Bound Gaussian adaptive routing with a paid public header

research/gaussian-adaptive-header-tax/PROOF.md

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+# Conditional transcripts with paid prefix codes
+
+Let the 60 ports be the Gaussian E8 lines of PR738. Each port t supplies an
+independent uniform b_t-bit string X_t, with b_t a fixed nonnegative integer.
+The receiver for t initially knows all X_u on lines orthogonal to t and does
+not know X_t. Source and receiver roles are distinct.
+
+A central encoder, optimistically given all inputs, chooses M=h(X). Distinct
+header values have distinct words in a finite prefix-free binary code.
+Every receiver and every payload encoder receives M. Header bits are charged
+the pairing rank of the **declared full source and full reader spans**, namely 4.
+This full public-observation contract is a premise, not an inference that all
+possible implementations of a header must cost 4 per bit.
+
+For each M=m there is a fixed finite ordered list of payloads. Payload j has
+fixed supports A_j(m),R_j(m), codebook K_j(m), and value
+
+    Y_j = f_j(m, X_(A_j(m))) in K_j(m).
+
+Every K_j(m) is a finite prefix-free set of binary strings, including the empty
+word only for a singleton codebook. Each reader receives the entire codeword.
+All choices of list, order, supports, functions and codebooks are determined
+by m and carry no further input-dependent metadata. Payloads do not relay
+previous payloads. Arbitrary nonlinear joint decoding is allowed and must
+recover every demand on every input. Declare
+
+    c(R,A) = rank_Q(V_R V_A^transpose)/2,
+    C(X) = 4 length(M) + sum_j c(R_j(M),A_j(M)) length(Y_j).
+
+The bases V span representatives and their J-images. Additive per-bit pricing
+and this geometry are explicit assumptions; encoding and decoding are free.
+
+PR738 supplies 640 nonorthogonal cliques Q of size 12, each weight 1/192. Each
+port lies in 128 cliques. For every support pair,
+
+    sum_Q [Q meets R and A]/192 <= c(R,A).                 (G)
+
+The full geometry replay covers all 2,579,856 nonempty flat pairs. Closing a
+support preserves its pairing rank and only increases clique intersections,
+so (G) covers arbitrary supports. For any individual nonorthogonal clique,
+meeting both supports implies c(R,A)>=1: choose one line in each intersection;
+their pairing is nonzero. These are retained geometric facts, not new results.
+
+Fix any such Q and fix X outside Q. Given m, retain only payloads with both
+supports meeting Q, in their m-determined order; call their tuple Z_Q. Then
+(M,Z_Q) determines X_Q. Any omitted payload received by Q has sources disjoint
+from Q and can be recomputed from m and the fixed outside inputs. All free
+orthogonal side information is outside Q. Equal retained transcripts would
+therefore give equal complete decoder inputs at every receiver in Q, forcing
+equal X_Q. This does not assume linearity or conditional independence after M.
+
+For every fixed m and outside assignment, each retained payload distribution
+is supported on a subset of its declared prefix-free codebook. Thus
+
+    H(Y_j | M=m, X_outside=u) <= E[length(Y_j) | m,u].
+
+Indeed, Kraft K=sum_y 2^-length(y)<=1 and q_y=2^-length(y)/K give, by
+nonnegativity of relative entropy, H(Y)<=E[length(Y)]+log2 K<=E[length(Y)].
+Zero-probability codewords do not matter. This also proves H(M)<=E[length(M)].
+Apply entropy subadditivity within each fixed branch, then average:
+
+    B_Q = sum_(t in Q) b_t
+       <= H(M | X_outside)
+          + E[sum_j length(Y_j) [Q meets R_j(M) and A_j(M)]]. (1)
+
+All variables have finite alphabets. Expectations use the original uniform
+independent inputs. The payload count can vary between branches because the
+branch is conditioned on before the sum.
+
+Put B=sum_t b_t, L=E[length(M)], H=H(M), and P=C-4 length(M).
+Multiply (1) by 1/192 and sum the 640 cliques. Use 128 incidences per port,
+H(M|X_outside)<=H, and (G) on each realized branch with nonnegative lengths:
+
+    (2/3)B <= (10/3)H + E[P].
+
+Consequently
+
+    E[C] >= (2/3)B + 4L -(10/3)H
+         >= (2/3)(B+L).                                  (2)
+
+For any one nonorthogonal clique, rank>=1 instead gives B_Q<=H+E[P], hence
+
+    E[C] >= B_Q+4L-H >= B_Q+3L.                           (3)
+
+Both (2) and (3) hold, so their maximum may be used. No integrality ceiling
+follows merely because realized charges are integral. A constant header with
+positive length has H=0, so the sharper entropy forms charge its full 4L;
+an empty constant header gives L=H=0. A nonconstant prefix-free code has L>0
+and incurs the strictly positive 2L/3 penalty. The theorem bounds the average,
+and therefore the maximum cost; it does not bound each individual execution.
+
+For uniform b-bit demands B=60b, charge is at least 40b+2L/3. Only with the
+separate Dfc stage equation and a worst-case rational copy allowance at least
+2 max_X C(X) does this give
+
+    D/T <= (s-1)120b -2s(40b+2L/3)
+        = 40b(s-3) -(4s/3)L.
+
+No physical realization or all-size implication follows from this entropy
+argument. The geometry and conditional-transcript/paid-header methods retain
+their credit. New coverage is the Gaussian header gap and conditional
+prefix-free payloads.

research/gaussian-adaptive-header-tax/README.md

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+# A public routing header pays a Gaussian copy-cost penalty
+
+The fixed-support nonlinear Gaussian bound in [PR888](https://github.com/CrocSwap/integer-mult-bounds/pull/888)
+explicitly excludes adaptive supports and variable-length messages. We extend
+its observation model to a paid public prefix-free header which selects the
+payload supports and their prefix-free codes. Payload encoders and receivers
+may be nonlinear. This is an expected-cost theorem under uniform independent
+input demands, not a pointwise bound or a native supplier.
+
+Write B = sum b_t, L = E[length of header], and H = entropy of the header.
+Charge a header bit four Gaussian units (the full-space pairing rank), and
+each payload bit its declared source/reader pairing rank. Then
+
+    E[C] >= (2/3)B + 4L - (10/3)H >= (2/3)(B + L).
+
+For every nonorthogonal clique Q there is also a concentration inequality:
+
+    E[C] >= sum_(t in Q) b_t + 4L - H
+         >= sum_(t in Q) b_t + 3L.
+
+Equal b-bit demands retain the floor 40b and incur at least 2L/3 extra Gaussian
+units when a nonempty public routing code is used. The entropy form charges
+a useless constant header its full 4L. An empty constant header recovers the
+existing bound. Expected charges need not be integers: no ceiling is applied.
+
+The distinction from Jerry Chen's Hurwitz adaptive-routing argument is the
+Gaussian geometry's strict gap: total weighted-clique weight 10/3 is less than
+the full-space header rank 4. The entropy method is credited, not claimed new.
+Variable prefix-free payloads are admitted as well as variable headers. The
+precise model and all-width proof are in [PROOF.md](PROOF.md).
+
+## Reproduce
+
+Python3.12, standard library, from the repository root:
+
+```
+python3.12 -I -O -B research/gaussian-adaptive-header-tax/verify.py
+python3.12 -B scripts/research_package.py --package research/gaussian-adaptive-header-tax
+```
+
+The replay checks exact SHA256/Git-blob source pins, runs PR738's unchanged
+full geometry calculation in a separate assertion-enabled process, compares
+its complete receipt, and checks all 640 weighted clique incidences. The child
+ignores inherited PYTHONOPTIMIZE. No historical selected-result entry point
+is used. The new actual-geometry fixture has 12 active independent bits, a
+nonlinear AND header, adaptive payload supports and a variable-length prefix
+payload. All 4096 inputs decode for all 12 receivers; every weighted clique's
+conditional transcript injection is checked.
+
+Separate rational elimination checks the fixture's source/reader prices.
+Controls reject omitted payloads, undeclared sources, illegal side information,
+uncharged headers and non-prefix payloads. The last is an actually decoded
+orthogonal-frame parity using an empty/one-bit code: apparent expected cost 2
+violates the 8/3 floor because termination is free. A valid variable header
+costs 4 on the all-zero input, below expected floor 8, preventing a false
+pointwise conclusion. These finite examples corroborate the written argument;
+they do not prove the entropy theorem by enumeration.
+
+Default replay is read-only and offline. `--write NEW.json` writes the verified
+receipt exclusively to a new file. The registered input list is the complete
+external replay closure. Source or receipt changes need explicit review.
+
+## Limits and selection
+
+The rank 4 tariff requires the header's declared source and reader spans to be
+the full four-dimensional Gaussian space. Localized or private headers have
+a different contract. Additive per-bit pairing pricing remains an explicit
+premise, not a physical cost theorem for arbitrary nonlinear circuits. Every
+input-dependent choice of payload count, order, supports or codebooks must
+come from the paid header. Payloads do not relay other payloads. Correlated
+inputs, errors, uncharged termination, private routing, interactive relays,
+sub-line operations and changed physical pricing are outside scope.
+
+Only with the additional inherited Gaussian Dfc stage identity does the
+worst-case budget imply D/T <=40b(s-3) -(4s/3)L. A nonempty header therefore
+makes the favorable three-stage screen strictly negative. This conditional
+screen is not a completed construction or exponent improvement. No hosting,
+all-size compilation, precision or whole multiplication theorem is admitted.
+
+Fresh base 0fdd7d82de03e43c9e5a1000dfbcd380d1b687d2, priority 4, picker
+822a27d6015294b31b90467ba9f189a5 selected complex mechanisms. The prescribed
+P104 binding and complex-ceiling commands passed before this changed
+helper-observation scope. P104 correction work was reserved by another worker.
+The original ceiling is not applied to this different geometry. TASK.json
+records the detour. The selected record is unchanged.
+
+## Open next questions
+
+Can localized paid headers beat this public-header penalty while preserving
+a valid weighted transcript bound? Start with a single header whose source
+and reader spans have rank below 4, and include every dependency of the payload
+schedule. Receivers cannot obtain private schedule information for free.
+The Gaussian nonlinear optimum between 40b and60b also remains open; this
+result supplies no attaining 40b construction.
+
+## Credit
+
+Sagar Thakrar's PR738 supplies the Gaussian geometry and full flat-pair
+certificate; ToikY's PR888 supplies its nonlinear transcript composition.
+Alexander Templeton's PR785 and Michael Fischler's PR814 supply the underlying
+conditional transcript method. Jerry Chen's Hurwitz adaptive-routing proof
+supplies the paid-header entropy argument. Joel Pulikkan, Douglas Colkitt,
+DreamingOfClouds and the retained lineage keep their original credit.
+PR2607's adaptive affine Eisenstein theorem is distinct: these payload queries
+may be nonlinear and use Gaussian costs.
+
+Prepared by ywz65 with substantial OpenAI Codex assistance in proof, code,
+verification and writing. Apache-2.0; no independent human review or proof
+assistant verification is claimed.
+
+## Revision5 expansion decision
+
+Before submission, main advanced to b4560d9e7216f72b73c5aa940ac099c05e33ccf9.
+The changed guidance and family record were read and the branch fast-forwarded;
+scientific inputs retain their unchanged historical pins. Decision: narrow and
+finish this interface obstruction, without a dependent implementation campaign.
+The bound rules out public adaptive routing as a repair of this stated
+three-stage budget even when all other nonnegative costs are omitted. Native
+tariff realization, consumer/hosting and restoration remain unknown; favorable
+local examples do not admit a complete supplier.