu(d,r)-nu(d,r) is constant in r for every d, and these row constants have zero total mass across d. At prime precision L_1=1+rho(p)(p-1), and the blind dimension is rho(p)-1.

Proof. On the indicated line, the d-th summand is constantly mu(d,r). For every other direction e, its linear form is bijective from that line to F_p, because the determinant is nonzero. Summing that term along the line therefore gives m_e. The entire line sum is p mu(d,r)+S-m_d, proving PT5 and the kernel assertion.

For equal-total-mass signed mu,nu, let

$$b_d(r)=\mu (d,r)-\nu (d,r)-(m_d^{\mu }-m_d^{\nu })/p.$$

Their centered rows have zero sums. Uniform (A,B) makes linear readings in two distinct directions independent and uniform, so cross terms vanish. Expanding the square yields the exact stability identity

$$\boxed{\frac{1}{p^2}\sum _{A,B}|f_{\mu }-f_{\nu }{|}^2=\frac{1}{p}\sum _{d,r}|b_d(r){|}^2.}\text{\hspace{1em}(PT6)}$$

This controls the observable centered component. It supplies no recovery of the invisible row totals. It is a finite linear-algebra statement; physical measurement noise and how the response table is estimated must be specified separately.

13.4 Two same-trajectory reads recover the entire probability law

We now ENLARGE the task by permitting two exact, non-destructive tests of whether the current residue is zero, on the same arithmetic run. Between them the reader consumes a chosen guarded word. Returning the weight clock by a word of length divisible by T is not a reset of the accumulated residue or of the hidden initial state. The two tests are correlated observations.

Theorem PT4. For every modulus M>=2 and each fixed-boundary quotient state (d,r), there is a specified pair of legal clock-restoring probes whose joint success indicator is exactly the point indicator of (d,r). Hence the FAMILY of these two-read experiments determines every mass mu(d,r), including for composite M. The span of their joint-success functions is the entire function space on the M rho(M) fixed-boundary states.

Proof. Choose e,f with eu_d+fv_d=1, and set

$$x=(-re,-rf),y=(v_d,-u_d).\text{\hspace{1em}(PT7)}$$

Implement x by ZP1, read Y_1, implement y by ZP1 from the returned clock, and read Y_2. For a candidate state (d’,r’), joint success means

$$r^{\prime }+x\cdot v_{d^{\prime }}=0,r^{\prime }+(x+y)\cdot v_{d^{\prime }}=0.$$

Subtracting gives v_d u_(d’)-u_d v_(d’)=0. Two unimodular rows with zero determinant over Z/M are unit-proportional: with a=e u_(d’)+f v_(d’), the Bezout relation gives v_(d’)=a v_d, and a has an inverse because the second row is unimodular. Thus d’=d as projective classes. With the chosen representative fixed, the first equation then says r’=r. Conversely the target clearly makes both equations zero. Therefore

$$\boxed{\underset{\mu }{\mathrm{Pr}}(Y_1=1,Y_2=1\mid x,y)=\mu (d,r).}\text{\hspace{1em}(PT8)}$$

Both guarded words are explicit finite legal words from ZP1. Each has length at most4T(M-1); adding0^T when necessary ensures positive separation between reads and boundary zero while leaving residue and clock unchanged. Thus no unimplemented arbitrary affine oracle is being introduced. Their point indicators prove the asserted full span and injectivity. This is not a claim that two observed bits reveal an arbitrary distribution: there is one designed experiment for each target, and its probability requires repeated samples or other specified statistical information.

Corollary. The one-terminal response state PT1 is generally NOT closed under conditioning on an intermediate divisibility answer. Input-update closure PT2 and Bayesian posterior closure are distinct requirements.

Proof by actual probability laws. Fix two different projective directions d,e, and take a uniform residue r in each, with known boundary zero. Both laws give f(A,B)=1/M for every terminal query. Conditional on the initial zero test succeeding, they instead become the distinct point states (d,0) and (e,0). A subsequent tangent probe from PT7 distinguishes them. Thus no posterior updater using only the common response table and the observed zero answer can produce all the correct subsequent predictions. This does not contradict ZP4, whose equivalence concerns individual initial states, or the general predictive-state representation theory with its FULL joint experiment family.

At M=7, the actual initial clock rows have projective representatives (1,2) and (1,5). Use the two tests with x=(0,0), y=(2,-1). Under the first uniform-residue law, the joint distribution has masses1/7 at11 and6/7 at00. Under the second, it has masses1/7 at10 and01 and5/7 at00. Both individual read marginals are Bernoulli(1/7), but the joint laws differ.

13.5 The observation budget is not the algebraic rank

For the preceding two-direction uniform-residue example at M>=3, the joint total-variation distance is exactly2/M, where total variation has its one-half-L1 convention. With equal prior odds and n INDEPENDENT repetitions of this specified two-read experiment, the optimal Bayes error is

$$\boxed{\frac{1}{2}(1-2/M{)}^n.}\text{\hspace{1em}(PT9)}$$

Proof. Choose a tangent to the first direction, which is not a tangent to the second. The first law only produces11 or00; the second only10,01, or00. Every record containing a non-00 trial identifies its law. The sole common n-trial record is all00, whose masses are (1-1/M)^n and (1-2/M)^n. The binary Bayes overlap formula gives PT9. The common-record calculation also gives distance2/M when n=1. Fixed small Bayes error therefore requires order M repetitions for this pair, despite having only two reads per run. This is the exact risk for this specified experiment, not a minimax claim over all adaptive protocols or arbitrary preparations.

If the input words are generated passively by the golden Markov source, the same identifying words have positive probability but can be rare. Dividing their joint event probability by their known word probability recovers PT8 in principle; no uniform sample-efficiency claim follows. Neither an independent-copy assumption nor a reset operation is silently added to the repository’s single-nonresettable-trajectory observation model.

13.6 Original WSS depth in linear probability dimension

For p>5 write R=rho(p), h=h_p and let L_0=1. Applying the classical rank-depth formula of Section7 to PT4 gives

$$\boxed{L_s=1+R(p-1)\sum _{j=1}^s{p}^{j-1+\max (0,j-h)}.}\text{\hspace{1em}(PT10)}$$

For s<=h this is1+R(p^s-1). For s>h it is

$$1+R(p^h-1)+\frac{R{p}^{h+1}(p^{2(s-h)}-1)}{p+1}.$$

The increments obey

$$\boxed{\frac{L_{s+1}-L_s}{L_s-L_{s-1}}=\{\begin{array}{ll}p,& s<h,\\ p^2,& s\ge h.\end{array}}\text{\hspace{1em}(PT11)}$$

At the decisive square scale,

$$L_2=\{\begin{array}{ll}1+R(p-1)(1+p^2),& h=1,\\ 1+R(p^2-1),& h\ge 2.\end{array}$$

Thus the single-terminal probability rank and the deterministic state count are different arithmetic invariants, although both retain the same initial-depth breakpoint. For p=7, the fixed-boundary state counts at7 and49 are56 and2744, but the terminal ranks are49 and2401; the invisible signed dimensions are7 and343. For p=11 the prime-level counts are110 and101, respectively. These are dimensions of a response matrix, not numbers of samples and not quantum Hilbert-space dimensions.

PT10 is an exact reformulation, not a new bound on h_p. Neither the rank calculation nor the two-read identity forces a prime with h_p>=2. Tests on composite prime powers do not create WSS examples. New arithmetic progress would require independent control of the actual response rank or its correlated point probabilities as p varies, without assuming the unknown rank-lifting breakpoint. No new WSS occurrence or unbounded exclusion family is concluded here.

13.7 Research and formalization scope

This continues the current spacetime work on task-relative state, probability, joint observations and finite-word Zeckendorf arithmetic. It adds an explicit response operator, the complete conductor-layer kernel, a prime-level stable inverse, and a two-read family recovering all masses. It does not identify positional digit generation with the arithmetic successor or its binomial mixing law. The abstract future-quotient theorem alone does not compute this probability kernel or guarantee Bayesian closure.

Primary source roles are recorded in Library/notes/singh2004zeckendorfprobability.md: Singh-James-Rudary, UAI2004, arXiv:1207.4167, for predictive-state/system-dynamics-matrix scope; Kingston, Signal Processing86(2006),2040-2050, DOI10.1016/j.sigpro.2005.09.024, for established prime-power Radon/Fourier redundancy; and Ben-Ari-Miller, arXiv:1405.2379, for conditioned finite-word probability models. The exact arithmetic proofs above are supplied here. The inspected abstracts do not establish priority for our combined result.

The explicit modular two-test separating event and its finite-mass reconstruction are retained as ordinary proofs in PT4. The literal word implementation is justified by the earlier guarded compiler. The Fourier rank, affine image dimension, prime inverse, sampling risk, and WSS formulas likewise remain ordinary proofs rather than claims of kernel certification. No new external open problem is marked resolved and no new problem entry is opened.